From de23d16f7a9d828d1c3400ec76ded9622127f79a Mon Sep 17 00:00:00 2001 From: "Documenter.jl" Date: Mon, 7 Oct 2024 18:14:30 +0000 Subject: [PATCH] build based on caf87f6 --- dev/.documenter-siteinfo.json | 2 +- dev/ae/index.html | 4 +- dev/assets/documenter.js | 53 +++++++++++--------- dev/assets/themes/catppuccin-frappe.css | 2 +- dev/assets/themes/catppuccin-latte.css | 2 +- dev/assets/themes/catppuccin-macchiato.css | 2 +- dev/assets/themes/catppuccin-mocha.css | 2 +- dev/assets/themes/documenter-dark.css | 2 +- dev/assets/themes/documenter-light.css | 2 +- dev/diffgeo/index.html | 14 +++--- dev/encoders/index.html | 58 +++++++++++----------- dev/guidelines/index.html | 2 +- dev/hvae/index.html | 30 +++++------ dev/index.html | 2 +- dev/infomaxvae/index.html | 14 +++--- dev/layers/index.html | 8 +-- dev/mmdvae/index.html | 12 ++--- dev/quickstart/index.html | 2 +- dev/rhvae/index.html | 54 ++++++++++---------- dev/utils/index.html | 16 +++--- dev/vae/index.html | 8 +-- 21 files changed, 148 insertions(+), 143 deletions(-) diff --git a/dev/.documenter-siteinfo.json b/dev/.documenter-siteinfo.json index 24c353b..418ff2a 100644 --- a/dev/.documenter-siteinfo.json +++ b/dev/.documenter-siteinfo.json @@ -1 +1 @@ -{"documenter":{"julia_version":"1.10.4","generation_timestamp":"2024-07-23T19:27:47","documenter_version":"1.5.0"}} \ No newline at end of file +{"documenter":{"julia_version":"1.10.5","generation_timestamp":"2024-10-07T18:14:25","documenter_version":"1.7.0"}} \ No newline at end of file diff --git a/dev/ae/index.html b/dev/ae/index.html index 39b0ae4..f59f8e2 100644 --- a/dev/ae/index.html +++ b/dev/ae/index.html @@ -1,5 +1,5 @@ -Deterministic Autoencoders · AutoEncoderToolkit

Deterministic Autoencoder

The deterministic autoencoders are a type of neural network that learns to embed high-dimensional data into a lower-dimensional space in a one-to-one fashion. The AEs module provides the necessary tools to train these networks. The main type is the AE struct, which is a simple feedforward neural network composed of two parts: an Encoder and a Decoder.

Autoencoder struct AE

AutoEncoderToolkit.AEs.AE — Type

struct AE{E<:AbstractDeterministicEncoder, D<:AbstractDeterministicDecoder}

Autoencoder (AE) model defined for Flux.jl

Fields

  • encoder::E: Neural network that encodes the input into the latent space. E is a subtype of AbstractDeterministicEncoder.
  • decoder::D: Neural network that decodes the latent representation back to the original input space. D is a subtype of AbstractDeterministicDecoder.

An AE consists of an encoder and decoder network with a bottleneck latent space in between. The encoder compresses the input into a low-dimensional representation. The decoder tries to reconstruct the original input from the point in the latent space.

Forward pass

AutoEncoderToolkit.AEs.AE — Method
(ae::AE{Encoder, Decoder})(x::AbstractArray; latent::Bool=false)

Processes the input data x through the autoencoder (AE) that consists of an encoder and a decoder.

Arguments

  • x::AbstractVecOrMat{Float32}: The data to be decoded. This can be a vector or a matrix where each column represents a separate sample.

Optional Keyword Arguments

  • latent::Bool: If set to true, returns a dictionary containing the latent representation alongside the reconstructed data. Defaults to false.

Returns

  • If latent=false: A Namedtuple with key :decoder that contains the reconstructed data after processing through the encoder and decoder.
  • If latent=true: A Namedtuplewith keys :encoder, and :decoder, containing the corresponding values.

Description

The function first encodes the input x using the encoder to get the encoded representation in the latent space. This latent representation is then decoded using the decoder to produce the reconstructed data. If latent is set to true, it also returns the latent representation.

Note

Ensure the input data x matches the expected input dimensionality for the encoder in the AE.

Loss function

MSE loss

AutoEncoderToolkit.AEs.mse_loss — Function
mse_loss(ae::AE, 
+Deterministic Autoencoders · AutoEncoderToolkit
Deterministic Autoencoder
The deterministic autoencoders are a type of neural network that learns to embed high-dimensional data into a lower-dimensional space in a one-to-one fashion. The AEs module provides the necessary tools to train these networks. The main type is the AE struct, which is a simple feedforward neural network composed of two parts: an Encoder and a Decoder.
Autoencoder struct AE
AutoEncoderToolkit.AEs.AE — Type
struct AE{E<:AbstractDeterministicEncoder, D<:AbstractDeterministicDecoder}
Autoencoder (AE) model defined for Flux.jl
Fields
  • encoder::E: Neural network that encodes the input into the latent space. E is a subtype of AbstractDeterministicEncoder.
  • decoder::D: Neural network that decodes the latent representation back to the original input space. D is a subtype of AbstractDeterministicDecoder.
An AE consists of an encoder and decoder network with a bottleneck latent space in between. The encoder compresses the input into a low-dimensional representation. The decoder tries to reconstruct the original input from the point in the latent space. 
Forward pass
AutoEncoderToolkit.AEs.AE — Method
(ae::AE{Encoder, Decoder})(x::AbstractArray; latent::Bool=false)
Processes the input data x through the autoencoder (AE) that consists of an encoder and a decoder.
Arguments
  • x::AbstractVecOrMat{Float32}: The data to be decoded. This can be a vector or a matrix where each column represents a separate sample.
Optional Keyword Arguments
  • latent::Bool: If set to true, returns a dictionary containing the latent representation alongside the reconstructed data. Defaults to false.
Returns
  • If latent=false: A Namedtuple with key :decoder that contains the reconstructed data after processing through the encoder and decoder.
  • If latent=true: A Namedtuplewith keys :encoder, and :decoder, containing the corresponding values.
Description
The function first encodes the input x using the encoder to get the encoded representation in the latent space. This latent representation is then decoded using the decoder to produce the reconstructed data. If latent is set to true, it also returns the latent representation.
Note
Ensure the input data x matches the expected input dimensionality for the encoder in the AE.
Loss function
MSE loss
AutoEncoderToolkit.AEs.mse_loss — Function
mse_loss(ae::AE, 
          x::AbstractArray; 
          regularization::Union{Function, Nothing}=nothing, 
          reg_strength::Float32=1.0f0
@@ -7,4 +7,4 @@
          x_in::AbstractArray, 
          x_out::AbstractArray;
          regularization::Union{Function, Nothing}=nothing, 
-         reg_strength::Float32=1.0f0)
Calculate the mean squared error (MSE) loss for an autoencoder (AE) using separate input and target output vectors.
The AE loss is computed as: loss = MSE(xout, x̂) + regstrength × reg_term
Where:
  • x_out is the target output vector.
  • x̂ is the reconstructed output from the AE given x_in as input.
  • regstrength × regterm is an optional regularization term.
Arguments
  • ae::AE: An AE model.
  • x_in::AbstractArray: Input vector to the AE encoder.
  • x_out::AbstractArray: Target output vector to compute the reconstruction error.
Optional Keyword Arguments
  • reg_function::Union{Function, Nothing}=nothing: A function that computes the regularization term based on the ae outputs. Should return a Float32. This function must take as input the ae outputs and the keyword arguments provided in reg_kwargs.
  • reg_kwargs::NamedTuple=NamedTuple(): Keyword arguments to pass to the regularization function.
  • reg_strength::Number=1.0f0: The strength of the regularization term.
Returns
  • The computed loss value between the target x_out and its reconstructed counterpart from x_in, including possible regularization terms.
Note
Ensure that the input data x_in matches the expected input dimensionality for the encoder in the AE.
Training
AutoEncoderToolkit.AEs.train! — Function
`train!(ae, x, opt; loss_function, loss_kwargs...)`
Customized training function to update parameters of an autoencoder given a specified loss function.
Arguments
  • ae::AE: A struct containing the elements of an autoencoder.
  • x::AbstractArray: Input data on which the autoencoder will be trained.
  • opt::NamedTuple: State of the optimizer for updating parameters. Typically initialized using Flux.Train.setup.
Optional Keyword Arguments
  • loss_function::Function: The loss function used for training. It should accept the autoencoder model and input data x, and return a loss value.
  • loss_kwargs::NamedTuple=NamedTuple(): Additional arguments for the loss function.
  • verbose::Bool=false: If true, the loss value will be printed during training.
  • loss_return::Bool=false: If true, the loss value will be returned after training.
Description
Trains the autoencoder by:
  1. Computing the gradient of the loss with respect to the autoencoder parameters.
  2. Updating the autoencoder parameters using the optimizer.
train!(ae, x_in, x_out, opt; loss_function, loss_kwargs...)
Customized training function to update parameters of an autoencoder given a specified loss function.
Arguments
  • ae::AE: A struct containing the elements of an autoencoder.
  • x_in::AbstractArray: Input data on which the autoencoder will be trained.
  • x_out::AbstractArray: Target output data for the autoencoder.
  • opt::NamedTuple: State of the optimizer for updating parameters. Typically initialized using Flux.Train.setup.
Optional Keyword Arguments
  • loss_function::Function: The loss function used for training. It should accept the autoencoder model and input data x, and return a loss value.
  • loss_kwargs::NamedTuple=NamedTuple(): Additional arguments for the loss function.
  • verbose::Bool=false: If true, the loss value will be printed during training.
  • loss_return::Bool=false: If true, the loss value will be returned after training.
Description
Trains the autoencoder by:
  1. Computing the gradient of the loss with respect to the autoencoder parameters.
  2. Updating the autoencoder parameters using the optimizer.

+         reg_strength::Float32=1.0f0)

Calculate the mean squared error (MSE) loss for an autoencoder (AE) using separate input and target output vectors.

The AE loss is computed as: loss = MSE(xout, x̂) + regstrength × reg_term

Where:

  • x_out is the target output vector.
  • x̂ is the reconstructed output from the AE given x_in as input.
  • regstrength × regterm is an optional regularization term.

Arguments

  • ae::AE: An AE model.
  • x_in::AbstractArray: Input vector to the AE encoder.
  • x_out::AbstractArray: Target output vector to compute the reconstruction error.

Optional Keyword Arguments

  • reg_function::Union{Function, Nothing}=nothing: A function that computes the regularization term based on the ae outputs. Should return a Float32. This function must take as input the ae outputs and the keyword arguments provided in reg_kwargs.
  • reg_kwargs::NamedTuple=NamedTuple(): Keyword arguments to pass to the regularization function.
  • reg_strength::Number=1.0f0: The strength of the regularization term.

Returns

  • The computed loss value between the target x_out and its reconstructed counterpart from x_in, including possible regularization terms.

Note

Ensure that the input data x_in matches the expected input dimensionality for the encoder in the AE.

Training

AutoEncoderToolkit.AEs.train! — Function
`train!(ae, x, opt; loss_function, loss_kwargs...)`

Customized training function to update parameters of an autoencoder given a specified loss function.

Arguments

  • ae::AE: A struct containing the elements of an autoencoder.
  • x::AbstractArray: Input data on which the autoencoder will be trained.
  • opt::NamedTuple: State of the optimizer for updating parameters. Typically initialized using Flux.Train.setup.

Optional Keyword Arguments

  • loss_function::Function: The loss function used for training. It should accept the autoencoder model and input data x, and return a loss value.
  • loss_kwargs::NamedTuple=NamedTuple(): Additional arguments for the loss function.
  • verbose::Bool=false: If true, the loss value will be printed during training.
  • loss_return::Bool=false: If true, the loss value will be returned after training.

Description

Trains the autoencoder by:

  1. Computing the gradient of the loss with respect to the autoencoder parameters.
  2. Updating the autoencoder parameters using the optimizer.
train!(ae, x_in, x_out, opt; loss_function, loss_kwargs...)

Customized training function to update parameters of an autoencoder given a specified loss function.

Arguments

  • ae::AE: A struct containing the elements of an autoencoder.
  • x_in::AbstractArray: Input data on which the autoencoder will be trained.
  • x_out::AbstractArray: Target output data for the autoencoder.
  • opt::NamedTuple: State of the optimizer for updating parameters. Typically initialized using Flux.Train.setup.

Optional Keyword Arguments

  • loss_function::Function: The loss function used for training. It should accept the autoencoder model and input data x, and return a loss value.
  • loss_kwargs::NamedTuple=NamedTuple(): Additional arguments for the loss function.
  • verbose::Bool=false: If true, the loss value will be printed during training.
  • loss_return::Bool=false: If true, the loss value will be returned after training.

Description

Trains the autoencoder by:

  1. Computing the gradient of the loss with respect to the autoencoder parameters.
  2. Updating the autoencoder parameters using the optimizer.
diff --git a/dev/assets/documenter.js b/dev/assets/documenter.js index b2bdd43..82252a1 100644 --- a/dev/assets/documenter.js +++ b/dev/assets/documenter.js @@ -77,30 +77,35 @@ require(['jquery'], function($) { let timer = 0; var isExpanded = true; -$(document).on("click", ".docstring header", function () { - let articleToggleTitle = "Expand docstring"; - - debounce(() => { - if ($(this).siblings("section").is(":visible")) { - $(this) - .find(".docstring-article-toggle-button") - .removeClass("fa-chevron-down") - .addClass("fa-chevron-right"); - } else { - $(this) - .find(".docstring-article-toggle-button") - .removeClass("fa-chevron-right") - .addClass("fa-chevron-down"); +$(document).on( + "click", + ".docstring .docstring-article-toggle-button", + function () { + let articleToggleTitle = "Expand docstring"; + const parent = $(this).parent(); + + debounce(() => { + if (parent.siblings("section").is(":visible")) { + parent + .find("a.docstring-article-toggle-button") + .removeClass("fa-chevron-down") + .addClass("fa-chevron-right"); + } else { + parent + .find("a.docstring-article-toggle-button") + .removeClass("fa-chevron-right") + .addClass("fa-chevron-down"); - articleToggleTitle = "Collapse docstring"; - } + articleToggleTitle = "Collapse docstring"; + } - $(this) - .find(".docstring-article-toggle-button") - .prop("title", articleToggleTitle); - $(this).siblings("section").slideToggle(); - }); -}); + parent + .children(".docstring-article-toggle-button") + .prop("title", articleToggleTitle); + parent.siblings("section").slideToggle(); + }); + } +); $(document).on("click", ".docs-article-toggle-button", function (event) { let articleToggleTitle = "Expand docstring"; @@ -110,7 +115,7 @@ $(document).on("click", ".docs-article-toggle-button", function (event) { debounce(() => { if (isExpanded) { $(this).removeClass("fa-chevron-up").addClass("fa-chevron-down"); - $(".docstring-article-toggle-button") + $("a.docstring-article-toggle-button") .removeClass("fa-chevron-down") .addClass("fa-chevron-right"); @@ -119,7 +124,7 @@ $(document).on("click", ".docs-article-toggle-button", function (event) { $(".docstring section").slideUp(animationSpeed); } else { $(this).removeClass("fa-chevron-down").addClass("fa-chevron-up"); - $(".docstring-article-toggle-button") + $("a.docstring-article-toggle-button") .removeClass("fa-chevron-right") .addClass("fa-chevron-down"); diff --git a/dev/assets/themes/catppuccin-frappe.css b/dev/assets/themes/catppuccin-frappe.css index 54a5b77..32e3f00 100644 --- a/dev/assets/themes/catppuccin-frappe.css +++ b/dev/assets/themes/catppuccin-frappe.css @@ -1 +1 @@ -html.theme--catppuccin-frappe .pagination-previous,html.theme--catppuccin-frappe .pagination-next,html.theme--catppuccin-frappe .pagination-link,html.theme--catppuccin-frappe .pagination-ellipsis,html.theme--catppuccin-frappe .file-cta,html.theme--catppuccin-frappe .file-name,html.theme--catppuccin-frappe .select 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.button.is-white.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(255,255,255,0.25)}html.theme--catppuccin-frappe .button.is-white:active,html.theme--catppuccin-frappe .button.is-white.is-active{background-color:#f2f2f2;border-color:transparent;color:#0a0a0a}html.theme--catppuccin-frappe .button.is-white[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-white{background-color:#fff;border-color:#fff;box-shadow:none}html.theme--catppuccin-frappe .button.is-white.is-inverted{background-color:#0a0a0a;color:#fff}html.theme--catppuccin-frappe .button.is-white.is-inverted:hover,html.theme--catppuccin-frappe .button.is-white.is-inverted.is-hovered{background-color:#000}html.theme--catppuccin-frappe .button.is-white.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-white.is-inverted{background-color:#0a0a0a;border-color:transparent;box-shadow:none;color:#fff}html.theme--catppuccin-frappe .button.is-white.is-loading::after{border-color:transparent transparent #0a0a0a #0a0a0a !important}html.theme--catppuccin-frappe .button.is-white.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-frappe .button.is-white.is-outlined:hover,html.theme--catppuccin-frappe .button.is-white.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-white.is-outlined:focus,html.theme--catppuccin-frappe .button.is-white.is-outlined.is-focused{background-color:#fff;border-color:#fff;color:#0a0a0a}html.theme--catppuccin-frappe .button.is-white.is-outlined.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-frappe .button.is-white.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-white.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-white.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-white.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #0a0a0a #0a0a0a !important}html.theme--catppuccin-frappe .button.is-white.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-white.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-frappe .button.is-white.is-inverted.is-outlined{background-color:transparent;border-color:#0a0a0a;color:#0a0a0a}html.theme--catppuccin-frappe .button.is-white.is-inverted.is-outlined:hover,html.theme--catppuccin-frappe .button.is-white.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-white.is-inverted.is-outlined:focus,html.theme--catppuccin-frappe .button.is-white.is-inverted.is-outlined.is-focused{background-color:#0a0a0a;color:#fff}html.theme--catppuccin-frappe .button.is-white.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-white.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-white.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-white.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-frappe .button.is-white.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-white.is-inverted.is-outlined{background-color:transparent;border-color:#0a0a0a;box-shadow:none;color:#0a0a0a}html.theme--catppuccin-frappe .button.is-black{background-color:#0a0a0a;border-color:transparent;color:#fff}html.theme--catppuccin-frappe .button.is-black:hover,html.theme--catppuccin-frappe .button.is-black.is-hovered{background-color:#040404;border-color:transparent;color:#fff}html.theme--catppuccin-frappe .button.is-black:focus,html.theme--catppuccin-frappe .button.is-black.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-frappe .button.is-black:focus:not(:active),html.theme--catppuccin-frappe .button.is-black.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(10,10,10,0.25)}html.theme--catppuccin-frappe .button.is-black:active,html.theme--catppuccin-frappe .button.is-black.is-active{background-color:#000;border-color:transparent;color:#fff}html.theme--catppuccin-frappe .button.is-black[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-black{background-color:#0a0a0a;border-color:#0a0a0a;box-shadow:none}html.theme--catppuccin-frappe .button.is-black.is-inverted{background-color:#fff;color:#0a0a0a}html.theme--catppuccin-frappe .button.is-black.is-inverted:hover,html.theme--catppuccin-frappe .button.is-black.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-frappe .button.is-black.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-black.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#0a0a0a}html.theme--catppuccin-frappe .button.is-black.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-frappe .button.is-black.is-outlined{background-color:transparent;border-color:#0a0a0a;color:#0a0a0a}html.theme--catppuccin-frappe .button.is-black.is-outlined:hover,html.theme--catppuccin-frappe .button.is-black.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-black.is-outlined:focus,html.theme--catppuccin-frappe .button.is-black.is-outlined.is-focused{background-color:#0a0a0a;border-color:#0a0a0a;color:#fff}html.theme--catppuccin-frappe .button.is-black.is-outlined.is-loading::after{border-color:transparent transparent #0a0a0a #0a0a0a !important}html.theme--catppuccin-frappe .button.is-black.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-black.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-black.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-black.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-frappe .button.is-black.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-black.is-outlined{background-color:transparent;border-color:#0a0a0a;box-shadow:none;color:#0a0a0a}html.theme--catppuccin-frappe .button.is-black.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-frappe .button.is-black.is-inverted.is-outlined:hover,html.theme--catppuccin-frappe .button.is-black.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-black.is-inverted.is-outlined:focus,html.theme--catppuccin-frappe .button.is-black.is-inverted.is-outlined.is-focused{background-color:#fff;color:#0a0a0a}html.theme--catppuccin-frappe .button.is-black.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-black.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-black.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-black.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #0a0a0a #0a0a0a !important}html.theme--catppuccin-frappe .button.is-black.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-black.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-frappe .button.is-light{background-color:#f5f5f5;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-light:hover,html.theme--catppuccin-frappe .button.is-light.is-hovered{background-color:#eee;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-light:focus,html.theme--catppuccin-frappe .button.is-light.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-light:focus:not(:active),html.theme--catppuccin-frappe .button.is-light.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(245,245,245,0.25)}html.theme--catppuccin-frappe .button.is-light:active,html.theme--catppuccin-frappe .button.is-light.is-active{background-color:#e8e8e8;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-light[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-light{background-color:#f5f5f5;border-color:#f5f5f5;box-shadow:none}html.theme--catppuccin-frappe .button.is-light.is-inverted{background-color:rgba(0,0,0,0.7);color:#f5f5f5}html.theme--catppuccin-frappe .button.is-light.is-inverted:hover,html.theme--catppuccin-frappe .button.is-light.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-light.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-light.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#f5f5f5}html.theme--catppuccin-frappe .button.is-light.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-frappe .button.is-light.is-outlined{background-color:transparent;border-color:#f5f5f5;color:#f5f5f5}html.theme--catppuccin-frappe .button.is-light.is-outlined:hover,html.theme--catppuccin-frappe .button.is-light.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-light.is-outlined:focus,html.theme--catppuccin-frappe .button.is-light.is-outlined.is-focused{background-color:#f5f5f5;border-color:#f5f5f5;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-light.is-outlined.is-loading::after{border-color:transparent transparent #f5f5f5 #f5f5f5 !important}html.theme--catppuccin-frappe .button.is-light.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-light.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-light.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-light.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-frappe .button.is-light.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-light.is-outlined{background-color:transparent;border-color:#f5f5f5;box-shadow:none;color:#f5f5f5}html.theme--catppuccin-frappe .button.is-light.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-light.is-inverted.is-outlined:hover,html.theme--catppuccin-frappe .button.is-light.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-light.is-inverted.is-outlined:focus,html.theme--catppuccin-frappe .button.is-light.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#f5f5f5}html.theme--catppuccin-frappe .button.is-light.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-light.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-light.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-light.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #f5f5f5 #f5f5f5 !important}html.theme--catppuccin-frappe .button.is-light.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-light.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-dark,html.theme--catppuccin-frappe .content kbd.button{background-color:#414559;border-color:transparent;color:#fff}html.theme--catppuccin-frappe .button.is-dark:hover,html.theme--catppuccin-frappe .content kbd.button:hover,html.theme--catppuccin-frappe .button.is-dark.is-hovered,html.theme--catppuccin-frappe .content kbd.button.is-hovered{background-color:#3c3f52;border-color:transparent;color:#fff}html.theme--catppuccin-frappe .button.is-dark:focus,html.theme--catppuccin-frappe .content kbd.button:focus,html.theme--catppuccin-frappe .button.is-dark.is-focused,html.theme--catppuccin-frappe .content kbd.button.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-frappe .button.is-dark:focus:not(:active),html.theme--catppuccin-frappe .content kbd.button:focus:not(:active),html.theme--catppuccin-frappe .button.is-dark.is-focused:not(:active),html.theme--catppuccin-frappe .content kbd.button.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(65,69,89,0.25)}html.theme--catppuccin-frappe .button.is-dark:active,html.theme--catppuccin-frappe .content kbd.button:active,html.theme--catppuccin-frappe .button.is-dark.is-active,html.theme--catppuccin-frappe .content kbd.button.is-active{background-color:#363a4a;border-color:transparent;color:#fff}html.theme--catppuccin-frappe .button.is-dark[disabled],html.theme--catppuccin-frappe .content kbd.button[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-dark,fieldset[disabled] html.theme--catppuccin-frappe .content kbd.button{background-color:#414559;border-color:#414559;box-shadow:none}html.theme--catppuccin-frappe .button.is-dark.is-inverted,html.theme--catppuccin-frappe .content kbd.button.is-inverted{background-color:#fff;color:#414559}html.theme--catppuccin-frappe .button.is-dark.is-inverted:hover,html.theme--catppuccin-frappe .content kbd.button.is-inverted:hover,html.theme--catppuccin-frappe .button.is-dark.is-inverted.is-hovered,html.theme--catppuccin-frappe .content kbd.button.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-frappe .button.is-dark.is-inverted[disabled],html.theme--catppuccin-frappe .content kbd.button.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-dark.is-inverted,fieldset[disabled] html.theme--catppuccin-frappe .content kbd.button.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#414559}html.theme--catppuccin-frappe .button.is-dark.is-loading::after,html.theme--catppuccin-frappe .content kbd.button.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-frappe .button.is-dark.is-outlined,html.theme--catppuccin-frappe .content kbd.button.is-outlined{background-color:transparent;border-color:#414559;color:#414559}html.theme--catppuccin-frappe .button.is-dark.is-outlined:hover,html.theme--catppuccin-frappe .content kbd.button.is-outlined:hover,html.theme--catppuccin-frappe .button.is-dark.is-outlined.is-hovered,html.theme--catppuccin-frappe .content kbd.button.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-dark.is-outlined:focus,html.theme--catppuccin-frappe .content kbd.button.is-outlined:focus,html.theme--catppuccin-frappe .button.is-dark.is-outlined.is-focused,html.theme--catppuccin-frappe .content kbd.button.is-outlined.is-focused{background-color:#414559;border-color:#414559;color:#fff}html.theme--catppuccin-frappe .button.is-dark.is-outlined.is-loading::after,html.theme--catppuccin-frappe .content kbd.button.is-outlined.is-loading::after{border-color:transparent transparent #414559 #414559 !important}html.theme--catppuccin-frappe .button.is-dark.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .content kbd.button.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-dark.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .content kbd.button.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-dark.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .content kbd.button.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-dark.is-outlined.is-loading.is-focused::after,html.theme--catppuccin-frappe .content kbd.button.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-frappe .button.is-dark.is-outlined[disabled],html.theme--catppuccin-frappe .content kbd.button.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-dark.is-outlined,fieldset[disabled] html.theme--catppuccin-frappe .content kbd.button.is-outlined{background-color:transparent;border-color:#414559;box-shadow:none;color:#414559}html.theme--catppuccin-frappe .button.is-dark.is-inverted.is-outlined,html.theme--catppuccin-frappe .content kbd.button.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-frappe .button.is-dark.is-inverted.is-outlined:hover,html.theme--catppuccin-frappe .content kbd.button.is-inverted.is-outlined:hover,html.theme--catppuccin-frappe .button.is-dark.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-frappe .content kbd.button.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-dark.is-inverted.is-outlined:focus,html.theme--catppuccin-frappe .content kbd.button.is-inverted.is-outlined:focus,html.theme--catppuccin-frappe .button.is-dark.is-inverted.is-outlined.is-focused,html.theme--catppuccin-frappe .content kbd.button.is-inverted.is-outlined.is-focused{background-color:#fff;color:#414559}html.theme--catppuccin-frappe .button.is-dark.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .content kbd.button.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-dark.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .content kbd.button.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-dark.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .content kbd.button.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-dark.is-inverted.is-outlined.is-loading.is-focused::after,html.theme--catppuccin-frappe .content kbd.button.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #414559 #414559 !important}html.theme--catppuccin-frappe .button.is-dark.is-inverted.is-outlined[disabled],html.theme--catppuccin-frappe .content kbd.button.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-dark.is-inverted.is-outlined,fieldset[disabled] html.theme--catppuccin-frappe .content kbd.button.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-frappe .button.is-primary,html.theme--catppuccin-frappe .docstring>section>a.button.docs-sourcelink{background-color:#8caaee;border-color:transparent;color:#fff}html.theme--catppuccin-frappe .button.is-primary:hover,html.theme--catppuccin-frappe .docstring>section>a.button.docs-sourcelink:hover,html.theme--catppuccin-frappe .button.is-primary.is-hovered,html.theme--catppuccin-frappe .docstring>section>a.button.is-hovered.docs-sourcelink{background-color:#81a2ec;border-color:transparent;color:#fff}html.theme--catppuccin-frappe .button.is-primary:focus,html.theme--catppuccin-frappe 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.button.is-info.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-info.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #81c8be #81c8be !important}html.theme--catppuccin-frappe .button.is-info.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-info.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-info.is-light{background-color:#f1f9f8;color:#2d675f}html.theme--catppuccin-frappe .button.is-info.is-light:hover,html.theme--catppuccin-frappe .button.is-info.is-light.is-hovered{background-color:#e8f5f3;border-color:transparent;color:#2d675f}html.theme--catppuccin-frappe .button.is-info.is-light:active,html.theme--catppuccin-frappe .button.is-info.is-light.is-active{background-color:#dff1ef;border-color:transparent;color:#2d675f}html.theme--catppuccin-frappe .button.is-success{background-color:#a6d189;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-success:hover,html.theme--catppuccin-frappe .button.is-success.is-hovered{background-color:#9fcd80;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-success:focus,html.theme--catppuccin-frappe .button.is-success.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-success:focus:not(:active),html.theme--catppuccin-frappe .button.is-success.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(166,209,137,0.25)}html.theme--catppuccin-frappe .button.is-success:active,html.theme--catppuccin-frappe .button.is-success.is-active{background-color:#98ca77;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-success[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-success{background-color:#a6d189;border-color:#a6d189;box-shadow:none}html.theme--catppuccin-frappe .button.is-success.is-inverted{background-color:rgba(0,0,0,0.7);color:#a6d189}html.theme--catppuccin-frappe .button.is-success.is-inverted:hover,html.theme--catppuccin-frappe .button.is-success.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-success.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-success.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#a6d189}html.theme--catppuccin-frappe .button.is-success.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-frappe .button.is-success.is-outlined{background-color:transparent;border-color:#a6d189;color:#a6d189}html.theme--catppuccin-frappe .button.is-success.is-outlined:hover,html.theme--catppuccin-frappe .button.is-success.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-success.is-outlined:focus,html.theme--catppuccin-frappe .button.is-success.is-outlined.is-focused{background-color:#a6d189;border-color:#a6d189;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-success.is-outlined.is-loading::after{border-color:transparent transparent #a6d189 #a6d189 !important}html.theme--catppuccin-frappe .button.is-success.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-success.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-success.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-success.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-frappe .button.is-success.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-success.is-outlined{background-color:transparent;border-color:#a6d189;box-shadow:none;color:#a6d189}html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined:hover,html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined:focus,html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#a6d189}html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #a6d189 #a6d189 !important}html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-success.is-light{background-color:#f4f9f0;color:#446a29}html.theme--catppuccin-frappe .button.is-success.is-light:hover,html.theme--catppuccin-frappe .button.is-success.is-light.is-hovered{background-color:#edf6e7;border-color:transparent;color:#446a29}html.theme--catppuccin-frappe .button.is-success.is-light:active,html.theme--catppuccin-frappe .button.is-success.is-light.is-active{background-color:#e6f2de;border-color:transparent;color:#446a29}html.theme--catppuccin-frappe .button.is-warning{background-color:#e5c890;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-warning:hover,html.theme--catppuccin-frappe .button.is-warning.is-hovered{background-color:#e3c386;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-warning:focus,html.theme--catppuccin-frappe .button.is-warning.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-warning:focus:not(:active),html.theme--catppuccin-frappe .button.is-warning.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(229,200,144,0.25)}html.theme--catppuccin-frappe .button.is-warning:active,html.theme--catppuccin-frappe .button.is-warning.is-active{background-color:#e0be7b;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-warning[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-warning{background-color:#e5c890;border-color:#e5c890;box-shadow:none}html.theme--catppuccin-frappe .button.is-warning.is-inverted{background-color:rgba(0,0,0,0.7);color:#e5c890}html.theme--catppuccin-frappe .button.is-warning.is-inverted:hover,html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-warning.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-warning.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#e5c890}html.theme--catppuccin-frappe .button.is-warning.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-frappe .button.is-warning.is-outlined{background-color:transparent;border-color:#e5c890;color:#e5c890}html.theme--catppuccin-frappe .button.is-warning.is-outlined:hover,html.theme--catppuccin-frappe .button.is-warning.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-warning.is-outlined:focus,html.theme--catppuccin-frappe .button.is-warning.is-outlined.is-focused{background-color:#e5c890;border-color:#e5c890;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-warning.is-outlined.is-loading::after{border-color:transparent transparent #e5c890 #e5c890 !important}html.theme--catppuccin-frappe .button.is-warning.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-warning.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-warning.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-warning.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-frappe .button.is-warning.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-warning.is-outlined{background-color:transparent;border-color:#e5c890;box-shadow:none;color:#e5c890}html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined:hover,html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined:focus,html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#e5c890}html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #e5c890 #e5c890 !important}html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-warning.is-light{background-color:#fbf7ee;color:#78591c}html.theme--catppuccin-frappe .button.is-warning.is-light:hover,html.theme--catppuccin-frappe .button.is-warning.is-light.is-hovered{background-color:#f9f2e4;border-color:transparent;color:#78591c}html.theme--catppuccin-frappe .button.is-warning.is-light:active,html.theme--catppuccin-frappe .button.is-warning.is-light.is-active{background-color:#f6edda;border-color:transparent;color:#78591c}html.theme--catppuccin-frappe .button.is-danger{background-color:#e78284;border-color:transparent;color:#fff}html.theme--catppuccin-frappe .button.is-danger:hover,html.theme--catppuccin-frappe .button.is-danger.is-hovered{background-color:#e57779;border-color:transparent;color:#fff}html.theme--catppuccin-frappe .button.is-danger:focus,html.theme--catppuccin-frappe .button.is-danger.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-frappe .button.is-danger:focus:not(:active),html.theme--catppuccin-frappe .button.is-danger.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(231,130,132,0.25)}html.theme--catppuccin-frappe .button.is-danger:active,html.theme--catppuccin-frappe .button.is-danger.is-active{background-color:#e36d6f;border-color:transparent;color:#fff}html.theme--catppuccin-frappe .button.is-danger[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-danger{background-color:#e78284;border-color:#e78284;box-shadow:none}html.theme--catppuccin-frappe .button.is-danger.is-inverted{background-color:#fff;color:#e78284}html.theme--catppuccin-frappe .button.is-danger.is-inverted:hover,html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-frappe .button.is-danger.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-danger.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#e78284}html.theme--catppuccin-frappe .button.is-danger.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-frappe .button.is-danger.is-outlined{background-color:transparent;border-color:#e78284;color:#e78284}html.theme--catppuccin-frappe .button.is-danger.is-outlined:hover,html.theme--catppuccin-frappe .button.is-danger.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-danger.is-outlined:focus,html.theme--catppuccin-frappe .button.is-danger.is-outlined.is-focused{background-color:#e78284;border-color:#e78284;color:#fff}html.theme--catppuccin-frappe .button.is-danger.is-outlined.is-loading::after{border-color:transparent transparent #e78284 #e78284 !important}html.theme--catppuccin-frappe .button.is-danger.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-danger.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-danger.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-danger.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-frappe .button.is-danger.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-danger.is-outlined{background-color:transparent;border-color:#e78284;box-shadow:none;color:#e78284}html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined:hover,html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined:focus,html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined.is-focused{background-color:#fff;color:#e78284}html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #e78284 #e78284 !important}html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-frappe .button.is-danger.is-light{background-color:#fceeee;color:#9a1e20}html.theme--catppuccin-frappe .button.is-danger.is-light:hover,html.theme--catppuccin-frappe .button.is-danger.is-light.is-hovered{background-color:#fae3e4;border-color:transparent;color:#9a1e20}html.theme--catppuccin-frappe .button.is-danger.is-light:active,html.theme--catppuccin-frappe .button.is-danger.is-light.is-active{background-color:#f8d8d9;border-color:transparent;color:#9a1e20}html.theme--catppuccin-frappe .button.is-small,html.theme--catppuccin-frappe #documenter .docs-sidebar form.docs-search>input.button{font-size:.75rem}html.theme--catppuccin-frappe .button.is-small:not(.is-rounded),html.theme--catppuccin-frappe #documenter .docs-sidebar form.docs-search>input.button:not(.is-rounded){border-radius:3px}html.theme--catppuccin-frappe .button.is-normal{font-size:1rem}html.theme--catppuccin-frappe .button.is-medium{font-size:1.25rem}html.theme--catppuccin-frappe .button.is-large{font-size:1.5rem}html.theme--catppuccin-frappe .button[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button{background-color:#737994;border-color:#626880;box-shadow:none;opacity:.5}html.theme--catppuccin-frappe .button.is-fullwidth{display:flex;width:100%}html.theme--catppuccin-frappe .button.is-loading{color:transparent !important;pointer-events:none}html.theme--catppuccin-frappe .button.is-loading::after{position:absolute;left:calc(50% - (1em * 0.5));top:calc(50% - (1em * 0.5));position:absolute !important}html.theme--catppuccin-frappe .button.is-static{background-color:#292c3c;border-color:#626880;color:#838ba7;box-shadow:none;pointer-events:none}html.theme--catppuccin-frappe .button.is-rounded,html.theme--catppuccin-frappe #documenter .docs-sidebar form.docs-search>input.button{border-radius:9999px;padding-left:calc(1em + 0.25em);padding-right:calc(1em + 0.25em)}html.theme--catppuccin-frappe .buttons{align-items:center;display:flex;flex-wrap:wrap;justify-content:flex-start}html.theme--catppuccin-frappe .buttons .button{margin-bottom:0.5rem}html.theme--catppuccin-frappe .buttons .button:not(:last-child):not(.is-fullwidth){margin-right:.5rem}html.theme--catppuccin-frappe .buttons:last-child{margin-bottom:-0.5rem}html.theme--catppuccin-frappe .buttons:not(:last-child){margin-bottom:1rem}html.theme--catppuccin-frappe .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large){font-size:.75rem}html.theme--catppuccin-frappe .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large):not(.is-rounded){border-radius:3px}html.theme--catppuccin-frappe .buttons.are-medium .button:not(.is-small):not(.is-normal):not(.is-large){font-size:1.25rem}html.theme--catppuccin-frappe .buttons.are-large .button:not(.is-small):not(.is-normal):not(.is-medium){font-size:1.5rem}html.theme--catppuccin-frappe .buttons.has-addons .button:not(:first-child){border-bottom-left-radius:0;border-top-left-radius:0}html.theme--catppuccin-frappe .buttons.has-addons .button:not(:last-child){border-bottom-right-radius:0;border-top-right-radius:0;margin-right:-1px}html.theme--catppuccin-frappe .buttons.has-addons .button:last-child{margin-right:0}html.theme--catppuccin-frappe .buttons.has-addons .button:hover,html.theme--catppuccin-frappe .buttons.has-addons .button.is-hovered{z-index:2}html.theme--catppuccin-frappe .buttons.has-addons .button:focus,html.theme--catppuccin-frappe .buttons.has-addons .button.is-focused,html.theme--catppuccin-frappe .buttons.has-addons .button:active,html.theme--catppuccin-frappe .buttons.has-addons .button.is-active,html.theme--catppuccin-frappe .buttons.has-addons .button.is-selected{z-index:3}html.theme--catppuccin-frappe .buttons.has-addons .button:focus:hover,html.theme--catppuccin-frappe .buttons.has-addons .button.is-focused:hover,html.theme--catppuccin-frappe .buttons.has-addons .button:active:hover,html.theme--catppuccin-frappe .buttons.has-addons .button.is-active:hover,html.theme--catppuccin-frappe .buttons.has-addons .button.is-selected:hover{z-index:4}html.theme--catppuccin-frappe .buttons.has-addons .button.is-expanded{flex-grow:1;flex-shrink:1}html.theme--catppuccin-frappe .buttons.is-centered{justify-content:center}html.theme--catppuccin-frappe .buttons.is-centered:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}html.theme--catppuccin-frappe .buttons.is-right{justify-content:flex-end}html.theme--catppuccin-frappe .buttons.is-right:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}@media screen and (max-width: 768px){html.theme--catppuccin-frappe .button.is-responsive.is-small,html.theme--catppuccin-frappe #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.5625rem}html.theme--catppuccin-frappe .button.is-responsive,html.theme--catppuccin-frappe .button.is-responsive.is-normal{font-size:.65625rem}html.theme--catppuccin-frappe .button.is-responsive.is-medium{font-size:.75rem}html.theme--catppuccin-frappe .button.is-responsive.is-large{font-size:1rem}}@media screen and (min-width: 769px) and (max-width: 1055px){html.theme--catppuccin-frappe .button.is-responsive.is-small,html.theme--catppuccin-frappe #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.65625rem}html.theme--catppuccin-frappe .button.is-responsive,html.theme--catppuccin-frappe .button.is-responsive.is-normal{font-size:.75rem}html.theme--catppuccin-frappe .button.is-responsive.is-medium{font-size:1rem}html.theme--catppuccin-frappe .button.is-responsive.is-large{font-size:1.25rem}}html.theme--catppuccin-frappe .container{flex-grow:1;margin:0 auto;position:relative;width:auto}html.theme--catppuccin-frappe 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h3{font-size:1.5em;margin-bottom:0.6666em}html.theme--catppuccin-frappe .content h3:not(:first-child){margin-top:1.3333em}html.theme--catppuccin-frappe .content h4{font-size:1.25em;margin-bottom:0.8em}html.theme--catppuccin-frappe .content h5{font-size:1.125em;margin-bottom:0.8888em}html.theme--catppuccin-frappe .content h6{font-size:1em;margin-bottom:1em}html.theme--catppuccin-frappe .content blockquote{background-color:#292c3c;border-left:5px solid #626880;padding:1.25em 1.5em}html.theme--catppuccin-frappe .content ol{list-style-position:outside;margin-left:2em;margin-top:1em}html.theme--catppuccin-frappe .content ol:not([type]){list-style-type:decimal}html.theme--catppuccin-frappe .content ol.is-lower-alpha:not([type]){list-style-type:lower-alpha}html.theme--catppuccin-frappe .content ol.is-lower-roman:not([type]){list-style-type:lower-roman}html.theme--catppuccin-frappe .content ol.is-upper-alpha:not([type]){list-style-type:upper-alpha}html.theme--catppuccin-frappe .content 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.has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-2by1 .has-ratio,html.theme--catppuccin-frappe .image.is-3by1 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-3by1 img,html.theme--catppuccin-frappe .image.is-3by1 .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-3by1 .has-ratio,html.theme--catppuccin-frappe .image.is-4by5 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-4by5 img,html.theme--catppuccin-frappe .image.is-4by5 .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-4by5 .has-ratio,html.theme--catppuccin-frappe .image.is-3by4 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-3by4 img,html.theme--catppuccin-frappe .image.is-3by4 .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-3by4 .has-ratio,html.theme--catppuccin-frappe .image.is-2by3 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-2by3 img,html.theme--catppuccin-frappe .image.is-2by3 .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-2by3 .has-ratio,html.theme--catppuccin-frappe .image.is-3by5 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-3by5 img,html.theme--catppuccin-frappe .image.is-3by5 .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-3by5 .has-ratio,html.theme--catppuccin-frappe .image.is-9by16 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-9by16 img,html.theme--catppuccin-frappe .image.is-9by16 .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-9by16 .has-ratio,html.theme--catppuccin-frappe .image.is-1by2 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-1by2 img,html.theme--catppuccin-frappe .image.is-1by2 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.button.is-white.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(255,255,255,0.25)}html.theme--catppuccin-frappe .button.is-white:active,html.theme--catppuccin-frappe .button.is-white.is-active{background-color:#f2f2f2;border-color:transparent;color:#0a0a0a}html.theme--catppuccin-frappe .button.is-white[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-white{background-color:#fff;border-color:#fff;box-shadow:none}html.theme--catppuccin-frappe .button.is-white.is-inverted{background-color:#0a0a0a;color:#fff}html.theme--catppuccin-frappe .button.is-white.is-inverted:hover,html.theme--catppuccin-frappe .button.is-white.is-inverted.is-hovered{background-color:#000}html.theme--catppuccin-frappe .button.is-white.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-white.is-inverted{background-color:#0a0a0a;border-color:transparent;box-shadow:none;color:#fff}html.theme--catppuccin-frappe .button.is-white.is-loading::after{border-color:transparent transparent #0a0a0a #0a0a0a !important}html.theme--catppuccin-frappe .button.is-white.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-frappe .button.is-white.is-outlined:hover,html.theme--catppuccin-frappe .button.is-white.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-white.is-outlined:focus,html.theme--catppuccin-frappe .button.is-white.is-outlined.is-focused{background-color:#fff;border-color:#fff;color:#0a0a0a}html.theme--catppuccin-frappe .button.is-white.is-outlined.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-frappe .button.is-white.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-white.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-white.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-white.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #0a0a0a #0a0a0a !important}html.theme--catppuccin-frappe .button.is-white.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-white.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-frappe .button.is-white.is-inverted.is-outlined{background-color:transparent;border-color:#0a0a0a;color:#0a0a0a}html.theme--catppuccin-frappe .button.is-white.is-inverted.is-outlined:hover,html.theme--catppuccin-frappe .button.is-white.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-white.is-inverted.is-outlined:focus,html.theme--catppuccin-frappe .button.is-white.is-inverted.is-outlined.is-focused{background-color:#0a0a0a;color:#fff}html.theme--catppuccin-frappe .button.is-white.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-white.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-white.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-white.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-frappe .button.is-white.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-white.is-inverted.is-outlined{background-color:transparent;border-color:#0a0a0a;box-shadow:none;color:#0a0a0a}html.theme--catppuccin-frappe .button.is-black{background-color:#0a0a0a;border-color:transparent;color:#fff}html.theme--catppuccin-frappe .button.is-black:hover,html.theme--catppuccin-frappe .button.is-black.is-hovered{background-color:#040404;border-color:transparent;color:#fff}html.theme--catppuccin-frappe .button.is-black:focus,html.theme--catppuccin-frappe .button.is-black.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-frappe .button.is-black:focus:not(:active),html.theme--catppuccin-frappe .button.is-black.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(10,10,10,0.25)}html.theme--catppuccin-frappe .button.is-black:active,html.theme--catppuccin-frappe .button.is-black.is-active{background-color:#000;border-color:transparent;color:#fff}html.theme--catppuccin-frappe .button.is-black[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-black{background-color:#0a0a0a;border-color:#0a0a0a;box-shadow:none}html.theme--catppuccin-frappe .button.is-black.is-inverted{background-color:#fff;color:#0a0a0a}html.theme--catppuccin-frappe .button.is-black.is-inverted:hover,html.theme--catppuccin-frappe .button.is-black.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-frappe .button.is-black.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-black.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#0a0a0a}html.theme--catppuccin-frappe .button.is-black.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-frappe .button.is-black.is-outlined{background-color:transparent;border-color:#0a0a0a;color:#0a0a0a}html.theme--catppuccin-frappe .button.is-black.is-outlined:hover,html.theme--catppuccin-frappe .button.is-black.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-black.is-outlined:focus,html.theme--catppuccin-frappe .button.is-black.is-outlined.is-focused{background-color:#0a0a0a;border-color:#0a0a0a;color:#fff}html.theme--catppuccin-frappe .button.is-black.is-outlined.is-loading::after{border-color:transparent transparent #0a0a0a #0a0a0a !important}html.theme--catppuccin-frappe .button.is-black.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-black.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-black.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-black.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-frappe .button.is-black.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-black.is-outlined{background-color:transparent;border-color:#0a0a0a;box-shadow:none;color:#0a0a0a}html.theme--catppuccin-frappe .button.is-black.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-frappe .button.is-black.is-inverted.is-outlined:hover,html.theme--catppuccin-frappe .button.is-black.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-black.is-inverted.is-outlined:focus,html.theme--catppuccin-frappe .button.is-black.is-inverted.is-outlined.is-focused{background-color:#fff;color:#0a0a0a}html.theme--catppuccin-frappe .button.is-black.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-black.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-black.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-black.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #0a0a0a #0a0a0a !important}html.theme--catppuccin-frappe .button.is-black.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-black.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-frappe .button.is-light{background-color:#f5f5f5;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-light:hover,html.theme--catppuccin-frappe .button.is-light.is-hovered{background-color:#eee;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-light:focus,html.theme--catppuccin-frappe .button.is-light.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-light:focus:not(:active),html.theme--catppuccin-frappe .button.is-light.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(245,245,245,0.25)}html.theme--catppuccin-frappe .button.is-light:active,html.theme--catppuccin-frappe .button.is-light.is-active{background-color:#e8e8e8;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-light[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-light{background-color:#f5f5f5;border-color:#f5f5f5;box-shadow:none}html.theme--catppuccin-frappe .button.is-light.is-inverted{background-color:rgba(0,0,0,0.7);color:#f5f5f5}html.theme--catppuccin-frappe .button.is-light.is-inverted:hover,html.theme--catppuccin-frappe .button.is-light.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-light.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-light.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#f5f5f5}html.theme--catppuccin-frappe .button.is-light.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-frappe .button.is-light.is-outlined{background-color:transparent;border-color:#f5f5f5;color:#f5f5f5}html.theme--catppuccin-frappe .button.is-light.is-outlined:hover,html.theme--catppuccin-frappe .button.is-light.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-light.is-outlined:focus,html.theme--catppuccin-frappe .button.is-light.is-outlined.is-focused{background-color:#f5f5f5;border-color:#f5f5f5;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-light.is-outlined.is-loading::after{border-color:transparent transparent #f5f5f5 #f5f5f5 !important}html.theme--catppuccin-frappe .button.is-light.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-light.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-light.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-light.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-frappe .button.is-light.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-light.is-outlined{background-color:transparent;border-color:#f5f5f5;box-shadow:none;color:#f5f5f5}html.theme--catppuccin-frappe .button.is-light.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-light.is-inverted.is-outlined:hover,html.theme--catppuccin-frappe .button.is-light.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-light.is-inverted.is-outlined:focus,html.theme--catppuccin-frappe .button.is-light.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#f5f5f5}html.theme--catppuccin-frappe .button.is-light.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-light.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-light.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-light.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #f5f5f5 #f5f5f5 !important}html.theme--catppuccin-frappe .button.is-light.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-light.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-dark,html.theme--catppuccin-frappe .content kbd.button{background-color:#414559;border-color:transparent;color:#fff}html.theme--catppuccin-frappe .button.is-dark:hover,html.theme--catppuccin-frappe .content kbd.button:hover,html.theme--catppuccin-frappe .button.is-dark.is-hovered,html.theme--catppuccin-frappe .content kbd.button.is-hovered{background-color:#3c3f52;border-color:transparent;color:#fff}html.theme--catppuccin-frappe .button.is-dark:focus,html.theme--catppuccin-frappe .content kbd.button:focus,html.theme--catppuccin-frappe .button.is-dark.is-focused,html.theme--catppuccin-frappe .content kbd.button.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-frappe .button.is-dark:focus:not(:active),html.theme--catppuccin-frappe .content kbd.button:focus:not(:active),html.theme--catppuccin-frappe .button.is-dark.is-focused:not(:active),html.theme--catppuccin-frappe .content kbd.button.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(65,69,89,0.25)}html.theme--catppuccin-frappe .button.is-dark:active,html.theme--catppuccin-frappe .content kbd.button:active,html.theme--catppuccin-frappe .button.is-dark.is-active,html.theme--catppuccin-frappe .content kbd.button.is-active{background-color:#363a4a;border-color:transparent;color:#fff}html.theme--catppuccin-frappe .button.is-dark[disabled],html.theme--catppuccin-frappe .content kbd.button[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-dark,fieldset[disabled] html.theme--catppuccin-frappe .content kbd.button{background-color:#414559;border-color:#414559;box-shadow:none}html.theme--catppuccin-frappe .button.is-dark.is-inverted,html.theme--catppuccin-frappe .content kbd.button.is-inverted{background-color:#fff;color:#414559}html.theme--catppuccin-frappe .button.is-dark.is-inverted:hover,html.theme--catppuccin-frappe .content kbd.button.is-inverted:hover,html.theme--catppuccin-frappe .button.is-dark.is-inverted.is-hovered,html.theme--catppuccin-frappe .content kbd.button.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-frappe .button.is-dark.is-inverted[disabled],html.theme--catppuccin-frappe .content kbd.button.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-dark.is-inverted,fieldset[disabled] html.theme--catppuccin-frappe .content kbd.button.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#414559}html.theme--catppuccin-frappe .button.is-dark.is-loading::after,html.theme--catppuccin-frappe .content kbd.button.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-frappe .button.is-dark.is-outlined,html.theme--catppuccin-frappe .content kbd.button.is-outlined{background-color:transparent;border-color:#414559;color:#414559}html.theme--catppuccin-frappe .button.is-dark.is-outlined:hover,html.theme--catppuccin-frappe .content kbd.button.is-outlined:hover,html.theme--catppuccin-frappe .button.is-dark.is-outlined.is-hovered,html.theme--catppuccin-frappe .content kbd.button.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-dark.is-outlined:focus,html.theme--catppuccin-frappe .content kbd.button.is-outlined:focus,html.theme--catppuccin-frappe .button.is-dark.is-outlined.is-focused,html.theme--catppuccin-frappe .content kbd.button.is-outlined.is-focused{background-color:#414559;border-color:#414559;color:#fff}html.theme--catppuccin-frappe .button.is-dark.is-outlined.is-loading::after,html.theme--catppuccin-frappe .content kbd.button.is-outlined.is-loading::after{border-color:transparent transparent #414559 #414559 !important}html.theme--catppuccin-frappe .button.is-dark.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .content kbd.button.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-dark.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .content kbd.button.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-dark.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .content kbd.button.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-dark.is-outlined.is-loading.is-focused::after,html.theme--catppuccin-frappe .content kbd.button.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-frappe .button.is-dark.is-outlined[disabled],html.theme--catppuccin-frappe .content kbd.button.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-dark.is-outlined,fieldset[disabled] html.theme--catppuccin-frappe .content kbd.button.is-outlined{background-color:transparent;border-color:#414559;box-shadow:none;color:#414559}html.theme--catppuccin-frappe .button.is-dark.is-inverted.is-outlined,html.theme--catppuccin-frappe .content kbd.button.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-frappe .button.is-dark.is-inverted.is-outlined:hover,html.theme--catppuccin-frappe .content kbd.button.is-inverted.is-outlined:hover,html.theme--catppuccin-frappe .button.is-dark.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-frappe .content kbd.button.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-dark.is-inverted.is-outlined:focus,html.theme--catppuccin-frappe .content kbd.button.is-inverted.is-outlined:focus,html.theme--catppuccin-frappe .button.is-dark.is-inverted.is-outlined.is-focused,html.theme--catppuccin-frappe .content kbd.button.is-inverted.is-outlined.is-focused{background-color:#fff;color:#414559}html.theme--catppuccin-frappe .button.is-dark.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .content kbd.button.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-dark.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .content kbd.button.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-dark.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .content kbd.button.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-dark.is-inverted.is-outlined.is-loading.is-focused::after,html.theme--catppuccin-frappe .content kbd.button.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #414559 #414559 !important}html.theme--catppuccin-frappe .button.is-dark.is-inverted.is-outlined[disabled],html.theme--catppuccin-frappe .content kbd.button.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-dark.is-inverted.is-outlined,fieldset[disabled] html.theme--catppuccin-frappe .content kbd.button.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-frappe .button.is-primary,html.theme--catppuccin-frappe .docstring>section>a.button.docs-sourcelink{background-color:#8caaee;border-color:transparent;color:#fff}html.theme--catppuccin-frappe .button.is-primary:hover,html.theme--catppuccin-frappe .docstring>section>a.button.docs-sourcelink:hover,html.theme--catppuccin-frappe .button.is-primary.is-hovered,html.theme--catppuccin-frappe .docstring>section>a.button.is-hovered.docs-sourcelink{background-color:#81a2ec;border-color:transparent;color:#fff}html.theme--catppuccin-frappe .button.is-primary:focus,html.theme--catppuccin-frappe 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.button.is-info.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-info.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #81c8be #81c8be !important}html.theme--catppuccin-frappe .button.is-info.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-info.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-info.is-light{background-color:#f1f9f8;color:#2d675f}html.theme--catppuccin-frappe .button.is-info.is-light:hover,html.theme--catppuccin-frappe .button.is-info.is-light.is-hovered{background-color:#e8f5f3;border-color:transparent;color:#2d675f}html.theme--catppuccin-frappe .button.is-info.is-light:active,html.theme--catppuccin-frappe .button.is-info.is-light.is-active{background-color:#dff1ef;border-color:transparent;color:#2d675f}html.theme--catppuccin-frappe .button.is-success{background-color:#a6d189;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-success:hover,html.theme--catppuccin-frappe .button.is-success.is-hovered{background-color:#9fcd80;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-success:focus,html.theme--catppuccin-frappe .button.is-success.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-success:focus:not(:active),html.theme--catppuccin-frappe .button.is-success.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(166,209,137,0.25)}html.theme--catppuccin-frappe .button.is-success:active,html.theme--catppuccin-frappe .button.is-success.is-active{background-color:#98ca77;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-success[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-success{background-color:#a6d189;border-color:#a6d189;box-shadow:none}html.theme--catppuccin-frappe .button.is-success.is-inverted{background-color:rgba(0,0,0,0.7);color:#a6d189}html.theme--catppuccin-frappe .button.is-success.is-inverted:hover,html.theme--catppuccin-frappe .button.is-success.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-success.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-success.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#a6d189}html.theme--catppuccin-frappe .button.is-success.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-frappe .button.is-success.is-outlined{background-color:transparent;border-color:#a6d189;color:#a6d189}html.theme--catppuccin-frappe .button.is-success.is-outlined:hover,html.theme--catppuccin-frappe .button.is-success.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-success.is-outlined:focus,html.theme--catppuccin-frappe .button.is-success.is-outlined.is-focused{background-color:#a6d189;border-color:#a6d189;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-success.is-outlined.is-loading::after{border-color:transparent transparent #a6d189 #a6d189 !important}html.theme--catppuccin-frappe .button.is-success.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-success.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-success.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-success.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-frappe .button.is-success.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-success.is-outlined{background-color:transparent;border-color:#a6d189;box-shadow:none;color:#a6d189}html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined:hover,html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined:focus,html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#a6d189}html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #a6d189 #a6d189 !important}html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-success.is-light{background-color:#f4f9f0;color:#446a29}html.theme--catppuccin-frappe .button.is-success.is-light:hover,html.theme--catppuccin-frappe .button.is-success.is-light.is-hovered{background-color:#edf6e7;border-color:transparent;color:#446a29}html.theme--catppuccin-frappe .button.is-success.is-light:active,html.theme--catppuccin-frappe .button.is-success.is-light.is-active{background-color:#e6f2de;border-color:transparent;color:#446a29}html.theme--catppuccin-frappe .button.is-warning{background-color:#e5c890;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-warning:hover,html.theme--catppuccin-frappe .button.is-warning.is-hovered{background-color:#e3c386;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-warning:focus,html.theme--catppuccin-frappe .button.is-warning.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-warning:focus:not(:active),html.theme--catppuccin-frappe .button.is-warning.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(229,200,144,0.25)}html.theme--catppuccin-frappe .button.is-warning:active,html.theme--catppuccin-frappe .button.is-warning.is-active{background-color:#e0be7b;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-warning[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-warning{background-color:#e5c890;border-color:#e5c890;box-shadow:none}html.theme--catppuccin-frappe .button.is-warning.is-inverted{background-color:rgba(0,0,0,0.7);color:#e5c890}html.theme--catppuccin-frappe .button.is-warning.is-inverted:hover,html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-warning.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-warning.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#e5c890}html.theme--catppuccin-frappe .button.is-warning.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-frappe .button.is-warning.is-outlined{background-color:transparent;border-color:#e5c890;color:#e5c890}html.theme--catppuccin-frappe .button.is-warning.is-outlined:hover,html.theme--catppuccin-frappe .button.is-warning.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-warning.is-outlined:focus,html.theme--catppuccin-frappe .button.is-warning.is-outlined.is-focused{background-color:#e5c890;border-color:#e5c890;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-warning.is-outlined.is-loading::after{border-color:transparent transparent #e5c890 #e5c890 !important}html.theme--catppuccin-frappe .button.is-warning.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-warning.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-warning.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-warning.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-frappe .button.is-warning.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-warning.is-outlined{background-color:transparent;border-color:#e5c890;box-shadow:none;color:#e5c890}html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined:hover,html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined:focus,html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#e5c890}html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #e5c890 #e5c890 !important}html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-warning.is-light{background-color:#fbf7ee;color:#78591c}html.theme--catppuccin-frappe .button.is-warning.is-light:hover,html.theme--catppuccin-frappe .button.is-warning.is-light.is-hovered{background-color:#f9f2e4;border-color:transparent;color:#78591c}html.theme--catppuccin-frappe .button.is-warning.is-light:active,html.theme--catppuccin-frappe .button.is-warning.is-light.is-active{background-color:#f6edda;border-color:transparent;color:#78591c}html.theme--catppuccin-frappe .button.is-danger{background-color:#e78284;border-color:transparent;color:#fff}html.theme--catppuccin-frappe .button.is-danger:hover,html.theme--catppuccin-frappe .button.is-danger.is-hovered{background-color:#e57779;border-color:transparent;color:#fff}html.theme--catppuccin-frappe .button.is-danger:focus,html.theme--catppuccin-frappe .button.is-danger.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-frappe .button.is-danger:focus:not(:active),html.theme--catppuccin-frappe .button.is-danger.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(231,130,132,0.25)}html.theme--catppuccin-frappe .button.is-danger:active,html.theme--catppuccin-frappe .button.is-danger.is-active{background-color:#e36d6f;border-color:transparent;color:#fff}html.theme--catppuccin-frappe .button.is-danger[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-danger{background-color:#e78284;border-color:#e78284;box-shadow:none}html.theme--catppuccin-frappe .button.is-danger.is-inverted{background-color:#fff;color:#e78284}html.theme--catppuccin-frappe .button.is-danger.is-inverted:hover,html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-frappe .button.is-danger.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-danger.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#e78284}html.theme--catppuccin-frappe .button.is-danger.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-frappe .button.is-danger.is-outlined{background-color:transparent;border-color:#e78284;color:#e78284}html.theme--catppuccin-frappe .button.is-danger.is-outlined:hover,html.theme--catppuccin-frappe .button.is-danger.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-danger.is-outlined:focus,html.theme--catppuccin-frappe .button.is-danger.is-outlined.is-focused{background-color:#e78284;border-color:#e78284;color:#fff}html.theme--catppuccin-frappe .button.is-danger.is-outlined.is-loading::after{border-color:transparent transparent #e78284 #e78284 !important}html.theme--catppuccin-frappe .button.is-danger.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-danger.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-danger.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-danger.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-frappe .button.is-danger.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-danger.is-outlined{background-color:transparent;border-color:#e78284;box-shadow:none;color:#e78284}html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined:hover,html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined:focus,html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined.is-focused{background-color:#fff;color:#e78284}html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #e78284 #e78284 !important}html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-frappe .button.is-danger.is-light{background-color:#fceeee;color:#9a1e20}html.theme--catppuccin-frappe .button.is-danger.is-light:hover,html.theme--catppuccin-frappe .button.is-danger.is-light.is-hovered{background-color:#fae3e4;border-color:transparent;color:#9a1e20}html.theme--catppuccin-frappe .button.is-danger.is-light:active,html.theme--catppuccin-frappe .button.is-danger.is-light.is-active{background-color:#f8d8d9;border-color:transparent;color:#9a1e20}html.theme--catppuccin-frappe .button.is-small,html.theme--catppuccin-frappe #documenter .docs-sidebar form.docs-search>input.button{font-size:.75rem}html.theme--catppuccin-frappe .button.is-small:not(.is-rounded),html.theme--catppuccin-frappe #documenter .docs-sidebar form.docs-search>input.button:not(.is-rounded){border-radius:3px}html.theme--catppuccin-frappe .button.is-normal{font-size:1rem}html.theme--catppuccin-frappe .button.is-medium{font-size:1.25rem}html.theme--catppuccin-frappe .button.is-large{font-size:1.5rem}html.theme--catppuccin-frappe .button[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button{background-color:#737994;border-color:#626880;box-shadow:none;opacity:.5}html.theme--catppuccin-frappe .button.is-fullwidth{display:flex;width:100%}html.theme--catppuccin-frappe .button.is-loading{color:transparent !important;pointer-events:none}html.theme--catppuccin-frappe .button.is-loading::after{position:absolute;left:calc(50% - (1em * 0.5));top:calc(50% - (1em * 0.5));position:absolute !important}html.theme--catppuccin-frappe .button.is-static{background-color:#292c3c;border-color:#626880;color:#838ba7;box-shadow:none;pointer-events:none}html.theme--catppuccin-frappe .button.is-rounded,html.theme--catppuccin-frappe #documenter .docs-sidebar form.docs-search>input.button{border-radius:9999px;padding-left:calc(1em + 0.25em);padding-right:calc(1em + 0.25em)}html.theme--catppuccin-frappe .buttons{align-items:center;display:flex;flex-wrap:wrap;justify-content:flex-start}html.theme--catppuccin-frappe .buttons .button{margin-bottom:0.5rem}html.theme--catppuccin-frappe .buttons .button:not(:last-child):not(.is-fullwidth){margin-right:.5rem}html.theme--catppuccin-frappe .buttons:last-child{margin-bottom:-0.5rem}html.theme--catppuccin-frappe .buttons:not(:last-child){margin-bottom:1rem}html.theme--catppuccin-frappe .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large){font-size:.75rem}html.theme--catppuccin-frappe .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large):not(.is-rounded){border-radius:3px}html.theme--catppuccin-frappe .buttons.are-medium .button:not(.is-small):not(.is-normal):not(.is-large){font-size:1.25rem}html.theme--catppuccin-frappe .buttons.are-large .button:not(.is-small):not(.is-normal):not(.is-medium){font-size:1.5rem}html.theme--catppuccin-frappe .buttons.has-addons .button:not(:first-child){border-bottom-left-radius:0;border-top-left-radius:0}html.theme--catppuccin-frappe .buttons.has-addons .button:not(:last-child){border-bottom-right-radius:0;border-top-right-radius:0;margin-right:-1px}html.theme--catppuccin-frappe .buttons.has-addons .button:last-child{margin-right:0}html.theme--catppuccin-frappe .buttons.has-addons .button:hover,html.theme--catppuccin-frappe .buttons.has-addons .button.is-hovered{z-index:2}html.theme--catppuccin-frappe .buttons.has-addons .button:focus,html.theme--catppuccin-frappe .buttons.has-addons .button.is-focused,html.theme--catppuccin-frappe .buttons.has-addons .button:active,html.theme--catppuccin-frappe .buttons.has-addons .button.is-active,html.theme--catppuccin-frappe .buttons.has-addons .button.is-selected{z-index:3}html.theme--catppuccin-frappe .buttons.has-addons .button:focus:hover,html.theme--catppuccin-frappe .buttons.has-addons .button.is-focused:hover,html.theme--catppuccin-frappe .buttons.has-addons .button:active:hover,html.theme--catppuccin-frappe .buttons.has-addons .button.is-active:hover,html.theme--catppuccin-frappe .buttons.has-addons .button.is-selected:hover{z-index:4}html.theme--catppuccin-frappe .buttons.has-addons .button.is-expanded{flex-grow:1;flex-shrink:1}html.theme--catppuccin-frappe .buttons.is-centered{justify-content:center}html.theme--catppuccin-frappe .buttons.is-centered:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}html.theme--catppuccin-frappe .buttons.is-right{justify-content:flex-end}html.theme--catppuccin-frappe .buttons.is-right:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}@media screen and (max-width: 768px){html.theme--catppuccin-frappe .button.is-responsive.is-small,html.theme--catppuccin-frappe #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.5625rem}html.theme--catppuccin-frappe .button.is-responsive,html.theme--catppuccin-frappe .button.is-responsive.is-normal{font-size:.65625rem}html.theme--catppuccin-frappe .button.is-responsive.is-medium{font-size:.75rem}html.theme--catppuccin-frappe .button.is-responsive.is-large{font-size:1rem}}@media screen and (min-width: 769px) and (max-width: 1055px){html.theme--catppuccin-frappe .button.is-responsive.is-small,html.theme--catppuccin-frappe #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.65625rem}html.theme--catppuccin-frappe .button.is-responsive,html.theme--catppuccin-frappe .button.is-responsive.is-normal{font-size:.75rem}html.theme--catppuccin-frappe .button.is-responsive.is-medium{font-size:1rem}html.theme--catppuccin-frappe .button.is-responsive.is-large{font-size:1.25rem}}html.theme--catppuccin-frappe .container{flex-grow:1;margin:0 auto;position:relative;width:auto}html.theme--catppuccin-frappe 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h3{font-size:1.5em;margin-bottom:0.6666em}html.theme--catppuccin-frappe .content h3:not(:first-child){margin-top:1.3333em}html.theme--catppuccin-frappe .content h4{font-size:1.25em;margin-bottom:0.8em}html.theme--catppuccin-frappe .content h5{font-size:1.125em;margin-bottom:0.8888em}html.theme--catppuccin-frappe .content h6{font-size:1em;margin-bottom:1em}html.theme--catppuccin-frappe .content blockquote{background-color:#292c3c;border-left:5px solid #626880;padding:1.25em 1.5em}html.theme--catppuccin-frappe .content ol{list-style-position:outside;margin-left:2em;margin-top:1em}html.theme--catppuccin-frappe .content ol:not([type]){list-style-type:decimal}html.theme--catppuccin-frappe .content ol.is-lower-alpha:not([type]){list-style-type:lower-alpha}html.theme--catppuccin-frappe .content ol.is-lower-roman:not([type]){list-style-type:lower-roman}html.theme--catppuccin-frappe .content ol.is-upper-alpha:not([type]){list-style-type:upper-alpha}html.theme--catppuccin-frappe .content 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.has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-2by1 .has-ratio,html.theme--catppuccin-frappe .image.is-3by1 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-3by1 img,html.theme--catppuccin-frappe .image.is-3by1 .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-3by1 .has-ratio,html.theme--catppuccin-frappe .image.is-4by5 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-4by5 img,html.theme--catppuccin-frappe .image.is-4by5 .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-4by5 .has-ratio,html.theme--catppuccin-frappe .image.is-3by4 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-3by4 img,html.theme--catppuccin-frappe .image.is-3by4 .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-3by4 .has-ratio,html.theme--catppuccin-frappe .image.is-2by3 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-2by3 img,html.theme--catppuccin-frappe .image.is-2by3 .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-2by3 .has-ratio,html.theme--catppuccin-frappe .image.is-3by5 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-3by5 img,html.theme--catppuccin-frappe .image.is-3by5 .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-3by5 .has-ratio,html.theme--catppuccin-frappe .image.is-9by16 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-9by16 img,html.theme--catppuccin-frappe .image.is-9by16 .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-9by16 .has-ratio,html.theme--catppuccin-frappe .image.is-1by2 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-1by2 img,html.theme--catppuccin-frappe .image.is-1by2 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.button.is-black.is-hovered{background-color:#040404;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-black:focus,html.theme--catppuccin-latte .button.is-black.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-black:focus:not(:active),html.theme--catppuccin-latte .button.is-black.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(10,10,10,0.25)}html.theme--catppuccin-latte .button.is-black:active,html.theme--catppuccin-latte .button.is-black.is-active{background-color:#000;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-black[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-black{background-color:#0a0a0a;border-color:#0a0a0a;box-shadow:none}html.theme--catppuccin-latte .button.is-black.is-inverted{background-color:#fff;color:#0a0a0a}html.theme--catppuccin-latte .button.is-black.is-inverted:hover,html.theme--catppuccin-latte .button.is-black.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-latte .button.is-black.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-black.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#0a0a0a}html.theme--catppuccin-latte .button.is-black.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-black.is-outlined{background-color:transparent;border-color:#0a0a0a;color:#0a0a0a}html.theme--catppuccin-latte .button.is-black.is-outlined:hover,html.theme--catppuccin-latte .button.is-black.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-black.is-outlined:focus,html.theme--catppuccin-latte .button.is-black.is-outlined.is-focused{background-color:#0a0a0a;border-color:#0a0a0a;color:#fff}html.theme--catppuccin-latte .button.is-black.is-outlined.is-loading::after{border-color:transparent transparent #0a0a0a #0a0a0a !important}html.theme--catppuccin-latte .button.is-black.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-black.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-black.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-black.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-black.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-black.is-outlined{background-color:transparent;border-color:#0a0a0a;box-shadow:none;color:#0a0a0a}html.theme--catppuccin-latte .button.is-black.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-latte .button.is-black.is-inverted.is-outlined:hover,html.theme--catppuccin-latte .button.is-black.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-black.is-inverted.is-outlined:focus,html.theme--catppuccin-latte .button.is-black.is-inverted.is-outlined.is-focused{background-color:#fff;color:#0a0a0a}html.theme--catppuccin-latte .button.is-black.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-black.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-black.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-black.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #0a0a0a #0a0a0a !important}html.theme--catppuccin-latte .button.is-black.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-black.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-latte .button.is-light{background-color:#f5f5f5;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-latte .button.is-light:hover,html.theme--catppuccin-latte .button.is-light.is-hovered{background-color:#eee;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-latte .button.is-light:focus,html.theme--catppuccin-latte .button.is-light.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-latte .button.is-light:focus:not(:active),html.theme--catppuccin-latte .button.is-light.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(245,245,245,0.25)}html.theme--catppuccin-latte .button.is-light:active,html.theme--catppuccin-latte .button.is-light.is-active{background-color:#e8e8e8;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-latte .button.is-light[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-light{background-color:#f5f5f5;border-color:#f5f5f5;box-shadow:none}html.theme--catppuccin-latte .button.is-light.is-inverted{background-color:rgba(0,0,0,0.7);color:#f5f5f5}html.theme--catppuccin-latte .button.is-light.is-inverted:hover,html.theme--catppuccin-latte .button.is-light.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-latte .button.is-light.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-light.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#f5f5f5}html.theme--catppuccin-latte .button.is-light.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-latte .button.is-light.is-outlined{background-color:transparent;border-color:#f5f5f5;color:#f5f5f5}html.theme--catppuccin-latte .button.is-light.is-outlined:hover,html.theme--catppuccin-latte .button.is-light.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-light.is-outlined:focus,html.theme--catppuccin-latte .button.is-light.is-outlined.is-focused{background-color:#f5f5f5;border-color:#f5f5f5;color:rgba(0,0,0,0.7)}html.theme--catppuccin-latte .button.is-light.is-outlined.is-loading::after{border-color:transparent transparent #f5f5f5 #f5f5f5 !important}html.theme--catppuccin-latte .button.is-light.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-light.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-light.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-light.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-latte .button.is-light.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-light.is-outlined{background-color:transparent;border-color:#f5f5f5;box-shadow:none;color:#f5f5f5}html.theme--catppuccin-latte 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.button.is-light.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-light.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-latte .button.is-dark,html.theme--catppuccin-latte .content kbd.button{background-color:#ccd0da;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-latte .button.is-dark:hover,html.theme--catppuccin-latte .content kbd.button:hover,html.theme--catppuccin-latte .button.is-dark.is-hovered,html.theme--catppuccin-latte .content kbd.button.is-hovered{background-color:#c5c9d5;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-latte .button.is-dark:focus,html.theme--catppuccin-latte .content kbd.button:focus,html.theme--catppuccin-latte .button.is-dark.is-focused,html.theme--catppuccin-latte .content kbd.button.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-latte 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kbd.button.is-inverted{background-color:rgba(0,0,0,0.7);color:#ccd0da}html.theme--catppuccin-latte .button.is-dark.is-inverted:hover,html.theme--catppuccin-latte .content kbd.button.is-inverted:hover,html.theme--catppuccin-latte .button.is-dark.is-inverted.is-hovered,html.theme--catppuccin-latte .content kbd.button.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-latte .button.is-dark.is-inverted[disabled],html.theme--catppuccin-latte .content kbd.button.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-dark.is-inverted,fieldset[disabled] html.theme--catppuccin-latte .content kbd.button.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#ccd0da}html.theme--catppuccin-latte .button.is-dark.is-loading::after,html.theme--catppuccin-latte .content kbd.button.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-latte .button.is-dark.is-outlined,html.theme--catppuccin-latte .content kbd.button.is-outlined{background-color:transparent;border-color:#ccd0da;color:#ccd0da}html.theme--catppuccin-latte .button.is-dark.is-outlined:hover,html.theme--catppuccin-latte .content kbd.button.is-outlined:hover,html.theme--catppuccin-latte .button.is-dark.is-outlined.is-hovered,html.theme--catppuccin-latte .content kbd.button.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-dark.is-outlined:focus,html.theme--catppuccin-latte .content kbd.button.is-outlined:focus,html.theme--catppuccin-latte .button.is-dark.is-outlined.is-focused,html.theme--catppuccin-latte .content kbd.button.is-outlined.is-focused{background-color:#ccd0da;border-color:#ccd0da;color:rgba(0,0,0,0.7)}html.theme--catppuccin-latte .button.is-dark.is-outlined.is-loading::after,html.theme--catppuccin-latte .content kbd.button.is-outlined.is-loading::after{border-color:transparent transparent #ccd0da #ccd0da !important}html.theme--catppuccin-latte .button.is-dark.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .content kbd.button.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-dark.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .content kbd.button.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-dark.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .content kbd.button.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-dark.is-outlined.is-loading.is-focused::after,html.theme--catppuccin-latte .content kbd.button.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-latte .button.is-dark.is-outlined[disabled],html.theme--catppuccin-latte .content kbd.button.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-dark.is-outlined,fieldset[disabled] html.theme--catppuccin-latte .content kbd.button.is-outlined{background-color:transparent;border-color:#ccd0da;box-shadow:none;color:#ccd0da}html.theme--catppuccin-latte .button.is-dark.is-inverted.is-outlined,html.theme--catppuccin-latte .content kbd.button.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-latte .button.is-dark.is-inverted.is-outlined:hover,html.theme--catppuccin-latte .content kbd.button.is-inverted.is-outlined:hover,html.theme--catppuccin-latte .button.is-dark.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-latte .content kbd.button.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-dark.is-inverted.is-outlined:focus,html.theme--catppuccin-latte .content kbd.button.is-inverted.is-outlined:focus,html.theme--catppuccin-latte .button.is-dark.is-inverted.is-outlined.is-focused,html.theme--catppuccin-latte .content kbd.button.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#ccd0da}html.theme--catppuccin-latte .button.is-dark.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .content kbd.button.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-dark.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .content kbd.button.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-dark.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .content kbd.button.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-dark.is-inverted.is-outlined.is-loading.is-focused::after,html.theme--catppuccin-latte .content kbd.button.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #ccd0da #ccd0da !important}html.theme--catppuccin-latte .button.is-dark.is-inverted.is-outlined[disabled],html.theme--catppuccin-latte .content kbd.button.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-dark.is-inverted.is-outlined,fieldset[disabled] html.theme--catppuccin-latte .content kbd.button.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-latte .button.is-primary,html.theme--catppuccin-latte .docstring>section>a.button.docs-sourcelink{background-color:#1e66f5;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-primary:hover,html.theme--catppuccin-latte .docstring>section>a.button.docs-sourcelink:hover,html.theme--catppuccin-latte .button.is-primary.is-hovered,html.theme--catppuccin-latte .docstring>section>a.button.is-hovered.docs-sourcelink{background-color:#125ef4;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-primary:focus,html.theme--catppuccin-latte .docstring>section>a.button.docs-sourcelink:focus,html.theme--catppuccin-latte .button.is-primary.is-focused,html.theme--catppuccin-latte .docstring>section>a.button.is-focused.docs-sourcelink{border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-primary:focus:not(:active),html.theme--catppuccin-latte .docstring>section>a.button.docs-sourcelink:focus:not(:active),html.theme--catppuccin-latte .button.is-primary.is-focused:not(:active),html.theme--catppuccin-latte .docstring>section>a.button.is-focused.docs-sourcelink:not(:active){box-shadow:0 0 0 0.125em rgba(30,102,245,0.25)}html.theme--catppuccin-latte .button.is-primary:active,html.theme--catppuccin-latte .docstring>section>a.button.docs-sourcelink:active,html.theme--catppuccin-latte .button.is-primary.is-active,html.theme--catppuccin-latte 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.button.is-primary.is-outlined:hover,html.theme--catppuccin-latte .docstring>section>a.button.is-outlined.docs-sourcelink:hover,html.theme--catppuccin-latte .button.is-primary.is-outlined.is-hovered,html.theme--catppuccin-latte .docstring>section>a.button.is-outlined.is-hovered.docs-sourcelink,html.theme--catppuccin-latte .button.is-primary.is-outlined:focus,html.theme--catppuccin-latte .docstring>section>a.button.is-outlined.docs-sourcelink:focus,html.theme--catppuccin-latte .button.is-primary.is-outlined.is-focused,html.theme--catppuccin-latte .docstring>section>a.button.is-outlined.is-focused.docs-sourcelink{background-color:#1e66f5;border-color:#1e66f5;color:#fff}html.theme--catppuccin-latte .button.is-primary.is-outlined.is-loading::after,html.theme--catppuccin-latte .docstring>section>a.button.is-outlined.is-loading.docs-sourcelink::after{border-color:transparent transparent #1e66f5 #1e66f5 !important}html.theme--catppuccin-latte .button.is-primary.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .docstring>section>a.button.is-outlined.is-loading.docs-sourcelink:hover::after,html.theme--catppuccin-latte .button.is-primary.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .docstring>section>a.button.is-outlined.is-loading.is-hovered.docs-sourcelink::after,html.theme--catppuccin-latte .button.is-primary.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .docstring>section>a.button.is-outlined.is-loading.docs-sourcelink:focus::after,html.theme--catppuccin-latte .button.is-primary.is-outlined.is-loading.is-focused::after,html.theme--catppuccin-latte .docstring>section>a.button.is-outlined.is-loading.is-focused.docs-sourcelink::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-primary.is-outlined[disabled],html.theme--catppuccin-latte .docstring>section>a.button.is-outlined.docs-sourcelink[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-primary.is-outlined,fieldset[disabled] html.theme--catppuccin-latte .docstring>section>a.button.is-outlined.docs-sourcelink{background-color:transparent;border-color:#1e66f5;box-shadow:none;color:#1e66f5}html.theme--catppuccin-latte .button.is-primary.is-inverted.is-outlined,html.theme--catppuccin-latte .docstring>section>a.button.is-inverted.is-outlined.docs-sourcelink{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-latte .button.is-primary.is-inverted.is-outlined:hover,html.theme--catppuccin-latte .docstring>section>a.button.is-inverted.is-outlined.docs-sourcelink:hover,html.theme--catppuccin-latte .button.is-primary.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-latte .docstring>section>a.button.is-inverted.is-outlined.is-hovered.docs-sourcelink,html.theme--catppuccin-latte 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.docstring>section>a.button.is-inverted.is-outlined.is-loading.docs-sourcelink:focus::after,html.theme--catppuccin-latte .button.is-primary.is-inverted.is-outlined.is-loading.is-focused::after,html.theme--catppuccin-latte .docstring>section>a.button.is-inverted.is-outlined.is-loading.is-focused.docs-sourcelink::after{border-color:transparent transparent #1e66f5 #1e66f5 !important}html.theme--catppuccin-latte .button.is-primary.is-inverted.is-outlined[disabled],html.theme--catppuccin-latte .docstring>section>a.button.is-inverted.is-outlined.docs-sourcelink[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-primary.is-inverted.is-outlined,fieldset[disabled] html.theme--catppuccin-latte .docstring>section>a.button.is-inverted.is-outlined.docs-sourcelink{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-latte .button.is-primary.is-light,html.theme--catppuccin-latte .docstring>section>a.button.is-light.docs-sourcelink{background-color:#ebf2fe;color:#0a52e1}html.theme--catppuccin-latte .button.is-primary.is-light:hover,html.theme--catppuccin-latte .docstring>section>a.button.is-light.docs-sourcelink:hover,html.theme--catppuccin-latte .button.is-primary.is-light.is-hovered,html.theme--catppuccin-latte .docstring>section>a.button.is-light.is-hovered.docs-sourcelink{background-color:#dfe9fe;border-color:transparent;color:#0a52e1}html.theme--catppuccin-latte .button.is-primary.is-light:active,html.theme--catppuccin-latte .docstring>section>a.button.is-light.docs-sourcelink:active,html.theme--catppuccin-latte .button.is-primary.is-light.is-active,html.theme--catppuccin-latte .docstring>section>a.button.is-light.is-active.docs-sourcelink{background-color:#d3e1fd;border-color:transparent;color:#0a52e1}html.theme--catppuccin-latte .button.is-link{background-color:#1e66f5;border-color:transparent;color:#fff}html.theme--catppuccin-latte 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.button.is-info.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-info.is-outlined{background-color:transparent;border-color:#179299;box-shadow:none;color:#179299}html.theme--catppuccin-latte .button.is-info.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-latte .button.is-info.is-inverted.is-outlined:hover,html.theme--catppuccin-latte .button.is-info.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-info.is-inverted.is-outlined:focus,html.theme--catppuccin-latte .button.is-info.is-inverted.is-outlined.is-focused{background-color:#fff;color:#179299}html.theme--catppuccin-latte .button.is-info.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-info.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-info.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-info.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #179299 #179299 !important}html.theme--catppuccin-latte .button.is-info.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-info.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-latte .button.is-info.is-light{background-color:#edfcfc;color:#1cb2ba}html.theme--catppuccin-latte .button.is-info.is-light:hover,html.theme--catppuccin-latte .button.is-info.is-light.is-hovered{background-color:#e2f9fb;border-color:transparent;color:#1cb2ba}html.theme--catppuccin-latte .button.is-info.is-light:active,html.theme--catppuccin-latte .button.is-info.is-light.is-active{background-color:#d7f7f9;border-color:transparent;color:#1cb2ba}html.theme--catppuccin-latte .button.is-success{background-color:#40a02b;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-success:hover,html.theme--catppuccin-latte .button.is-success.is-hovered{background-color:#3c9628;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-success:focus,html.theme--catppuccin-latte .button.is-success.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-success:focus:not(:active),html.theme--catppuccin-latte .button.is-success.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(64,160,43,0.25)}html.theme--catppuccin-latte .button.is-success:active,html.theme--catppuccin-latte .button.is-success.is-active{background-color:#388c26;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-success[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-success{background-color:#40a02b;border-color:#40a02b;box-shadow:none}html.theme--catppuccin-latte .button.is-success.is-inverted{background-color:#fff;color:#40a02b}html.theme--catppuccin-latte .button.is-success.is-inverted:hover,html.theme--catppuccin-latte .button.is-success.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-latte .button.is-success.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-success.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#40a02b}html.theme--catppuccin-latte .button.is-success.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-success.is-outlined{background-color:transparent;border-color:#40a02b;color:#40a02b}html.theme--catppuccin-latte .button.is-success.is-outlined:hover,html.theme--catppuccin-latte .button.is-success.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-success.is-outlined:focus,html.theme--catppuccin-latte .button.is-success.is-outlined.is-focused{background-color:#40a02b;border-color:#40a02b;color:#fff}html.theme--catppuccin-latte .button.is-success.is-outlined.is-loading::after{border-color:transparent transparent #40a02b #40a02b !important}html.theme--catppuccin-latte .button.is-success.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-success.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-success.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-success.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-success.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-success.is-outlined{background-color:transparent;border-color:#40a02b;box-shadow:none;color:#40a02b}html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined:hover,html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined:focus,html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined.is-focused{background-color:#fff;color:#40a02b}html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #40a02b #40a02b !important}html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-latte .button.is-success.is-light{background-color:#f1fbef;color:#40a12b}html.theme--catppuccin-latte .button.is-success.is-light:hover,html.theme--catppuccin-latte .button.is-success.is-light.is-hovered{background-color:#e8f8e5;border-color:transparent;color:#40a12b}html.theme--catppuccin-latte .button.is-success.is-light:active,html.theme--catppuccin-latte .button.is-success.is-light.is-active{background-color:#e0f5db;border-color:transparent;color:#40a12b}html.theme--catppuccin-latte .button.is-warning{background-color:#df8e1d;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-warning:hover,html.theme--catppuccin-latte .button.is-warning.is-hovered{background-color:#d4871c;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-warning:focus,html.theme--catppuccin-latte .button.is-warning.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-warning:focus:not(:active),html.theme--catppuccin-latte .button.is-warning.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(223,142,29,0.25)}html.theme--catppuccin-latte .button.is-warning:active,html.theme--catppuccin-latte .button.is-warning.is-active{background-color:#c8801a;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-warning[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-warning{background-color:#df8e1d;border-color:#df8e1d;box-shadow:none}html.theme--catppuccin-latte .button.is-warning.is-inverted{background-color:#fff;color:#df8e1d}html.theme--catppuccin-latte .button.is-warning.is-inverted:hover,html.theme--catppuccin-latte .button.is-warning.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-latte .button.is-warning.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-warning.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#df8e1d}html.theme--catppuccin-latte .button.is-warning.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-warning.is-outlined{background-color:transparent;border-color:#df8e1d;color:#df8e1d}html.theme--catppuccin-latte .button.is-warning.is-outlined:hover,html.theme--catppuccin-latte .button.is-warning.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-warning.is-outlined:focus,html.theme--catppuccin-latte .button.is-warning.is-outlined.is-focused{background-color:#df8e1d;border-color:#df8e1d;color:#fff}html.theme--catppuccin-latte .button.is-warning.is-outlined.is-loading::after{border-color:transparent transparent #df8e1d #df8e1d !important}html.theme--catppuccin-latte .button.is-warning.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-warning.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-warning.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-warning.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-warning.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-warning.is-outlined{background-color:transparent;border-color:#df8e1d;box-shadow:none;color:#df8e1d}html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined:hover,html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined:focus,html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined.is-focused{background-color:#fff;color:#df8e1d}html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #df8e1d #df8e1d !important}html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-latte .button.is-warning.is-light{background-color:#fdf6ed;color:#9e6515}html.theme--catppuccin-latte .button.is-warning.is-light:hover,html.theme--catppuccin-latte .button.is-warning.is-light.is-hovered{background-color:#fbf1e2;border-color:transparent;color:#9e6515}html.theme--catppuccin-latte .button.is-warning.is-light:active,html.theme--catppuccin-latte .button.is-warning.is-light.is-active{background-color:#faebd6;border-color:transparent;color:#9e6515}html.theme--catppuccin-latte .button.is-danger{background-color:#d20f39;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-danger:hover,html.theme--catppuccin-latte .button.is-danger.is-hovered{background-color:#c60e36;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-danger:focus,html.theme--catppuccin-latte .button.is-danger.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-danger:focus:not(:active),html.theme--catppuccin-latte .button.is-danger.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(210,15,57,0.25)}html.theme--catppuccin-latte .button.is-danger:active,html.theme--catppuccin-latte .button.is-danger.is-active{background-color:#ba0d33;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-danger[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-danger{background-color:#d20f39;border-color:#d20f39;box-shadow:none}html.theme--catppuccin-latte .button.is-danger.is-inverted{background-color:#fff;color:#d20f39}html.theme--catppuccin-latte .button.is-danger.is-inverted:hover,html.theme--catppuccin-latte .button.is-danger.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-latte .button.is-danger.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-danger.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#d20f39}html.theme--catppuccin-latte .button.is-danger.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-danger.is-outlined{background-color:transparent;border-color:#d20f39;color:#d20f39}html.theme--catppuccin-latte .button.is-danger.is-outlined:hover,html.theme--catppuccin-latte .button.is-danger.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-danger.is-outlined:focus,html.theme--catppuccin-latte .button.is-danger.is-outlined.is-focused{background-color:#d20f39;border-color:#d20f39;color:#fff}html.theme--catppuccin-latte .button.is-danger.is-outlined.is-loading::after{border-color:transparent transparent #d20f39 #d20f39 !important}html.theme--catppuccin-latte .button.is-danger.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-danger.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-danger.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-danger.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-danger.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-danger.is-outlined{background-color:transparent;border-color:#d20f39;box-shadow:none;color:#d20f39}html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined:hover,html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined:focus,html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined.is-focused{background-color:#fff;color:#d20f39}html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #d20f39 #d20f39 !important}html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-latte .button.is-danger.is-light{background-color:#feecf0;color:#e9113f}html.theme--catppuccin-latte .button.is-danger.is-light:hover,html.theme--catppuccin-latte .button.is-danger.is-light.is-hovered{background-color:#fde0e6;border-color:transparent;color:#e9113f}html.theme--catppuccin-latte .button.is-danger.is-light:active,html.theme--catppuccin-latte .button.is-danger.is-light.is-active{background-color:#fcd4dd;border-color:transparent;color:#e9113f}html.theme--catppuccin-latte .button.is-small,html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.button{font-size:.75rem}html.theme--catppuccin-latte .button.is-small:not(.is-rounded),html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.button:not(.is-rounded){border-radius:3px}html.theme--catppuccin-latte .button.is-normal{font-size:1rem}html.theme--catppuccin-latte .button.is-medium{font-size:1.25rem}html.theme--catppuccin-latte .button.is-large{font-size:1.5rem}html.theme--catppuccin-latte .button[disabled],fieldset[disabled] html.theme--catppuccin-latte .button{background-color:#9ca0b0;border-color:#acb0be;box-shadow:none;opacity:.5}html.theme--catppuccin-latte .button.is-fullwidth{display:flex;width:100%}html.theme--catppuccin-latte .button.is-loading{color:transparent !important;pointer-events:none}html.theme--catppuccin-latte .button.is-loading::after{position:absolute;left:calc(50% - (1em * 0.5));top:calc(50% - (1em * 0.5));position:absolute !important}html.theme--catppuccin-latte .button.is-static{background-color:#e6e9ef;border-color:#acb0be;color:#8c8fa1;box-shadow:none;pointer-events:none}html.theme--catppuccin-latte .button.is-rounded,html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.button{border-radius:9999px;padding-left:calc(1em + 0.25em);padding-right:calc(1em + 0.25em)}html.theme--catppuccin-latte .buttons{align-items:center;display:flex;flex-wrap:wrap;justify-content:flex-start}html.theme--catppuccin-latte .buttons .button{margin-bottom:0.5rem}html.theme--catppuccin-latte .buttons .button:not(:last-child):not(.is-fullwidth){margin-right:.5rem}html.theme--catppuccin-latte .buttons:last-child{margin-bottom:-0.5rem}html.theme--catppuccin-latte .buttons:not(:last-child){margin-bottom:1rem}html.theme--catppuccin-latte .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large){font-size:.75rem}html.theme--catppuccin-latte .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large):not(.is-rounded){border-radius:3px}html.theme--catppuccin-latte .buttons.are-medium .button:not(.is-small):not(.is-normal):not(.is-large){font-size:1.25rem}html.theme--catppuccin-latte .buttons.are-large .button:not(.is-small):not(.is-normal):not(.is-medium){font-size:1.5rem}html.theme--catppuccin-latte .buttons.has-addons .button:not(:first-child){border-bottom-left-radius:0;border-top-left-radius:0}html.theme--catppuccin-latte .buttons.has-addons .button:not(:last-child){border-bottom-right-radius:0;border-top-right-radius:0;margin-right:-1px}html.theme--catppuccin-latte .buttons.has-addons .button:last-child{margin-right:0}html.theme--catppuccin-latte .buttons.has-addons .button:hover,html.theme--catppuccin-latte .buttons.has-addons .button.is-hovered{z-index:2}html.theme--catppuccin-latte .buttons.has-addons .button:focus,html.theme--catppuccin-latte .buttons.has-addons .button.is-focused,html.theme--catppuccin-latte .buttons.has-addons .button:active,html.theme--catppuccin-latte .buttons.has-addons .button.is-active,html.theme--catppuccin-latte .buttons.has-addons .button.is-selected{z-index:3}html.theme--catppuccin-latte .buttons.has-addons .button:focus:hover,html.theme--catppuccin-latte .buttons.has-addons .button.is-focused:hover,html.theme--catppuccin-latte .buttons.has-addons .button:active:hover,html.theme--catppuccin-latte .buttons.has-addons .button.is-active:hover,html.theme--catppuccin-latte .buttons.has-addons .button.is-selected:hover{z-index:4}html.theme--catppuccin-latte .buttons.has-addons .button.is-expanded{flex-grow:1;flex-shrink:1}html.theme--catppuccin-latte .buttons.is-centered{justify-content:center}html.theme--catppuccin-latte .buttons.is-centered:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}html.theme--catppuccin-latte .buttons.is-right{justify-content:flex-end}html.theme--catppuccin-latte .buttons.is-right:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}@media screen and (max-width: 768px){html.theme--catppuccin-latte .button.is-responsive.is-small,html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.5625rem}html.theme--catppuccin-latte .button.is-responsive,html.theme--catppuccin-latte .button.is-responsive.is-normal{font-size:.65625rem}html.theme--catppuccin-latte .button.is-responsive.is-medium{font-size:.75rem}html.theme--catppuccin-latte .button.is-responsive.is-large{font-size:1rem}}@media screen and (min-width: 769px) and (max-width: 1055px){html.theme--catppuccin-latte .button.is-responsive.is-small,html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.65625rem}html.theme--catppuccin-latte .button.is-responsive,html.theme--catppuccin-latte .button.is-responsive.is-normal{font-size:.75rem}html.theme--catppuccin-latte .button.is-responsive.is-medium{font-size:1rem}html.theme--catppuccin-latte .button.is-responsive.is-large{font-size:1.25rem}}html.theme--catppuccin-latte .container{flex-grow:1;margin:0 auto;position:relative;width:auto}html.theme--catppuccin-latte 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.file.is-link.is-active .file-cta{background-color:#0b57ef;border-color:transparent;color:#fff}html.theme--catppuccin-latte .file.is-info .file-cta{background-color:#179299;border-color:transparent;color:#fff}html.theme--catppuccin-latte .file.is-info:hover .file-cta,html.theme--catppuccin-latte .file.is-info.is-hovered .file-cta{background-color:#15878e;border-color:transparent;color:#fff}html.theme--catppuccin-latte .file.is-info:focus .file-cta,html.theme--catppuccin-latte .file.is-info.is-focused .file-cta{border-color:transparent;box-shadow:0 0 0.5em rgba(23,146,153,0.25);color:#fff}html.theme--catppuccin-latte .file.is-info:active .file-cta,html.theme--catppuccin-latte .file.is-info.is-active .file-cta{background-color:#147d83;border-color:transparent;color:#fff}html.theme--catppuccin-latte .file.is-success .file-cta{background-color:#40a02b;border-color:transparent;color:#fff}html.theme--catppuccin-latte .file.is-success:hover .file-cta,html.theme--catppuccin-latte .file.is-success.is-hovered .file-cta{background-color:#3c9628;border-color:transparent;color:#fff}html.theme--catppuccin-latte .file.is-success:focus .file-cta,html.theme--catppuccin-latte .file.is-success.is-focused .file-cta{border-color:transparent;box-shadow:0 0 0.5em rgba(64,160,43,0.25);color:#fff}html.theme--catppuccin-latte .file.is-success:active .file-cta,html.theme--catppuccin-latte .file.is-success.is-active .file-cta{background-color:#388c26;border-color:transparent;color:#fff}html.theme--catppuccin-latte .file.is-warning .file-cta{background-color:#df8e1d;border-color:transparent;color:#fff}html.theme--catppuccin-latte .file.is-warning:hover .file-cta,html.theme--catppuccin-latte .file.is-warning.is-hovered .file-cta{background-color:#d4871c;border-color:transparent;color:#fff}html.theme--catppuccin-latte .file.is-warning:focus .file-cta,html.theme--catppuccin-latte .file.is-warning.is-focused .file-cta{border-color:transparent;box-shadow:0 0 0.5em rgba(223,142,29,0.25);color:#fff}html.theme--catppuccin-latte .file.is-warning:active .file-cta,html.theme--catppuccin-latte .file.is-warning.is-active .file-cta{background-color:#c8801a;border-color:transparent;color:#fff}html.theme--catppuccin-latte .file.is-danger .file-cta{background-color:#d20f39;border-color:transparent;color:#fff}html.theme--catppuccin-latte .file.is-danger:hover .file-cta,html.theme--catppuccin-latte .file.is-danger.is-hovered .file-cta{background-color:#c60e36;border-color:transparent;color:#fff}html.theme--catppuccin-latte .file.is-danger:focus .file-cta,html.theme--catppuccin-latte .file.is-danger.is-focused .file-cta{border-color:transparent;box-shadow:0 0 0.5em rgba(210,15,57,0.25);color:#fff}html.theme--catppuccin-latte .file.is-danger:active .file-cta,html.theme--catppuccin-latte .file.is-danger.is-active .file-cta{background-color:#ba0d33;border-color:transparent;color:#fff}html.theme--catppuccin-latte .file.is-small,html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.file{font-size:.75rem}html.theme--catppuccin-latte .file.is-normal{font-size:1rem}html.theme--catppuccin-latte .file.is-medium{font-size:1.25rem}html.theme--catppuccin-latte .file.is-medium .file-icon .fa{font-size:21px}html.theme--catppuccin-latte .file.is-large{font-size:1.5rem}html.theme--catppuccin-latte .file.is-large .file-icon .fa{font-size:28px}html.theme--catppuccin-latte .file.has-name .file-cta{border-bottom-right-radius:0;border-top-right-radius:0}html.theme--catppuccin-latte .file.has-name .file-name{border-bottom-left-radius:0;border-top-left-radius:0}html.theme--catppuccin-latte .file.has-name.is-empty .file-cta{border-radius:.4em}html.theme--catppuccin-latte .file.has-name.is-empty .file-name{display:none}html.theme--catppuccin-latte .file.is-boxed .file-label{flex-direction:column}html.theme--catppuccin-latte .file.is-boxed .file-cta{flex-direction:column;height:auto;padding:1em 3em}html.theme--catppuccin-latte .file.is-boxed .file-name{border-width:0 1px 1px}html.theme--catppuccin-latte .file.is-boxed .file-icon{height:1.5em;width:1.5em}html.theme--catppuccin-latte .file.is-boxed .file-icon .fa{font-size:21px}html.theme--catppuccin-latte .file.is-boxed.is-small .file-icon .fa,html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.is-boxed .file-icon .fa{font-size:14px}html.theme--catppuccin-latte .file.is-boxed.is-medium .file-icon .fa{font-size:28px}html.theme--catppuccin-latte .file.is-boxed.is-large .file-icon .fa{font-size:35px}html.theme--catppuccin-latte .file.is-boxed.has-name .file-cta{border-radius:.4em .4em 0 0}html.theme--catppuccin-latte .file.is-boxed.has-name .file-name{border-radius:0 0 .4em .4em;border-width:0 1px 1px}html.theme--catppuccin-latte .file.is-centered{justify-content:center}html.theme--catppuccin-latte .file.is-fullwidth .file-label{width:100%}html.theme--catppuccin-latte .file.is-fullwidth .file-name{flex-grow:1;max-width:none}html.theme--catppuccin-latte .file.is-right{justify-content:flex-end}html.theme--catppuccin-latte .file.is-right .file-cta{border-radius:0 .4em .4em 0}html.theme--catppuccin-latte .file.is-right .file-name{border-radius:.4em 0 0 .4em;border-width:1px 0 1px 1px;order:-1}html.theme--catppuccin-latte .file-label{align-items:stretch;display:flex;cursor:pointer;justify-content:flex-start;overflow:hidden;position:relative}html.theme--catppuccin-latte .file-label:hover .file-cta{background-color:#c5c9d5;color:#41445a}html.theme--catppuccin-latte .file-label:hover .file-name{border-color:#a5a9b8}html.theme--catppuccin-latte .file-label:active .file-cta{background-color:#bdc2cf;color:#41445a}html.theme--catppuccin-latte .file-label:active .file-name{border-color:#9ea2b3}html.theme--catppuccin-latte .file-input{height:100%;left:0;opacity:0;outline:none;position:absolute;top:0;width:100%}html.theme--catppuccin-latte .file-cta,html.theme--catppuccin-latte .file-name{border-color:#acb0be;border-radius:.4em;font-size:1em;padding-left:1em;padding-right:1em;white-space:nowrap}html.theme--catppuccin-latte .file-cta{background-color:#ccd0da;color:#4c4f69}html.theme--catppuccin-latte .file-name{border-color:#acb0be;border-style:solid;border-width:1px 1px 1px 0;display:block;max-width:16em;overflow:hidden;text-align:inherit;text-overflow:ellipsis}html.theme--catppuccin-latte .file-icon{align-items:center;display:flex;height:1em;justify-content:center;margin-right:.5em;width:1em}html.theme--catppuccin-latte .file-icon .fa{font-size:14px}html.theme--catppuccin-latte .label{color:#41445a;display:block;font-size:1rem;font-weight:700}html.theme--catppuccin-latte .label:not(:last-child){margin-bottom:0.5em}html.theme--catppuccin-latte .label.is-small,html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.label{font-size:.75rem}html.theme--catppuccin-latte .label.is-medium{font-size:1.25rem}html.theme--catppuccin-latte .label.is-large{font-size:1.5rem}html.theme--catppuccin-latte .help{display:block;font-size:.75rem;margin-top:0.25rem}html.theme--catppuccin-latte .help.is-white{color:#fff}html.theme--catppuccin-latte .help.is-black{color:#0a0a0a}html.theme--catppuccin-latte .help.is-light{color:#f5f5f5}html.theme--catppuccin-latte .help.is-dark,html.theme--catppuccin-latte .content kbd.help{color:#ccd0da}html.theme--catppuccin-latte .help.is-primary,html.theme--catppuccin-latte .docstring>section>a.help.docs-sourcelink{color:#1e66f5}html.theme--catppuccin-latte .help.is-link{color:#1e66f5}html.theme--catppuccin-latte .help.is-info{color:#179299}html.theme--catppuccin-latte .help.is-success{color:#40a02b}html.theme--catppuccin-latte .help.is-warning{color:#df8e1d}html.theme--catppuccin-latte .help.is-danger{color:#d20f39}html.theme--catppuccin-latte .field:not(:last-child){margin-bottom:0.75rem}html.theme--catppuccin-latte .field.has-addons{display:flex;justify-content:flex-start}html.theme--catppuccin-latte .field.has-addons .control:not(:last-child){margin-right:-1px}html.theme--catppuccin-latte .field.has-addons .control:not(:first-child):not(:last-child) .button,html.theme--catppuccin-latte .field.has-addons .control:not(:first-child):not(:last-child) .input,html.theme--catppuccin-latte .field.has-addons .control:not(:first-child):not(:last-child) #documenter .docs-sidebar form.docs-search>input,html.theme--catppuccin-latte #documenter .docs-sidebar .field.has-addons .control:not(:first-child):not(:last-child) form.docs-search>input,html.theme--catppuccin-latte .field.has-addons .control:not(:first-child):not(:last-child) .select select{border-radius:0}html.theme--catppuccin-latte .field.has-addons .control:first-child:not(:only-child) .button,html.theme--catppuccin-latte .field.has-addons .control:first-child:not(:only-child) .input,html.theme--catppuccin-latte .field.has-addons .control:first-child:not(:only-child) #documenter .docs-sidebar form.docs-search>input,html.theme--catppuccin-latte #documenter .docs-sidebar .field.has-addons .control:first-child:not(:only-child) form.docs-search>input,html.theme--catppuccin-latte .field.has-addons .control:first-child:not(:only-child) .select select{border-bottom-right-radius:0;border-top-right-radius:0}html.theme--catppuccin-latte .field.has-addons .control:last-child:not(:only-child) .button,html.theme--catppuccin-latte .field.has-addons .control:last-child:not(:only-child) .input,html.theme--catppuccin-latte .field.has-addons .control:last-child:not(:only-child) #documenter .docs-sidebar form.docs-search>input,html.theme--catppuccin-latte #documenter .docs-sidebar .field.has-addons .control:last-child:not(:only-child) form.docs-search>input,html.theme--catppuccin-latte .field.has-addons .control:last-child:not(:only-child) .select select{border-bottom-left-radius:0;border-top-left-radius:0}html.theme--catppuccin-latte .field.has-addons .control .button:not([disabled]):hover,html.theme--catppuccin-latte .field.has-addons .control .button.is-hovered:not([disabled]),html.theme--catppuccin-latte .field.has-addons .control .input:not([disabled]):hover,html.theme--catppuccin-latte .field.has-addons .control #documenter .docs-sidebar form.docs-search>input:not([disabled]):hover,html.theme--catppuccin-latte #documenter .docs-sidebar .field.has-addons .control form.docs-search>input:not([disabled]):hover,html.theme--catppuccin-latte .field.has-addons .control .input.is-hovered:not([disabled]),html.theme--catppuccin-latte .field.has-addons .control #documenter .docs-sidebar form.docs-search>input.is-hovered:not([disabled]),html.theme--catppuccin-latte #documenter .docs-sidebar .field.has-addons .control form.docs-search>input.is-hovered:not([disabled]),html.theme--catppuccin-latte .field.has-addons .control .select select:not([disabled]):hover,html.theme--catppuccin-latte .field.has-addons .control .select select.is-hovered:not([disabled]){z-index:2}html.theme--catppuccin-latte .field.has-addons .control .button:not([disabled]):focus,html.theme--catppuccin-latte .field.has-addons .control .button.is-focused:not([disabled]),html.theme--catppuccin-latte .field.has-addons .control .button:not([disabled]):active,html.theme--catppuccin-latte .field.has-addons .control .button.is-active:not([disabled]),html.theme--catppuccin-latte .field.has-addons .control .input:not([disabled]):focus,html.theme--catppuccin-latte .field.has-addons .control #documenter .docs-sidebar form.docs-search>input:not([disabled]):focus,html.theme--catppuccin-latte #documenter .docs-sidebar .field.has-addons .control form.docs-search>input:not([disabled]):focus,html.theme--catppuccin-latte .field.has-addons .control .input.is-focused:not([disabled]),html.theme--catppuccin-latte .field.has-addons .control #documenter .docs-sidebar form.docs-search>input.is-focused:not([disabled]),html.theme--catppuccin-latte #documenter .docs-sidebar .field.has-addons .control form.docs-search>input.is-focused:not([disabled]),html.theme--catppuccin-latte .field.has-addons .control .input:not([disabled]):active,html.theme--catppuccin-latte .field.has-addons .control #documenter .docs-sidebar form.docs-search>input:not([disabled]):active,html.theme--catppuccin-latte #documenter .docs-sidebar .field.has-addons .control form.docs-search>input:not([disabled]):active,html.theme--catppuccin-latte .field.has-addons .control .input.is-active:not([disabled]),html.theme--catppuccin-latte .field.has-addons .control #documenter .docs-sidebar form.docs-search>input.is-active:not([disabled]),html.theme--catppuccin-latte #documenter .docs-sidebar .field.has-addons .control form.docs-search>input.is-active:not([disabled]),html.theme--catppuccin-latte .field.has-addons .control .select select:not([disabled]):focus,html.theme--catppuccin-latte .field.has-addons .control .select select.is-focused:not([disabled]),html.theme--catppuccin-latte .field.has-addons .control .select select:not([disabled]):active,html.theme--catppuccin-latte .field.has-addons .control .select select.is-active:not([disabled]){z-index:3}html.theme--catppuccin-latte .field.has-addons .control .button:not([disabled]):focus:hover,html.theme--catppuccin-latte .field.has-addons .control .button.is-focused:not([disabled]):hover,html.theme--catppuccin-latte .field.has-addons .control .button:not([disabled]):active:hover,html.theme--catppuccin-latte .field.has-addons .control .button.is-active:not([disabled]):hover,html.theme--catppuccin-latte .field.has-addons .control .input:not([disabled]):focus:hover,html.theme--catppuccin-latte .field.has-addons .control #documenter .docs-sidebar form.docs-search>input:not([disabled]):focus:hover,html.theme--catppuccin-latte #documenter .docs-sidebar .field.has-addons .control form.docs-search>input:not([disabled]):focus:hover,html.theme--catppuccin-latte .field.has-addons .control .input.is-focused:not([disabled]):hover,html.theme--catppuccin-latte .field.has-addons .control #documenter .docs-sidebar form.docs-search>input.is-focused:not([disabled]):hover,html.theme--catppuccin-latte #documenter .docs-sidebar .field.has-addons .control form.docs-search>input.is-focused:not([disabled]):hover,html.theme--catppuccin-latte .field.has-addons .control .input:not([disabled]):active:hover,html.theme--catppuccin-latte .field.has-addons .control #documenter .docs-sidebar form.docs-search>input:not([disabled]):active:hover,html.theme--catppuccin-latte #documenter .docs-sidebar .field.has-addons .control form.docs-search>input:not([disabled]):active:hover,html.theme--catppuccin-latte .field.has-addons .control .input.is-active:not([disabled]):hover,html.theme--catppuccin-latte .field.has-addons .control #documenter .docs-sidebar form.docs-search>input.is-active:not([disabled]):hover,html.theme--catppuccin-latte #documenter .docs-sidebar .field.has-addons .control form.docs-search>input.is-active:not([disabled]):hover,html.theme--catppuccin-latte .field.has-addons .control .select select:not([disabled]):focus:hover,html.theme--catppuccin-latte .field.has-addons .control .select select.is-focused:not([disabled]):hover,html.theme--catppuccin-latte .field.has-addons .control .select select:not([disabled]):active:hover,html.theme--catppuccin-latte .field.has-addons .control .select select.is-active:not([disabled]):hover{z-index:4}html.theme--catppuccin-latte .field.has-addons .control.is-expanded{flex-grow:1;flex-shrink:1}html.theme--catppuccin-latte .field.has-addons.has-addons-centered{justify-content:center}html.theme--catppuccin-latte .field.has-addons.has-addons-right{justify-content:flex-end}html.theme--catppuccin-latte .field.has-addons.has-addons-fullwidth .control{flex-grow:1;flex-shrink:0}html.theme--catppuccin-latte .field.is-grouped{display:flex;justify-content:flex-start}html.theme--catppuccin-latte .field.is-grouped>.control{flex-shrink:0}html.theme--catppuccin-latte .field.is-grouped>.control:not(:last-child){margin-bottom:0;margin-right:.75rem}html.theme--catppuccin-latte .field.is-grouped>.control.is-expanded{flex-grow:1;flex-shrink:1}html.theme--catppuccin-latte .field.is-grouped.is-grouped-centered{justify-content:center}html.theme--catppuccin-latte .field.is-grouped.is-grouped-right{justify-content:flex-end}html.theme--catppuccin-latte .field.is-grouped.is-grouped-multiline{flex-wrap:wrap}html.theme--catppuccin-latte .field.is-grouped.is-grouped-multiline>.control:last-child,html.theme--catppuccin-latte .field.is-grouped.is-grouped-multiline>.control:not(:last-child){margin-bottom:0.75rem}html.theme--catppuccin-latte .field.is-grouped.is-grouped-multiline:last-child{margin-bottom:-0.75rem}html.theme--catppuccin-latte 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.icon:only-child,html.theme--catppuccin-latte .navbar-link .icon:only-child{margin-left:-0.25rem;margin-right:-0.25rem}html.theme--catppuccin-latte a.navbar-item,html.theme--catppuccin-latte .navbar-link{cursor:pointer}html.theme--catppuccin-latte a.navbar-item:focus,html.theme--catppuccin-latte a.navbar-item:focus-within,html.theme--catppuccin-latte a.navbar-item:hover,html.theme--catppuccin-latte a.navbar-item.is-active,html.theme--catppuccin-latte .navbar-link:focus,html.theme--catppuccin-latte .navbar-link:focus-within,html.theme--catppuccin-latte .navbar-link:hover,html.theme--catppuccin-latte .navbar-link.is-active{background-color:rgba(0,0,0,0);color:#1e66f5}html.theme--catppuccin-latte .navbar-item{flex-grow:0;flex-shrink:0}html.theme--catppuccin-latte .navbar-item img{max-height:1.75rem}html.theme--catppuccin-latte .navbar-item.has-dropdown{padding:0}html.theme--catppuccin-latte .navbar-item.is-expanded{flex-grow:1;flex-shrink:1}html.theme--catppuccin-latte .navbar-item.is-tab{border-bottom:1px solid transparent;min-height:4rem;padding-bottom:calc(0.5rem - 1px)}html.theme--catppuccin-latte .navbar-item.is-tab:focus,html.theme--catppuccin-latte .navbar-item.is-tab:hover{background-color:rgba(0,0,0,0);border-bottom-color:#1e66f5}html.theme--catppuccin-latte .navbar-item.is-tab.is-active{background-color:rgba(0,0,0,0);border-bottom-color:#1e66f5;border-bottom-style:solid;border-bottom-width:3px;color:#1e66f5;padding-bottom:calc(0.5rem - 3px)}html.theme--catppuccin-latte .navbar-content{flex-grow:1;flex-shrink:1}html.theme--catppuccin-latte .navbar-link:not(.is-arrowless){padding-right:2.5em}html.theme--catppuccin-latte .navbar-link:not(.is-arrowless)::after{border-color:#fff;margin-top:-0.375em;right:1.125em}html.theme--catppuccin-latte .navbar-dropdown{font-size:0.875rem;padding-bottom:0.5rem;padding-top:0.5rem}html.theme--catppuccin-latte .navbar-dropdown .navbar-item{padding-left:1.5rem;padding-right:1.5rem}html.theme--catppuccin-latte .navbar-divider{background-color:rgba(0,0,0,0.2);border:none;display:none;height:2px;margin:0.5rem 0}@media screen and (max-width: 1055px){html.theme--catppuccin-latte .navbar>.container{display:block}html.theme--catppuccin-latte .navbar-brand .navbar-item,html.theme--catppuccin-latte .navbar-tabs .navbar-item{align-items:center;display:flex}html.theme--catppuccin-latte .navbar-link::after{display:none}html.theme--catppuccin-latte .navbar-menu{background-color:#1e66f5;box-shadow:0 8px 16px rgba(10,10,10,0.1);padding:0.5rem 0}html.theme--catppuccin-latte .navbar-menu.is-active{display:block}html.theme--catppuccin-latte .navbar.is-fixed-bottom-touch,html.theme--catppuccin-latte .navbar.is-fixed-top-touch{left:0;position:fixed;right:0;z-index:30}html.theme--catppuccin-latte .navbar.is-fixed-bottom-touch{bottom:0}html.theme--catppuccin-latte .navbar.is-fixed-bottom-touch.has-shadow{box-shadow:0 -2px 3px rgba(10,10,10,0.1)}html.theme--catppuccin-latte .navbar.is-fixed-top-touch{top:0}html.theme--catppuccin-latte .navbar.is-fixed-top .navbar-menu,html.theme--catppuccin-latte .navbar.is-fixed-top-touch .navbar-menu{-webkit-overflow-scrolling:touch;max-height:calc(100vh - 4rem);overflow:auto}html.theme--catppuccin-latte html.has-navbar-fixed-top-touch,html.theme--catppuccin-latte body.has-navbar-fixed-top-touch{padding-top:4rem}html.theme--catppuccin-latte html.has-navbar-fixed-bottom-touch,html.theme--catppuccin-latte body.has-navbar-fixed-bottom-touch{padding-bottom:4rem}}@media screen and (min-width: 1056px){html.theme--catppuccin-latte .navbar,html.theme--catppuccin-latte .navbar-menu,html.theme--catppuccin-latte .navbar-start,html.theme--catppuccin-latte .navbar-end{align-items:stretch;display:flex}html.theme--catppuccin-latte .navbar{min-height:4rem}html.theme--catppuccin-latte .navbar.is-spaced{padding:1rem 2rem}html.theme--catppuccin-latte .navbar.is-spaced .navbar-start,html.theme--catppuccin-latte .navbar.is-spaced .navbar-end{align-items:center}html.theme--catppuccin-latte .navbar.is-spaced a.navbar-item,html.theme--catppuccin-latte .navbar.is-spaced .navbar-link{border-radius:.4em}html.theme--catppuccin-latte .navbar.is-transparent a.navbar-item:focus,html.theme--catppuccin-latte .navbar.is-transparent a.navbar-item:hover,html.theme--catppuccin-latte .navbar.is-transparent a.navbar-item.is-active,html.theme--catppuccin-latte .navbar.is-transparent .navbar-link:focus,html.theme--catppuccin-latte .navbar.is-transparent .navbar-link:hover,html.theme--catppuccin-latte .navbar.is-transparent .navbar-link.is-active{background-color:transparent !important}html.theme--catppuccin-latte .navbar.is-transparent .navbar-item.has-dropdown.is-active .navbar-link,html.theme--catppuccin-latte .navbar.is-transparent .navbar-item.has-dropdown.is-hoverable:focus .navbar-link,html.theme--catppuccin-latte .navbar.is-transparent .navbar-item.has-dropdown.is-hoverable:focus-within 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.navbar-dropdown.is-boxed,.navbar.is-spaced html.theme--catppuccin-latte .navbar-item.is-hoverable:hover .navbar-dropdown,html.theme--catppuccin-latte .navbar-item.is-hoverable:hover .navbar-dropdown.is-boxed{opacity:1;pointer-events:auto;transform:translateY(0)}html.theme--catppuccin-latte .navbar-menu{flex-grow:1;flex-shrink:0}html.theme--catppuccin-latte .navbar-start{justify-content:flex-start;margin-right:auto}html.theme--catppuccin-latte .navbar-end{justify-content:flex-end;margin-left:auto}html.theme--catppuccin-latte .navbar-dropdown{background-color:#1e66f5;border-bottom-left-radius:8px;border-bottom-right-radius:8px;border-top:1px solid rgba(0,0,0,0.2);box-shadow:0 8px 8px rgba(10,10,10,0.1);display:none;font-size:0.875rem;left:0;min-width:100%;position:absolute;top:100%;z-index:20}html.theme--catppuccin-latte .navbar-dropdown .navbar-item{padding:0.375rem 1rem;white-space:nowrap}html.theme--catppuccin-latte .navbar-dropdown a.navbar-item{padding-right:3rem}html.theme--catppuccin-latte .navbar-dropdown a.navbar-item:focus,html.theme--catppuccin-latte .navbar-dropdown a.navbar-item:hover{background-color:rgba(0,0,0,0);color:#8c8fa1}html.theme--catppuccin-latte .navbar-dropdown a.navbar-item.is-active{background-color:rgba(0,0,0,0);color:#1e66f5}.navbar.is-spaced html.theme--catppuccin-latte .navbar-dropdown,html.theme--catppuccin-latte .navbar-dropdown.is-boxed{border-radius:8px;border-top:none;box-shadow:0 8px 8px rgba(10,10,10,0.1), 0 0 0 1px rgba(10,10,10,0.1);display:block;opacity:0;pointer-events:none;top:calc(100% + (-4px));transform:translateY(-5px);transition-duration:86ms;transition-property:opacity, transform}html.theme--catppuccin-latte .navbar-dropdown.is-right{left:auto;right:0}html.theme--catppuccin-latte .navbar-divider{display:block}html.theme--catppuccin-latte .navbar>.container .navbar-brand,html.theme--catppuccin-latte .container>.navbar .navbar-brand{margin-left:-.75rem}html.theme--catppuccin-latte .navbar>.container .navbar-menu,html.theme--catppuccin-latte .container>.navbar .navbar-menu{margin-right:-.75rem}html.theme--catppuccin-latte .navbar.is-fixed-bottom-desktop,html.theme--catppuccin-latte .navbar.is-fixed-top-desktop{left:0;position:fixed;right:0;z-index:30}html.theme--catppuccin-latte .navbar.is-fixed-bottom-desktop{bottom:0}html.theme--catppuccin-latte .navbar.is-fixed-bottom-desktop.has-shadow{box-shadow:0 -2px 3px rgba(10,10,10,0.1)}html.theme--catppuccin-latte .navbar.is-fixed-top-desktop{top:0}html.theme--catppuccin-latte html.has-navbar-fixed-top-desktop,html.theme--catppuccin-latte body.has-navbar-fixed-top-desktop{padding-top:4rem}html.theme--catppuccin-latte html.has-navbar-fixed-bottom-desktop,html.theme--catppuccin-latte body.has-navbar-fixed-bottom-desktop{padding-bottom:4rem}html.theme--catppuccin-latte html.has-spaced-navbar-fixed-top,html.theme--catppuccin-latte body.has-spaced-navbar-fixed-top{padding-top:6rem}html.theme--catppuccin-latte html.has-spaced-navbar-fixed-bottom,html.theme--catppuccin-latte body.has-spaced-navbar-fixed-bottom{padding-bottom:6rem}html.theme--catppuccin-latte a.navbar-item.is-active,html.theme--catppuccin-latte .navbar-link.is-active{color:#1e66f5}html.theme--catppuccin-latte a.navbar-item.is-active:not(:focus):not(:hover),html.theme--catppuccin-latte .navbar-link.is-active:not(:focus):not(:hover){background-color:rgba(0,0,0,0)}html.theme--catppuccin-latte .navbar-item.has-dropdown:focus .navbar-link,html.theme--catppuccin-latte .navbar-item.has-dropdown:hover .navbar-link,html.theme--catppuccin-latte .navbar-item.has-dropdown.is-active .navbar-link{background-color:rgba(0,0,0,0)}}html.theme--catppuccin-latte .hero.is-fullheight-with-navbar{min-height:calc(100vh - 4rem)}html.theme--catppuccin-latte .pagination{font-size:1rem;margin:-.25rem}html.theme--catppuccin-latte .pagination.is-small,html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.pagination{font-size:.75rem}html.theme--catppuccin-latte .pagination.is-medium{font-size:1.25rem}html.theme--catppuccin-latte .pagination.is-large{font-size:1.5rem}html.theme--catppuccin-latte .pagination.is-rounded .pagination-previous,html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.pagination .pagination-previous,html.theme--catppuccin-latte .pagination.is-rounded .pagination-next,html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.pagination .pagination-next{padding-left:1em;padding-right:1em;border-radius:9999px}html.theme--catppuccin-latte .pagination.is-rounded .pagination-link,html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.pagination .pagination-link{border-radius:9999px}html.theme--catppuccin-latte .pagination,html.theme--catppuccin-latte .pagination-list{align-items:center;display:flex;justify-content:center;text-align:center}html.theme--catppuccin-latte .pagination-previous,html.theme--catppuccin-latte .pagination-next,html.theme--catppuccin-latte .pagination-link,html.theme--catppuccin-latte .pagination-ellipsis{font-size:1em;justify-content:center;margin:.25rem;padding-left:.5em;padding-right:.5em;text-align:center}html.theme--catppuccin-latte .pagination-previous,html.theme--catppuccin-latte .pagination-next,html.theme--catppuccin-latte .pagination-link{border-color:#acb0be;color:#1e66f5;min-width:2.5em}html.theme--catppuccin-latte .pagination-previous:hover,html.theme--catppuccin-latte .pagination-next:hover,html.theme--catppuccin-latte .pagination-link:hover{border-color:#9ca0b0;color:#04a5e5}html.theme--catppuccin-latte .pagination-previous:focus,html.theme--catppuccin-latte .pagination-next:focus,html.theme--catppuccin-latte .pagination-link:focus{border-color:#9ca0b0}html.theme--catppuccin-latte .pagination-previous:active,html.theme--catppuccin-latte .pagination-next:active,html.theme--catppuccin-latte .pagination-link:active{box-shadow:inset 0 1px 2px rgba(10,10,10,0.2)}html.theme--catppuccin-latte .pagination-previous[disabled],html.theme--catppuccin-latte .pagination-previous.is-disabled,html.theme--catppuccin-latte .pagination-next[disabled],html.theme--catppuccin-latte .pagination-next.is-disabled,html.theme--catppuccin-latte .pagination-link[disabled],html.theme--catppuccin-latte .pagination-link.is-disabled{background-color:#acb0be;border-color:#acb0be;box-shadow:none;color:#616587;opacity:0.5}html.theme--catppuccin-latte .pagination-previous,html.theme--catppuccin-latte .pagination-next{padding-left:.75em;padding-right:.75em;white-space:nowrap}html.theme--catppuccin-latte .pagination-link.is-current{background-color:#1e66f5;border-color:#1e66f5;color:#fff}html.theme--catppuccin-latte .pagination-ellipsis{color:#9ca0b0;pointer-events:none}html.theme--catppuccin-latte .pagination-list{flex-wrap:wrap}html.theme--catppuccin-latte .pagination-list li{list-style:none}@media screen and (max-width: 768px){html.theme--catppuccin-latte .pagination{flex-wrap:wrap}html.theme--catppuccin-latte .pagination-previous,html.theme--catppuccin-latte .pagination-next{flex-grow:1;flex-shrink:1}html.theme--catppuccin-latte .pagination-list li{flex-grow:1;flex-shrink:1}}@media screen and (min-width: 769px),print{html.theme--catppuccin-latte .pagination-list{flex-grow:1;flex-shrink:1;justify-content:flex-start;order:1}html.theme--catppuccin-latte .pagination-previous,html.theme--catppuccin-latte .pagination-next,html.theme--catppuccin-latte .pagination-link,html.theme--catppuccin-latte .pagination-ellipsis{margin-bottom:0;margin-top:0}html.theme--catppuccin-latte .pagination-previous{order:2}html.theme--catppuccin-latte .pagination-next{order:3}html.theme--catppuccin-latte .pagination{justify-content:space-between;margin-bottom:0;margin-top:0}html.theme--catppuccin-latte .pagination.is-centered .pagination-previous{order:1}html.theme--catppuccin-latte .pagination.is-centered .pagination-list{justify-content:center;order:2}html.theme--catppuccin-latte .pagination.is-centered .pagination-next{order:3}html.theme--catppuccin-latte .pagination.is-right .pagination-previous{order:1}html.theme--catppuccin-latte .pagination.is-right .pagination-next{order:2}html.theme--catppuccin-latte .pagination.is-right .pagination-list{justify-content:flex-end;order:3}}html.theme--catppuccin-latte .panel{border-radius:8px;box-shadow:#171717;font-size:1rem}html.theme--catppuccin-latte .panel:not(:last-child){margin-bottom:1.5rem}html.theme--catppuccin-latte .panel.is-white .panel-heading{background-color:#fff;color:#0a0a0a}html.theme--catppuccin-latte .panel.is-white .panel-tabs a.is-active{border-bottom-color:#fff}html.theme--catppuccin-latte .panel.is-white .panel-block.is-active .panel-icon{color:#fff}html.theme--catppuccin-latte .panel.is-black .panel-heading{background-color:#0a0a0a;color:#fff}html.theme--catppuccin-latte .panel.is-black .panel-tabs a.is-active{border-bottom-color:#0a0a0a}html.theme--catppuccin-latte .panel.is-black .panel-block.is-active .panel-icon{color:#0a0a0a}html.theme--catppuccin-latte .panel.is-light .panel-heading{background-color:#f5f5f5;color:rgba(0,0,0,0.7)}html.theme--catppuccin-latte .panel.is-light .panel-tabs a.is-active{border-bottom-color:#f5f5f5}html.theme--catppuccin-latte .panel.is-light .panel-block.is-active .panel-icon{color:#f5f5f5}html.theme--catppuccin-latte .panel.is-dark .panel-heading,html.theme--catppuccin-latte .content kbd.panel .panel-heading{background-color:#ccd0da;color:rgba(0,0,0,0.7)}html.theme--catppuccin-latte .panel.is-dark .panel-tabs a.is-active,html.theme--catppuccin-latte .content kbd.panel .panel-tabs a.is-active{border-bottom-color:#ccd0da}html.theme--catppuccin-latte .panel.is-dark .panel-block.is-active .panel-icon,html.theme--catppuccin-latte .content kbd.panel .panel-block.is-active .panel-icon{color:#ccd0da}html.theme--catppuccin-latte .panel.is-primary .panel-heading,html.theme--catppuccin-latte .docstring>section>a.panel.docs-sourcelink .panel-heading{background-color:#1e66f5;color:#fff}html.theme--catppuccin-latte .panel.is-primary .panel-tabs a.is-active,html.theme--catppuccin-latte .docstring>section>a.panel.docs-sourcelink .panel-tabs a.is-active{border-bottom-color:#1e66f5}html.theme--catppuccin-latte .panel.is-primary .panel-block.is-active .panel-icon,html.theme--catppuccin-latte .docstring>section>a.panel.docs-sourcelink .panel-block.is-active .panel-icon{color:#1e66f5}html.theme--catppuccin-latte .panel.is-link .panel-heading{background-color:#1e66f5;color:#fff}html.theme--catppuccin-latte .panel.is-link .panel-tabs a.is-active{border-bottom-color:#1e66f5}html.theme--catppuccin-latte .panel.is-link .panel-block.is-active .panel-icon{color:#1e66f5}html.theme--catppuccin-latte .panel.is-info 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kbd.button.is-active{background-color:#bdc2cf;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-latte .button.is-dark[disabled],html.theme--catppuccin-latte .content kbd.button[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-dark,fieldset[disabled] html.theme--catppuccin-latte .content kbd.button{background-color:#ccd0da;border-color:#ccd0da;box-shadow:none}html.theme--catppuccin-latte .button.is-dark.is-inverted,html.theme--catppuccin-latte .content kbd.button.is-inverted{background-color:rgba(0,0,0,0.7);color:#ccd0da}html.theme--catppuccin-latte .button.is-dark.is-inverted:hover,html.theme--catppuccin-latte .content kbd.button.is-inverted:hover,html.theme--catppuccin-latte .button.is-dark.is-inverted.is-hovered,html.theme--catppuccin-latte .content kbd.button.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-latte .button.is-dark.is-inverted[disabled],html.theme--catppuccin-latte .content kbd.button.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-dark.is-inverted,fieldset[disabled] html.theme--catppuccin-latte .content kbd.button.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#ccd0da}html.theme--catppuccin-latte .button.is-dark.is-loading::after,html.theme--catppuccin-latte .content kbd.button.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-latte .button.is-dark.is-outlined,html.theme--catppuccin-latte .content kbd.button.is-outlined{background-color:transparent;border-color:#ccd0da;color:#ccd0da}html.theme--catppuccin-latte .button.is-dark.is-outlined:hover,html.theme--catppuccin-latte .content kbd.button.is-outlined:hover,html.theme--catppuccin-latte .button.is-dark.is-outlined.is-hovered,html.theme--catppuccin-latte .content kbd.button.is-outlined.is-hovered,html.theme--catppuccin-latte 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kbd.button.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-dark.is-outlined.is-loading.is-focused::after,html.theme--catppuccin-latte .content kbd.button.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-latte .button.is-dark.is-outlined[disabled],html.theme--catppuccin-latte .content kbd.button.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-dark.is-outlined,fieldset[disabled] html.theme--catppuccin-latte .content kbd.button.is-outlined{background-color:transparent;border-color:#ccd0da;box-shadow:none;color:#ccd0da}html.theme--catppuccin-latte .button.is-dark.is-inverted.is-outlined,html.theme--catppuccin-latte .content kbd.button.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-latte .button.is-dark.is-inverted.is-outlined:hover,html.theme--catppuccin-latte .content kbd.button.is-inverted.is-outlined:hover,html.theme--catppuccin-latte .button.is-dark.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-latte .content kbd.button.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-dark.is-inverted.is-outlined:focus,html.theme--catppuccin-latte .content kbd.button.is-inverted.is-outlined:focus,html.theme--catppuccin-latte .button.is-dark.is-inverted.is-outlined.is-focused,html.theme--catppuccin-latte .content kbd.button.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#ccd0da}html.theme--catppuccin-latte .button.is-dark.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .content kbd.button.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-dark.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .content kbd.button.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-dark.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .content kbd.button.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-dark.is-inverted.is-outlined.is-loading.is-focused::after,html.theme--catppuccin-latte .content kbd.button.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #ccd0da #ccd0da !important}html.theme--catppuccin-latte .button.is-dark.is-inverted.is-outlined[disabled],html.theme--catppuccin-latte .content kbd.button.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-dark.is-inverted.is-outlined,fieldset[disabled] html.theme--catppuccin-latte .content kbd.button.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-latte 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.button.is-link.is-outlined{background-color:transparent;border-color:#1e66f5;box-shadow:none;color:#1e66f5}html.theme--catppuccin-latte .button.is-link.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-latte .button.is-link.is-inverted.is-outlined:hover,html.theme--catppuccin-latte .button.is-link.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-link.is-inverted.is-outlined:focus,html.theme--catppuccin-latte .button.is-link.is-inverted.is-outlined.is-focused{background-color:#fff;color:#1e66f5}html.theme--catppuccin-latte .button.is-link.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-link.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-link.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-link.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #1e66f5 #1e66f5 !important}html.theme--catppuccin-latte .button.is-link.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-link.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-latte .button.is-link.is-light{background-color:#ebf2fe;color:#0a52e1}html.theme--catppuccin-latte .button.is-link.is-light:hover,html.theme--catppuccin-latte .button.is-link.is-light.is-hovered{background-color:#dfe9fe;border-color:transparent;color:#0a52e1}html.theme--catppuccin-latte .button.is-link.is-light:active,html.theme--catppuccin-latte .button.is-link.is-light.is-active{background-color:#d3e1fd;border-color:transparent;color:#0a52e1}html.theme--catppuccin-latte .button.is-info{background-color:#179299;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-info:hover,html.theme--catppuccin-latte .button.is-info.is-hovered{background-color:#15878e;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-info:focus,html.theme--catppuccin-latte .button.is-info.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-info:focus:not(:active),html.theme--catppuccin-latte .button.is-info.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(23,146,153,0.25)}html.theme--catppuccin-latte .button.is-info:active,html.theme--catppuccin-latte .button.is-info.is-active{background-color:#147d83;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-info[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-info{background-color:#179299;border-color:#179299;box-shadow:none}html.theme--catppuccin-latte .button.is-info.is-inverted{background-color:#fff;color:#179299}html.theme--catppuccin-latte .button.is-info.is-inverted:hover,html.theme--catppuccin-latte .button.is-info.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-latte .button.is-info.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-info.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#179299}html.theme--catppuccin-latte .button.is-info.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-info.is-outlined{background-color:transparent;border-color:#179299;color:#179299}html.theme--catppuccin-latte .button.is-info.is-outlined:hover,html.theme--catppuccin-latte .button.is-info.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-info.is-outlined:focus,html.theme--catppuccin-latte .button.is-info.is-outlined.is-focused{background-color:#179299;border-color:#179299;color:#fff}html.theme--catppuccin-latte .button.is-info.is-outlined.is-loading::after{border-color:transparent transparent #179299 #179299 !important}html.theme--catppuccin-latte .button.is-info.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-info.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-info.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-info.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-info.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-info.is-outlined{background-color:transparent;border-color:#179299;box-shadow:none;color:#179299}html.theme--catppuccin-latte .button.is-info.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-latte .button.is-info.is-inverted.is-outlined:hover,html.theme--catppuccin-latte .button.is-info.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-info.is-inverted.is-outlined:focus,html.theme--catppuccin-latte .button.is-info.is-inverted.is-outlined.is-focused{background-color:#fff;color:#179299}html.theme--catppuccin-latte .button.is-info.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-info.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-info.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-info.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #179299 #179299 !important}html.theme--catppuccin-latte .button.is-info.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-info.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-latte .button.is-info.is-light{background-color:#edfcfc;color:#1cb2ba}html.theme--catppuccin-latte .button.is-info.is-light:hover,html.theme--catppuccin-latte .button.is-info.is-light.is-hovered{background-color:#e2f9fb;border-color:transparent;color:#1cb2ba}html.theme--catppuccin-latte .button.is-info.is-light:active,html.theme--catppuccin-latte .button.is-info.is-light.is-active{background-color:#d7f7f9;border-color:transparent;color:#1cb2ba}html.theme--catppuccin-latte .button.is-success{background-color:#40a02b;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-success:hover,html.theme--catppuccin-latte .button.is-success.is-hovered{background-color:#3c9628;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-success:focus,html.theme--catppuccin-latte .button.is-success.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-success:focus:not(:active),html.theme--catppuccin-latte .button.is-success.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(64,160,43,0.25)}html.theme--catppuccin-latte .button.is-success:active,html.theme--catppuccin-latte .button.is-success.is-active{background-color:#388c26;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-success[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-success{background-color:#40a02b;border-color:#40a02b;box-shadow:none}html.theme--catppuccin-latte .button.is-success.is-inverted{background-color:#fff;color:#40a02b}html.theme--catppuccin-latte .button.is-success.is-inverted:hover,html.theme--catppuccin-latte .button.is-success.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-latte .button.is-success.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-success.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#40a02b}html.theme--catppuccin-latte .button.is-success.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-success.is-outlined{background-color:transparent;border-color:#40a02b;color:#40a02b}html.theme--catppuccin-latte .button.is-success.is-outlined:hover,html.theme--catppuccin-latte .button.is-success.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-success.is-outlined:focus,html.theme--catppuccin-latte .button.is-success.is-outlined.is-focused{background-color:#40a02b;border-color:#40a02b;color:#fff}html.theme--catppuccin-latte .button.is-success.is-outlined.is-loading::after{border-color:transparent transparent #40a02b #40a02b !important}html.theme--catppuccin-latte .button.is-success.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-success.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-success.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-success.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-success.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-success.is-outlined{background-color:transparent;border-color:#40a02b;box-shadow:none;color:#40a02b}html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined:hover,html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined:focus,html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined.is-focused{background-color:#fff;color:#40a02b}html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #40a02b #40a02b !important}html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-latte .button.is-success.is-light{background-color:#f1fbef;color:#40a12b}html.theme--catppuccin-latte .button.is-success.is-light:hover,html.theme--catppuccin-latte .button.is-success.is-light.is-hovered{background-color:#e8f8e5;border-color:transparent;color:#40a12b}html.theme--catppuccin-latte .button.is-success.is-light:active,html.theme--catppuccin-latte .button.is-success.is-light.is-active{background-color:#e0f5db;border-color:transparent;color:#40a12b}html.theme--catppuccin-latte .button.is-warning{background-color:#df8e1d;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-warning:hover,html.theme--catppuccin-latte .button.is-warning.is-hovered{background-color:#d4871c;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-warning:focus,html.theme--catppuccin-latte .button.is-warning.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-warning:focus:not(:active),html.theme--catppuccin-latte .button.is-warning.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(223,142,29,0.25)}html.theme--catppuccin-latte .button.is-warning:active,html.theme--catppuccin-latte .button.is-warning.is-active{background-color:#c8801a;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-warning[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-warning{background-color:#df8e1d;border-color:#df8e1d;box-shadow:none}html.theme--catppuccin-latte .button.is-warning.is-inverted{background-color:#fff;color:#df8e1d}html.theme--catppuccin-latte .button.is-warning.is-inverted:hover,html.theme--catppuccin-latte .button.is-warning.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-latte .button.is-warning.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-warning.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#df8e1d}html.theme--catppuccin-latte .button.is-warning.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-warning.is-outlined{background-color:transparent;border-color:#df8e1d;color:#df8e1d}html.theme--catppuccin-latte .button.is-warning.is-outlined:hover,html.theme--catppuccin-latte .button.is-warning.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-warning.is-outlined:focus,html.theme--catppuccin-latte .button.is-warning.is-outlined.is-focused{background-color:#df8e1d;border-color:#df8e1d;color:#fff}html.theme--catppuccin-latte .button.is-warning.is-outlined.is-loading::after{border-color:transparent transparent #df8e1d #df8e1d !important}html.theme--catppuccin-latte .button.is-warning.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-warning.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-warning.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-warning.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-warning.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-warning.is-outlined{background-color:transparent;border-color:#df8e1d;box-shadow:none;color:#df8e1d}html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined:hover,html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined:focus,html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined.is-focused{background-color:#fff;color:#df8e1d}html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #df8e1d #df8e1d !important}html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-latte .button.is-warning.is-light{background-color:#fdf6ed;color:#9e6515}html.theme--catppuccin-latte .button.is-warning.is-light:hover,html.theme--catppuccin-latte .button.is-warning.is-light.is-hovered{background-color:#fbf1e2;border-color:transparent;color:#9e6515}html.theme--catppuccin-latte .button.is-warning.is-light:active,html.theme--catppuccin-latte .button.is-warning.is-light.is-active{background-color:#faebd6;border-color:transparent;color:#9e6515}html.theme--catppuccin-latte .button.is-danger{background-color:#d20f39;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-danger:hover,html.theme--catppuccin-latte .button.is-danger.is-hovered{background-color:#c60e36;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-danger:focus,html.theme--catppuccin-latte .button.is-danger.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-danger:focus:not(:active),html.theme--catppuccin-latte .button.is-danger.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(210,15,57,0.25)}html.theme--catppuccin-latte .button.is-danger:active,html.theme--catppuccin-latte .button.is-danger.is-active{background-color:#ba0d33;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-danger[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-danger{background-color:#d20f39;border-color:#d20f39;box-shadow:none}html.theme--catppuccin-latte .button.is-danger.is-inverted{background-color:#fff;color:#d20f39}html.theme--catppuccin-latte .button.is-danger.is-inverted:hover,html.theme--catppuccin-latte .button.is-danger.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-latte .button.is-danger.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-danger.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#d20f39}html.theme--catppuccin-latte .button.is-danger.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-danger.is-outlined{background-color:transparent;border-color:#d20f39;color:#d20f39}html.theme--catppuccin-latte .button.is-danger.is-outlined:hover,html.theme--catppuccin-latte .button.is-danger.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-danger.is-outlined:focus,html.theme--catppuccin-latte .button.is-danger.is-outlined.is-focused{background-color:#d20f39;border-color:#d20f39;color:#fff}html.theme--catppuccin-latte .button.is-danger.is-outlined.is-loading::after{border-color:transparent transparent #d20f39 #d20f39 !important}html.theme--catppuccin-latte .button.is-danger.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-danger.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-danger.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-danger.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-danger.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-danger.is-outlined{background-color:transparent;border-color:#d20f39;box-shadow:none;color:#d20f39}html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined:hover,html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined:focus,html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined.is-focused{background-color:#fff;color:#d20f39}html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #d20f39 #d20f39 !important}html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-latte .button.is-danger.is-light{background-color:#feecf0;color:#e9113f}html.theme--catppuccin-latte .button.is-danger.is-light:hover,html.theme--catppuccin-latte .button.is-danger.is-light.is-hovered{background-color:#fde0e6;border-color:transparent;color:#e9113f}html.theme--catppuccin-latte .button.is-danger.is-light:active,html.theme--catppuccin-latte .button.is-danger.is-light.is-active{background-color:#fcd4dd;border-color:transparent;color:#e9113f}html.theme--catppuccin-latte .button.is-small,html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.button{font-size:.75rem}html.theme--catppuccin-latte .button.is-small:not(.is-rounded),html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.button:not(.is-rounded){border-radius:3px}html.theme--catppuccin-latte .button.is-normal{font-size:1rem}html.theme--catppuccin-latte .button.is-medium{font-size:1.25rem}html.theme--catppuccin-latte .button.is-large{font-size:1.5rem}html.theme--catppuccin-latte .button[disabled],fieldset[disabled] html.theme--catppuccin-latte .button{background-color:#9ca0b0;border-color:#acb0be;box-shadow:none;opacity:.5}html.theme--catppuccin-latte .button.is-fullwidth{display:flex;width:100%}html.theme--catppuccin-latte .button.is-loading{color:transparent !important;pointer-events:none}html.theme--catppuccin-latte .button.is-loading::after{position:absolute;left:calc(50% - (1em * 0.5));top:calc(50% - (1em * 0.5));position:absolute !important}html.theme--catppuccin-latte .button.is-static{background-color:#e6e9ef;border-color:#acb0be;color:#8c8fa1;box-shadow:none;pointer-events:none}html.theme--catppuccin-latte .button.is-rounded,html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.button{border-radius:9999px;padding-left:calc(1em + 0.25em);padding-right:calc(1em + 0.25em)}html.theme--catppuccin-latte .buttons{align-items:center;display:flex;flex-wrap:wrap;justify-content:flex-start}html.theme--catppuccin-latte .buttons .button{margin-bottom:0.5rem}html.theme--catppuccin-latte .buttons .button:not(:last-child):not(.is-fullwidth){margin-right:.5rem}html.theme--catppuccin-latte .buttons:last-child{margin-bottom:-0.5rem}html.theme--catppuccin-latte .buttons:not(:last-child){margin-bottom:1rem}html.theme--catppuccin-latte .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large){font-size:.75rem}html.theme--catppuccin-latte .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large):not(.is-rounded){border-radius:3px}html.theme--catppuccin-latte .buttons.are-medium .button:not(.is-small):not(.is-normal):not(.is-large){font-size:1.25rem}html.theme--catppuccin-latte .buttons.are-large .button:not(.is-small):not(.is-normal):not(.is-medium){font-size:1.5rem}html.theme--catppuccin-latte .buttons.has-addons .button:not(:first-child){border-bottom-left-radius:0;border-top-left-radius:0}html.theme--catppuccin-latte .buttons.has-addons .button:not(:last-child){border-bottom-right-radius:0;border-top-right-radius:0;margin-right:-1px}html.theme--catppuccin-latte .buttons.has-addons .button:last-child{margin-right:0}html.theme--catppuccin-latte .buttons.has-addons .button:hover,html.theme--catppuccin-latte .buttons.has-addons .button.is-hovered{z-index:2}html.theme--catppuccin-latte .buttons.has-addons .button:focus,html.theme--catppuccin-latte .buttons.has-addons .button.is-focused,html.theme--catppuccin-latte .buttons.has-addons .button:active,html.theme--catppuccin-latte .buttons.has-addons .button.is-active,html.theme--catppuccin-latte .buttons.has-addons .button.is-selected{z-index:3}html.theme--catppuccin-latte .buttons.has-addons .button:focus:hover,html.theme--catppuccin-latte .buttons.has-addons .button.is-focused:hover,html.theme--catppuccin-latte .buttons.has-addons .button:active:hover,html.theme--catppuccin-latte .buttons.has-addons .button.is-active:hover,html.theme--catppuccin-latte .buttons.has-addons .button.is-selected:hover{z-index:4}html.theme--catppuccin-latte .buttons.has-addons .button.is-expanded{flex-grow:1;flex-shrink:1}html.theme--catppuccin-latte .buttons.is-centered{justify-content:center}html.theme--catppuccin-latte .buttons.is-centered:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}html.theme--catppuccin-latte .buttons.is-right{justify-content:flex-end}html.theme--catppuccin-latte .buttons.is-right:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}@media screen and (max-width: 768px){html.theme--catppuccin-latte .button.is-responsive.is-small,html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.5625rem}html.theme--catppuccin-latte .button.is-responsive,html.theme--catppuccin-latte .button.is-responsive.is-normal{font-size:.65625rem}html.theme--catppuccin-latte .button.is-responsive.is-medium{font-size:.75rem}html.theme--catppuccin-latte .button.is-responsive.is-large{font-size:1rem}}@media screen and (min-width: 769px) and (max-width: 1055px){html.theme--catppuccin-latte .button.is-responsive.is-small,html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.65625rem}html.theme--catppuccin-latte .button.is-responsive,html.theme--catppuccin-latte .button.is-responsive.is-normal{font-size:.75rem}html.theme--catppuccin-latte .button.is-responsive.is-medium{font-size:1rem}html.theme--catppuccin-latte .button.is-responsive.is-large{font-size:1.25rem}}html.theme--catppuccin-latte .container{flex-grow:1;margin:0 auto;position:relative;width:auto}html.theme--catppuccin-latte .container.is-fluid{max-width:none !important;padding-left:32px;padding-right:32px;width:100%}@media screen and (min-width: 1056px){html.theme--catppuccin-latte .container{max-width:992px}}@media screen and (max-width: 1215px){html.theme--catppuccin-latte 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.content blockquote{background-color:#e6e9ef;border-left:5px solid #acb0be;padding:1.25em 1.5em}html.theme--catppuccin-latte .content ol{list-style-position:outside;margin-left:2em;margin-top:1em}html.theme--catppuccin-latte .content ol:not([type]){list-style-type:decimal}html.theme--catppuccin-latte .content ol.is-lower-alpha:not([type]){list-style-type:lower-alpha}html.theme--catppuccin-latte .content ol.is-lower-roman:not([type]){list-style-type:lower-roman}html.theme--catppuccin-latte .content ol.is-upper-alpha:not([type]){list-style-type:upper-alpha}html.theme--catppuccin-latte .content ol.is-upper-roman:not([type]){list-style-type:upper-roman}html.theme--catppuccin-latte .content ul{list-style:disc outside;margin-left:2em;margin-top:1em}html.theme--catppuccin-latte .content ul ul{list-style-type:circle;margin-top:0.5em}html.theme--catppuccin-latte .content ul ul ul{list-style-type:square}html.theme--catppuccin-latte .content dd{margin-left:2em}html.theme--catppuccin-latte .content figure{margin-left:2em;margin-right:2em;text-align:center}html.theme--catppuccin-latte .content figure:not(:first-child){margin-top:2em}html.theme--catppuccin-latte .content figure:not(:last-child){margin-bottom:2em}html.theme--catppuccin-latte .content figure img{display:inline-block}html.theme--catppuccin-latte .content figure figcaption{font-style:italic}html.theme--catppuccin-latte .content pre{-webkit-overflow-scrolling:touch;overflow-x:auto;padding:0;white-space:pre;word-wrap:normal}html.theme--catppuccin-latte .content sup,html.theme--catppuccin-latte .content sub{font-size:75%}html.theme--catppuccin-latte .content table{width:100%}html.theme--catppuccin-latte .content table td,html.theme--catppuccin-latte .content table th{border:1px solid #acb0be;border-width:0 0 1px;padding:0.5em 0.75em;vertical-align:top}html.theme--catppuccin-latte .content table th{color:#41445a}html.theme--catppuccin-latte .content table 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.image.is-5by4 img,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-5by4 img,html.theme--catppuccin-latte .image.is-5by4 .has-ratio,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-5by4 .has-ratio,html.theme--catppuccin-latte .image.is-4by3 img,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-4by3 img,html.theme--catppuccin-latte .image.is-4by3 .has-ratio,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-4by3 .has-ratio,html.theme--catppuccin-latte .image.is-3by2 img,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-3by2 img,html.theme--catppuccin-latte .image.is-3by2 .has-ratio,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-3by2 .has-ratio,html.theme--catppuccin-latte .image.is-5by3 img,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-5by3 img,html.theme--catppuccin-latte .image.is-5by3 .has-ratio,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-5by3 .has-ratio,html.theme--catppuccin-latte .image.is-16by9 img,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-16by9 img,html.theme--catppuccin-latte .image.is-16by9 .has-ratio,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-16by9 .has-ratio,html.theme--catppuccin-latte .image.is-2by1 img,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-2by1 img,html.theme--catppuccin-latte .image.is-2by1 .has-ratio,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-2by1 .has-ratio,html.theme--catppuccin-latte .image.is-3by1 img,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-3by1 img,html.theme--catppuccin-latte .image.is-3by1 .has-ratio,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-3by1 .has-ratio,html.theme--catppuccin-latte .image.is-4by5 img,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-4by5 img,html.theme--catppuccin-latte .image.is-4by5 .has-ratio,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-4by5 .has-ratio,html.theme--catppuccin-latte .image.is-3by4 img,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-3by4 img,html.theme--catppuccin-latte .image.is-3by4 .has-ratio,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-3by4 .has-ratio,html.theme--catppuccin-latte .image.is-2by3 img,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-2by3 img,html.theme--catppuccin-latte .image.is-2by3 .has-ratio,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-2by3 .has-ratio,html.theme--catppuccin-latte .image.is-3by5 img,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-3by5 img,html.theme--catppuccin-latte .image.is-3by5 .has-ratio,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-3by5 .has-ratio,html.theme--catppuccin-latte .image.is-9by16 img,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-9by16 img,html.theme--catppuccin-latte .image.is-9by16 .has-ratio,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-9by16 .has-ratio,html.theme--catppuccin-latte .image.is-1by2 img,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-1by2 img,html.theme--catppuccin-latte .image.is-1by2 .has-ratio,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-1by2 .has-ratio,html.theme--catppuccin-latte .image.is-1by3 img,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-1by3 img,html.theme--catppuccin-latte .image.is-1by3 .has-ratio,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-1by3 .has-ratio{height:100%;width:100%}html.theme--catppuccin-latte .image.is-square,html.theme--catppuccin-latte #documenter 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.docs-logo>img.is-2by1{padding-top:50%}html.theme--catppuccin-latte .image.is-3by1,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-3by1{padding-top:33.3333%}html.theme--catppuccin-latte .image.is-4by5,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-4by5{padding-top:125%}html.theme--catppuccin-latte .image.is-3by4,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-3by4{padding-top:133.3333%}html.theme--catppuccin-latte .image.is-2by3,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-2by3{padding-top:150%}html.theme--catppuccin-latte .image.is-3by5,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-3by5{padding-top:166.6666%}html.theme--catppuccin-latte .image.is-9by16,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-9by16{padding-top:177.7777%}html.theme--catppuccin-latte .image.is-1by2,html.theme--catppuccin-latte #documenter .docs-sidebar 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.file-name{flex-grow:1;max-width:none}html.theme--catppuccin-latte .file.is-right{justify-content:flex-end}html.theme--catppuccin-latte .file.is-right .file-cta{border-radius:0 .4em .4em 0}html.theme--catppuccin-latte .file.is-right .file-name{border-radius:.4em 0 0 .4em;border-width:1px 0 1px 1px;order:-1}html.theme--catppuccin-latte .file-label{align-items:stretch;display:flex;cursor:pointer;justify-content:flex-start;overflow:hidden;position:relative}html.theme--catppuccin-latte .file-label:hover .file-cta{background-color:#c5c9d5;color:#41445a}html.theme--catppuccin-latte .file-label:hover .file-name{border-color:#a5a9b8}html.theme--catppuccin-latte .file-label:active .file-cta{background-color:#bdc2cf;color:#41445a}html.theme--catppuccin-latte .file-label:active .file-name{border-color:#9ea2b3}html.theme--catppuccin-latte .file-input{height:100%;left:0;opacity:0;outline:none;position:absolute;top:0;width:100%}html.theme--catppuccin-latte .file-cta,html.theme--catppuccin-latte 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.label.is-large{font-size:1.5rem}html.theme--catppuccin-latte .help{display:block;font-size:.75rem;margin-top:0.25rem}html.theme--catppuccin-latte .help.is-white{color:#fff}html.theme--catppuccin-latte .help.is-black{color:#0a0a0a}html.theme--catppuccin-latte .help.is-light{color:#f5f5f5}html.theme--catppuccin-latte .help.is-dark,html.theme--catppuccin-latte .content kbd.help{color:#ccd0da}html.theme--catppuccin-latte .help.is-primary,html.theme--catppuccin-latte .docstring>section>a.help.docs-sourcelink{color:#1e66f5}html.theme--catppuccin-latte .help.is-link{color:#1e66f5}html.theme--catppuccin-latte .help.is-info{color:#179299}html.theme--catppuccin-latte .help.is-success{color:#40a02b}html.theme--catppuccin-latte .help.is-warning{color:#df8e1d}html.theme--catppuccin-latte .help.is-danger{color:#d20f39}html.theme--catppuccin-latte .field:not(:last-child){margin-bottom:0.75rem}html.theme--catppuccin-latte 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.control:first-child:not(:only-child) #documenter .docs-sidebar form.docs-search>input,html.theme--catppuccin-latte #documenter .docs-sidebar .field.has-addons .control:first-child:not(:only-child) form.docs-search>input,html.theme--catppuccin-latte .field.has-addons .control:first-child:not(:only-child) .select select{border-bottom-right-radius:0;border-top-right-radius:0}html.theme--catppuccin-latte .field.has-addons .control:last-child:not(:only-child) .button,html.theme--catppuccin-latte .field.has-addons .control:last-child:not(:only-child) .input,html.theme--catppuccin-latte .field.has-addons .control:last-child:not(:only-child) #documenter .docs-sidebar form.docs-search>input,html.theme--catppuccin-latte #documenter .docs-sidebar .field.has-addons .control:last-child:not(:only-child) form.docs-search>input,html.theme--catppuccin-latte .field.has-addons .control:last-child:not(:only-child) .select 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.icon:only-child,html.theme--catppuccin-latte .navbar-link .icon:only-child{margin-left:-0.25rem;margin-right:-0.25rem}html.theme--catppuccin-latte a.navbar-item,html.theme--catppuccin-latte .navbar-link{cursor:pointer}html.theme--catppuccin-latte a.navbar-item:focus,html.theme--catppuccin-latte a.navbar-item:focus-within,html.theme--catppuccin-latte a.navbar-item:hover,html.theme--catppuccin-latte a.navbar-item.is-active,html.theme--catppuccin-latte .navbar-link:focus,html.theme--catppuccin-latte .navbar-link:focus-within,html.theme--catppuccin-latte .navbar-link:hover,html.theme--catppuccin-latte .navbar-link.is-active{background-color:rgba(0,0,0,0);color:#1e66f5}html.theme--catppuccin-latte .navbar-item{flex-grow:0;flex-shrink:0}html.theme--catppuccin-latte .navbar-item img{max-height:1.75rem}html.theme--catppuccin-latte .navbar-item.has-dropdown{padding:0}html.theme--catppuccin-latte .navbar-item.is-expanded{flex-grow:1;flex-shrink:1}html.theme--catppuccin-latte .navbar-item.is-tab{border-bottom:1px solid transparent;min-height:4rem;padding-bottom:calc(0.5rem - 1px)}html.theme--catppuccin-latte .navbar-item.is-tab:focus,html.theme--catppuccin-latte .navbar-item.is-tab:hover{background-color:rgba(0,0,0,0);border-bottom-color:#1e66f5}html.theme--catppuccin-latte .navbar-item.is-tab.is-active{background-color:rgba(0,0,0,0);border-bottom-color:#1e66f5;border-bottom-style:solid;border-bottom-width:3px;color:#1e66f5;padding-bottom:calc(0.5rem - 3px)}html.theme--catppuccin-latte .navbar-content{flex-grow:1;flex-shrink:1}html.theme--catppuccin-latte .navbar-link:not(.is-arrowless){padding-right:2.5em}html.theme--catppuccin-latte .navbar-link:not(.is-arrowless)::after{border-color:#fff;margin-top:-0.375em;right:1.125em}html.theme--catppuccin-latte .navbar-dropdown{font-size:0.875rem;padding-bottom:0.5rem;padding-top:0.5rem}html.theme--catppuccin-latte .navbar-dropdown .navbar-item{padding-left:1.5rem;padding-right:1.5rem}html.theme--catppuccin-latte .navbar-divider{background-color:rgba(0,0,0,0.2);border:none;display:none;height:2px;margin:0.5rem 0}@media screen and (max-width: 1055px){html.theme--catppuccin-latte .navbar>.container{display:block}html.theme--catppuccin-latte .navbar-brand .navbar-item,html.theme--catppuccin-latte .navbar-tabs .navbar-item{align-items:center;display:flex}html.theme--catppuccin-latte .navbar-link::after{display:none}html.theme--catppuccin-latte .navbar-menu{background-color:#1e66f5;box-shadow:0 8px 16px rgba(10,10,10,0.1);padding:0.5rem 0}html.theme--catppuccin-latte .navbar-menu.is-active{display:block}html.theme--catppuccin-latte .navbar.is-fixed-bottom-touch,html.theme--catppuccin-latte .navbar.is-fixed-top-touch{left:0;position:fixed;right:0;z-index:30}html.theme--catppuccin-latte .navbar.is-fixed-bottom-touch{bottom:0}html.theme--catppuccin-latte .navbar.is-fixed-bottom-touch.has-shadow{box-shadow:0 -2px 3px rgba(10,10,10,0.1)}html.theme--catppuccin-latte 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a.navbar-item{padding-right:3rem}html.theme--catppuccin-latte .navbar-dropdown a.navbar-item:focus,html.theme--catppuccin-latte .navbar-dropdown a.navbar-item:hover{background-color:rgba(0,0,0,0);color:#8c8fa1}html.theme--catppuccin-latte .navbar-dropdown a.navbar-item.is-active{background-color:rgba(0,0,0,0);color:#1e66f5}.navbar.is-spaced html.theme--catppuccin-latte .navbar-dropdown,html.theme--catppuccin-latte .navbar-dropdown.is-boxed{border-radius:8px;border-top:none;box-shadow:0 8px 8px rgba(10,10,10,0.1), 0 0 0 1px rgba(10,10,10,0.1);display:block;opacity:0;pointer-events:none;top:calc(100% + (-4px));transform:translateY(-5px);transition-duration:86ms;transition-property:opacity, transform}html.theme--catppuccin-latte .navbar-dropdown.is-right{left:auto;right:0}html.theme--catppuccin-latte .navbar-divider{display:block}html.theme--catppuccin-latte .navbar>.container .navbar-brand,html.theme--catppuccin-latte .container>.navbar .navbar-brand{margin-left:-.75rem}html.theme--catppuccin-latte .navbar>.container .navbar-menu,html.theme--catppuccin-latte .container>.navbar .navbar-menu{margin-right:-.75rem}html.theme--catppuccin-latte .navbar.is-fixed-bottom-desktop,html.theme--catppuccin-latte .navbar.is-fixed-top-desktop{left:0;position:fixed;right:0;z-index:30}html.theme--catppuccin-latte .navbar.is-fixed-bottom-desktop{bottom:0}html.theme--catppuccin-latte .navbar.is-fixed-bottom-desktop.has-shadow{box-shadow:0 -2px 3px rgba(10,10,10,0.1)}html.theme--catppuccin-latte .navbar.is-fixed-top-desktop{top:0}html.theme--catppuccin-latte html.has-navbar-fixed-top-desktop,html.theme--catppuccin-latte body.has-navbar-fixed-top-desktop{padding-top:4rem}html.theme--catppuccin-latte html.has-navbar-fixed-bottom-desktop,html.theme--catppuccin-latte body.has-navbar-fixed-bottom-desktop{padding-bottom:4rem}html.theme--catppuccin-latte html.has-spaced-navbar-fixed-top,html.theme--catppuccin-latte body.has-spaced-navbar-fixed-top{padding-top:6rem}html.theme--catppuccin-latte html.has-spaced-navbar-fixed-bottom,html.theme--catppuccin-latte body.has-spaced-navbar-fixed-bottom{padding-bottom:6rem}html.theme--catppuccin-latte a.navbar-item.is-active,html.theme--catppuccin-latte .navbar-link.is-active{color:#1e66f5}html.theme--catppuccin-latte a.navbar-item.is-active:not(:focus):not(:hover),html.theme--catppuccin-latte .navbar-link.is-active:not(:focus):not(:hover){background-color:rgba(0,0,0,0)}html.theme--catppuccin-latte .navbar-item.has-dropdown:focus .navbar-link,html.theme--catppuccin-latte .navbar-item.has-dropdown:hover .navbar-link,html.theme--catppuccin-latte .navbar-item.has-dropdown.is-active .navbar-link{background-color:rgba(0,0,0,0)}}html.theme--catppuccin-latte .hero.is-fullheight-with-navbar{min-height:calc(100vh - 4rem)}html.theme--catppuccin-latte .pagination{font-size:1rem;margin:-.25rem}html.theme--catppuccin-latte .pagination.is-small,html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.pagination{font-size:.75rem}html.theme--catppuccin-latte .pagination.is-medium{font-size:1.25rem}html.theme--catppuccin-latte .pagination.is-large{font-size:1.5rem}html.theme--catppuccin-latte .pagination.is-rounded .pagination-previous,html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.pagination .pagination-previous,html.theme--catppuccin-latte .pagination.is-rounded .pagination-next,html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.pagination .pagination-next{padding-left:1em;padding-right:1em;border-radius:9999px}html.theme--catppuccin-latte .pagination.is-rounded .pagination-link,html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.pagination .pagination-link{border-radius:9999px}html.theme--catppuccin-latte .pagination,html.theme--catppuccin-latte 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.button.is-black[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-black{background-color:#0a0a0a;border-color:#0a0a0a;box-shadow:none}html.theme--catppuccin-macchiato .button.is-black.is-inverted{background-color:#fff;color:#0a0a0a}html.theme--catppuccin-macchiato .button.is-black.is-inverted:hover,html.theme--catppuccin-macchiato .button.is-black.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-macchiato .button.is-black.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-black.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#0a0a0a}html.theme--catppuccin-macchiato .button.is-black.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-macchiato .button.is-black.is-outlined{background-color:transparent;border-color:#0a0a0a;color:#0a0a0a}html.theme--catppuccin-macchiato .button.is-black.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-black.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-black.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-black.is-outlined.is-focused{background-color:#0a0a0a;border-color:#0a0a0a;color:#fff}html.theme--catppuccin-macchiato .button.is-black.is-outlined.is-loading::after{border-color:transparent transparent #0a0a0a #0a0a0a !important}html.theme--catppuccin-macchiato .button.is-black.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-black.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-black.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-black.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-macchiato .button.is-black.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-black.is-outlined{background-color:transparent;border-color:#0a0a0a;box-shadow:none;color:#0a0a0a}html.theme--catppuccin-macchiato .button.is-black.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-macchiato .button.is-black.is-inverted.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-black.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-black.is-inverted.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-black.is-inverted.is-outlined.is-focused{background-color:#fff;color:#0a0a0a}html.theme--catppuccin-macchiato .button.is-black.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-black.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-black.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-black.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #0a0a0a #0a0a0a !important}html.theme--catppuccin-macchiato .button.is-black.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-black.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-macchiato .button.is-light{background-color:#f5f5f5;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-light:hover,html.theme--catppuccin-macchiato .button.is-light.is-hovered{background-color:#eee;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-light:focus,html.theme--catppuccin-macchiato .button.is-light.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-light:focus:not(:active),html.theme--catppuccin-macchiato .button.is-light.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(245,245,245,0.25)}html.theme--catppuccin-macchiato .button.is-light:active,html.theme--catppuccin-macchiato .button.is-light.is-active{background-color:#e8e8e8;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-light[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-light{background-color:#f5f5f5;border-color:#f5f5f5;box-shadow:none}html.theme--catppuccin-macchiato .button.is-light.is-inverted{background-color:rgba(0,0,0,0.7);color:#f5f5f5}html.theme--catppuccin-macchiato .button.is-light.is-inverted:hover,html.theme--catppuccin-macchiato .button.is-light.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-light.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-light.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#f5f5f5}html.theme--catppuccin-macchiato .button.is-light.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-macchiato .button.is-light.is-outlined{background-color:transparent;border-color:#f5f5f5;color:#f5f5f5}html.theme--catppuccin-macchiato .button.is-light.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-light.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-light.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-light.is-outlined.is-focused{background-color:#f5f5f5;border-color:#f5f5f5;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-light.is-outlined.is-loading::after{border-color:transparent transparent #f5f5f5 #f5f5f5 !important}html.theme--catppuccin-macchiato .button.is-light.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-light.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-light.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-light.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-macchiato .button.is-light.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-light.is-outlined{background-color:transparent;border-color:#f5f5f5;box-shadow:none;color:#f5f5f5}html.theme--catppuccin-macchiato .button.is-light.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-light.is-inverted.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-light.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-macchiato 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.content kbd.button{background-color:#363a4f;border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-dark:hover,html.theme--catppuccin-macchiato .content kbd.button:hover,html.theme--catppuccin-macchiato .button.is-dark.is-hovered,html.theme--catppuccin-macchiato .content kbd.button.is-hovered{background-color:#313447;border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-dark:focus,html.theme--catppuccin-macchiato .content kbd.button:focus,html.theme--catppuccin-macchiato .button.is-dark.is-focused,html.theme--catppuccin-macchiato .content kbd.button.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-dark:focus:not(:active),html.theme--catppuccin-macchiato .content kbd.button:focus:not(:active),html.theme--catppuccin-macchiato .button.is-dark.is-focused:not(:active),html.theme--catppuccin-macchiato .content kbd.button.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(54,58,79,0.25)}html.theme--catppuccin-macchiato .button.is-dark:active,html.theme--catppuccin-macchiato .content kbd.button:active,html.theme--catppuccin-macchiato .button.is-dark.is-active,html.theme--catppuccin-macchiato .content kbd.button.is-active{background-color:#2c2f40;border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-dark[disabled],html.theme--catppuccin-macchiato .content kbd.button[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-dark,fieldset[disabled] html.theme--catppuccin-macchiato .content kbd.button{background-color:#363a4f;border-color:#363a4f;box-shadow:none}html.theme--catppuccin-macchiato .button.is-dark.is-inverted,html.theme--catppuccin-macchiato .content kbd.button.is-inverted{background-color:#fff;color:#363a4f}html.theme--catppuccin-macchiato .button.is-dark.is-inverted:hover,html.theme--catppuccin-macchiato .content kbd.button.is-inverted:hover,html.theme--catppuccin-macchiato 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kbd.button.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-dark.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .content kbd.button.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-dark.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .content kbd.button.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-dark.is-outlined.is-loading.is-focused::after,html.theme--catppuccin-macchiato .content kbd.button.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-macchiato .button.is-dark.is-outlined[disabled],html.theme--catppuccin-macchiato .content kbd.button.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-dark.is-outlined,fieldset[disabled] html.theme--catppuccin-macchiato .content kbd.button.is-outlined{background-color:transparent;border-color:#363a4f;box-shadow:none;color:#363a4f}html.theme--catppuccin-macchiato .button.is-dark.is-inverted.is-outlined,html.theme--catppuccin-macchiato .content kbd.button.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-macchiato .button.is-dark.is-inverted.is-outlined:hover,html.theme--catppuccin-macchiato .content kbd.button.is-inverted.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-dark.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-macchiato .content kbd.button.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-dark.is-inverted.is-outlined:focus,html.theme--catppuccin-macchiato .content kbd.button.is-inverted.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-dark.is-inverted.is-outlined.is-focused,html.theme--catppuccin-macchiato .content kbd.button.is-inverted.is-outlined.is-focused{background-color:#fff;color:#363a4f}html.theme--catppuccin-macchiato .button.is-dark.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .content kbd.button.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-dark.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .content kbd.button.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-dark.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .content kbd.button.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-dark.is-inverted.is-outlined.is-loading.is-focused::after,html.theme--catppuccin-macchiato .content kbd.button.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #363a4f #363a4f !important}html.theme--catppuccin-macchiato 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.button.is-success.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-success.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-success.is-outlined.is-focused{background-color:#a6da95;border-color:#a6da95;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-success.is-outlined.is-loading::after{border-color:transparent transparent #a6da95 #a6da95 !important}html.theme--catppuccin-macchiato .button.is-success.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-success.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-success.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-success.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-macchiato .button.is-success.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-success.is-outlined{background-color:transparent;border-color:#a6da95;box-shadow:none;color:#a6da95}html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#a6da95}html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #a6da95 #a6da95 !important}html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-success.is-light{background-color:#f2faf0;color:#386e26}html.theme--catppuccin-macchiato .button.is-success.is-light:hover,html.theme--catppuccin-macchiato .button.is-success.is-light.is-hovered{background-color:#eaf6e6;border-color:transparent;color:#386e26}html.theme--catppuccin-macchiato .button.is-success.is-light:active,html.theme--catppuccin-macchiato .button.is-success.is-light.is-active{background-color:#e2f3dd;border-color:transparent;color:#386e26}html.theme--catppuccin-macchiato .button.is-warning{background-color:#eed49f;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning:hover,html.theme--catppuccin-macchiato .button.is-warning.is-hovered{background-color:#eccf94;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning:focus,html.theme--catppuccin-macchiato .button.is-warning.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning:focus:not(:active),html.theme--catppuccin-macchiato .button.is-warning.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(238,212,159,0.25)}html.theme--catppuccin-macchiato .button.is-warning:active,html.theme--catppuccin-macchiato .button.is-warning.is-active{background-color:#eaca89;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-warning{background-color:#eed49f;border-color:#eed49f;box-shadow:none}html.theme--catppuccin-macchiato .button.is-warning.is-inverted{background-color:rgba(0,0,0,0.7);color:#eed49f}html.theme--catppuccin-macchiato .button.is-warning.is-inverted:hover,html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-warning.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#eed49f}html.theme--catppuccin-macchiato .button.is-warning.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-macchiato .button.is-warning.is-outlined{background-color:transparent;border-color:#eed49f;color:#eed49f}html.theme--catppuccin-macchiato .button.is-warning.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-warning.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-warning.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-warning.is-outlined.is-focused{background-color:#eed49f;border-color:#eed49f;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning.is-outlined.is-loading::after{border-color:transparent transparent #eed49f #eed49f !important}html.theme--catppuccin-macchiato .button.is-warning.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-warning.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-warning.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-warning.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-macchiato .button.is-warning.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-warning.is-outlined{background-color:transparent;border-color:#eed49f;box-shadow:none;color:#eed49f}html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#eed49f}html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #eed49f #eed49f !important}html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning.is-light{background-color:#fcf7ee;color:#7e5c16}html.theme--catppuccin-macchiato .button.is-warning.is-light:hover,html.theme--catppuccin-macchiato .button.is-warning.is-light.is-hovered{background-color:#faf2e3;border-color:transparent;color:#7e5c16}html.theme--catppuccin-macchiato .button.is-warning.is-light:active,html.theme--catppuccin-macchiato .button.is-warning.is-light.is-active{background-color:#f8eed8;border-color:transparent;color:#7e5c16}html.theme--catppuccin-macchiato .button.is-danger{background-color:#ed8796;border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-danger:hover,html.theme--catppuccin-macchiato .button.is-danger.is-hovered{background-color:#eb7c8c;border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-danger:focus,html.theme--catppuccin-macchiato .button.is-danger.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-danger:focus:not(:active),html.theme--catppuccin-macchiato .button.is-danger.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(237,135,150,0.25)}html.theme--catppuccin-macchiato .button.is-danger:active,html.theme--catppuccin-macchiato .button.is-danger.is-active{background-color:#ea7183;border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-danger[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-danger{background-color:#ed8796;border-color:#ed8796;box-shadow:none}html.theme--catppuccin-macchiato .button.is-danger.is-inverted{background-color:#fff;color:#ed8796}html.theme--catppuccin-macchiato .button.is-danger.is-inverted:hover,html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-macchiato .button.is-danger.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-danger.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#ed8796}html.theme--catppuccin-macchiato .button.is-danger.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-macchiato .button.is-danger.is-outlined{background-color:transparent;border-color:#ed8796;color:#ed8796}html.theme--catppuccin-macchiato .button.is-danger.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-danger.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-danger.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-danger.is-outlined.is-focused{background-color:#ed8796;border-color:#ed8796;color:#fff}html.theme--catppuccin-macchiato .button.is-danger.is-outlined.is-loading::after{border-color:transparent transparent #ed8796 #ed8796 !important}html.theme--catppuccin-macchiato .button.is-danger.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-danger.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-danger.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-danger.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-macchiato .button.is-danger.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-danger.is-outlined{background-color:transparent;border-color:#ed8796;box-shadow:none;color:#ed8796}html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined.is-focused{background-color:#fff;color:#ed8796}html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #ed8796 #ed8796 !important}html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-macchiato .button.is-danger.is-light{background-color:#fcedef;color:#971729}html.theme--catppuccin-macchiato .button.is-danger.is-light:hover,html.theme--catppuccin-macchiato .button.is-danger.is-light.is-hovered{background-color:#fbe2e6;border-color:transparent;color:#971729}html.theme--catppuccin-macchiato .button.is-danger.is-light:active,html.theme--catppuccin-macchiato .button.is-danger.is-light.is-active{background-color:#f9d7dc;border-color:transparent;color:#971729}html.theme--catppuccin-macchiato .button.is-small,html.theme--catppuccin-macchiato #documenter .docs-sidebar form.docs-search>input.button{font-size:.75rem}html.theme--catppuccin-macchiato .button.is-small:not(.is-rounded),html.theme--catppuccin-macchiato #documenter .docs-sidebar form.docs-search>input.button:not(.is-rounded){border-radius:3px}html.theme--catppuccin-macchiato .button.is-normal{font-size:1rem}html.theme--catppuccin-macchiato .button.is-medium{font-size:1.25rem}html.theme--catppuccin-macchiato .button.is-large{font-size:1.5rem}html.theme--catppuccin-macchiato .button[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button{background-color:#6e738d;border-color:#5b6078;box-shadow:none;opacity:.5}html.theme--catppuccin-macchiato .button.is-fullwidth{display:flex;width:100%}html.theme--catppuccin-macchiato .button.is-loading{color:transparent !important;pointer-events:none}html.theme--catppuccin-macchiato .button.is-loading::after{position:absolute;left:calc(50% - (1em * 0.5));top:calc(50% - (1em * 0.5));position:absolute !important}html.theme--catppuccin-macchiato .button.is-static{background-color:#1e2030;border-color:#5b6078;color:#8087a2;box-shadow:none;pointer-events:none}html.theme--catppuccin-macchiato .button.is-rounded,html.theme--catppuccin-macchiato #documenter .docs-sidebar form.docs-search>input.button{border-radius:9999px;padding-left:calc(1em + 0.25em);padding-right:calc(1em + 0.25em)}html.theme--catppuccin-macchiato .buttons{align-items:center;display:flex;flex-wrap:wrap;justify-content:flex-start}html.theme--catppuccin-macchiato .buttons .button{margin-bottom:0.5rem}html.theme--catppuccin-macchiato .buttons .button:not(:last-child):not(.is-fullwidth){margin-right:.5rem}html.theme--catppuccin-macchiato .buttons:last-child{margin-bottom:-0.5rem}html.theme--catppuccin-macchiato .buttons:not(:last-child){margin-bottom:1rem}html.theme--catppuccin-macchiato .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large){font-size:.75rem}html.theme--catppuccin-macchiato .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large):not(.is-rounded){border-radius:3px}html.theme--catppuccin-macchiato .buttons.are-medium .button:not(.is-small):not(.is-normal):not(.is-large){font-size:1.25rem}html.theme--catppuccin-macchiato .buttons.are-large .button:not(.is-small):not(.is-normal):not(.is-medium){font-size:1.5rem}html.theme--catppuccin-macchiato .buttons.has-addons .button:not(:first-child){border-bottom-left-radius:0;border-top-left-radius:0}html.theme--catppuccin-macchiato .buttons.has-addons .button:not(:last-child){border-bottom-right-radius:0;border-top-right-radius:0;margin-right:-1px}html.theme--catppuccin-macchiato .buttons.has-addons .button:last-child{margin-right:0}html.theme--catppuccin-macchiato .buttons.has-addons .button:hover,html.theme--catppuccin-macchiato .buttons.has-addons .button.is-hovered{z-index:2}html.theme--catppuccin-macchiato .buttons.has-addons .button:focus,html.theme--catppuccin-macchiato .buttons.has-addons .button.is-focused,html.theme--catppuccin-macchiato .buttons.has-addons .button:active,html.theme--catppuccin-macchiato .buttons.has-addons .button.is-active,html.theme--catppuccin-macchiato .buttons.has-addons .button.is-selected{z-index:3}html.theme--catppuccin-macchiato .buttons.has-addons .button:focus:hover,html.theme--catppuccin-macchiato .buttons.has-addons .button.is-focused:hover,html.theme--catppuccin-macchiato .buttons.has-addons .button:active:hover,html.theme--catppuccin-macchiato .buttons.has-addons .button.is-active:hover,html.theme--catppuccin-macchiato .buttons.has-addons .button.is-selected:hover{z-index:4}html.theme--catppuccin-macchiato .buttons.has-addons .button.is-expanded{flex-grow:1;flex-shrink:1}html.theme--catppuccin-macchiato .buttons.is-centered{justify-content:center}html.theme--catppuccin-macchiato .buttons.is-centered:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}html.theme--catppuccin-macchiato .buttons.is-right{justify-content:flex-end}html.theme--catppuccin-macchiato .buttons.is-right:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}@media screen and (max-width: 768px){html.theme--catppuccin-macchiato .button.is-responsive.is-small,html.theme--catppuccin-macchiato #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.5625rem}html.theme--catppuccin-macchiato .button.is-responsive,html.theme--catppuccin-macchiato .button.is-responsive.is-normal{font-size:.65625rem}html.theme--catppuccin-macchiato .button.is-responsive.is-medium{font-size:.75rem}html.theme--catppuccin-macchiato .button.is-responsive.is-large{font-size:1rem}}@media screen and (min-width: 769px) and (max-width: 1055px){html.theme--catppuccin-macchiato .button.is-responsive.is-small,html.theme--catppuccin-macchiato #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.65625rem}html.theme--catppuccin-macchiato 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.button.is-white.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-white.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-white.is-outlined.is-focused{background-color:#fff;border-color:#fff;color:#0a0a0a}html.theme--catppuccin-macchiato .button.is-white.is-outlined.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-macchiato .button.is-white.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-white.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-white.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-white.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #0a0a0a #0a0a0a !important}html.theme--catppuccin-macchiato .button.is-white.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-white.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-macchiato .button.is-white.is-inverted.is-outlined{background-color:transparent;border-color:#0a0a0a;color:#0a0a0a}html.theme--catppuccin-macchiato .button.is-white.is-inverted.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-white.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-white.is-inverted.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-white.is-inverted.is-outlined.is-focused{background-color:#0a0a0a;color:#fff}html.theme--catppuccin-macchiato .button.is-white.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-white.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-white.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-white.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-macchiato .button.is-white.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-white.is-inverted.is-outlined{background-color:transparent;border-color:#0a0a0a;box-shadow:none;color:#0a0a0a}html.theme--catppuccin-macchiato .button.is-black{background-color:#0a0a0a;border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-black:hover,html.theme--catppuccin-macchiato .button.is-black.is-hovered{background-color:#040404;border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-black:focus,html.theme--catppuccin-macchiato .button.is-black.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-black:focus:not(:active),html.theme--catppuccin-macchiato .button.is-black.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(10,10,10,0.25)}html.theme--catppuccin-macchiato .button.is-black:active,html.theme--catppuccin-macchiato .button.is-black.is-active{background-color:#000;border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-black[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-black{background-color:#0a0a0a;border-color:#0a0a0a;box-shadow:none}html.theme--catppuccin-macchiato .button.is-black.is-inverted{background-color:#fff;color:#0a0a0a}html.theme--catppuccin-macchiato .button.is-black.is-inverted:hover,html.theme--catppuccin-macchiato .button.is-black.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-macchiato .button.is-black.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-black.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#0a0a0a}html.theme--catppuccin-macchiato .button.is-black.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-macchiato .button.is-black.is-outlined{background-color:transparent;border-color:#0a0a0a;color:#0a0a0a}html.theme--catppuccin-macchiato .button.is-black.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-black.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-black.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-black.is-outlined.is-focused{background-color:#0a0a0a;border-color:#0a0a0a;color:#fff}html.theme--catppuccin-macchiato .button.is-black.is-outlined.is-loading::after{border-color:transparent transparent #0a0a0a #0a0a0a !important}html.theme--catppuccin-macchiato .button.is-black.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-black.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-black.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-black.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-macchiato .button.is-black.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-black.is-outlined{background-color:transparent;border-color:#0a0a0a;box-shadow:none;color:#0a0a0a}html.theme--catppuccin-macchiato .button.is-black.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-macchiato .button.is-black.is-inverted.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-black.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-black.is-inverted.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-black.is-inverted.is-outlined.is-focused{background-color:#fff;color:#0a0a0a}html.theme--catppuccin-macchiato .button.is-black.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-black.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-black.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-black.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #0a0a0a #0a0a0a !important}html.theme--catppuccin-macchiato .button.is-black.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-black.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-macchiato .button.is-light{background-color:#f5f5f5;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-light:hover,html.theme--catppuccin-macchiato .button.is-light.is-hovered{background-color:#eee;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-light:focus,html.theme--catppuccin-macchiato .button.is-light.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-light:focus:not(:active),html.theme--catppuccin-macchiato .button.is-light.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(245,245,245,0.25)}html.theme--catppuccin-macchiato .button.is-light:active,html.theme--catppuccin-macchiato .button.is-light.is-active{background-color:#e8e8e8;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-light[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-light{background-color:#f5f5f5;border-color:#f5f5f5;box-shadow:none}html.theme--catppuccin-macchiato .button.is-light.is-inverted{background-color:rgba(0,0,0,0.7);color:#f5f5f5}html.theme--catppuccin-macchiato .button.is-light.is-inverted:hover,html.theme--catppuccin-macchiato .button.is-light.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-light.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-light.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#f5f5f5}html.theme--catppuccin-macchiato .button.is-light.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-macchiato .button.is-light.is-outlined{background-color:transparent;border-color:#f5f5f5;color:#f5f5f5}html.theme--catppuccin-macchiato .button.is-light.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-light.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-light.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-light.is-outlined.is-focused{background-color:#f5f5f5;border-color:#f5f5f5;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-light.is-outlined.is-loading::after{border-color:transparent transparent #f5f5f5 #f5f5f5 !important}html.theme--catppuccin-macchiato .button.is-light.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-light.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-light.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-light.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-macchiato .button.is-light.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-light.is-outlined{background-color:transparent;border-color:#f5f5f5;box-shadow:none;color:#f5f5f5}html.theme--catppuccin-macchiato .button.is-light.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-light.is-inverted.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-light.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-light.is-inverted.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-light.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#f5f5f5}html.theme--catppuccin-macchiato .button.is-light.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-light.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-light.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-light.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #f5f5f5 #f5f5f5 !important}html.theme--catppuccin-macchiato .button.is-light.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-light.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-dark,html.theme--catppuccin-macchiato .content kbd.button{background-color:#363a4f;border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-dark:hover,html.theme--catppuccin-macchiato .content kbd.button:hover,html.theme--catppuccin-macchiato .button.is-dark.is-hovered,html.theme--catppuccin-macchiato .content kbd.button.is-hovered{background-color:#313447;border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-dark:focus,html.theme--catppuccin-macchiato .content kbd.button:focus,html.theme--catppuccin-macchiato .button.is-dark.is-focused,html.theme--catppuccin-macchiato .content kbd.button.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-dark:focus:not(:active),html.theme--catppuccin-macchiato .content kbd.button:focus:not(:active),html.theme--catppuccin-macchiato .button.is-dark.is-focused:not(:active),html.theme--catppuccin-macchiato .content kbd.button.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(54,58,79,0.25)}html.theme--catppuccin-macchiato .button.is-dark:active,html.theme--catppuccin-macchiato .content kbd.button:active,html.theme--catppuccin-macchiato .button.is-dark.is-active,html.theme--catppuccin-macchiato .content kbd.button.is-active{background-color:#2c2f40;border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-dark[disabled],html.theme--catppuccin-macchiato .content kbd.button[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-dark,fieldset[disabled] html.theme--catppuccin-macchiato .content kbd.button{background-color:#363a4f;border-color:#363a4f;box-shadow:none}html.theme--catppuccin-macchiato .button.is-dark.is-inverted,html.theme--catppuccin-macchiato .content kbd.button.is-inverted{background-color:#fff;color:#363a4f}html.theme--catppuccin-macchiato .button.is-dark.is-inverted:hover,html.theme--catppuccin-macchiato .content kbd.button.is-inverted:hover,html.theme--catppuccin-macchiato .button.is-dark.is-inverted.is-hovered,html.theme--catppuccin-macchiato .content kbd.button.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-macchiato .button.is-dark.is-inverted[disabled],html.theme--catppuccin-macchiato .content kbd.button.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-dark.is-inverted,fieldset[disabled] html.theme--catppuccin-macchiato .content kbd.button.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#363a4f}html.theme--catppuccin-macchiato .button.is-dark.is-loading::after,html.theme--catppuccin-macchiato .content kbd.button.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-macchiato .button.is-dark.is-outlined,html.theme--catppuccin-macchiato .content kbd.button.is-outlined{background-color:transparent;border-color:#363a4f;color:#363a4f}html.theme--catppuccin-macchiato .button.is-dark.is-outlined:hover,html.theme--catppuccin-macchiato .content kbd.button.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-dark.is-outlined.is-hovered,html.theme--catppuccin-macchiato .content kbd.button.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-dark.is-outlined:focus,html.theme--catppuccin-macchiato .content kbd.button.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-dark.is-outlined.is-focused,html.theme--catppuccin-macchiato .content kbd.button.is-outlined.is-focused{background-color:#363a4f;border-color:#363a4f;color:#fff}html.theme--catppuccin-macchiato .button.is-dark.is-outlined.is-loading::after,html.theme--catppuccin-macchiato .content kbd.button.is-outlined.is-loading::after{border-color:transparent transparent #363a4f #363a4f !important}html.theme--catppuccin-macchiato .button.is-dark.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .content kbd.button.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-dark.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .content kbd.button.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-dark.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .content kbd.button.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-dark.is-outlined.is-loading.is-focused::after,html.theme--catppuccin-macchiato .content kbd.button.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-macchiato .button.is-dark.is-outlined[disabled],html.theme--catppuccin-macchiato .content kbd.button.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-dark.is-outlined,fieldset[disabled] html.theme--catppuccin-macchiato .content kbd.button.is-outlined{background-color:transparent;border-color:#363a4f;box-shadow:none;color:#363a4f}html.theme--catppuccin-macchiato .button.is-dark.is-inverted.is-outlined,html.theme--catppuccin-macchiato .content kbd.button.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-macchiato .button.is-dark.is-inverted.is-outlined:hover,html.theme--catppuccin-macchiato .content kbd.button.is-inverted.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-dark.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-macchiato .content kbd.button.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-dark.is-inverted.is-outlined:focus,html.theme--catppuccin-macchiato .content kbd.button.is-inverted.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-dark.is-inverted.is-outlined.is-focused,html.theme--catppuccin-macchiato .content kbd.button.is-inverted.is-outlined.is-focused{background-color:#fff;color:#363a4f}html.theme--catppuccin-macchiato .button.is-dark.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .content kbd.button.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-dark.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .content kbd.button.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-dark.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .content kbd.button.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-dark.is-inverted.is-outlined.is-loading.is-focused::after,html.theme--catppuccin-macchiato .content kbd.button.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #363a4f #363a4f !important}html.theme--catppuccin-macchiato .button.is-dark.is-inverted.is-outlined[disabled],html.theme--catppuccin-macchiato .content kbd.button.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-dark.is-inverted.is-outlined,fieldset[disabled] html.theme--catppuccin-macchiato .content kbd.button.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-macchiato .button.is-primary,html.theme--catppuccin-macchiato .docstring>section>a.button.docs-sourcelink{background-color:#8aadf4;border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-primary:hover,html.theme--catppuccin-macchiato .docstring>section>a.button.docs-sourcelink:hover,html.theme--catppuccin-macchiato .button.is-primary.is-hovered,html.theme--catppuccin-macchiato .docstring>section>a.button.is-hovered.docs-sourcelink{background-color:#7ea5f3;border-color:transparent;color:#fff}html.theme--catppuccin-macchiato 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.button.is-success.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-success.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-success.is-outlined.is-focused{background-color:#a6da95;border-color:#a6da95;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-success.is-outlined.is-loading::after{border-color:transparent transparent #a6da95 #a6da95 !important}html.theme--catppuccin-macchiato .button.is-success.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-success.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-success.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-success.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-macchiato .button.is-success.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-success.is-outlined{background-color:transparent;border-color:#a6da95;box-shadow:none;color:#a6da95}html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#a6da95}html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #a6da95 #a6da95 !important}html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-success.is-light{background-color:#f2faf0;color:#386e26}html.theme--catppuccin-macchiato .button.is-success.is-light:hover,html.theme--catppuccin-macchiato .button.is-success.is-light.is-hovered{background-color:#eaf6e6;border-color:transparent;color:#386e26}html.theme--catppuccin-macchiato .button.is-success.is-light:active,html.theme--catppuccin-macchiato .button.is-success.is-light.is-active{background-color:#e2f3dd;border-color:transparent;color:#386e26}html.theme--catppuccin-macchiato .button.is-warning{background-color:#eed49f;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning:hover,html.theme--catppuccin-macchiato .button.is-warning.is-hovered{background-color:#eccf94;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning:focus,html.theme--catppuccin-macchiato .button.is-warning.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning:focus:not(:active),html.theme--catppuccin-macchiato .button.is-warning.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(238,212,159,0.25)}html.theme--catppuccin-macchiato .button.is-warning:active,html.theme--catppuccin-macchiato .button.is-warning.is-active{background-color:#eaca89;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-warning{background-color:#eed49f;border-color:#eed49f;box-shadow:none}html.theme--catppuccin-macchiato .button.is-warning.is-inverted{background-color:rgba(0,0,0,0.7);color:#eed49f}html.theme--catppuccin-macchiato .button.is-warning.is-inverted:hover,html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-warning.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#eed49f}html.theme--catppuccin-macchiato .button.is-warning.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-macchiato .button.is-warning.is-outlined{background-color:transparent;border-color:#eed49f;color:#eed49f}html.theme--catppuccin-macchiato .button.is-warning.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-warning.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-warning.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-warning.is-outlined.is-focused{background-color:#eed49f;border-color:#eed49f;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning.is-outlined.is-loading::after{border-color:transparent transparent #eed49f #eed49f !important}html.theme--catppuccin-macchiato .button.is-warning.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-warning.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-warning.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-warning.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-macchiato .button.is-warning.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-warning.is-outlined{background-color:transparent;border-color:#eed49f;box-shadow:none;color:#eed49f}html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#eed49f}html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #eed49f #eed49f !important}html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning.is-light{background-color:#fcf7ee;color:#7e5c16}html.theme--catppuccin-macchiato .button.is-warning.is-light:hover,html.theme--catppuccin-macchiato .button.is-warning.is-light.is-hovered{background-color:#faf2e3;border-color:transparent;color:#7e5c16}html.theme--catppuccin-macchiato .button.is-warning.is-light:active,html.theme--catppuccin-macchiato .button.is-warning.is-light.is-active{background-color:#f8eed8;border-color:transparent;color:#7e5c16}html.theme--catppuccin-macchiato .button.is-danger{background-color:#ed8796;border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-danger:hover,html.theme--catppuccin-macchiato .button.is-danger.is-hovered{background-color:#eb7c8c;border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-danger:focus,html.theme--catppuccin-macchiato .button.is-danger.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-danger:focus:not(:active),html.theme--catppuccin-macchiato .button.is-danger.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(237,135,150,0.25)}html.theme--catppuccin-macchiato .button.is-danger:active,html.theme--catppuccin-macchiato .button.is-danger.is-active{background-color:#ea7183;border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-danger[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-danger{background-color:#ed8796;border-color:#ed8796;box-shadow:none}html.theme--catppuccin-macchiato .button.is-danger.is-inverted{background-color:#fff;color:#ed8796}html.theme--catppuccin-macchiato .button.is-danger.is-inverted:hover,html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-macchiato .button.is-danger.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-danger.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#ed8796}html.theme--catppuccin-macchiato .button.is-danger.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-macchiato .button.is-danger.is-outlined{background-color:transparent;border-color:#ed8796;color:#ed8796}html.theme--catppuccin-macchiato .button.is-danger.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-danger.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-danger.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-danger.is-outlined.is-focused{background-color:#ed8796;border-color:#ed8796;color:#fff}html.theme--catppuccin-macchiato .button.is-danger.is-outlined.is-loading::after{border-color:transparent transparent #ed8796 #ed8796 !important}html.theme--catppuccin-macchiato .button.is-danger.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-danger.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-danger.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-danger.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-macchiato .button.is-danger.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-danger.is-outlined{background-color:transparent;border-color:#ed8796;box-shadow:none;color:#ed8796}html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined.is-focused{background-color:#fff;color:#ed8796}html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #ed8796 #ed8796 !important}html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-macchiato .button.is-danger.is-light{background-color:#fcedef;color:#971729}html.theme--catppuccin-macchiato .button.is-danger.is-light:hover,html.theme--catppuccin-macchiato .button.is-danger.is-light.is-hovered{background-color:#fbe2e6;border-color:transparent;color:#971729}html.theme--catppuccin-macchiato .button.is-danger.is-light:active,html.theme--catppuccin-macchiato .button.is-danger.is-light.is-active{background-color:#f9d7dc;border-color:transparent;color:#971729}html.theme--catppuccin-macchiato .button.is-small,html.theme--catppuccin-macchiato #documenter .docs-sidebar form.docs-search>input.button{font-size:.75rem}html.theme--catppuccin-macchiato .button.is-small:not(.is-rounded),html.theme--catppuccin-macchiato #documenter .docs-sidebar form.docs-search>input.button:not(.is-rounded){border-radius:3px}html.theme--catppuccin-macchiato .button.is-normal{font-size:1rem}html.theme--catppuccin-macchiato .button.is-medium{font-size:1.25rem}html.theme--catppuccin-macchiato .button.is-large{font-size:1.5rem}html.theme--catppuccin-macchiato .button[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button{background-color:#6e738d;border-color:#5b6078;box-shadow:none;opacity:.5}html.theme--catppuccin-macchiato .button.is-fullwidth{display:flex;width:100%}html.theme--catppuccin-macchiato .button.is-loading{color:transparent !important;pointer-events:none}html.theme--catppuccin-macchiato .button.is-loading::after{position:absolute;left:calc(50% - (1em * 0.5));top:calc(50% - (1em * 0.5));position:absolute !important}html.theme--catppuccin-macchiato .button.is-static{background-color:#1e2030;border-color:#5b6078;color:#8087a2;box-shadow:none;pointer-events:none}html.theme--catppuccin-macchiato .button.is-rounded,html.theme--catppuccin-macchiato #documenter .docs-sidebar form.docs-search>input.button{border-radius:9999px;padding-left:calc(1em + 0.25em);padding-right:calc(1em + 0.25em)}html.theme--catppuccin-macchiato .buttons{align-items:center;display:flex;flex-wrap:wrap;justify-content:flex-start}html.theme--catppuccin-macchiato .buttons .button{margin-bottom:0.5rem}html.theme--catppuccin-macchiato .buttons .button:not(:last-child):not(.is-fullwidth){margin-right:.5rem}html.theme--catppuccin-macchiato .buttons:last-child{margin-bottom:-0.5rem}html.theme--catppuccin-macchiato .buttons:not(:last-child){margin-bottom:1rem}html.theme--catppuccin-macchiato .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large){font-size:.75rem}html.theme--catppuccin-macchiato .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large):not(.is-rounded){border-radius:3px}html.theme--catppuccin-macchiato .buttons.are-medium .button:not(.is-small):not(.is-normal):not(.is-large){font-size:1.25rem}html.theme--catppuccin-macchiato .buttons.are-large .button:not(.is-small):not(.is-normal):not(.is-medium){font-size:1.5rem}html.theme--catppuccin-macchiato .buttons.has-addons .button:not(:first-child){border-bottom-left-radius:0;border-top-left-radius:0}html.theme--catppuccin-macchiato .buttons.has-addons .button:not(:last-child){border-bottom-right-radius:0;border-top-right-radius:0;margin-right:-1px}html.theme--catppuccin-macchiato .buttons.has-addons .button:last-child{margin-right:0}html.theme--catppuccin-macchiato .buttons.has-addons .button:hover,html.theme--catppuccin-macchiato .buttons.has-addons .button.is-hovered{z-index:2}html.theme--catppuccin-macchiato .buttons.has-addons .button:focus,html.theme--catppuccin-macchiato .buttons.has-addons .button.is-focused,html.theme--catppuccin-macchiato .buttons.has-addons .button:active,html.theme--catppuccin-macchiato .buttons.has-addons .button.is-active,html.theme--catppuccin-macchiato .buttons.has-addons .button.is-selected{z-index:3}html.theme--catppuccin-macchiato .buttons.has-addons .button:focus:hover,html.theme--catppuccin-macchiato .buttons.has-addons .button.is-focused:hover,html.theme--catppuccin-macchiato .buttons.has-addons .button:active:hover,html.theme--catppuccin-macchiato .buttons.has-addons .button.is-active:hover,html.theme--catppuccin-macchiato .buttons.has-addons .button.is-selected:hover{z-index:4}html.theme--catppuccin-macchiato .buttons.has-addons .button.is-expanded{flex-grow:1;flex-shrink:1}html.theme--catppuccin-macchiato .buttons.is-centered{justify-content:center}html.theme--catppuccin-macchiato .buttons.is-centered:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}html.theme--catppuccin-macchiato .buttons.is-right{justify-content:flex-end}html.theme--catppuccin-macchiato .buttons.is-right:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}@media screen and (max-width: 768px){html.theme--catppuccin-macchiato .button.is-responsive.is-small,html.theme--catppuccin-macchiato #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.5625rem}html.theme--catppuccin-macchiato .button.is-responsive,html.theme--catppuccin-macchiato .button.is-responsive.is-normal{font-size:.65625rem}html.theme--catppuccin-macchiato .button.is-responsive.is-medium{font-size:.75rem}html.theme--catppuccin-macchiato .button.is-responsive.is-large{font-size:1rem}}@media screen and (min-width: 769px) and (max-width: 1055px){html.theme--catppuccin-macchiato .button.is-responsive.is-small,html.theme--catppuccin-macchiato #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.65625rem}html.theme--catppuccin-macchiato 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.button.is-white.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-white.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-mocha .button.is-white.is-inverted.is-outlined{background-color:transparent;border-color:#0a0a0a;color:#0a0a0a}html.theme--catppuccin-mocha .button.is-white.is-inverted.is-outlined:hover,html.theme--catppuccin-mocha .button.is-white.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-white.is-inverted.is-outlined:focus,html.theme--catppuccin-mocha .button.is-white.is-inverted.is-outlined.is-focused{background-color:#0a0a0a;color:#fff}html.theme--catppuccin-mocha .button.is-white.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-white.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-white.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-white.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-mocha .button.is-white.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-white.is-inverted.is-outlined{background-color:transparent;border-color:#0a0a0a;box-shadow:none;color:#0a0a0a}html.theme--catppuccin-mocha .button.is-black{background-color:#0a0a0a;border-color:transparent;color:#fff}html.theme--catppuccin-mocha .button.is-black:hover,html.theme--catppuccin-mocha .button.is-black.is-hovered{background-color:#040404;border-color:transparent;color:#fff}html.theme--catppuccin-mocha .button.is-black:focus,html.theme--catppuccin-mocha .button.is-black.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-mocha .button.is-black:focus:not(:active),html.theme--catppuccin-mocha .button.is-black.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(10,10,10,0.25)}html.theme--catppuccin-mocha .button.is-black:active,html.theme--catppuccin-mocha .button.is-black.is-active{background-color:#000;border-color:transparent;color:#fff}html.theme--catppuccin-mocha .button.is-black[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-black{background-color:#0a0a0a;border-color:#0a0a0a;box-shadow:none}html.theme--catppuccin-mocha .button.is-black.is-inverted{background-color:#fff;color:#0a0a0a}html.theme--catppuccin-mocha .button.is-black.is-inverted:hover,html.theme--catppuccin-mocha .button.is-black.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-mocha .button.is-black.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-black.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#0a0a0a}html.theme--catppuccin-mocha .button.is-black.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-mocha .button.is-black.is-outlined{background-color:transparent;border-color:#0a0a0a;color:#0a0a0a}html.theme--catppuccin-mocha .button.is-black.is-outlined:hover,html.theme--catppuccin-mocha .button.is-black.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-black.is-outlined:focus,html.theme--catppuccin-mocha .button.is-black.is-outlined.is-focused{background-color:#0a0a0a;border-color:#0a0a0a;color:#fff}html.theme--catppuccin-mocha .button.is-black.is-outlined.is-loading::after{border-color:transparent transparent #0a0a0a #0a0a0a !important}html.theme--catppuccin-mocha .button.is-black.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-black.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-black.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-black.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-mocha .button.is-black.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-black.is-outlined{background-color:transparent;border-color:#0a0a0a;box-shadow:none;color:#0a0a0a}html.theme--catppuccin-mocha .button.is-black.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-mocha .button.is-black.is-inverted.is-outlined:hover,html.theme--catppuccin-mocha .button.is-black.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-black.is-inverted.is-outlined:focus,html.theme--catppuccin-mocha .button.is-black.is-inverted.is-outlined.is-focused{background-color:#fff;color:#0a0a0a}html.theme--catppuccin-mocha .button.is-black.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-black.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-black.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-black.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #0a0a0a #0a0a0a !important}html.theme--catppuccin-mocha .button.is-black.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-black.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-mocha .button.is-light{background-color:#f5f5f5;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-light:hover,html.theme--catppuccin-mocha .button.is-light.is-hovered{background-color:#eee;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-light:focus,html.theme--catppuccin-mocha .button.is-light.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-light:focus:not(:active),html.theme--catppuccin-mocha .button.is-light.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(245,245,245,0.25)}html.theme--catppuccin-mocha .button.is-light:active,html.theme--catppuccin-mocha .button.is-light.is-active{background-color:#e8e8e8;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-light[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-light{background-color:#f5f5f5;border-color:#f5f5f5;box-shadow:none}html.theme--catppuccin-mocha .button.is-light.is-inverted{background-color:rgba(0,0,0,0.7);color:#f5f5f5}html.theme--catppuccin-mocha .button.is-light.is-inverted:hover,html.theme--catppuccin-mocha .button.is-light.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-light.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-light.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#f5f5f5}html.theme--catppuccin-mocha .button.is-light.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-mocha .button.is-light.is-outlined{background-color:transparent;border-color:#f5f5f5;color:#f5f5f5}html.theme--catppuccin-mocha .button.is-light.is-outlined:hover,html.theme--catppuccin-mocha .button.is-light.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-light.is-outlined:focus,html.theme--catppuccin-mocha .button.is-light.is-outlined.is-focused{background-color:#f5f5f5;border-color:#f5f5f5;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-light.is-outlined.is-loading::after{border-color:transparent transparent #f5f5f5 #f5f5f5 !important}html.theme--catppuccin-mocha .button.is-light.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-light.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-light.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-light.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-mocha .button.is-light.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-light.is-outlined{background-color:transparent;border-color:#f5f5f5;box-shadow:none;color:#f5f5f5}html.theme--catppuccin-mocha .button.is-light.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-light.is-inverted.is-outlined:hover,html.theme--catppuccin-mocha .button.is-light.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-light.is-inverted.is-outlined:focus,html.theme--catppuccin-mocha .button.is-light.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#f5f5f5}html.theme--catppuccin-mocha .button.is-light.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-light.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-light.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-light.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #f5f5f5 #f5f5f5 !important}html.theme--catppuccin-mocha .button.is-light.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-light.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-dark,html.theme--catppuccin-mocha .content kbd.button{background-color:#313244;border-color:transparent;color:#fff}html.theme--catppuccin-mocha .button.is-dark:hover,html.theme--catppuccin-mocha .content kbd.button:hover,html.theme--catppuccin-mocha .button.is-dark.is-hovered,html.theme--catppuccin-mocha .content kbd.button.is-hovered{background-color:#2c2d3d;border-color:transparent;color:#fff}html.theme--catppuccin-mocha .button.is-dark:focus,html.theme--catppuccin-mocha .content kbd.button:focus,html.theme--catppuccin-mocha .button.is-dark.is-focused,html.theme--catppuccin-mocha .content kbd.button.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-mocha .button.is-dark:focus:not(:active),html.theme--catppuccin-mocha .content kbd.button:focus:not(:active),html.theme--catppuccin-mocha .button.is-dark.is-focused:not(:active),html.theme--catppuccin-mocha .content kbd.button.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(49,50,68,0.25)}html.theme--catppuccin-mocha .button.is-dark:active,html.theme--catppuccin-mocha .content kbd.button:active,html.theme--catppuccin-mocha .button.is-dark.is-active,html.theme--catppuccin-mocha .content kbd.button.is-active{background-color:#262735;border-color:transparent;color:#fff}html.theme--catppuccin-mocha .button.is-dark[disabled],html.theme--catppuccin-mocha .content kbd.button[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-dark,fieldset[disabled] html.theme--catppuccin-mocha .content kbd.button{background-color:#313244;border-color:#313244;box-shadow:none}html.theme--catppuccin-mocha .button.is-dark.is-inverted,html.theme--catppuccin-mocha .content kbd.button.is-inverted{background-color:#fff;color:#313244}html.theme--catppuccin-mocha .button.is-dark.is-inverted:hover,html.theme--catppuccin-mocha .content kbd.button.is-inverted:hover,html.theme--catppuccin-mocha .button.is-dark.is-inverted.is-hovered,html.theme--catppuccin-mocha .content kbd.button.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-mocha .button.is-dark.is-inverted[disabled],html.theme--catppuccin-mocha .content kbd.button.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-dark.is-inverted,fieldset[disabled] html.theme--catppuccin-mocha .content kbd.button.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#313244}html.theme--catppuccin-mocha .button.is-dark.is-loading::after,html.theme--catppuccin-mocha .content kbd.button.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-mocha .button.is-dark.is-outlined,html.theme--catppuccin-mocha .content kbd.button.is-outlined{background-color:transparent;border-color:#313244;color:#313244}html.theme--catppuccin-mocha .button.is-dark.is-outlined:hover,html.theme--catppuccin-mocha .content kbd.button.is-outlined:hover,html.theme--catppuccin-mocha .button.is-dark.is-outlined.is-hovered,html.theme--catppuccin-mocha .content kbd.button.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-dark.is-outlined:focus,html.theme--catppuccin-mocha .content kbd.button.is-outlined:focus,html.theme--catppuccin-mocha .button.is-dark.is-outlined.is-focused,html.theme--catppuccin-mocha .content kbd.button.is-outlined.is-focused{background-color:#313244;border-color:#313244;color:#fff}html.theme--catppuccin-mocha .button.is-dark.is-outlined.is-loading::after,html.theme--catppuccin-mocha .content kbd.button.is-outlined.is-loading::after{border-color:transparent transparent #313244 #313244 !important}html.theme--catppuccin-mocha .button.is-dark.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .content kbd.button.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-dark.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .content kbd.button.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-dark.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .content kbd.button.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-dark.is-outlined.is-loading.is-focused::after,html.theme--catppuccin-mocha .content kbd.button.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-mocha .button.is-dark.is-outlined[disabled],html.theme--catppuccin-mocha .content kbd.button.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-dark.is-outlined,fieldset[disabled] html.theme--catppuccin-mocha .content kbd.button.is-outlined{background-color:transparent;border-color:#313244;box-shadow:none;color:#313244}html.theme--catppuccin-mocha .button.is-dark.is-inverted.is-outlined,html.theme--catppuccin-mocha .content kbd.button.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-mocha .button.is-dark.is-inverted.is-outlined:hover,html.theme--catppuccin-mocha .content kbd.button.is-inverted.is-outlined:hover,html.theme--catppuccin-mocha .button.is-dark.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-mocha .content kbd.button.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-dark.is-inverted.is-outlined:focus,html.theme--catppuccin-mocha .content kbd.button.is-inverted.is-outlined:focus,html.theme--catppuccin-mocha .button.is-dark.is-inverted.is-outlined.is-focused,html.theme--catppuccin-mocha .content kbd.button.is-inverted.is-outlined.is-focused{background-color:#fff;color:#313244}html.theme--catppuccin-mocha .button.is-dark.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .content kbd.button.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-dark.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .content kbd.button.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-dark.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .content kbd.button.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-dark.is-inverted.is-outlined.is-loading.is-focused::after,html.theme--catppuccin-mocha .content kbd.button.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #313244 #313244 !important}html.theme--catppuccin-mocha .button.is-dark.is-inverted.is-outlined[disabled],html.theme--catppuccin-mocha .content kbd.button.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-dark.is-inverted.is-outlined,fieldset[disabled] html.theme--catppuccin-mocha .content kbd.button.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-mocha .button.is-primary,html.theme--catppuccin-mocha .docstring>section>a.button.docs-sourcelink{background-color:#89b4fa;border-color:transparent;color:#fff}html.theme--catppuccin-mocha .button.is-primary:hover,html.theme--catppuccin-mocha .docstring>section>a.button.docs-sourcelink:hover,html.theme--catppuccin-mocha .button.is-primary.is-hovered,html.theme--catppuccin-mocha .docstring>section>a.button.is-hovered.docs-sourcelink{background-color:#7dacf9;border-color:transparent;color:#fff}html.theme--catppuccin-mocha .button.is-primary:focus,html.theme--catppuccin-mocha .docstring>section>a.button.docs-sourcelink:focus,html.theme--catppuccin-mocha .button.is-primary.is-focused,html.theme--catppuccin-mocha .docstring>section>a.button.is-focused.docs-sourcelink{border-color:transparent;color:#fff}html.theme--catppuccin-mocha .button.is-primary:focus:not(:active),html.theme--catppuccin-mocha .docstring>section>a.button.docs-sourcelink:focus:not(:active),html.theme--catppuccin-mocha .button.is-primary.is-focused:not(:active),html.theme--catppuccin-mocha .docstring>section>a.button.is-focused.docs-sourcelink:not(:active){box-shadow:0 0 0 0.125em rgba(137,180,250,0.25)}html.theme--catppuccin-mocha .button.is-primary:active,html.theme--catppuccin-mocha .docstring>section>a.button.docs-sourcelink:active,html.theme--catppuccin-mocha .button.is-primary.is-active,html.theme--catppuccin-mocha .docstring>section>a.button.is-active.docs-sourcelink{background-color:#71a4f9;border-color:transparent;color:#fff}html.theme--catppuccin-mocha .button.is-primary[disabled],html.theme--catppuccin-mocha .docstring>section>a.button.docs-sourcelink[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-primary,fieldset[disabled] html.theme--catppuccin-mocha .docstring>section>a.button.docs-sourcelink{background-color:#89b4fa;border-color:#89b4fa;box-shadow:none}html.theme--catppuccin-mocha .button.is-primary.is-inverted,html.theme--catppuccin-mocha .docstring>section>a.button.is-inverted.docs-sourcelink{background-color:#fff;color:#89b4fa}html.theme--catppuccin-mocha .button.is-primary.is-inverted:hover,html.theme--catppuccin-mocha .docstring>section>a.button.is-inverted.docs-sourcelink:hover,html.theme--catppuccin-mocha .button.is-primary.is-inverted.is-hovered,html.theme--catppuccin-mocha .docstring>section>a.button.is-inverted.is-hovered.docs-sourcelink{background-color:#f2f2f2}html.theme--catppuccin-mocha .button.is-primary.is-inverted[disabled],html.theme--catppuccin-mocha .docstring>section>a.button.is-inverted.docs-sourcelink[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-primary.is-inverted,fieldset[disabled] html.theme--catppuccin-mocha .docstring>section>a.button.is-inverted.docs-sourcelink{background-color:#fff;border-color:transparent;box-shadow:none;color:#89b4fa}html.theme--catppuccin-mocha .button.is-primary.is-loading::after,html.theme--catppuccin-mocha .docstring>section>a.button.is-loading.docs-sourcelink::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-mocha .button.is-primary.is-outlined,html.theme--catppuccin-mocha 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html.theme--catppuccin-mocha .button.is-link.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-mocha .button.is-link.is-light{background-color:#ebf3fe;color:#063c93}html.theme--catppuccin-mocha .button.is-link.is-light:hover,html.theme--catppuccin-mocha .button.is-link.is-light.is-hovered{background-color:#dfebfe;border-color:transparent;color:#063c93}html.theme--catppuccin-mocha .button.is-link.is-light:active,html.theme--catppuccin-mocha .button.is-link.is-light.is-active{background-color:#d3e3fd;border-color:transparent;color:#063c93}html.theme--catppuccin-mocha .button.is-info{background-color:#94e2d5;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-info:hover,html.theme--catppuccin-mocha .button.is-info.is-hovered{background-color:#8adfd1;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-info:focus,html.theme--catppuccin-mocha 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!important}html.theme--catppuccin-mocha .button.is-success.is-outlined{background-color:transparent;border-color:#a6e3a1;color:#a6e3a1}html.theme--catppuccin-mocha .button.is-success.is-outlined:hover,html.theme--catppuccin-mocha .button.is-success.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-success.is-outlined:focus,html.theme--catppuccin-mocha .button.is-success.is-outlined.is-focused{background-color:#a6e3a1;border-color:#a6e3a1;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-success.is-outlined.is-loading::after{border-color:transparent transparent #a6e3a1 #a6e3a1 !important}html.theme--catppuccin-mocha .button.is-success.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-success.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-success.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-success.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-mocha .button.is-success.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-success.is-outlined{background-color:transparent;border-color:#a6e3a1;box-shadow:none;color:#a6e3a1}html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined:hover,html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined:focus,html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#a6e3a1}html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #a6e3a1 #a6e3a1 !important}html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-success.is-light{background-color:#f0faef;color:#287222}html.theme--catppuccin-mocha .button.is-success.is-light:hover,html.theme--catppuccin-mocha .button.is-success.is-light.is-hovered{background-color:#e7f7e5;border-color:transparent;color:#287222}html.theme--catppuccin-mocha .button.is-success.is-light:active,html.theme--catppuccin-mocha .button.is-success.is-light.is-active{background-color:#def4dc;border-color:transparent;color:#287222}html.theme--catppuccin-mocha .button.is-warning{background-color:#f9e2af;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning:hover,html.theme--catppuccin-mocha .button.is-warning.is-hovered{background-color:#f8dea3;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning:focus,html.theme--catppuccin-mocha .button.is-warning.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning:focus:not(:active),html.theme--catppuccin-mocha .button.is-warning.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(249,226,175,0.25)}html.theme--catppuccin-mocha .button.is-warning:active,html.theme--catppuccin-mocha .button.is-warning.is-active{background-color:#f7d997;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-warning{background-color:#f9e2af;border-color:#f9e2af;box-shadow:none}html.theme--catppuccin-mocha .button.is-warning.is-inverted{background-color:rgba(0,0,0,0.7);color:#f9e2af}html.theme--catppuccin-mocha .button.is-warning.is-inverted:hover,html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-warning.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#f9e2af}html.theme--catppuccin-mocha .button.is-warning.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-mocha .button.is-warning.is-outlined{background-color:transparent;border-color:#f9e2af;color:#f9e2af}html.theme--catppuccin-mocha .button.is-warning.is-outlined:hover,html.theme--catppuccin-mocha .button.is-warning.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-warning.is-outlined:focus,html.theme--catppuccin-mocha .button.is-warning.is-outlined.is-focused{background-color:#f9e2af;border-color:#f9e2af;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning.is-outlined.is-loading::after{border-color:transparent transparent #f9e2af #f9e2af !important}html.theme--catppuccin-mocha .button.is-warning.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-warning.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-warning.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-warning.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-mocha .button.is-warning.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-warning.is-outlined{background-color:transparent;border-color:#f9e2af;box-shadow:none;color:#f9e2af}html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined:hover,html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined:focus,html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#f9e2af}html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #f9e2af #f9e2af !important}html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning.is-light{background-color:#fef8ec;color:#8a620a}html.theme--catppuccin-mocha .button.is-warning.is-light:hover,html.theme--catppuccin-mocha .button.is-warning.is-light.is-hovered{background-color:#fdf4e0;border-color:transparent;color:#8a620a}html.theme--catppuccin-mocha .button.is-warning.is-light:active,html.theme--catppuccin-mocha .button.is-warning.is-light.is-active{background-color:#fcf0d4;border-color:transparent;color:#8a620a}html.theme--catppuccin-mocha .button.is-danger{background-color:#f38ba8;border-color:transparent;color:#fff}html.theme--catppuccin-mocha .button.is-danger:hover,html.theme--catppuccin-mocha .button.is-danger.is-hovered{background-color:#f27f9f;border-color:transparent;color:#fff}html.theme--catppuccin-mocha .button.is-danger:focus,html.theme--catppuccin-mocha .button.is-danger.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-mocha .button.is-danger:focus:not(:active),html.theme--catppuccin-mocha .button.is-danger.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(243,139,168,0.25)}html.theme--catppuccin-mocha .button.is-danger:active,html.theme--catppuccin-mocha .button.is-danger.is-active{background-color:#f17497;border-color:transparent;color:#fff}html.theme--catppuccin-mocha .button.is-danger[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-danger{background-color:#f38ba8;border-color:#f38ba8;box-shadow:none}html.theme--catppuccin-mocha .button.is-danger.is-inverted{background-color:#fff;color:#f38ba8}html.theme--catppuccin-mocha .button.is-danger.is-inverted:hover,html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-mocha .button.is-danger.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-danger.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#f38ba8}html.theme--catppuccin-mocha .button.is-danger.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-mocha .button.is-danger.is-outlined{background-color:transparent;border-color:#f38ba8;color:#f38ba8}html.theme--catppuccin-mocha .button.is-danger.is-outlined:hover,html.theme--catppuccin-mocha .button.is-danger.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-danger.is-outlined:focus,html.theme--catppuccin-mocha .button.is-danger.is-outlined.is-focused{background-color:#f38ba8;border-color:#f38ba8;color:#fff}html.theme--catppuccin-mocha .button.is-danger.is-outlined.is-loading::after{border-color:transparent transparent #f38ba8 #f38ba8 !important}html.theme--catppuccin-mocha .button.is-danger.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-danger.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-danger.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-danger.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-mocha .button.is-danger.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-danger.is-outlined{background-color:transparent;border-color:#f38ba8;box-shadow:none;color:#f38ba8}html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined:hover,html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined:focus,html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined.is-focused{background-color:#fff;color:#f38ba8}html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #f38ba8 #f38ba8 !important}html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-mocha .button.is-danger.is-light{background-color:#fdedf1;color:#991036}html.theme--catppuccin-mocha .button.is-danger.is-light:hover,html.theme--catppuccin-mocha .button.is-danger.is-light.is-hovered{background-color:#fce1e8;border-color:transparent;color:#991036}html.theme--catppuccin-mocha .button.is-danger.is-light:active,html.theme--catppuccin-mocha .button.is-danger.is-light.is-active{background-color:#fbd5e0;border-color:transparent;color:#991036}html.theme--catppuccin-mocha .button.is-small,html.theme--catppuccin-mocha #documenter .docs-sidebar form.docs-search>input.button{font-size:.75rem}html.theme--catppuccin-mocha .button.is-small:not(.is-rounded),html.theme--catppuccin-mocha #documenter .docs-sidebar form.docs-search>input.button:not(.is-rounded){border-radius:3px}html.theme--catppuccin-mocha .button.is-normal{font-size:1rem}html.theme--catppuccin-mocha .button.is-medium{font-size:1.25rem}html.theme--catppuccin-mocha .button.is-large{font-size:1.5rem}html.theme--catppuccin-mocha .button[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button{background-color:#6c7086;border-color:#585b70;box-shadow:none;opacity:.5}html.theme--catppuccin-mocha .button.is-fullwidth{display:flex;width:100%}html.theme--catppuccin-mocha .button.is-loading{color:transparent !important;pointer-events:none}html.theme--catppuccin-mocha .button.is-loading::after{position:absolute;left:calc(50% - (1em * 0.5));top:calc(50% - (1em * 0.5));position:absolute !important}html.theme--catppuccin-mocha .button.is-static{background-color:#181825;border-color:#585b70;color:#7f849c;box-shadow:none;pointer-events:none}html.theme--catppuccin-mocha .button.is-rounded,html.theme--catppuccin-mocha #documenter .docs-sidebar form.docs-search>input.button{border-radius:9999px;padding-left:calc(1em + 0.25em);padding-right:calc(1em + 0.25em)}html.theme--catppuccin-mocha .buttons{align-items:center;display:flex;flex-wrap:wrap;justify-content:flex-start}html.theme--catppuccin-mocha .buttons .button{margin-bottom:0.5rem}html.theme--catppuccin-mocha .buttons .button:not(:last-child):not(.is-fullwidth){margin-right:.5rem}html.theme--catppuccin-mocha .buttons:last-child{margin-bottom:-0.5rem}html.theme--catppuccin-mocha .buttons:not(:last-child){margin-bottom:1rem}html.theme--catppuccin-mocha .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large){font-size:.75rem}html.theme--catppuccin-mocha .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large):not(.is-rounded){border-radius:3px}html.theme--catppuccin-mocha .buttons.are-medium .button:not(.is-small):not(.is-normal):not(.is-large){font-size:1.25rem}html.theme--catppuccin-mocha .buttons.are-large .button:not(.is-small):not(.is-normal):not(.is-medium){font-size:1.5rem}html.theme--catppuccin-mocha .buttons.has-addons .button:not(:first-child){border-bottom-left-radius:0;border-top-left-radius:0}html.theme--catppuccin-mocha .buttons.has-addons .button:not(:last-child){border-bottom-right-radius:0;border-top-right-radius:0;margin-right:-1px}html.theme--catppuccin-mocha .buttons.has-addons .button:last-child{margin-right:0}html.theme--catppuccin-mocha .buttons.has-addons .button:hover,html.theme--catppuccin-mocha .buttons.has-addons .button.is-hovered{z-index:2}html.theme--catppuccin-mocha .buttons.has-addons .button:focus,html.theme--catppuccin-mocha .buttons.has-addons .button.is-focused,html.theme--catppuccin-mocha .buttons.has-addons .button:active,html.theme--catppuccin-mocha .buttons.has-addons .button.is-active,html.theme--catppuccin-mocha .buttons.has-addons .button.is-selected{z-index:3}html.theme--catppuccin-mocha .buttons.has-addons .button:focus:hover,html.theme--catppuccin-mocha .buttons.has-addons .button.is-focused:hover,html.theme--catppuccin-mocha .buttons.has-addons .button:active:hover,html.theme--catppuccin-mocha .buttons.has-addons .button.is-active:hover,html.theme--catppuccin-mocha .buttons.has-addons .button.is-selected:hover{z-index:4}html.theme--catppuccin-mocha .buttons.has-addons .button.is-expanded{flex-grow:1;flex-shrink:1}html.theme--catppuccin-mocha .buttons.is-centered{justify-content:center}html.theme--catppuccin-mocha .buttons.is-centered:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}html.theme--catppuccin-mocha .buttons.is-right{justify-content:flex-end}html.theme--catppuccin-mocha .buttons.is-right:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}@media screen and (max-width: 768px){html.theme--catppuccin-mocha .button.is-responsive.is-small,html.theme--catppuccin-mocha #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.5625rem}html.theme--catppuccin-mocha .button.is-responsive,html.theme--catppuccin-mocha .button.is-responsive.is-normal{font-size:.65625rem}html.theme--catppuccin-mocha .button.is-responsive.is-medium{font-size:.75rem}html.theme--catppuccin-mocha .button.is-responsive.is-large{font-size:1rem}}@media screen and (min-width: 769px) and (max-width: 1055px){html.theme--catppuccin-mocha .button.is-responsive.is-small,html.theme--catppuccin-mocha #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.65625rem}html.theme--catppuccin-mocha .button.is-responsive,html.theme--catppuccin-mocha .button.is-responsive.is-normal{font-size:.75rem}html.theme--catppuccin-mocha .button.is-responsive.is-medium{font-size:1rem}html.theme--catppuccin-mocha .button.is-responsive.is-large{font-size:1.25rem}}html.theme--catppuccin-mocha .container{flex-grow:1;margin:0 auto;position:relative;width:auto}html.theme--catppuccin-mocha .container.is-fluid{max-width:none !important;padding-left:32px;padding-right:32px;width:100%}@media screen and (min-width: 1056px){html.theme--catppuccin-mocha .container{max-width:992px}}@media screen and (max-width: 1215px){html.theme--catppuccin-mocha .container.is-widescreen:not(.is-max-desktop){max-width:1152px}}@media screen and (max-width: 1407px){html.theme--catppuccin-mocha .container.is-fullhd:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}@media screen and (min-width: 1216px){html.theme--catppuccin-mocha .container:not(.is-max-desktop){max-width:1152px}}@media screen and (min-width: 1408px){html.theme--catppuccin-mocha .container:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}html.theme--catppuccin-mocha .content li+li{margin-top:0.25em}html.theme--catppuccin-mocha .content p:not(:last-child),html.theme--catppuccin-mocha .content dl:not(:last-child),html.theme--catppuccin-mocha .content ol:not(:last-child),html.theme--catppuccin-mocha .content ul:not(:last-child),html.theme--catppuccin-mocha .content blockquote:not(:last-child),html.theme--catppuccin-mocha .content pre:not(:last-child),html.theme--catppuccin-mocha .content table:not(:last-child){margin-bottom:1em}html.theme--catppuccin-mocha .content h1,html.theme--catppuccin-mocha .content h2,html.theme--catppuccin-mocha .content h3,html.theme--catppuccin-mocha .content h4,html.theme--catppuccin-mocha .content h5,html.theme--catppuccin-mocha .content h6{color:#cdd6f4;font-weight:600;line-height:1.125}html.theme--catppuccin-mocha .content h1{font-size:2em;margin-bottom:0.5em}html.theme--catppuccin-mocha .content h1:not(:first-child){margin-top:1em}html.theme--catppuccin-mocha .content h2{font-size:1.75em;margin-bottom:0.5714em}html.theme--catppuccin-mocha .content h2:not(:first-child){margin-top:1.1428em}html.theme--catppuccin-mocha .content h3{font-size:1.5em;margin-bottom:0.6666em}html.theme--catppuccin-mocha .content 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html.theme--catppuccin-mocha .button.is-link.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-mocha .button.is-link.is-light{background-color:#ebf3fe;color:#063c93}html.theme--catppuccin-mocha .button.is-link.is-light:hover,html.theme--catppuccin-mocha .button.is-link.is-light.is-hovered{background-color:#dfebfe;border-color:transparent;color:#063c93}html.theme--catppuccin-mocha .button.is-link.is-light:active,html.theme--catppuccin-mocha .button.is-link.is-light.is-active{background-color:#d3e3fd;border-color:transparent;color:#063c93}html.theme--catppuccin-mocha .button.is-info{background-color:#94e2d5;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-info:hover,html.theme--catppuccin-mocha .button.is-info.is-hovered{background-color:#8adfd1;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-info:focus,html.theme--catppuccin-mocha .button.is-info.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-info:focus:not(:active),html.theme--catppuccin-mocha .button.is-info.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(148,226,213,0.25)}html.theme--catppuccin-mocha .button.is-info:active,html.theme--catppuccin-mocha .button.is-info.is-active{background-color:#80ddcd;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-info[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-info{background-color:#94e2d5;border-color:#94e2d5;box-shadow:none}html.theme--catppuccin-mocha .button.is-info.is-inverted{background-color:rgba(0,0,0,0.7);color:#94e2d5}html.theme--catppuccin-mocha .button.is-info.is-inverted:hover,html.theme--catppuccin-mocha .button.is-info.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-info.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-info.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#94e2d5}html.theme--catppuccin-mocha .button.is-info.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-mocha .button.is-info.is-outlined{background-color:transparent;border-color:#94e2d5;color:#94e2d5}html.theme--catppuccin-mocha .button.is-info.is-outlined:hover,html.theme--catppuccin-mocha .button.is-info.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-info.is-outlined:focus,html.theme--catppuccin-mocha .button.is-info.is-outlined.is-focused{background-color:#94e2d5;border-color:#94e2d5;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-info.is-outlined.is-loading::after{border-color:transparent transparent #94e2d5 #94e2d5 !important}html.theme--catppuccin-mocha .button.is-info.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-info.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-info.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-info.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-mocha .button.is-info.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-info.is-outlined{background-color:transparent;border-color:#94e2d5;box-shadow:none;color:#94e2d5}html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined:hover,html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined:focus,html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#94e2d5}html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #94e2d5 #94e2d5 !important}html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-info.is-light{background-color:#effbf9;color:#207466}html.theme--catppuccin-mocha .button.is-info.is-light:hover,html.theme--catppuccin-mocha .button.is-info.is-light.is-hovered{background-color:#e5f8f5;border-color:transparent;color:#207466}html.theme--catppuccin-mocha .button.is-info.is-light:active,html.theme--catppuccin-mocha .button.is-info.is-light.is-active{background-color:#dbf5f1;border-color:transparent;color:#207466}html.theme--catppuccin-mocha .button.is-success{background-color:#a6e3a1;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-success:hover,html.theme--catppuccin-mocha .button.is-success.is-hovered{background-color:#9de097;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-success:focus,html.theme--catppuccin-mocha .button.is-success.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-success:focus:not(:active),html.theme--catppuccin-mocha .button.is-success.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(166,227,161,0.25)}html.theme--catppuccin-mocha .button.is-success:active,html.theme--catppuccin-mocha .button.is-success.is-active{background-color:#93dd8d;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-success[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-success{background-color:#a6e3a1;border-color:#a6e3a1;box-shadow:none}html.theme--catppuccin-mocha .button.is-success.is-inverted{background-color:rgba(0,0,0,0.7);color:#a6e3a1}html.theme--catppuccin-mocha .button.is-success.is-inverted:hover,html.theme--catppuccin-mocha .button.is-success.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-success.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-success.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#a6e3a1}html.theme--catppuccin-mocha .button.is-success.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-mocha .button.is-success.is-outlined{background-color:transparent;border-color:#a6e3a1;color:#a6e3a1}html.theme--catppuccin-mocha .button.is-success.is-outlined:hover,html.theme--catppuccin-mocha .button.is-success.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-success.is-outlined:focus,html.theme--catppuccin-mocha .button.is-success.is-outlined.is-focused{background-color:#a6e3a1;border-color:#a6e3a1;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-success.is-outlined.is-loading::after{border-color:transparent transparent #a6e3a1 #a6e3a1 !important}html.theme--catppuccin-mocha .button.is-success.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-success.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-success.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-success.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-mocha .button.is-success.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-success.is-outlined{background-color:transparent;border-color:#a6e3a1;box-shadow:none;color:#a6e3a1}html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined:hover,html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined:focus,html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#a6e3a1}html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #a6e3a1 #a6e3a1 !important}html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-success.is-light{background-color:#f0faef;color:#287222}html.theme--catppuccin-mocha .button.is-success.is-light:hover,html.theme--catppuccin-mocha .button.is-success.is-light.is-hovered{background-color:#e7f7e5;border-color:transparent;color:#287222}html.theme--catppuccin-mocha .button.is-success.is-light:active,html.theme--catppuccin-mocha .button.is-success.is-light.is-active{background-color:#def4dc;border-color:transparent;color:#287222}html.theme--catppuccin-mocha .button.is-warning{background-color:#f9e2af;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning:hover,html.theme--catppuccin-mocha .button.is-warning.is-hovered{background-color:#f8dea3;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning:focus,html.theme--catppuccin-mocha .button.is-warning.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning:focus:not(:active),html.theme--catppuccin-mocha .button.is-warning.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(249,226,175,0.25)}html.theme--catppuccin-mocha .button.is-warning:active,html.theme--catppuccin-mocha .button.is-warning.is-active{background-color:#f7d997;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-warning{background-color:#f9e2af;border-color:#f9e2af;box-shadow:none}html.theme--catppuccin-mocha .button.is-warning.is-inverted{background-color:rgba(0,0,0,0.7);color:#f9e2af}html.theme--catppuccin-mocha .button.is-warning.is-inverted:hover,html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-warning.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#f9e2af}html.theme--catppuccin-mocha .button.is-warning.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-mocha .button.is-warning.is-outlined{background-color:transparent;border-color:#f9e2af;color:#f9e2af}html.theme--catppuccin-mocha .button.is-warning.is-outlined:hover,html.theme--catppuccin-mocha .button.is-warning.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-warning.is-outlined:focus,html.theme--catppuccin-mocha .button.is-warning.is-outlined.is-focused{background-color:#f9e2af;border-color:#f9e2af;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning.is-outlined.is-loading::after{border-color:transparent transparent #f9e2af #f9e2af !important}html.theme--catppuccin-mocha .button.is-warning.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-warning.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-warning.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-warning.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-mocha .button.is-warning.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-warning.is-outlined{background-color:transparent;border-color:#f9e2af;box-shadow:none;color:#f9e2af}html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined:hover,html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined:focus,html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#f9e2af}html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #f9e2af #f9e2af !important}html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning.is-light{background-color:#fef8ec;color:#8a620a}html.theme--catppuccin-mocha .button.is-warning.is-light:hover,html.theme--catppuccin-mocha .button.is-warning.is-light.is-hovered{background-color:#fdf4e0;border-color:transparent;color:#8a620a}html.theme--catppuccin-mocha .button.is-warning.is-light:active,html.theme--catppuccin-mocha .button.is-warning.is-light.is-active{background-color:#fcf0d4;border-color:transparent;color:#8a620a}html.theme--catppuccin-mocha .button.is-danger{background-color:#f38ba8;border-color:transparent;color:#fff}html.theme--catppuccin-mocha .button.is-danger:hover,html.theme--catppuccin-mocha .button.is-danger.is-hovered{background-color:#f27f9f;border-color:transparent;color:#fff}html.theme--catppuccin-mocha .button.is-danger:focus,html.theme--catppuccin-mocha .button.is-danger.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-mocha .button.is-danger:focus:not(:active),html.theme--catppuccin-mocha .button.is-danger.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(243,139,168,0.25)}html.theme--catppuccin-mocha .button.is-danger:active,html.theme--catppuccin-mocha .button.is-danger.is-active{background-color:#f17497;border-color:transparent;color:#fff}html.theme--catppuccin-mocha .button.is-danger[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-danger{background-color:#f38ba8;border-color:#f38ba8;box-shadow:none}html.theme--catppuccin-mocha .button.is-danger.is-inverted{background-color:#fff;color:#f38ba8}html.theme--catppuccin-mocha .button.is-danger.is-inverted:hover,html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-mocha .button.is-danger.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-danger.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#f38ba8}html.theme--catppuccin-mocha .button.is-danger.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-mocha .button.is-danger.is-outlined{background-color:transparent;border-color:#f38ba8;color:#f38ba8}html.theme--catppuccin-mocha .button.is-danger.is-outlined:hover,html.theme--catppuccin-mocha .button.is-danger.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-danger.is-outlined:focus,html.theme--catppuccin-mocha .button.is-danger.is-outlined.is-focused{background-color:#f38ba8;border-color:#f38ba8;color:#fff}html.theme--catppuccin-mocha .button.is-danger.is-outlined.is-loading::after{border-color:transparent transparent #f38ba8 #f38ba8 !important}html.theme--catppuccin-mocha .button.is-danger.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-danger.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-danger.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-danger.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-mocha .button.is-danger.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-danger.is-outlined{background-color:transparent;border-color:#f38ba8;box-shadow:none;color:#f38ba8}html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined:hover,html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined:focus,html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined.is-focused{background-color:#fff;color:#f38ba8}html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #f38ba8 #f38ba8 !important}html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-mocha .button.is-danger.is-light{background-color:#fdedf1;color:#991036}html.theme--catppuccin-mocha .button.is-danger.is-light:hover,html.theme--catppuccin-mocha .button.is-danger.is-light.is-hovered{background-color:#fce1e8;border-color:transparent;color:#991036}html.theme--catppuccin-mocha .button.is-danger.is-light:active,html.theme--catppuccin-mocha .button.is-danger.is-light.is-active{background-color:#fbd5e0;border-color:transparent;color:#991036}html.theme--catppuccin-mocha .button.is-small,html.theme--catppuccin-mocha #documenter .docs-sidebar form.docs-search>input.button{font-size:.75rem}html.theme--catppuccin-mocha .button.is-small:not(.is-rounded),html.theme--catppuccin-mocha #documenter .docs-sidebar form.docs-search>input.button:not(.is-rounded){border-radius:3px}html.theme--catppuccin-mocha .button.is-normal{font-size:1rem}html.theme--catppuccin-mocha .button.is-medium{font-size:1.25rem}html.theme--catppuccin-mocha .button.is-large{font-size:1.5rem}html.theme--catppuccin-mocha .button[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button{background-color:#6c7086;border-color:#585b70;box-shadow:none;opacity:.5}html.theme--catppuccin-mocha .button.is-fullwidth{display:flex;width:100%}html.theme--catppuccin-mocha .button.is-loading{color:transparent !important;pointer-events:none}html.theme--catppuccin-mocha .button.is-loading::after{position:absolute;left:calc(50% - (1em * 0.5));top:calc(50% - (1em * 0.5));position:absolute !important}html.theme--catppuccin-mocha .button.is-static{background-color:#181825;border-color:#585b70;color:#7f849c;box-shadow:none;pointer-events:none}html.theme--catppuccin-mocha .button.is-rounded,html.theme--catppuccin-mocha #documenter .docs-sidebar form.docs-search>input.button{border-radius:9999px;padding-left:calc(1em + 0.25em);padding-right:calc(1em + 0.25em)}html.theme--catppuccin-mocha .buttons{align-items:center;display:flex;flex-wrap:wrap;justify-content:flex-start}html.theme--catppuccin-mocha .buttons .button{margin-bottom:0.5rem}html.theme--catppuccin-mocha .buttons .button:not(:last-child):not(.is-fullwidth){margin-right:.5rem}html.theme--catppuccin-mocha .buttons:last-child{margin-bottom:-0.5rem}html.theme--catppuccin-mocha .buttons:not(:last-child){margin-bottom:1rem}html.theme--catppuccin-mocha .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large){font-size:.75rem}html.theme--catppuccin-mocha .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large):not(.is-rounded){border-radius:3px}html.theme--catppuccin-mocha .buttons.are-medium .button:not(.is-small):not(.is-normal):not(.is-large){font-size:1.25rem}html.theme--catppuccin-mocha .buttons.are-large .button:not(.is-small):not(.is-normal):not(.is-medium){font-size:1.5rem}html.theme--catppuccin-mocha .buttons.has-addons .button:not(:first-child){border-bottom-left-radius:0;border-top-left-radius:0}html.theme--catppuccin-mocha .buttons.has-addons .button:not(:last-child){border-bottom-right-radius:0;border-top-right-radius:0;margin-right:-1px}html.theme--catppuccin-mocha .buttons.has-addons .button:last-child{margin-right:0}html.theme--catppuccin-mocha .buttons.has-addons .button:hover,html.theme--catppuccin-mocha .buttons.has-addons .button.is-hovered{z-index:2}html.theme--catppuccin-mocha .buttons.has-addons .button:focus,html.theme--catppuccin-mocha .buttons.has-addons .button.is-focused,html.theme--catppuccin-mocha .buttons.has-addons .button:active,html.theme--catppuccin-mocha .buttons.has-addons .button.is-active,html.theme--catppuccin-mocha .buttons.has-addons .button.is-selected{z-index:3}html.theme--catppuccin-mocha .buttons.has-addons .button:focus:hover,html.theme--catppuccin-mocha .buttons.has-addons .button.is-focused:hover,html.theme--catppuccin-mocha .buttons.has-addons .button:active:hover,html.theme--catppuccin-mocha .buttons.has-addons .button.is-active:hover,html.theme--catppuccin-mocha .buttons.has-addons .button.is-selected:hover{z-index:4}html.theme--catppuccin-mocha .buttons.has-addons .button.is-expanded{flex-grow:1;flex-shrink:1}html.theme--catppuccin-mocha .buttons.is-centered{justify-content:center}html.theme--catppuccin-mocha .buttons.is-centered:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}html.theme--catppuccin-mocha .buttons.is-right{justify-content:flex-end}html.theme--catppuccin-mocha .buttons.is-right:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}@media screen and (max-width: 768px){html.theme--catppuccin-mocha .button.is-responsive.is-small,html.theme--catppuccin-mocha #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.5625rem}html.theme--catppuccin-mocha .button.is-responsive,html.theme--catppuccin-mocha .button.is-responsive.is-normal{font-size:.65625rem}html.theme--catppuccin-mocha .button.is-responsive.is-medium{font-size:.75rem}html.theme--catppuccin-mocha .button.is-responsive.is-large{font-size:1rem}}@media screen and (min-width: 769px) and (max-width: 1055px){html.theme--catppuccin-mocha .button.is-responsive.is-small,html.theme--catppuccin-mocha #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.65625rem}html.theme--catppuccin-mocha .button.is-responsive,html.theme--catppuccin-mocha .button.is-responsive.is-normal{font-size:.75rem}html.theme--catppuccin-mocha .button.is-responsive.is-medium{font-size:1rem}html.theme--catppuccin-mocha .button.is-responsive.is-large{font-size:1.25rem}}html.theme--catppuccin-mocha .container{flex-grow:1;margin:0 auto;position:relative;width:auto}html.theme--catppuccin-mocha .container.is-fluid{max-width:none !important;padding-left:32px;padding-right:32px;width:100%}@media screen and (min-width: 1056px){html.theme--catppuccin-mocha .container{max-width:992px}}@media screen and (max-width: 1215px){html.theme--catppuccin-mocha .container.is-widescreen:not(.is-max-desktop){max-width:1152px}}@media screen and (max-width: 1407px){html.theme--catppuccin-mocha .container.is-fullhd:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}@media screen and (min-width: 1216px){html.theme--catppuccin-mocha .container:not(.is-max-desktop){max-width:1152px}}@media screen and (min-width: 1408px){html.theme--catppuccin-mocha .container:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}html.theme--catppuccin-mocha .content li+li{margin-top:0.25em}html.theme--catppuccin-mocha .content p:not(:last-child),html.theme--catppuccin-mocha .content dl:not(:last-child),html.theme--catppuccin-mocha .content ol:not(:last-child),html.theme--catppuccin-mocha .content ul:not(:last-child),html.theme--catppuccin-mocha .content blockquote:not(:last-child),html.theme--catppuccin-mocha .content pre:not(:last-child),html.theme--catppuccin-mocha .content table:not(:last-child){margin-bottom:1em}html.theme--catppuccin-mocha .content h1,html.theme--catppuccin-mocha .content h2,html.theme--catppuccin-mocha .content h3,html.theme--catppuccin-mocha .content h4,html.theme--catppuccin-mocha .content h5,html.theme--catppuccin-mocha .content h6{color:#cdd6f4;font-weight:600;line-height:1.125}html.theme--catppuccin-mocha .content h1{font-size:2em;margin-bottom:0.5em}html.theme--catppuccin-mocha .content h1:not(:first-child){margin-top:1em}html.theme--catppuccin-mocha .content h2{font-size:1.75em;margin-bottom:0.5714em}html.theme--catppuccin-mocha .content h2:not(:first-child){margin-top:1.1428em}html.theme--catppuccin-mocha .content h3{font-size:1.5em;margin-bottom:0.6666em}html.theme--catppuccin-mocha .content h3:not(:first-child){margin-top:1.3333em}html.theme--catppuccin-mocha .content h4{font-size:1.25em;margin-bottom:0.8em}html.theme--catppuccin-mocha .content h5{font-size:1.125em;margin-bottom:0.8888em}html.theme--catppuccin-mocha .content h6{font-size:1em;margin-bottom:1em}html.theme--catppuccin-mocha .content blockquote{background-color:#181825;border-left:5px solid #585b70;padding:1.25em 1.5em}html.theme--catppuccin-mocha .content ol{list-style-position:outside;margin-left:2em;margin-top:1em}html.theme--catppuccin-mocha .content ol:not([type]){list-style-type:decimal}html.theme--catppuccin-mocha .content ol.is-lower-alpha:not([type]){list-style-type:lower-alpha}html.theme--catppuccin-mocha .content ol.is-lower-roman:not([type]){list-style-type:lower-roman}html.theme--catppuccin-mocha .content ol.is-upper-alpha:not([type]){list-style-type:upper-alpha}html.theme--catppuccin-mocha .content ol.is-upper-roman:not([type]){list-style-type:upper-roman}html.theme--catppuccin-mocha .content ul{list-style:disc outside;margin-left:2em;margin-top:1em}html.theme--catppuccin-mocha .content ul ul{list-style-type:circle;margin-top:0.5em}html.theme--catppuccin-mocha .content ul ul ul{list-style-type:square}html.theme--catppuccin-mocha .content dd{margin-left:2em}html.theme--catppuccin-mocha .content figure{margin-left:2em;margin-right:2em;text-align:center}html.theme--catppuccin-mocha .content figure:not(:first-child){margin-top:2em}html.theme--catppuccin-mocha .content figure:not(:last-child){margin-bottom:2em}html.theme--catppuccin-mocha .content figure img{display:inline-block}html.theme--catppuccin-mocha .content figure figcaption{font-style:italic}html.theme--catppuccin-mocha .content pre{-webkit-overflow-scrolling:touch;overflow-x:auto;padding:0;white-space:pre;word-wrap:normal}html.theme--catppuccin-mocha .content sup,html.theme--catppuccin-mocha .content sub{font-size:75%}html.theme--catppuccin-mocha .content table{width:100%}html.theme--catppuccin-mocha .content table td,html.theme--catppuccin-mocha .content table th{border:1px solid #585b70;border-width:0 0 1px;padding:0.5em 0.75em;vertical-align:top}html.theme--catppuccin-mocha .content table th{color:#b8c5ef}html.theme--catppuccin-mocha .content table th:not([align]){text-align:inherit}html.theme--catppuccin-mocha .content table thead td,html.theme--catppuccin-mocha .content table thead th{border-width:0 0 2px;color:#b8c5ef}html.theme--catppuccin-mocha .content table tfoot td,html.theme--catppuccin-mocha .content table tfoot th{border-width:2px 0 0;color:#b8c5ef}html.theme--catppuccin-mocha .content table tbody tr:last-child td,html.theme--catppuccin-mocha .content table tbody tr:last-child th{border-bottom-width:0}html.theme--catppuccin-mocha .content .tabs li+li{margin-top:0}html.theme--catppuccin-mocha .content.is-small,html.theme--catppuccin-mocha #documenter .docs-sidebar form.docs-search>input.content{font-size:.75rem}html.theme--catppuccin-mocha .content.is-normal{font-size:1rem}html.theme--catppuccin-mocha .content.is-medium{font-size:1.25rem}html.theme--catppuccin-mocha .content.is-large{font-size:1.5rem}html.theme--catppuccin-mocha .icon{align-items:center;display:inline-flex;justify-content:center;height:1.5rem;width:1.5rem}html.theme--catppuccin-mocha .icon.is-small,html.theme--catppuccin-mocha #documenter .docs-sidebar form.docs-search>input.icon{height:1rem;width:1rem}html.theme--catppuccin-mocha .icon.is-medium{height:2rem;width:2rem}html.theme--catppuccin-mocha .icon.is-large{height:3rem;width:3rem}html.theme--catppuccin-mocha .icon-text{align-items:flex-start;color:inherit;display:inline-flex;flex-wrap:wrap;line-height:1.5rem;vertical-align:top}html.theme--catppuccin-mocha .icon-text .icon{flex-grow:0;flex-shrink:0}html.theme--catppuccin-mocha .icon-text .icon:not(:last-child){margin-right:.25em}html.theme--catppuccin-mocha .icon-text .icon:not(:first-child){margin-left:.25em}html.theme--catppuccin-mocha div.icon-text{display:flex}html.theme--catppuccin-mocha .image,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img{display:block;position:relative}html.theme--catppuccin-mocha .image img,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img img{display:block;height:auto;width:100%}html.theme--catppuccin-mocha .image img.is-rounded,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img img.is-rounded{border-radius:9999px}html.theme--catppuccin-mocha .image.is-fullwidth,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-fullwidth{width:100%}html.theme--catppuccin-mocha .image.is-square img,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-square img,html.theme--catppuccin-mocha .image.is-square .has-ratio,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-square .has-ratio,html.theme--catppuccin-mocha .image.is-1by1 img,html.theme--catppuccin-mocha #documenter .docs-sidebar 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.has-ratio,html.theme--catppuccin-mocha .image.is-5by3 img,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-5by3 img,html.theme--catppuccin-mocha .image.is-5by3 .has-ratio,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-5by3 .has-ratio,html.theme--catppuccin-mocha .image.is-16by9 img,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-16by9 img,html.theme--catppuccin-mocha .image.is-16by9 .has-ratio,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-16by9 .has-ratio,html.theme--catppuccin-mocha .image.is-2by1 img,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-2by1 img,html.theme--catppuccin-mocha .image.is-2by1 .has-ratio,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-2by1 .has-ratio,html.theme--catppuccin-mocha .image.is-3by1 img,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-3by1 img,html.theme--catppuccin-mocha 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img,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-3by5 img,html.theme--catppuccin-mocha .image.is-3by5 .has-ratio,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-3by5 .has-ratio,html.theme--catppuccin-mocha .image.is-9by16 img,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-9by16 img,html.theme--catppuccin-mocha .image.is-9by16 .has-ratio,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-9by16 .has-ratio,html.theme--catppuccin-mocha .image.is-1by2 img,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-1by2 img,html.theme--catppuccin-mocha .image.is-1by2 .has-ratio,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-1by2 .has-ratio,html.theme--catppuccin-mocha .image.is-1by3 img,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-1by3 img,html.theme--catppuccin-mocha .image.is-1by3 .has-ratio,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-1by3 .has-ratio{height:100%;width:100%}html.theme--catppuccin-mocha .image.is-square,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-square,html.theme--catppuccin-mocha .image.is-1by1,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-1by1{padding-top:100%}html.theme--catppuccin-mocha .image.is-5by4,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-5by4{padding-top:80%}html.theme--catppuccin-mocha .image.is-4by3,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-4by3{padding-top:75%}html.theme--catppuccin-mocha .image.is-3by2,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-3by2{padding-top:66.6666%}html.theme--catppuccin-mocha .image.is-5by3,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-5by3{padding-top:60%}html.theme--catppuccin-mocha .image.is-16by9,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-16by9{padding-top:56.25%}html.theme--catppuccin-mocha .image.is-2by1,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-2by1{padding-top:50%}html.theme--catppuccin-mocha .image.is-3by1,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-3by1{padding-top:33.3333%}html.theme--catppuccin-mocha .image.is-4by5,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-4by5{padding-top:125%}html.theme--catppuccin-mocha .image.is-3by4,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-3by4{padding-top:133.3333%}html.theme--catppuccin-mocha .image.is-2by3,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-2by3{padding-top:150%}html.theme--catppuccin-mocha .image.is-3by5,html.theme--catppuccin-mocha #documenter .docs-sidebar .docs-logo>img.is-3by5{padding-top:166.6666%}html.theme--catppuccin-mocha .image.is-9by16,html.theme--catppuccin-mocha #documenter 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html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--documenter-dark .button.is-danger.is-light{background-color:#fbefef;color:#c03930}html.theme--documenter-dark .button.is-danger.is-light:hover,html.theme--documenter-dark .button.is-danger.is-light.is-hovered{background-color:#f8e6e5;border-color:transparent;color:#c03930}html.theme--documenter-dark .button.is-danger.is-light:active,html.theme--documenter-dark .button.is-danger.is-light.is-active{background-color:#f6dcda;border-color:transparent;color:#c03930}html.theme--documenter-dark .button.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.button{font-size:.75rem}html.theme--documenter-dark .button.is-small:not(.is-rounded),html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.button:not(.is-rounded){border-radius:3px}html.theme--documenter-dark 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.button.is-responsive.is-normal{font-size:.65625rem}html.theme--documenter-dark .button.is-responsive.is-medium{font-size:.75rem}html.theme--documenter-dark .button.is-responsive.is-large{font-size:1rem}}@media screen and (min-width: 769px) and (max-width: 1055px){html.theme--documenter-dark .button.is-responsive.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.65625rem}html.theme--documenter-dark .button.is-responsive,html.theme--documenter-dark .button.is-responsive.is-normal{font-size:.75rem}html.theme--documenter-dark .button.is-responsive.is-medium{font-size:1rem}html.theme--documenter-dark .button.is-responsive.is-large{font-size:1.25rem}}html.theme--documenter-dark .container{flex-grow:1;margin:0 auto;position:relative;width:auto}html.theme--documenter-dark .container.is-fluid{max-width:none !important;padding-left:32px;padding-right:32px;width:100%}@media screen and (min-width: 1056px){html.theme--documenter-dark 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h5{font-size:1.125em;margin-bottom:0.8888em}html.theme--documenter-dark .content h6{font-size:1em;margin-bottom:1em}html.theme--documenter-dark .content blockquote{background-color:#282f2f;border-left:5px solid #5e6d6f;padding:1.25em 1.5em}html.theme--documenter-dark .content ol{list-style-position:outside;margin-left:2em;margin-top:1em}html.theme--documenter-dark .content ol:not([type]){list-style-type:decimal}html.theme--documenter-dark .content ol.is-lower-alpha:not([type]){list-style-type:lower-alpha}html.theme--documenter-dark .content ol.is-lower-roman:not([type]){list-style-type:lower-roman}html.theme--documenter-dark .content ol.is-upper-alpha:not([type]){list-style-type:upper-alpha}html.theme--documenter-dark .content ol.is-upper-roman:not([type]){list-style-type:upper-roman}html.theme--documenter-dark .content ul{list-style:disc outside;margin-left:2em;margin-top:1em}html.theme--documenter-dark .content ul ul{list-style-type:circle;margin-top:0.5em}html.theme--documenter-dark .content ul ul ul{list-style-type:square}html.theme--documenter-dark .content dd{margin-left:2em}html.theme--documenter-dark .content figure{margin-left:2em;margin-right:2em;text-align:center}html.theme--documenter-dark .content figure:not(:first-child){margin-top:2em}html.theme--documenter-dark .content figure:not(:last-child){margin-bottom:2em}html.theme--documenter-dark .content figure img{display:inline-block}html.theme--documenter-dark .content figure figcaption{font-style:italic}html.theme--documenter-dark .content pre{-webkit-overflow-scrolling:touch;overflow-x:auto;padding:0;white-space:pre;word-wrap:normal}html.theme--documenter-dark .content sup,html.theme--documenter-dark .content sub{font-size:75%}html.theme--documenter-dark .content table{width:100%}html.theme--documenter-dark .content table td,html.theme--documenter-dark .content table th{border:1px solid #5e6d6f;border-width:0 0 1px;padding:0.5em 0.75em;vertical-align:top}html.theme--documenter-dark .content table th{color:#f2f2f2}html.theme--documenter-dark .content table th:not([align]){text-align:inherit}html.theme--documenter-dark .content table thead td,html.theme--documenter-dark .content table thead th{border-width:0 0 2px;color:#f2f2f2}html.theme--documenter-dark .content table tfoot td,html.theme--documenter-dark .content table tfoot th{border-width:2px 0 0;color:#f2f2f2}html.theme--documenter-dark .content table tbody tr:last-child td,html.theme--documenter-dark .content table tbody tr:last-child th{border-bottom-width:0}html.theme--documenter-dark .content .tabs li+li{margin-top:0}html.theme--documenter-dark .content.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.content{font-size:.75rem}html.theme--documenter-dark .content.is-normal{font-size:1rem}html.theme--documenter-dark .content.is-medium{font-size:1.25rem}html.theme--documenter-dark .content.is-large{font-size:1.5rem}html.theme--documenter-dark 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span.sgr30{color:#242424}html.theme--documenter-dark .ansi span.sgr31{color:#f6705f}html.theme--documenter-dark .ansi span.sgr32{color:#4fb43a}html.theme--documenter-dark .ansi span.sgr33{color:#f4c72f}html.theme--documenter-dark .ansi span.sgr34{color:#7587f0}html.theme--documenter-dark .ansi span.sgr35{color:#bc89d3}html.theme--documenter-dark .ansi span.sgr36{color:#49b6ca}html.theme--documenter-dark .ansi span.sgr37{color:#b3bdbe}html.theme--documenter-dark .ansi span.sgr40{background-color:#242424}html.theme--documenter-dark .ansi span.sgr41{background-color:#f6705f}html.theme--documenter-dark .ansi span.sgr42{background-color:#4fb43a}html.theme--documenter-dark .ansi span.sgr43{background-color:#f4c72f}html.theme--documenter-dark .ansi span.sgr44{background-color:#7587f0}html.theme--documenter-dark .ansi span.sgr45{background-color:#bc89d3}html.theme--documenter-dark .ansi span.sgr46{background-color:#49b6ca}html.theme--documenter-dark .ansi span.sgr47{background-color:#b3bdbe}html.theme--documenter-dark .ansi span.sgr90{color:#92a0a2}html.theme--documenter-dark .ansi span.sgr91{color:#ff8674}html.theme--documenter-dark .ansi span.sgr92{color:#79d462}html.theme--documenter-dark .ansi span.sgr93{color:#ffe76b}html.theme--documenter-dark .ansi span.sgr94{color:#8a98ff}html.theme--documenter-dark .ansi span.sgr95{color:#d2a4e6}html.theme--documenter-dark .ansi span.sgr96{color:#6bc8db}html.theme--documenter-dark .ansi span.sgr97{color:#ecf0f1}html.theme--documenter-dark .ansi span.sgr100{background-color:#92a0a2}html.theme--documenter-dark .ansi span.sgr101{background-color:#ff8674}html.theme--documenter-dark .ansi span.sgr102{background-color:#79d462}html.theme--documenter-dark .ansi span.sgr103{background-color:#ffe76b}html.theme--documenter-dark .ansi span.sgr104{background-color:#8a98ff}html.theme--documenter-dark .ansi span.sgr105{background-color:#d2a4e6}html.theme--documenter-dark .ansi 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footer,html.theme--documenter-dark header,html.theme--documenter-dark hgroup,html.theme--documenter-dark section{display:block}html.theme--documenter-dark body,html.theme--documenter-dark button,html.theme--documenter-dark input,html.theme--documenter-dark optgroup,html.theme--documenter-dark select,html.theme--documenter-dark textarea{font-family:"Lato Medium",-apple-system,BlinkMacSystemFont,"Segoe UI","Helvetica Neue","Helvetica","Arial",sans-serif}html.theme--documenter-dark code,html.theme--documenter-dark pre{-moz-osx-font-smoothing:auto;-webkit-font-smoothing:auto;font-family:"JuliaMono","SFMono-Regular","Menlo","Consolas","Liberation Mono","DejaVu Sans Mono",monospace}html.theme--documenter-dark body{color:#fff;font-size:1em;font-weight:400;line-height:1.5}html.theme--documenter-dark a{color:#1abc9c;cursor:pointer;text-decoration:none}html.theme--documenter-dark a strong{color:currentColor}html.theme--documenter-dark a:hover{color:#1dd2af}html.theme--documenter-dark 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.button.is-light.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-light.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #ecf0f1 #ecf0f1 !important}html.theme--documenter-dark .button.is-light.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-light.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-dark,html.theme--documenter-dark .content kbd.button{background-color:#282f2f;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-dark:hover,html.theme--documenter-dark .content kbd.button:hover,html.theme--documenter-dark .button.is-dark.is-hovered,html.theme--documenter-dark .content kbd.button.is-hovered{background-color:#232829;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-dark:focus,html.theme--documenter-dark .content kbd.button:focus,html.theme--documenter-dark .button.is-dark.is-focused,html.theme--documenter-dark .content kbd.button.is-focused{border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-dark:focus:not(:active),html.theme--documenter-dark .content kbd.button:focus:not(:active),html.theme--documenter-dark .button.is-dark.is-focused:not(:active),html.theme--documenter-dark .content kbd.button.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(40,47,47,0.25)}html.theme--documenter-dark .button.is-dark:active,html.theme--documenter-dark .content kbd.button:active,html.theme--documenter-dark .button.is-dark.is-active,html.theme--documenter-dark .content kbd.button.is-active{background-color:#1d2122;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-dark[disabled],html.theme--documenter-dark .content kbd.button[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-dark,fieldset[disabled] html.theme--documenter-dark .content 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kbd.button.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-dark.is-outlined,html.theme--documenter-dark .content kbd.button.is-outlined{background-color:transparent;border-color:#282f2f;color:#282f2f}html.theme--documenter-dark .button.is-dark.is-outlined:hover,html.theme--documenter-dark .content kbd.button.is-outlined:hover,html.theme--documenter-dark .button.is-dark.is-outlined.is-hovered,html.theme--documenter-dark .content kbd.button.is-outlined.is-hovered,html.theme--documenter-dark .button.is-dark.is-outlined:focus,html.theme--documenter-dark .content kbd.button.is-outlined:focus,html.theme--documenter-dark .button.is-dark.is-outlined.is-focused,html.theme--documenter-dark .content kbd.button.is-outlined.is-focused{background-color:#282f2f;border-color:#282f2f;color:#fff}html.theme--documenter-dark .button.is-dark.is-outlined.is-loading::after,html.theme--documenter-dark .content kbd.button.is-outlined.is-loading::after{border-color:transparent transparent #282f2f #282f2f !important}html.theme--documenter-dark .button.is-dark.is-outlined.is-loading:hover::after,html.theme--documenter-dark .content kbd.button.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-dark.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .content kbd.button.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-dark.is-outlined.is-loading:focus::after,html.theme--documenter-dark .content kbd.button.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-dark.is-outlined.is-loading.is-focused::after,html.theme--documenter-dark .content kbd.button.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-dark.is-outlined[disabled],html.theme--documenter-dark .content kbd.button.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-dark.is-outlined,fieldset[disabled] html.theme--documenter-dark .content kbd.button.is-outlined{background-color:transparent;border-color:#282f2f;box-shadow:none;color:#282f2f}html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined:hover,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined:focus,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-focused,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-focused{background-color:#fff;color:#282f2f}html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-loading.is-focused::after,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #282f2f #282f2f !important}html.theme--documenter-dark 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.docstring>section>a.button.is-outlined.is-hovered.docs-sourcelink,html.theme--documenter-dark .button.is-primary.is-outlined:focus,html.theme--documenter-dark .docstring>section>a.button.is-outlined.docs-sourcelink:focus,html.theme--documenter-dark .button.is-primary.is-outlined.is-focused,html.theme--documenter-dark .docstring>section>a.button.is-outlined.is-focused.docs-sourcelink{background-color:#375a7f;border-color:#375a7f;color:#fff}html.theme--documenter-dark .button.is-primary.is-outlined.is-loading::after,html.theme--documenter-dark .docstring>section>a.button.is-outlined.is-loading.docs-sourcelink::after{border-color:transparent transparent #375a7f #375a7f !important}html.theme--documenter-dark .button.is-primary.is-outlined.is-loading:hover::after,html.theme--documenter-dark .docstring>section>a.button.is-outlined.is-loading.docs-sourcelink:hover::after,html.theme--documenter-dark .button.is-primary.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark 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html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--documenter-dark .button.is-danger.is-light{background-color:#fbefef;color:#c03930}html.theme--documenter-dark .button.is-danger.is-light:hover,html.theme--documenter-dark .button.is-danger.is-light.is-hovered{background-color:#f8e6e5;border-color:transparent;color:#c03930}html.theme--documenter-dark .button.is-danger.is-light:active,html.theme--documenter-dark .button.is-danger.is-light.is-active{background-color:#f6dcda;border-color:transparent;color:#c03930}html.theme--documenter-dark .button.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.button{font-size:.75rem}html.theme--documenter-dark .button.is-small:not(.is-rounded),html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.button:not(.is-rounded){border-radius:3px}html.theme--documenter-dark 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.button.is-responsive.is-normal{font-size:.65625rem}html.theme--documenter-dark .button.is-responsive.is-medium{font-size:.75rem}html.theme--documenter-dark .button.is-responsive.is-large{font-size:1rem}}@media screen and (min-width: 769px) and (max-width: 1055px){html.theme--documenter-dark .button.is-responsive.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.65625rem}html.theme--documenter-dark .button.is-responsive,html.theme--documenter-dark .button.is-responsive.is-normal{font-size:.75rem}html.theme--documenter-dark .button.is-responsive.is-medium{font-size:1rem}html.theme--documenter-dark .button.is-responsive.is-large{font-size:1.25rem}}html.theme--documenter-dark .container{flex-grow:1;margin:0 auto;position:relative;width:auto}html.theme--documenter-dark .container.is-fluid{max-width:none !important;padding-left:32px;padding-right:32px;width:100%}@media screen and (min-width: 1056px){html.theme--documenter-dark 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h5{font-size:1.125em;margin-bottom:0.8888em}html.theme--documenter-dark .content h6{font-size:1em;margin-bottom:1em}html.theme--documenter-dark .content blockquote{background-color:#282f2f;border-left:5px solid #5e6d6f;padding:1.25em 1.5em}html.theme--documenter-dark .content ol{list-style-position:outside;margin-left:2em;margin-top:1em}html.theme--documenter-dark .content ol:not([type]){list-style-type:decimal}html.theme--documenter-dark .content ol.is-lower-alpha:not([type]){list-style-type:lower-alpha}html.theme--documenter-dark .content ol.is-lower-roman:not([type]){list-style-type:lower-roman}html.theme--documenter-dark .content ol.is-upper-alpha:not([type]){list-style-type:upper-alpha}html.theme--documenter-dark .content ol.is-upper-roman:not([type]){list-style-type:upper-roman}html.theme--documenter-dark .content ul{list-style:disc outside;margin-left:2em;margin-top:1em}html.theme--documenter-dark .content ul ul{list-style-type:circle;margin-top:0.5em}html.theme--documenter-dark .content ul ul ul{list-style-type:square}html.theme--documenter-dark .content dd{margin-left:2em}html.theme--documenter-dark .content figure{margin-left:2em;margin-right:2em;text-align:center}html.theme--documenter-dark .content figure:not(:first-child){margin-top:2em}html.theme--documenter-dark .content figure:not(:last-child){margin-bottom:2em}html.theme--documenter-dark .content figure img{display:inline-block}html.theme--documenter-dark .content figure figcaption{font-style:italic}html.theme--documenter-dark .content pre{-webkit-overflow-scrolling:touch;overflow-x:auto;padding:0;white-space:pre;word-wrap:normal}html.theme--documenter-dark .content sup,html.theme--documenter-dark .content sub{font-size:75%}html.theme--documenter-dark .content table{width:100%}html.theme--documenter-dark .content table td,html.theme--documenter-dark .content table th{border:1px solid #5e6d6f;border-width:0 0 1px;padding:0.5em 0.75em;vertical-align:top}html.theme--documenter-dark .content table th{color:#f2f2f2}html.theme--documenter-dark .content table th:not([align]){text-align:inherit}html.theme--documenter-dark .content table thead td,html.theme--documenter-dark .content table thead th{border-width:0 0 2px;color:#f2f2f2}html.theme--documenter-dark .content table tfoot td,html.theme--documenter-dark .content table tfoot th{border-width:2px 0 0;color:#f2f2f2}html.theme--documenter-dark .content table tbody tr:last-child td,html.theme--documenter-dark .content table tbody tr:last-child th{border-bottom-width:0}html.theme--documenter-dark .content .tabs li+li{margin-top:0}html.theme--documenter-dark .content.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.content{font-size:.75rem}html.theme--documenter-dark .content.is-normal{font-size:1rem}html.theme--documenter-dark .content.is-medium{font-size:1.25rem}html.theme--documenter-dark .content.is-large{font-size:1.5rem}html.theme--documenter-dark 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Theme: Default Description: Original highlight.js style Author: (c) Ivan Sagalaev diff --git a/dev/diffgeo/index.html b/dev/diffgeo/index.html index a40c797..64d1111 100644 --- a/dev/diffgeo/index.html +++ b/dev/diffgeo/index.html @@ -18,13 +18,13 @@ \tag{10}\]

i.e., the neural network takes a number between zero and one and maps it to the dimensionality of the latent space. The intention is to have $g_\omega \approx \gamma$, where $\omega$ are the parameters of the neural network we are free to optimize.

We approximate the integral defining the length of the curve in the latent space with $n$ equidistantly sampled points $t_i$ between zero and one. The length of the curve is then approximated by

\[L(g_\gamma(t)) \approx \frac{1}{n} \sum_{i=1}^n \sqrt{ \dot{g}_\omega(t_i)^T \mathbf{G}(g_\omega(t_i)) \dot{g}_\omega(t_i) -},\]

By setting the loss function to be this approximation of the length of the curve, we can train the neural network to approximate the geodesic curve.

AutoEncoderToolkit.jl provides the NeuralGeodesic struct to implement this idea. The struct takes three inputs:

NeuralGeodesic struct

AutoEncoderToolkit.diffgeo.NeuralGeodesics.NeuralGeodesic — Type
NeuralGeodesic

Type to define a neural network that approximates a geodesic curve on a Riemanian manifold. If a curve γ̲(t) represents a geodesic curve on a manifold, i.e.,

L(γ̲) = min_γ ∫ dt √(⟨γ̲̇(t), M̲̲ γ̲̇(t)⟩),

where M̲̲ is the Riemmanian metric, then this type defines a neural network g_ω(t) such that

γ̲(t) ≈ g_ω(t).

This neural network must have a single input (1D). The dimensionality of the output must match the dimensionality of the manifold.

Fields

  • mlp::Flux.Chain: Neural network that approximates the geodesic curve. The dimensionality of the input must be one.
  • z_init::AbstractVector: Initial position of the geodesic curve on the latent space.
  • z_end::AbstractVector: Final position of the geodesic curve on the latent space.

Citation

Chen, N. et al. Metrics for Deep Generative Models. in Proceedings of the Twenty-First International Conference on Artificial Intelligence and Statistics 1540–1550 (PMLR, 2018).

NeuralGeodesic forward pass

AutoEncoderToolkit.diffgeo.NeuralGeodesics.NeuralGeodesic — Method
    (g::NeuralGeodesic)(t::AbstractArray)

Computes the output of the NeuralGeodesic at each given time in t by scaling and shifting the output of the neural network.

Arguments

  • t::AbstractArray: An array of times at which the output of the NeuralGeodesic is to be computed. This must be within the interval [0, 1].

Returns

  • output::Array: The computed output of the NeuralGeodesic at each time in t.

Description

The function computes the output of the NeuralGeodesic at each given time in t. The steps are:

  1. Compute the output of the neural network at each time in t.
  2. Compute the output of the neural network at time 0 and 1.
  3. Compute scale and shift parameters based on the initial and end points of the geodesic and the neural network outputs at times 0 and 1.
  4. Scale and shift the output of the neural network at each time in t according to these parameters. The result is the output of the NeuralGeodesic at each time in t.

Scale and shift parameters are defined as:

  • scale = (zinit - zend) / (ẑinit - ẑend)
  • shift = (zinit * ẑend - zend * ẑinit) / (ẑinit - ẑend)

where zinit and zend are the initial and end points of the geodesic, and ẑinit and ẑend are the outputs of the neural network at times 0 and 1, respectively.

Note

Ensure that each t in the array is within the interval [0, 1].

NeuralGeodesic loss function

AutoEncoderToolkit.diffgeo.NeuralGeodesics.loss — Function
loss(
+},\]
By setting the loss function to be this approximation of the length of the curve, we can train the neural network to approximate the geodesic curve.
AutoEncoderToolkit.jl provides the NeuralGeodesic struct to implement this idea. The struct takes three inputs:
  • The multi-layer perceptron (MLP) that approximates the geodesic curve.
  • The initial point in latent space.
  • The final point in latent space.
NeuralGeodesic struct
AutoEncoderToolkit.diffgeo.NeuralGeodesics.NeuralGeodesic — Type
NeuralGeodesic
Type to define a neural network that approximates a geodesic curve on a Riemanian manifold. If a curve γ̲(t) represents a geodesic curve on a manifold, i.e.,
L(γ̲) = min_γ ∫ dt √(⟨γ̲̇(t), M̲̲ γ̲̇(t)⟩),
where M̲̲ is the Riemmanian metric, then this type defines a neural network g_ω(t) such that
γ̲(t) ≈ g_ω(t).
This neural network must have a single input (1D). The dimensionality of the output must match the dimensionality of the manifold.
Fields
  • mlp::Flux.Chain: Neural network that approximates the geodesic curve. The dimensionality of the input must be one.
  • z_init::AbstractVector: Initial position of the geodesic curve on the latent space.
  • z_end::AbstractVector: Final position of the geodesic curve on the latent space.
Citation
Chen, N. et al. Metrics for Deep Generative Models. in Proceedings of the Twenty-First International Conference on Artificial Intelligence and Statistics 1540–1550 (PMLR, 2018).
NeuralGeodesic forward pass
AutoEncoderToolkit.diffgeo.NeuralGeodesics.NeuralGeodesic — Method
    (g::NeuralGeodesic)(t::AbstractArray)
Computes the output of the NeuralGeodesic at each given time in t by scaling and shifting the output of the neural network.
Arguments
  • t::AbstractArray: An array of times at which the output of the NeuralGeodesic is to be computed. This must be within the interval [0, 1].
Returns
  • output::Array: The computed output of the NeuralGeodesic at each time in t.
Description
The function computes the output of the NeuralGeodesic at each given time in t. The steps are:
  1. Compute the output of the neural network at each time in t.
  2. Compute the output of the neural network at time 0 and 1.
  3. Compute scale and shift parameters based on the initial and end points of the geodesic and the neural network outputs at times 0 and 1.
  4. Scale and shift the output of the neural network at each time in t according to these parameters. The result is the output of the NeuralGeodesic at each time in t.
Scale and shift parameters are defined as:
  • scale = (zinit - zend) / (ẑinit - ẑend)
  • shift = (zinit * ẑend - zend * ẑinit) / (ẑinit - ẑend)
where zinit and zend are the initial and end points of the geodesic, and ẑinit and ẑend are the outputs of the neural network at times 0 and 1, respectively.
Note
Ensure that each t in the array is within the interval [0, 1].
NeuralGeodesic loss function
AutoEncoderToolkit.diffgeo.NeuralGeodesics.loss — Function
loss(
     curve::NeuralGeodesic,
     rhvae::RHVAE,
     t::AbstractVector;
     curve_velocity::Function=curve_velocity_TaylorDiff,
     curve_integral::Function=curve_length,
-)
Function to compute the loss for a given curve on a Riemmanian manifold. The loss is defined as the integral over the curve, computed using the provided curve_integral function (either length or energy).
Arguments
  • curve::NeuralGeodesic: The curve on the Riemmanian manifold.
  • rhvae::RHVAE: The Riemmanian Hamiltonian Variational AutoEncoder used to compute the Riemmanian metric tensor.
  • t::AbstractVector: Vector of time points at which the curve is sampled.
Optional Keyword Arguments
  • curve_velocity::Function=curve_velocity_TaylorDiff: Function to compute the velocity of the curve. Default is curve_velocity_TaylorDiff. Also accepts curve_velocity_finitediff.
  • curve_integral::Function=curve_length: Function to compute the integral over the curve. Default is curve_energy. Also accepts curve_length.
Returns
  • Loss::Number: The computed loss for the given curve.
Notes
This function first computes the geodesic curve using the provided curve function. It then computes the Riemmanian metric tensor using the metric_tensor function from the RHVAE module with the computed curve and the provided rhvae. The velocity of the curve is then computed using the provided curve_velocity function. Finally, the integral over the curve is computed using the provided curve_integral function and returned as the loss.
NeuralGeodesic training
AutoEncoderToolkit.diffgeo.NeuralGeodesics.train! — Function
train!(
+)
Function to compute the loss for a given curve on a Riemmanian manifold. The loss is defined as the integral over the curve, computed using the provided curve_integral function (either length or energy).
Arguments
  • curve::NeuralGeodesic: The curve on the Riemmanian manifold.
  • rhvae::RHVAE: The Riemmanian Hamiltonian Variational AutoEncoder used to compute the Riemmanian metric tensor.
  • t::AbstractVector: Vector of time points at which the curve is sampled.
Optional Keyword Arguments
  • curve_velocity::Function=curve_velocity_TaylorDiff: Function to compute the velocity of the curve. Default is curve_velocity_TaylorDiff. Also accepts curve_velocity_finitediff.
  • curve_integral::Function=curve_length: Function to compute the integral over the curve. Default is curve_energy. Also accepts curve_length.
Returns
  • Loss::Number: The computed loss for the given curve.
Notes
This function first computes the geodesic curve using the provided curve function. It then computes the Riemmanian metric tensor using the metric_tensor function from the RHVAE module with the computed curve and the provided rhvae. The velocity of the curve is then computed using the provided curve_velocity function. Finally, the integral over the curve is computed using the provided curve_integral function and returned as the loss.
NeuralGeodesic training
AutoEncoderToolkit.diffgeo.NeuralGeodesics.train! — Function
train!(
     curve::NeuralGeodesic,
     rhvae::RHVAE,
     t::AbstractVector,
@@ -33,22 +33,22 @@
     loss_kwargs::NamedTuple=NamedTuple(),
     verbose::Bool=false,
     loss_return::Bool=false,
-)
Function to train a NeuralGeodesic model using a Riemmanian Hamiltonian Variational AutoEncoder (RHVAE). The training process involves computing the gradient of the loss function and updating the model parameters accordingly.
Arguments
  • curve::NeuralGeodesic: The curve on the Riemmanian manifold.
  • rhvae::RHVAE: The Riemmanian Hamiltonian Variational AutoEncoder used to compute the Riemmanian metric tensor.
  • t::AbstractVector: Vector of time points at which the curve is sampled. These must be equally spaced.
  • opt::NamedTuple: The optimization parameters.
Optional Keyword Arguments
  • loss_function::Function=loss: The loss function to be minimized during training. Default is loss.
  • loss_kwargs::NamedTuple=NamedTuple(): Additional keyword arguments to be passed to the loss function.
  • verbose::Bool=false: If true, the loss value is printed at each iteration.
  • loss_return::Bool=false: If true, the function returns the loss value.
Returns
  • Loss::Number: The computed loss for the given curve. This is only returned if loss_return is true.
Notes
This function first computes the gradient of the loss function with respect to the model parameters. It then updates the model parameters using the computed gradient and the provided optimization parameters. If verbose is true, the loss value is printed at each iteration. If loss_return is true, the function returns the loss value.
Other functions for NeuralGeodesic
AutoEncoderToolkit.diffgeo.NeuralGeodesics.curve_velocity_TaylorDiff — Function
curve_velocity_TaylorDiff(
+)
Function to train a NeuralGeodesic model using a Riemmanian Hamiltonian Variational AutoEncoder (RHVAE). The training process involves computing the gradient of the loss function and updating the model parameters accordingly.
Arguments
  • curve::NeuralGeodesic: The curve on the Riemmanian manifold.
  • rhvae::RHVAE: The Riemmanian Hamiltonian Variational AutoEncoder used to compute the Riemmanian metric tensor.
  • t::AbstractVector: Vector of time points at which the curve is sampled. These must be equally spaced.
  • opt::NamedTuple: The optimization parameters.
Optional Keyword Arguments
  • loss_function::Function=loss: The loss function to be minimized during training. Default is loss.
  • loss_kwargs::NamedTuple=NamedTuple(): Additional keyword arguments to be passed to the loss function.
  • verbose::Bool=false: If true, the loss value is printed at each iteration.
  • loss_return::Bool=false: If true, the function returns the loss value.
Returns
  • Loss::Number: The computed loss for the given curve. This is only returned if loss_return is true.
Notes
This function first computes the gradient of the loss function with respect to the model parameters. It then updates the model parameters using the computed gradient and the provided optimization parameters. If verbose is true, the loss value is printed at each iteration. If loss_return is true, the function returns the loss value.
Other functions for NeuralGeodesic
AutoEncoderToolkit.diffgeo.NeuralGeodesics.curve_velocity_TaylorDiff — Function
curve_velocity_TaylorDiff(
     curve::NeuralGeodesic,
     t
 )
Compute the velocity of a neural geodesic curve at a given time using Taylor differentiation.
This function takes a NeuralGeodesic instance and a time t, and computes the velocity of the curve at that time using Taylor differentiation. The computation is performed for each dimension of the latent space.
Arguments
  • curve::NeuralGeodesic: The neural geodesic curve.
  • t: The time at which to compute the velocity.
Returns
A vector representing the velocity of the curve at time t.
Notes
This function uses the TaylorDiff package to compute derivatives. Please note that TaylorDiff has limited support for certain activation functions. If you encounter an error while using this function, it may be due to the activation function used in your NeuralGeodesic instance.
curve_velocity_TaylorDiff(
     curve::NeuralGeodesic,
     t::AbstractVector
-)
Compute the velocity of a neural geodesic curve at each time in a vector of times using Taylor differentiation.
This function takes a NeuralGeodesic instance and a vector of times t, and computes the velocity of the curve at each time using Taylor differentiation. The computation is performed for each dimension of the latent space and each time in t.
Arguments
  • curve::NeuralGeodesic: The neural geodesic curve.
  • t::AbstractVector: The vector of times at which to compute the velocity.
Returns
A matrix where each column represents the velocity of the curve at a time in t.
Notes
This function uses the TaylorDiff package to compute derivatives. Please note that TaylorDiff has limited support for certain activation functions. If you encounter an error while using this function, it may be due to the activation function used in your NeuralGeodesic instance.
AutoEncoderToolkit.diffgeo.NeuralGeodesics.curve_velocity_finitediff — Function
curve_velocity_finitediff(
+)
Compute the velocity of a neural geodesic curve at each time in a vector of times using Taylor differentiation.
This function takes a NeuralGeodesic instance and a vector of times t, and computes the velocity of the curve at each time using Taylor differentiation. The computation is performed for each dimension of the latent space and each time in t.
Arguments
  • curve::NeuralGeodesic: The neural geodesic curve.
  • t::AbstractVector: The vector of times at which to compute the velocity.
Returns
A matrix where each column represents the velocity of the curve at a time in t.
Notes
This function uses the TaylorDiff package to compute derivatives. Please note that TaylorDiff has limited support for certain activation functions. If you encounter an error while using this function, it may be due to the activation function used in your NeuralGeodesic instance.
AutoEncoderToolkit.diffgeo.NeuralGeodesics.curve_velocity_finitediff — Function
curve_velocity_finitediff(
     curve::NeuralGeodesic,
     t::AbstractVector;
     fdtype::Symbol=:central,
-)
Compute the velocity of a neural geodesic curve at each time in a vector of times using finite difference methods.
This function takes a NeuralGeodesic instance, a vector of times t, and an optional finite difference type fdtype (which can be either :forward or :central), and computes the velocity of the curve at each time using the specified finite difference method. The computation is performed for each dimension of the latent space and each time in t.
Arguments
  • curve::NeuralGeodesic: The neural geodesic curve.
  • t::AbstractVector: The vector of times at which to compute the velocity.
  • fdtype::Symbol=:central: The type of finite difference method to use. Can be either :forward or :central. Default is :central.
Returns
A matrix where each column represents the velocity of the curve at a time in t.
Notes
This function uses finite difference methods to compute derivatives. Please note that the accuracy of the computed velocities depends on the chosen finite difference method and the step size used, which is determined by the machine epsilon of the type of t.
AutoEncoderToolkit.diffgeo.NeuralGeodesics.curve_length — Function
curve_length(
+)
Compute the velocity of a neural geodesic curve at each time in a vector of times using finite difference methods.
This function takes a NeuralGeodesic instance, a vector of times t, and an optional finite difference type fdtype (which can be either :forward or :central), and computes the velocity of the curve at each time using the specified finite difference method. The computation is performed for each dimension of the latent space and each time in t.
Arguments
  • curve::NeuralGeodesic: The neural geodesic curve.
  • t::AbstractVector: The vector of times at which to compute the velocity.
  • fdtype::Symbol=:central: The type of finite difference method to use. Can be either :forward or :central. Default is :central.
Returns
A matrix where each column represents the velocity of the curve at a time in t.
Notes
This function uses finite difference methods to compute derivatives. Please note that the accuracy of the computed velocities depends on the chosen finite difference method and the step size used, which is determined by the machine epsilon of the type of t.
AutoEncoderToolkit.diffgeo.NeuralGeodesics.curve_length — Function
curve_length(
     riemannian_metric::AbstractArray,
     curve_velocity::AbstractArray,
     t::AbstractVector;
-)
Function to compute the (discretized) integral defining the length of a curve γ̲ on a Riemmanina manifold. The length is defined as
L(γ̲) = ∫ dt √(⟨γ̲̇(t), G̲̲ γ̲̇(t)⟩),
where γ̲̇(t) defines the velocity of the parametric curve, and G̲̲ is the Riemmanian metric tensor. For this function, we approximate the integral as
L(γ̲) ≈ ∑ᵢ Δt √(⟨γ̲̇(tᵢ)ᵀ G̲̲ (γ̲(tᵢ+1)) γ̲̇(tᵢ))⟩),
where Δt is the time step between points. Note that this Δt is assumed to be constant, thus, the time points t must be equally spaced.
Arguments
  • riemannian_metric::AbstractArray: d×d×N tensor where d is the dimension of the manifold on which the curve lies and N is the number of sampled time points along the curve. Each slice of the array represents the Riemmanian metric tensor for the curve at the corresponding time point.
  • curve_velocity::AbstractArray: d×N Matrix where d is the dimension of the manifold on which the curve lies and N is the number of sampled time points along the curve. Each column represents the velocity of the curve at the corresponding time point.
  • t::AbstractVector: Vector of time points at which the curve is sampled.
Returns
  • Length::Number: Approximation of the Length for the path on the manifold.
AutoEncoderToolkit.diffgeo.NeuralGeodesics.curve_energy — Function
curve_energy(
+)
Function to compute the (discretized) integral defining the length of a curve γ̲ on a Riemmanina manifold. The length is defined as
L(γ̲) = ∫ dt √(⟨γ̲̇(t), G̲̲ γ̲̇(t)⟩),
where γ̲̇(t) defines the velocity of the parametric curve, and G̲̲ is the Riemmanian metric tensor. For this function, we approximate the integral as
L(γ̲) ≈ ∑ᵢ Δt √(⟨γ̲̇(tᵢ)ᵀ G̲̲ (γ̲(tᵢ+1)) γ̲̇(tᵢ))⟩),
where Δt is the time step between points. Note that this Δt is assumed to be constant, thus, the time points t must be equally spaced.
Arguments
  • riemannian_metric::AbstractArray: d×d×N tensor where d is the dimension of the manifold on which the curve lies and N is the number of sampled time points along the curve. Each slice of the array represents the Riemmanian metric tensor for the curve at the corresponding time point.
  • curve_velocity::AbstractArray: d×N Matrix where d is the dimension of the manifold on which the curve lies and N is the number of sampled time points along the curve. Each column represents the velocity of the curve at the corresponding time point.
  • t::AbstractVector: Vector of time points at which the curve is sampled.
Returns
  • Length::Number: Approximation of the Length for the path on the manifold.
AutoEncoderToolkit.diffgeo.NeuralGeodesics.curve_energy — Function
curve_energy(
     riemannian_metric::AbstractArray,
     curve_velocity::AbstractArray,
     t::AbstractVector;
-)
Function to compute the (discretized) integral defining the energy of a curve γ̲ on a Riemmanina manifold. The energy is defined as
    E(γ̲) = ∫ dt ⟨γ̲̇(t), G̲̲ γ̲̇(t)⟩,
where γ̲̇(t) defines the velocity of the parametric curve, and G̲̲ is the Riemmanian metric tensor. For this function, we approximate the integral as
    E(γ̲) ≈ ∑ᵢ Δt ⟨γ̲̇(tᵢ)ᵀ G̲̲ (γ̲(tᵢ+1) γ̲̇(tᵢ))⟩,
where Δt is the time step between points. Note that this Δt is assumed to be constant, thus, the time points t must be equally spaced.
Arguments
  • riemannian_metric::AbstractArray: d×d×N tensor where d is the dimension of the manifold on which the curve lies and N is the number of sampled time points along the curve. Each slice of the array represents the Riemmanian metric tensor for the curve at the corresponding time point.
  • curve_velocity::AbstractArray: d×N Matrix where d is the dimension of the manifold on which the curve lies and N is the number of sampled time points along the curve. Each column represents the velocity of the curve at the corresponding time point.
  • t::AbstractVector: Vector of time points at which the curve is sampled.
Returns
  • Energy::Number: Approximation of the Energy for the path on the manifold.
References
  1. Chen, N. et al. Metrics for Deep Generative Models. in Proceedings of the Twenty-First International Conference on Artificial Intelligence and Statistics 1540–1550 (PMLR, 2018).
  2. Chadebec, C. & Allassonnière, S. A Geometric Perspective on Variational Autoencoders. Preprint at http://arxiv.org/abs/2209.07370 (2022).
  3. Chadebec, C., Mantoux, C. & Allassonnière, S. Geometry-Aware Hamiltonian Variational Auto-Encoder. Preprint at http://arxiv.org/abs/2010.11518 (2020).
  4. Arvanitidis, G., Hauberg, S., Hennig, P. & Schober, M. Fast and Robust Shortest Paths on Manifolds Learned from Data. in Proceedings of the Twenty-Second International Conference on Artificial Intelligence and Statistics 1506–1515 (PMLR, 2019).
  5. Arvanitidis, G., Hauberg, S. & Schölkopf, B. Geometrically Enriched Latent Spaces. Preprint at https://doi.org/10.48550/arXiv.2008.00565 (2020).
  6. Arvanitidis, G., González-Duque, M., Pouplin, A., Kalatzis, D. & Hauberg, S. Pulling back information geometry. Preprint at http://arxiv.org/abs/2106.05367 (2022).
  7. Fröhlich, C., Gessner, A., Hennig, P., Schölkopf, B. & Arvanitidis, G. Bayesian Quadrature on Riemannian Data Manifolds.
  8. Kalatzis, D., Eklund, D., Arvanitidis, G. & Hauberg, S. Variational Autoencoders with Riemannian Brownian Motion Priors. Preprint at http://arxiv.org/abs/2002.05227 (2020).
  9. Arvanitidis, G., Hansen, L. K. & Hauberg, S. Latent Space Oddity: on the Curvature of Deep Generative Models. Preprint at http://arxiv.org/abs/1710.11379 (2021).
+)

Function to compute the (discretized) integral defining the energy of a curve γ̲ on a Riemmanina manifold. The energy is defined as

    E(γ̲) = ∫ dt ⟨γ̲̇(t), G̲̲ γ̲̇(t)⟩,

where γ̲̇(t) defines the velocity of the parametric curve, and G̲̲ is the Riemmanian metric tensor. For this function, we approximate the integral as

    E(γ̲) ≈ ∑ᵢ Δt ⟨γ̲̇(tᵢ)ᵀ G̲̲ (γ̲(tᵢ+1) γ̲̇(tᵢ))⟩,

where Δt is the time step between points. Note that this Δt is assumed to be constant, thus, the time points t must be equally spaced.

Arguments

Returns

References

  1. Chen, N. et al. Metrics for Deep Generative Models. in Proceedings of the Twenty-First International Conference on Artificial Intelligence and Statistics 1540–1550 (PMLR, 2018).
  2. Chadebec, C. & Allassonnière, S. A Geometric Perspective on Variational Autoencoders. Preprint at http://arxiv.org/abs/2209.07370 (2022).
  3. Chadebec, C., Mantoux, C. & Allassonnière, S. Geometry-Aware Hamiltonian Variational Auto-Encoder. Preprint at http://arxiv.org/abs/2010.11518 (2020).
  4. Arvanitidis, G., Hauberg, S., Hennig, P. & Schober, M. Fast and Robust Shortest Paths on Manifolds Learned from Data. in Proceedings of the Twenty-Second International Conference on Artificial Intelligence and Statistics 1506–1515 (PMLR, 2019).
  5. Arvanitidis, G., Hauberg, S. & Schölkopf, B. Geometrically Enriched Latent Spaces. Preprint at https://doi.org/10.48550/arXiv.2008.00565 (2020).
  6. Arvanitidis, G., González-Duque, M., Pouplin, A., Kalatzis, D. & Hauberg, S. Pulling back information geometry. Preprint at http://arxiv.org/abs/2106.05367 (2022).
  7. Fröhlich, C., Gessner, A., Hennig, P., Schölkopf, B. & Arvanitidis, G. Bayesian Quadrature on Riemannian Data Manifolds.
  8. Kalatzis, D., Eklund, D., Arvanitidis, G. & Hauberg, S. Variational Autoencoders with Riemannian Brownian Motion Priors. Preprint at http://arxiv.org/abs/2002.05227 (2020).
  9. Arvanitidis, G., Hansen, L. K. & Hauberg, S. Latent Space Oddity: on the Curvature of Deep Generative Models. Preprint at http://arxiv.org/abs/1710.11379 (2021).
diff --git a/dev/encoders/index.html b/dev/encoders/index.html index 1dc804c..dc68767 100644 --- a/dev/encoders/index.html +++ b/dev/encoders/index.html @@ -1,27 +1,27 @@ -Encoders & Decoders · AutoEncoderToolkit

Encoders & Decoders

AutoEncoderToolkit.jl provides a set of predefined encoders and decoders that can be used to define custom (variational) autoencoder architectures.

Encoders

The tree structure of the encoder types looks like this (🧱 represents concrete types):

Encoder

AutoEncoderToolkit.Encoder — Type

struct Encoder{E<:Union{Flux.Chain,Flux.Dense}} <: AbstractDeterministicEncoder

Default encoder function for deterministic autoencoders. The encoder network is used to map the input data directly into the latent space representation.

Fields

  • encoder::Union{Flux.Chain,Flux.Dense}: The primary neural network used to process input data and map it into a latent space representation.

Example

enc = Encoder(Flux.Chain(Dense(784, 400, relu), Dense(400, 20)))
AutoEncoderToolkit.Encoder — Method
(encoder::Encoder)(x)

Forward propagate the input x through the Encoder to obtain the encoded representation in the latent space.

Arguments

  • x::Array: Input data to be encoded.

Returns

  • z: Encoded representation of the input data in the latent space.

Description

This method allows for a direct call on an instance of Encoder with the input data x. It runs the input through the encoder network and outputs the encoded representation in the latent space.

Example

enc = Encoder(...)
-z = enc(some_input)

Note

Ensure that the input x matches the expected dimensionality of the encoder's input layer.

JointGaussianEncoder

AutoEncoderToolkit.JointGaussianEncoder — Type

struct JointGaussianEncoder <: AbstractGaussianLinearEncoder

Encoder function for variational autoencoders where the same encoder network is used to map to the latent space mean µ and standard deviation σ.

Fields

  • encoder::Flux.Chain: The primary neural network used to process input data and map it into a latent space representation.
  • µ::Flux.Dense: A dense layer mapping from the output of the encoder to the mean of the latent space.
  • σ::Flux.Dense: A dense layer mapping from the output of the encoder to the standard deviation of the latent space.

Example

enc = JointGaussianEncoder(
+Encoders & Decoders · AutoEncoderToolkit
Encoders & Decoders
AutoEncoderToolkit.jl provides a set of predefined encoders and decoders that can be used to define custom (variational) autoencoder architectures.
Encoders
The tree structure of the encoder types looks like this (🧱 represents concrete types):
Encoder
AutoEncoderToolkit.Encoder — Type
struct Encoder{E<:Union{Flux.Chain,Flux.Dense}} <: AbstractDeterministicEncoder
Default encoder function for deterministic autoencoders. The encoder network is used to map the input data directly into the latent space representation.
Fields
  • encoder::Union{Flux.Chain,Flux.Dense}: The primary neural network used to process input data and map it into a latent space representation.
Example
enc = Encoder(Flux.Chain(Dense(784, 400, relu), Dense(400, 20)))
AutoEncoderToolkit.Encoder — Method
(encoder::Encoder)(x)
Forward propagate the input x through the Encoder to obtain the encoded representation in the latent space.
Arguments
  • x::Array: Input data to be encoded.
Returns
  • z: Encoded representation of the input data in the latent space.
Description
This method allows for a direct call on an instance of Encoder with the input data x. It runs the input through the encoder network and outputs the encoded representation in the latent space.
Example
enc = Encoder(...)
+z = enc(some_input)
Note
Ensure that the input x matches the expected dimensionality of the encoder's input layer.
JointGaussianEncoder
AutoEncoderToolkit.JointGaussianEncoder — Type
struct JointGaussianEncoder <: AbstractGaussianLinearEncoder
Encoder function for variational autoencoders where the same encoder network is used to map to the latent space mean µ and standard deviation σ.
Fields
  • encoder::Flux.Chain: The primary neural network used to process input data and map it into a latent space representation.
  • µ::Flux.Dense: A dense layer mapping from the output of the encoder to the mean of the latent space.
  • σ::Flux.Dense: A dense layer mapping from the output of the encoder to the standard deviation of the latent space.
Example
enc = JointGaussianEncoder(
     Flux.Chain(Dense(784, 400, relu)), Flux.Dense(400, 20), Flux.Dense(400, 20)
-)
AutoEncoderToolkit.JointGaussianEncoder — Method
    (encoder::JointGaussianEncoder)(x::AbstractArray)
Forward propagate the input x through the JointGaussianEncoder to obtain the mean (µ) and standard deviation (σ) of the latent space.
Arguments
  • x::AbstractArray: Input data to be encoded.
Returns
  • A NamedTuple (µ=µ, σ=σ,) where:
    • µ: Mean of the latent space after passing the input through the encoder and subsequently through the µ layer.
    • σ: Standard deviation of the latent space after passing the input through the encoder and subsequently through the σ layer.
Description
This method allows for a direct call on an instance of JointGaussianEncoder with the input data x. It first runs the input through the encoder network, then maps the output of the last encoder layer to both the mean and standard deviation of the latent space.
Example
je = JointGaussianEncoder(...)
-µ, σ = je(some_input)
Note
Ensure that the input x matches the expected dimensionality of the encoder's input layer.
JointGaussianLogEncoder
AutoEncoderToolkit.JointGaussianLogEncoder — Type
struct JointGaussianLogEncoder <: AbstractGaussianLogEncoder
Default encoder function for variational autoencoders where the same encoder network is used to map to the latent space mean µ and log standard deviation logσ.
Fields
  • encoder::Flux.Chain: The primary neural network used to process input data and map it into a latent space representation.
  • µ::Union{Flux.Dense,Flux.Chain}: A dense layer or a chain of layers mapping from the output of the encoder to the mean of the latent space.
  • logσ::Union{Flux.Dense,Flux.Chain}: A dense layer or a chain of layers mapping from the output of the encoder to the log standard deviation of the latent space.
Example
enc = JointGaussianLogEncoder(
+)
AutoEncoderToolkit.JointGaussianEncoder — Method
    (encoder::JointGaussianEncoder)(x::AbstractArray)
Forward propagate the input x through the JointGaussianEncoder to obtain the mean (µ) and standard deviation (σ) of the latent space.
Arguments
  • x::AbstractArray: Input data to be encoded.
Returns
  • A NamedTuple (µ=µ, σ=σ,) where:
    • µ: Mean of the latent space after passing the input through the encoder and subsequently through the µ layer.
    • σ: Standard deviation of the latent space after passing the input through the encoder and subsequently through the σ layer.
Description
This method allows for a direct call on an instance of JointGaussianEncoder with the input data x. It first runs the input through the encoder network, then maps the output of the last encoder layer to both the mean and standard deviation of the latent space.
Example
je = JointGaussianEncoder(...)
+µ, σ = je(some_input)
Note
Ensure that the input x matches the expected dimensionality of the encoder's input layer.
JointGaussianLogEncoder
AutoEncoderToolkit.JointGaussianLogEncoder — Type
struct JointGaussianLogEncoder <: AbstractGaussianLogEncoder
Default encoder function for variational autoencoders where the same encoder network is used to map to the latent space mean µ and log standard deviation logσ.
Fields
  • encoder::Flux.Chain: The primary neural network used to process input data and map it into a latent space representation.
  • µ::Union{Flux.Dense,Flux.Chain}: A dense layer or a chain of layers mapping from the output of the encoder to the mean of the latent space.
  • logσ::Union{Flux.Dense,Flux.Chain}: A dense layer or a chain of layers mapping from the output of the encoder to the log standard deviation of the latent space.
Example
enc = JointGaussianLogEncoder(
         Flux.Chain(Dense(784, 400, relu)), Flux.Dense(400, 20), Flux.Dense(400, 20)
-)
AutoEncoderToolkit.JointGaussianLogEncoder — Method
    (encoder::JointGaussianLogEncoder)(x)
This method forward propagates the input x through the JointGaussianLogEncoder to compute the mean (mu) and log standard deviation (logσ) of the latent space.
Arguments
  • x::Array{Float32}: The input data to be encoded.
Returns
  • A NamedTuple (µ=µ, logσ=logσ,) where:
    • µ: The mean of the latent space. This is computed by passing the input through the encoder and subsequently through the µ layer.  
    • logσ: The log standard deviation of the latent space. This is computed by passing the input through the encoder and subsequently through the logσ layer.
Description
This method allows for a direct call on an instance of JointGaussianLogEncoder with the input data x. It first processes the input through the encoder network, then maps the output of the last encoder layer to both the mean and log standard deviation of the latent space.
Example
je = JointGaussianLogEncoder(...)
-mu, logσ = je(some_input)
Note
Ensure that the input x matches the expected dimensionality of the encoder's input layer.
Decoders
The tree structure of the decoder types looks like this (🧱 represents concrete types):
Decoder
AutoEncoderToolkit.Decoder — Type
struct Decoder{D<:Flux.Chain} <: AbstractDeterministicDecoder
Default decoder function for deterministic autoencoders. The decoder network is used to map the latent space representation directly back to the original data space.
Fields
  • decoder::Flux.Chain: The primary neural network used to process the latent space representation and map it back to the data space.
Example
dec = Decoder(Flux.Chain(Dense(20, 400, relu), Dense(400, 784)))
AutoEncoderToolkit.Decoder — Method
(decoder::Decoder)(z::AbstractArray)
Forward propagate the encoded representation z through the Decoder to obtain the reconstructed input data.
Arguments
  • z::AbstractArray: Encoded representation in the latent space.
Returns
  • x_reconstructed: Reconstructed version of the original input data after decoding from the latent space.
Description
This method allows for a direct call on an instance of Decoder with the encoded data z. It runs the encoded representation through the decoder network and outputs the reconstructed version of the original input data.
Example
julia dec = Decoder(...) x_reconstructed = dec(encoded_representation)`
Note
Ensure that the input z matches the expected dimensionality of the decoder's input layer.
BernoulliDecoder
AutoEncoderToolkit.BernoulliDecoder — Type
    BernoulliDecoder{D<:Flux.Chain} <: AbstractVariationalDecoder
A decoder structure for variational autoencoders (VAEs) that models the output data as a Bernoulli distribution. This is typically used when the outputs of the decoder are probabilities.
Fields
  • decoder::Flux.Chain: The primary neural network used to process the latent   space and map it to the output (or reconstructed) space.
Description
BernoulliDecoder represents a VAE decoder that models the output data as a Bernoulli distribution. It's commonly used when the outputs of the decoder are probabilities, such as in a binary classification task or when modeling binary data. Unlike a Gaussian decoder, there's no need for separate paths or operations on the mean or log standard deviation.
Note
Ensure the last layer of the decoder outputs a value between 0 and 1, as this is required for a Bernoulli distribution.
AutoEncoderToolkit.BernoulliDecoder — Method
    (decoder::BernoulliDecoder)(z::AbstractArray)
Maps the given latent representation z through the BernoulliDecoder network to reconstruct the original input.
Arguments
  • z::AbstractArray: The latent space representation to be decoded.   This can be a vector or a matrix, where each column represents a separate   sample from the latent space of a VAE.
Returns
  • A NamedTuple (p=p,) where p is an array representing the output of the   decoder, which should resemble the original input to the VAE (post encoding   and sampling from the latent space).
Description
This function processes the latent space representation z using the neural network defined in the BernoulliDecoder struct. The aim is to decode or reconstruct the original input from this representation.
Note
Ensure that the latent space representation z matches the expected input dimensionality for the BernoulliDecoder.
CategoricalDecoder
AutoEncoderToolkit.CategoricalDecoder — Type
CategoricalDecoder{D<:Flux.Chain} <: AbstractVariationalDecoder
A decoder structure for variational autoencoders (VAEs) that models the output data as a categorical distribution. This is typically used when the outputs of the decoder are categorical variables encoded as one-hot vectors.
Fields
  • decoder::Flux.Chain: The primary neural network used to process the latent space and map it to the output (or reconstructed) space.
Description
CategoricalDecoder represents a VAE decoder that models the output data as a categorical distribution. It's commonly used when the outputs of the decoder are categorical variables, such as in a multi-class one-hot encoded vectors. Unlike a Gaussian decoder, there's no need for separate paths or operations on the mean or log standard deviation.
Note
Ensure the last layer of the decoder outputs a probability distribution over the categories, as this is required for a categorical distribution. This can be done using a softmax activation function, for example.
AutoEncoderToolkit.CategoricalDecoder — Method
(decoder::CategoricalDecoder)(z::AbstractArray)
Maps the given latent representation z through the CategoricalDecoder network to reconstruct the original input.
Arguments
  • z::AbstractArray: The latent space representation to be decoded.  This can be a vector or a matrix, where each column represents a separate sample from the latent space of a VAE.
Returns
  • A NamedTuple (p=p,) where p is an array representing the output of the decoder, which should resemble the original input to the VAE (post encoding and sampling from the latent space).
Description
This function processes the latent space representation z using the neural network defined in the CategoricalDecoder struct. The aim is to decode or reconstruct the original input from this representation.
Note
Ensure that the latent space representation z matches the expected input dimensionality for the CategoricalDecoder.
SimpleGaussianDecoder
AutoEncoderToolkit.SimpleGaussianDecoder — Type
SimpleGaussianDecoder{D} <: AbstractGaussianDecoder
A straightforward decoder structure for variational autoencoders (VAEs) that contains only a single decoder network.
Fields
  • decoder::Flux.Chain: The primary neural network used to process the latent space and map it to the output (or reconstructed) space.
Description
SimpleGaussianDecoder represents a basic VAE decoder without explicit components for the latent space's mean (µ) or log standard deviation (logσ). It's commonly used when the VAE's latent space distribution is implicitly defined, and there's no need for separate paths or operations on the mean or log standard deviation.
AutoEncoderToolkit.SimpleGaussianDecoder — Method
(decoder::SimpleGaussianDecoder)(z::AbstractVecOrMat)
Maps the given latent representation z through the SimpleGaussianDecoder network to reconstruct the original input.
Arguments
  • z::AbstractArray: The latent space representation to be decoded. This can be a vector or a matrix, where each column represents a separate sample from the latent space of a VAE.
Returns
  • A NamedTuple (µ=µ,) where µ is an array representing the output of the decoder, which should resemble the original input to the VAE (post encoding and sampling from the latent space).
Description
This function processes the latent space representation z using the neural network defined in the SimpleGaussianDecoder struct. The aim is to decode or reconstruct the original input from this representation.
Example
decoder = SimpleGaussianDecoder(...)
+)
AutoEncoderToolkit.JointGaussianLogEncoder — Method
    (encoder::JointGaussianLogEncoder)(x)
This method forward propagates the input x through the JointGaussianLogEncoder to compute the mean (mu) and log standard deviation (logσ) of the latent space.
Arguments
  • x::Array{Float32}: The input data to be encoded.
Returns
  • A NamedTuple (µ=µ, logσ=logσ,) where:
    • µ: The mean of the latent space. This is computed by passing the input through the encoder and subsequently through the µ layer.  
    • logσ: The log standard deviation of the latent space. This is computed by passing the input through the encoder and subsequently through the logσ layer.
Description
This method allows for a direct call on an instance of JointGaussianLogEncoder with the input data x. It first processes the input through the encoder network, then maps the output of the last encoder layer to both the mean and log standard deviation of the latent space.
Example
je = JointGaussianLogEncoder(...)
+mu, logσ = je(some_input)
Note
Ensure that the input x matches the expected dimensionality of the encoder's input layer.
Decoders
The tree structure of the decoder types looks like this (🧱 represents concrete types):
Decoder
AutoEncoderToolkit.Decoder — Type
struct Decoder{D<:Flux.Chain} <: AbstractDeterministicDecoder
Default decoder function for deterministic autoencoders. The decoder network is used to map the latent space representation directly back to the original data space.
Fields
  • decoder::Flux.Chain: The primary neural network used to process the latent space representation and map it back to the data space.
Example
dec = Decoder(Flux.Chain(Dense(20, 400, relu), Dense(400, 784)))
AutoEncoderToolkit.Decoder — Method
(decoder::Decoder)(z::AbstractArray)
Forward propagate the encoded representation z through the Decoder to obtain the reconstructed input data.
Arguments
  • z::AbstractArray: Encoded representation in the latent space.
Returns
  • x_reconstructed: Reconstructed version of the original input data after decoding from the latent space.
Description
This method allows for a direct call on an instance of Decoder with the encoded data z. It runs the encoded representation through the decoder network and outputs the reconstructed version of the original input data.
Example
julia dec = Decoder(...) x_reconstructed = dec(encoded_representation)`
Note
Ensure that the input z matches the expected dimensionality of the decoder's input layer.
BernoulliDecoder
AutoEncoderToolkit.BernoulliDecoder — Type
    BernoulliDecoder{D<:Flux.Chain} <: AbstractVariationalDecoder
A decoder structure for variational autoencoders (VAEs) that models the output data as a Bernoulli distribution. This is typically used when the outputs of the decoder are probabilities.
Fields
  • decoder::Flux.Chain: The primary neural network used to process the latent   space and map it to the output (or reconstructed) space.
Description
BernoulliDecoder represents a VAE decoder that models the output data as a Bernoulli distribution. It's commonly used when the outputs of the decoder are probabilities, such as in a binary classification task or when modeling binary data. Unlike a Gaussian decoder, there's no need for separate paths or operations on the mean or log standard deviation.
Note
Ensure the last layer of the decoder outputs a value between 0 and 1, as this is required for a Bernoulli distribution.
AutoEncoderToolkit.BernoulliDecoder — Method
    (decoder::BernoulliDecoder)(z::AbstractArray)
Maps the given latent representation z through the BernoulliDecoder network to reconstruct the original input.
Arguments
  • z::AbstractArray: The latent space representation to be decoded.   This can be a vector or a matrix, where each column represents a separate   sample from the latent space of a VAE.
Returns
  • A NamedTuple (p=p,) where p is an array representing the output of the   decoder, which should resemble the original input to the VAE (post encoding   and sampling from the latent space).
Description
This function processes the latent space representation z using the neural network defined in the BernoulliDecoder struct. The aim is to decode or reconstruct the original input from this representation.
Note
Ensure that the latent space representation z matches the expected input dimensionality for the BernoulliDecoder.
CategoricalDecoder
AutoEncoderToolkit.CategoricalDecoder — Type
CategoricalDecoder{D<:Flux.Chain} <: AbstractVariationalDecoder
A decoder structure for variational autoencoders (VAEs) that models the output data as a categorical distribution. This is typically used when the outputs of the decoder are categorical variables encoded as one-hot vectors.
Fields
  • decoder::Flux.Chain: The primary neural network used to process the latent space and map it to the output (or reconstructed) space.
Description
CategoricalDecoder represents a VAE decoder that models the output data as a categorical distribution. It's commonly used when the outputs of the decoder are categorical variables, such as in a multi-class one-hot encoded vectors. Unlike a Gaussian decoder, there's no need for separate paths or operations on the mean or log standard deviation.
Note
Ensure the last layer of the decoder outputs a probability distribution over the categories, as this is required for a categorical distribution. This can be done using a softmax activation function, for example.
AutoEncoderToolkit.CategoricalDecoder — Method
(decoder::CategoricalDecoder)(z::AbstractArray)
Maps the given latent representation z through the CategoricalDecoder network to reconstruct the original input.
Arguments
  • z::AbstractArray: The latent space representation to be decoded.  This can be a vector or a matrix, where each column represents a separate sample from the latent space of a VAE.
Returns
  • A NamedTuple (p=p,) where p is an array representing the output of the decoder, which should resemble the original input to the VAE (post encoding and sampling from the latent space).
Description
This function processes the latent space representation z using the neural network defined in the CategoricalDecoder struct. The aim is to decode or reconstruct the original input from this representation.
Note
Ensure that the latent space representation z matches the expected input dimensionality for the CategoricalDecoder.
SimpleGaussianDecoder
AutoEncoderToolkit.SimpleGaussianDecoder — Type
SimpleGaussianDecoder{D} <: AbstractGaussianDecoder
A straightforward decoder structure for variational autoencoders (VAEs) that contains only a single decoder network.
Fields
  • decoder::Flux.Chain: The primary neural network used to process the latent space and map it to the output (or reconstructed) space.
Description
SimpleGaussianDecoder represents a basic VAE decoder without explicit components for the latent space's mean (µ) or log standard deviation (logσ). It's commonly used when the VAE's latent space distribution is implicitly defined, and there's no need for separate paths or operations on the mean or log standard deviation.
AutoEncoderToolkit.SimpleGaussianDecoder — Method
(decoder::SimpleGaussianDecoder)(z::AbstractVecOrMat)
Maps the given latent representation z through the SimpleGaussianDecoder network to reconstruct the original input.
Arguments
  • z::AbstractArray: The latent space representation to be decoded. This can be a vector or a matrix, where each column represents a separate sample from the latent space of a VAE.
Returns
  • A NamedTuple (µ=µ,) where µ is an array representing the output of the decoder, which should resemble the original input to the VAE (post encoding and sampling from the latent space).
Description
This function processes the latent space representation z using the neural network defined in the SimpleGaussianDecoder struct. The aim is to decode or reconstruct the original input from this representation.
Example
decoder = SimpleGaussianDecoder(...)
 z = ... # some latent space representation
-output = decoder(z)
Note
Ensure that the latent space representation z matches the expected input dimensionality for the SimpleGaussianDecoder.
JointGaussianDecoder
AutoEncoderToolkit.JointGaussianDecoder — Type
JointGaussianDecoder{D<:Flux.Chain,L<:Flux.Dense} <: AbstractGaussianLinearDecoder
An extended decoder structure for VAEs that incorporates separate layers for mapping from the latent space to both its mean (µ) and standard deviation (σ).
Fields
  • decoder::Flux.Chain: The primary neural network used to process the latent space before determining its mean and log standard deviation.
  • µ::Flux.Dense: A dense layer that maps from the output of the decoder to the mean of the latent space.
  • σ::Flux.Dense: A dense layer that maps from the output of the decoder to the standard deviation of the latent space.
Description
JointGaussianDecoder is tailored for VAE architectures where the same decoder network is used initially, and then splits into two separate paths for determining both the mean and standard deviation of the latent space.
AutoEncoderToolkit.JointGaussianDecoder — Method
    (decoder::JointGaussianDecoder)(z::AbstractArray)
Maps the given latent representation z through the JointGaussianDecoder network to produce both the mean (µ) and standard deviation (σ).
Arguments
  • z::AbstractArray: The latent space representation to be decoded. If array, the last dimension contains each of the latent space representations to be decoded.
Returns
  • A NamedTuple (µ=µ, σ=σ,) where:
    • µ::AbstractArray: The mean representation obtained from the decoder.
    • σ::AbstractArray: The standard deviation representation obtained from the decoder.
Description
This function processes the latent space representation z using the primary neural network of the JointGaussianDecoder struct. It then separately maps the output of this network to the mean and standard deviation using the µ and σ dense layers, respectively.
Example
decoder = JointGaussianDecoder(...)
+output = decoder(z)
Note
Ensure that the latent space representation z matches the expected input dimensionality for the SimpleGaussianDecoder.
JointGaussianDecoder
AutoEncoderToolkit.JointGaussianDecoder — Type
JointGaussianDecoder{D<:Flux.Chain,L<:Flux.Dense} <: AbstractGaussianLinearDecoder
An extended decoder structure for VAEs that incorporates separate layers for mapping from the latent space to both its mean (µ) and standard deviation (σ).
Fields
  • decoder::Flux.Chain: The primary neural network used to process the latent space before determining its mean and log standard deviation.
  • µ::Flux.Dense: A dense layer that maps from the output of the decoder to the mean of the latent space.
  • σ::Flux.Dense: A dense layer that maps from the output of the decoder to the standard deviation of the latent space.
Description
JointGaussianDecoder is tailored for VAE architectures where the same decoder network is used initially, and then splits into two separate paths for determining both the mean and standard deviation of the latent space.
AutoEncoderToolkit.JointGaussianDecoder — Method
    (decoder::JointGaussianDecoder)(z::AbstractArray)
Maps the given latent representation z through the JointGaussianDecoder network to produce both the mean (µ) and standard deviation (σ).
Arguments
  • z::AbstractArray: The latent space representation to be decoded. If array, the last dimension contains each of the latent space representations to be decoded.
Returns
  • A NamedTuple (µ=µ, σ=σ,) where:
    • µ::AbstractArray: The mean representation obtained from the decoder.
    • σ::AbstractArray: The standard deviation representation obtained from the decoder.
Description
This function processes the latent space representation z using the primary neural network of the JointGaussianDecoder struct. It then separately maps the output of this network to the mean and standard deviation using the µ and σ dense layers, respectively.
Example
decoder = JointGaussianDecoder(...)
 z = ... # some latent space representation
-output = decoder(z)
Note
Ensure that the latent space representation z matches the expected input dimensionality for the JointGaussianDecoder.
JointGaussianLogDecoder
AutoEncoderToolkit.JointGaussianLogDecoder — Type
JointGaussianLogDecoder{D<:Flux.Chain,L<:Flux.Dense} <: AbstractGaussianLogDecoder
An extended decoder structure for VAEs that incorporates separate layers for mapping from the latent space to both its mean (µ) and log standard deviation (logσ).
Fields
  • decoder::Flux.Chain: The primary neural network used to process the latent space before determining its mean and log standard deviation.
  • µ::Flux.Dense: A dense layer that maps from the output of the decoder to the mean of the latent space.
  • logσ::Flux.Dense: A dense layer that maps from the output of the decoder to the log standard deviation of the latent space.
Description
JointGaussianLogDecoder is tailored for VAE architectures where the same decoder network is used initially, and then splits into two separate paths for determining both the mean and log standard deviation of the latent space.
AutoEncoderToolkit.JointGaussianLogDecoder — Method
    (decoder::JointGaussianLogDecoder)(z::AbstractArray)
Maps the given latent representation z through the JointGaussianLogDecoder network to produce both the mean (µ) and log standard deviation (logσ).
Arguments
  • z::AbstractArray: The latent space representation to be decoded. If array, the last dimension contains each of the latent space representations.
Returns
  • A NamedTuple (µ=µ, logσ=logσ,) where:
    • µ::Array: The mean representation obtained from the decoder.
    • logσ::Array: The log standard deviation representation obtained from the decoder.
Description
This function processes the latent space representation z using the primary neural network of the JointGaussianLogDecoder struct. It then separately maps the output of this network to the mean and log standard deviation using the µ and logσ dense layers, respectively.
Example
decoder = JointGaussianLogDecoder(...)
+output = decoder(z)
Note
Ensure that the latent space representation z matches the expected input dimensionality for the JointGaussianDecoder.
JointGaussianLogDecoder
AutoEncoderToolkit.JointGaussianLogDecoder — Type
JointGaussianLogDecoder{D<:Flux.Chain,L<:Flux.Dense} <: AbstractGaussianLogDecoder
An extended decoder structure for VAEs that incorporates separate layers for mapping from the latent space to both its mean (µ) and log standard deviation (logσ).
Fields
  • decoder::Flux.Chain: The primary neural network used to process the latent space before determining its mean and log standard deviation.
  • µ::Flux.Dense: A dense layer that maps from the output of the decoder to the mean of the latent space.
  • logσ::Flux.Dense: A dense layer that maps from the output of the decoder to the log standard deviation of the latent space.
Description
JointGaussianLogDecoder is tailored for VAE architectures where the same decoder network is used initially, and then splits into two separate paths for determining both the mean and log standard deviation of the latent space.
AutoEncoderToolkit.JointGaussianLogDecoder — Method
    (decoder::JointGaussianLogDecoder)(z::AbstractArray)
Maps the given latent representation z through the JointGaussianLogDecoder network to produce both the mean (µ) and log standard deviation (logσ).
Arguments
  • z::AbstractArray: The latent space representation to be decoded. If array, the last dimension contains each of the latent space representations.
Returns
  • A NamedTuple (µ=µ, logσ=logσ,) where:
    • µ::Array: The mean representation obtained from the decoder.
    • logσ::Array: The log standard deviation representation obtained from the decoder.
Description
This function processes the latent space representation z using the primary neural network of the JointGaussianLogDecoder struct. It then separately maps the output of this network to the mean and log standard deviation using the µ and logσ dense layers, respectively.
Example
decoder = JointGaussianLogDecoder(...)
 z = ... # some latent space representation
-output = decoder(z)
Note
Ensure that the latent space representation z matches the expected input dimensionality for the JointGaussianLogDecoder.
SplitGaussianDecoder
AutoEncoderToolkit.SplitGaussianDecoder — Type
SplitGaussianDecoder{D<:Flux.Chain} <: AbstractGaussianLinearDecoder
A specialized decoder structure for VAEs that uses distinct neural networks for determining the mean (µ) and standard deviation (logσ) of the latent space.
Fields
  • decoder_µ::Flux.Chain: A neural network dedicated to processing the latent space and mapping it to its mean.
  • decoder_σ::Flux.Chain: A neural network dedicated to processing the latent space and mapping it to its standard deviation.
Description
SplitGaussianDecoder is designed for VAE architectures where separate decoder networks are preferred for computing the mean and log standard deviation, ensuring that each has its own distinct set of parameters and transformation logic.
AutoEncoderToolkit.SplitGaussianDecoder — Method
    (decoder::SplitGaussianDecoder)(z::AbstractArray)
Maps the given latent representation z through the separate networks of the SplitGaussianDecoder to produce both the mean (µ) and standard deviation (σ).
Arguments
  • z::AbstractArray: The latent space representation to be decoded. If array, the last dimension contains each of the latent space representations to be decoded.
Returns
  • A NamedTuple (µ=µ, σ=σ,) where:
    • µ::AbstractArray: The mean representation obtained using the dedicated decoder_µ network.
    • σ::AbstractArray: The standard deviation representation obtained using the dedicated decoder_σ network.
Description
This function processes the latent space representation z through two distinct neural networks within the SplitGaussianDecoder struct. The decoder_µ network is used to produce the mean representation, while the decoder_σ network is utilized for the standard deviation.
Example
decoder = SplitGaussianDecoder(...)
+output = decoder(z)
Note
Ensure that the latent space representation z matches the expected input dimensionality for the JointGaussianLogDecoder.
SplitGaussianDecoder
AutoEncoderToolkit.SplitGaussianDecoder — Type
SplitGaussianDecoder{D<:Flux.Chain} <: AbstractGaussianLinearDecoder
A specialized decoder structure for VAEs that uses distinct neural networks for determining the mean (µ) and standard deviation (logσ) of the latent space.
Fields
  • decoder_µ::Flux.Chain: A neural network dedicated to processing the latent space and mapping it to its mean.
  • decoder_σ::Flux.Chain: A neural network dedicated to processing the latent space and mapping it to its standard deviation.
Description
SplitGaussianDecoder is designed for VAE architectures where separate decoder networks are preferred for computing the mean and log standard deviation, ensuring that each has its own distinct set of parameters and transformation logic.
AutoEncoderToolkit.SplitGaussianDecoder — Method
    (decoder::SplitGaussianDecoder)(z::AbstractArray)
Maps the given latent representation z through the separate networks of the SplitGaussianDecoder to produce both the mean (µ) and standard deviation (σ).
Arguments
  • z::AbstractArray: The latent space representation to be decoded. If array, the last dimension contains each of the latent space representations to be decoded.
Returns
  • A NamedTuple (µ=µ, σ=σ,) where:
    • µ::AbstractArray: The mean representation obtained using the dedicated decoder_µ network.
    • σ::AbstractArray: The standard deviation representation obtained using the dedicated decoder_σ network.
Description
This function processes the latent space representation z through two distinct neural networks within the SplitGaussianDecoder struct. The decoder_µ network is used to produce the mean representation, while the decoder_σ network is utilized for the standard deviation.
Example
decoder = SplitGaussianDecoder(...)
 z = ... # some latent space representation
-output = decoder(z)
Note
Ensure that the latent space representation z matches the expected input dimensionality for both networks in the SplitGaussianDecoder.
SplitGaussianLogDecoder
AutoEncoderToolkit.SplitGaussianLogDecoder — Type
SplitGaussianLogDecoder{D<:Flux.Chain} <: AbstractGaussianLogDecoder
A specialized decoder structure for VAEs that uses distinct neural networks for determining the mean (µ) and log standard deviation (logσ) of the latent space.
Fields
  • decoder_µ::Flux.Chain: A neural network dedicated to processing the latent space and mapping it to its mean.
  • decoder_logσ::Flux.Chain: A neural network dedicated to processing the latent space and mapping it to its log standard deviation.
Description
SplitGaussianLogDecoder is designed for VAE architectures where separate decoder networks are preferred for computing the mean and log standard deviation, ensuring that each has its own distinct set of parameters and transformation logic.
AutoEncoderToolkit.SplitGaussianLogDecoder — Method
    (decoder::SplitGaussianLogDecoder)(z::AbstractArray)
Maps the given latent representation z through the separate networks of the SplitGaussianLogDecoder to produce both the mean (µ) and log standard deviation (logσ).
Arguments
  • z::AbstractArray: The latent space representation to be decoded. If array, the last dimension contains each of the latent space representations to be decoded.
Returns
  • A NamedTuple (µ=µ, logσ=logσ,) where:
    • µ::AbstractArray: The mean representation obtained using the dedicated decoder_µ network.
    • logσ::AbstractArray: The log standard deviation representation obtained using the dedicated decoder_logσ network.
Description
This function processes the latent space representation z through two distinct neural networks within the SplitGaussianLogDecoder struct. The decoder_µ network is used to produce the mean representation, while the decoder_logσ network is utilized for the log standard deviation.
Example
decoder = SplitGaussianLogDecoder(...)
+output = decoder(z)
Note
Ensure that the latent space representation z matches the expected input dimensionality for both networks in the SplitGaussianDecoder.
SplitGaussianLogDecoder
AutoEncoderToolkit.SplitGaussianLogDecoder — Type
SplitGaussianLogDecoder{D<:Flux.Chain} <: AbstractGaussianLogDecoder
A specialized decoder structure for VAEs that uses distinct neural networks for determining the mean (µ) and log standard deviation (logσ) of the latent space.
Fields
  • decoder_µ::Flux.Chain: A neural network dedicated to processing the latent space and mapping it to its mean.
  • decoder_logσ::Flux.Chain: A neural network dedicated to processing the latent space and mapping it to its log standard deviation.
Description
SplitGaussianLogDecoder is designed for VAE architectures where separate decoder networks are preferred for computing the mean and log standard deviation, ensuring that each has its own distinct set of parameters and transformation logic.
AutoEncoderToolkit.SplitGaussianLogDecoder — Method
    (decoder::SplitGaussianLogDecoder)(z::AbstractArray)
Maps the given latent representation z through the separate networks of the SplitGaussianLogDecoder to produce both the mean (µ) and log standard deviation (logσ).
Arguments
  • z::AbstractArray: The latent space representation to be decoded. If array, the last dimension contains each of the latent space representations to be decoded.
Returns
  • A NamedTuple (µ=µ, logσ=logσ,) where:
    • µ::AbstractArray: The mean representation obtained using the dedicated decoder_µ network.
    • logσ::AbstractArray: The log standard deviation representation obtained using the dedicated decoder_logσ network.
Description
This function processes the latent space representation z through two distinct neural networks within the SplitGaussianLogDecoder struct. The decoder_µ network is used to produce the mean representation, while the decoder_logσ network is utilized for the log standard deviation.
Example
decoder = SplitGaussianLogDecoder(...)
 z = ... # some latent space representation
-output = decoder(z))
Note
Ensure that the latent space representation z matches the expected input dimensionality for both networks in the SplitGaussianLogDecoder.
Default initializations
The package provides a set of functions to initialize encoder and decoder architectures. Although it gives the user less flexibility, it can be useful for quick prototyping.
Encoder initializations
AutoEncoderToolkit.Encoder — Method
Encoder(n_input, n_latent, latent_activation, encoder_neurons, 
-        encoder_activation; init=Flux.glorot_uniform)
Construct and initialize an Encoder struct that defines an encoder network for a deterministic autoencoder.
Arguments
  • n_input::Int: The dimensionality of the input data.
  • n_latent::Int: The dimensionality of the latent space.
  • encoder_neurons::Vector{<:Int}: A vector specifying the number of neurons in each layer of the encoder network.
  • encoder_activation::Vector{<:Function}: Activation functions corresponding to each layer in the encoder_neurons.
  • latent_activation::Function: Activation function for the latent space layer.
Optional Keyword Arguments
  • init::Function=Flux.glorot_uniform: The initialization function used for the neural network weights.
Returns
  • An Encoder struct initialized based on the provided arguments.
Examples
julia encoder = Encoder(784, 20, tanh, [400], [relu])`
Notes
The length of encoderneurons should match the length of encoderactivation, ensuring that each layer in the encoder has a corresponding activation function.
AutoEncoderToolkit.JointGaussianLogEncoder — Method
JointGaussianLogEncoder(n_input, n_latent, encoder_neurons, encoder_activation, 
-             latent_activation; init=Flux.glorot_uniform)
Construct and initialize a JointGaussianLogEncoder struct that defines an encoder network for a variational autoencoder.
Arguments
  • n_input::Int: The dimensionality of the input data.
  • n_latent::Int: The dimensionality of the latent space.
  • encoder_neurons::Vector{<:Int}: A vector specifying the number of neurons in each layer of the encoder network.
  • encoder_activation::Vector{<:Function}: Activation functions corresponding to each layer in the encoder_neurons.
  • latent_activation::Function: Activation function for the latent space layers (both µ and logσ).
Optional Keyword Arguments
  • init::Function=Flux.glorot_uniform: The initialization function used for the neural network weights.
Returns
  • A JointGaussianLogEncoder struct initialized based on the provided arguments.
Examples
encoder = JointGaussianLogEncoder(784, 20, [400], [relu], tanh)
Notes
The length of encoderneurons should match the length of encoderactivation, ensuring that each layer in the encoder has a corresponding activation function.
AutoEncoderToolkit.JointGaussianLogEncoder — Method
JointGaussianLogEncoder(n_input, n_latent, encoder_neurons, encoder_activation, 
-             latent_activation; init=Flux.glorot_uniform)
Construct and initialize a JointGaussianLogEncoder struct that defines an encoder network for a variational autoencoder.
Arguments
  • n_input::Int: The dimensionality of the input data.
  • n_latent::Int: The dimensionality of the latent space.
  • encoder_neurons::Vector{<:Int}: A vector specifying the number of neurons in each layer of the encoder network.
  • encoder_activation::Vector{<:Function}: Activation functions corresponding to each layer in the encoder_neurons.
  • latent_activation::Vector{<:Function}: Activation functions for the latent space layers (both µ and logσ).
Optional Keyword Arguments
  • init::Function=Flux.glorot_uniform: The initialization function used for the neural network weights.
Returns
  • A JointGaussianLogEncoder struct initialized based on the provided arguments.
Examples
encoder = JointGaussianLogEncoder(784, 20, [400], [relu], tanh)
Notes
The length of encoderneurons should match the length of encoderactivation, ensuring that each layer in the encoder has a corresponding activation function.
AutoEncoderToolkit.JointGaussianEncoder — Method
JointGaussianEncoder(n_input, n_latent, encoder_neurons, encoder_activation, 
-             latent_activation; init=Flux.glorot_uniform)
Construct and initialize a JointGaussianLogEncoder struct that defines an encoder network for a variational autoencoder.
Arguments
  • n_input::Int: The dimensionality of the input data.
  • n_latent::Int: The dimensionality of the latent space.
  • encoder_neurons::Vector{<:Int}: A vector specifying the number of neurons in each layer of the encoder network.
  • encoder_activation::Vector{<:Function}: Activation functions corresponding to each layer in the encoder_neurons.
  • latent_activation::Vector{<:Function}: Activation function for the latent space layers. This vector must contain the activation for both µ and logσ.
Optional Keyword Arguments
  • init::Function=Flux.glorot_uniform: The initialization function used for the neural network weights.
Returns
  • A JointGaussianEncoder struct initialized based on the provided arguments.
Examples
encoder = JointGaussianEncoder(784, 20, [400], [relu], [tanh, softplus])
Notes
The length of encoderneurons should match the length of encoderactivation, ensuring that each layer in the encoder has a corresponding activation function.
Decoder initializations
AutoEncoderToolkit.Decoder — Method
Decoder(n_input, n_latent, decoder_neurons, decoder_activation, 
-        output_activation; init=Flux.glorot_uniform)
Construct and initialize a Decoder struct that defines a decoder network for a deterministic autoencoder.
Arguments
  • n_input::Int: The dimensionality of the output data (which typically matches the input data dimensionality of the autoencoder).
  • n_latent::Int: The dimensionality of the latent space.
  • decoder_neurons::Vector{<:Int}: A vector specifying the number of neurons in each layer of the decoder network.
  • decoder_activation::Vector{<:Function}: Activation functions corresponding to each layer in the decoder_neurons.
  • output_activation::Function: Activation function for the final output layer.
Optional Keyword Arguments
  • init::Function=Flux.glorot_uniform: The initialization function used for the neural network weights.
Returns
  • A Decoder struct initialized based on the provided arguments.
Examples
decoder = Decoder(784, 20, sigmoid, [400], [relu])
Notes
The length of decoderneurons should match the length of decoderactivation, ensuring that each layer in the decoder has a corresponding activation function.
AutoEncoderToolkit.SimpleGaussianDecoder — Method
SimpleGaussianDecoder(
+output = decoder(z))
Note
Ensure that the latent space representation z matches the expected input dimensionality for both networks in the SplitGaussianLogDecoder.
Default initializations
The package provides a set of functions to initialize encoder and decoder architectures. Although it gives the user less flexibility, it can be useful for quick prototyping.
Encoder initializations
AutoEncoderToolkit.Encoder — Method
Encoder(n_input, n_latent, latent_activation, encoder_neurons, 
+        encoder_activation; init=Flux.glorot_uniform)
Construct and initialize an Encoder struct that defines an encoder network for a deterministic autoencoder.
Arguments
  • n_input::Int: The dimensionality of the input data.
  • n_latent::Int: The dimensionality of the latent space.
  • encoder_neurons::Vector{<:Int}: A vector specifying the number of neurons in each layer of the encoder network.
  • encoder_activation::Vector{<:Function}: Activation functions corresponding to each layer in the encoder_neurons.
  • latent_activation::Function: Activation function for the latent space layer.
Optional Keyword Arguments
  • init::Function=Flux.glorot_uniform: The initialization function used for the neural network weights.
Returns
  • An Encoder struct initialized based on the provided arguments.
Examples
julia encoder = Encoder(784, 20, tanh, [400], [relu])`
Notes
The length of encoderneurons should match the length of encoderactivation, ensuring that each layer in the encoder has a corresponding activation function.
AutoEncoderToolkit.JointGaussianLogEncoder — Method
JointGaussianLogEncoder(n_input, n_latent, encoder_neurons, encoder_activation, 
+             latent_activation; init=Flux.glorot_uniform)
Construct and initialize a JointGaussianLogEncoder struct that defines an encoder network for a variational autoencoder.
Arguments
  • n_input::Int: The dimensionality of the input data.
  • n_latent::Int: The dimensionality of the latent space.
  • encoder_neurons::Vector{<:Int}: A vector specifying the number of neurons in each layer of the encoder network.
  • encoder_activation::Vector{<:Function}: Activation functions corresponding to each layer in the encoder_neurons.
  • latent_activation::Function: Activation function for the latent space layers (both µ and logσ).
Optional Keyword Arguments
  • init::Function=Flux.glorot_uniform: The initialization function used for the neural network weights.
Returns
  • A JointGaussianLogEncoder struct initialized based on the provided arguments.
Examples
encoder = JointGaussianLogEncoder(784, 20, [400], [relu], tanh)
Notes
The length of encoderneurons should match the length of encoderactivation, ensuring that each layer in the encoder has a corresponding activation function.
AutoEncoderToolkit.JointGaussianLogEncoder — Method
JointGaussianLogEncoder(n_input, n_latent, encoder_neurons, encoder_activation, 
+             latent_activation; init=Flux.glorot_uniform)
Construct and initialize a JointGaussianLogEncoder struct that defines an encoder network for a variational autoencoder.
Arguments
  • n_input::Int: The dimensionality of the input data.
  • n_latent::Int: The dimensionality of the latent space.
  • encoder_neurons::Vector{<:Int}: A vector specifying the number of neurons in each layer of the encoder network.
  • encoder_activation::Vector{<:Function}: Activation functions corresponding to each layer in the encoder_neurons.
  • latent_activation::Vector{<:Function}: Activation functions for the latent space layers (both µ and logσ).
Optional Keyword Arguments
  • init::Function=Flux.glorot_uniform: The initialization function used for the neural network weights.
Returns
  • A JointGaussianLogEncoder struct initialized based on the provided arguments.
Examples
encoder = JointGaussianLogEncoder(784, 20, [400], [relu], tanh)
Notes
The length of encoderneurons should match the length of encoderactivation, ensuring that each layer in the encoder has a corresponding activation function.
AutoEncoderToolkit.JointGaussianEncoder — Method
JointGaussianEncoder(n_input, n_latent, encoder_neurons, encoder_activation, 
+             latent_activation; init=Flux.glorot_uniform)
Construct and initialize a JointGaussianLogEncoder struct that defines an encoder network for a variational autoencoder.
Arguments
  • n_input::Int: The dimensionality of the input data.
  • n_latent::Int: The dimensionality of the latent space.
  • encoder_neurons::Vector{<:Int}: A vector specifying the number of neurons in each layer of the encoder network.
  • encoder_activation::Vector{<:Function}: Activation functions corresponding to each layer in the encoder_neurons.
  • latent_activation::Vector{<:Function}: Activation function for the latent space layers. This vector must contain the activation for both µ and logσ.
Optional Keyword Arguments
  • init::Function=Flux.glorot_uniform: The initialization function used for the neural network weights.
Returns
  • A JointGaussianEncoder struct initialized based on the provided arguments.
Examples
encoder = JointGaussianEncoder(784, 20, [400], [relu], [tanh, softplus])
Notes
The length of encoderneurons should match the length of encoderactivation, ensuring that each layer in the encoder has a corresponding activation function.
Decoder initializations
AutoEncoderToolkit.Decoder — Method
Decoder(n_input, n_latent, decoder_neurons, decoder_activation, 
+        output_activation; init=Flux.glorot_uniform)
Construct and initialize a Decoder struct that defines a decoder network for a deterministic autoencoder.
Arguments
  • n_input::Int: The dimensionality of the output data (which typically matches the input data dimensionality of the autoencoder).
  • n_latent::Int: The dimensionality of the latent space.
  • decoder_neurons::Vector{<:Int}: A vector specifying the number of neurons in each layer of the decoder network.
  • decoder_activation::Vector{<:Function}: Activation functions corresponding to each layer in the decoder_neurons.
  • output_activation::Function: Activation function for the final output layer.
Optional Keyword Arguments
  • init::Function=Flux.glorot_uniform: The initialization function used for the neural network weights.
Returns
  • A Decoder struct initialized based on the provided arguments.
Examples
decoder = Decoder(784, 20, sigmoid, [400], [relu])
Notes
The length of decoderneurons should match the length of decoderactivation, ensuring that each layer in the decoder has a corresponding activation function.
AutoEncoderToolkit.SimpleGaussianDecoder — Method
SimpleGaussianDecoder(
     n_input, n_latent, decoder_neurons, 
     decoder_activation, output_activation; 
     init=Flux.glorot_uniform
@@ -32,7 +32,7 @@
 output_activation = sigmoid
 decoder = SimpleGaussianDecoder(
     n_input, n_latent, decoder_neurons, decoder_activation, output_activation
-)
Note
Ensure that the lengths of decoderneurons and decoderactivation match, excluding the output layer.
AutoEncoderToolkit.JointGaussianLogDecoder — Method
JointGaussianLogDecoder(n_input, n_latent, decoder_neurons, decoder_activation, 
+)
Note
Ensure that the lengths of decoderneurons and decoderactivation match, excluding the output layer.
AutoEncoderToolkit.JointGaussianLogDecoder — Method
JointGaussianLogDecoder(n_input, n_latent, decoder_neurons, decoder_activation, 
             latent_activation; init=Flux.glorot_uniform)
Constructs and initializes a JointGaussianLogDecoder object for variational autoencoders (VAEs). This function sets up a decoder network that first processes the latent space and then maps it separately to both its mean (µ) and log standard deviation (logσ).
Arguments
  • n_input::Int: Dimensionality of the output data (or the data to be reconstructed).
  • n_latent::Int: Dimensionality of the latent space.
  • decoder_neurons::Vector{<:Int}: Vector of layer sizes for the primary decoder network, not including the input latent layer.
  • decoder_activation::Vector{<:Function}: Activation functions for each primary decoder layer.
  • output_activation::Function: Activation function for the mean (µ) and log standard deviation (logσ) layers.
Optional Keyword Arguments
  • init::Function=Flux.glorot_uniform: Initialization function for the network parameters.
Returns
A JointGaussianLogDecoder object with the specified architecture and initialized weights.
Description
This function constructs a JointGaussianLogDecoder object, setting up its primary decoder network based on the provided specifications. The architecture begins with a dense layer mapping from the latent space and goes through a sequence of middle layers if specified. After processing the latent space through the primary decoder, it then maps separately to both its mean (µ) and log standard deviation (logσ).
Example
n_input = 28*28
 n_latent = 64
 decoder_neurons = [128, 256]
@@ -40,7 +40,7 @@
 output_activation = tanh
 decoder = JointGaussianLogDecoder(
     n_input, n_latent, decoder_neurons, decoder_activation, output_activation
-)
Note
Ensure that the lengths of decoderneurons and decoderactivation match.
AutoEncoderToolkit.JointGaussianLogDecoder — Method
JointGaussianLogDecoder(n_input, n_latent, decoder_neurons, decoder_activation, 
             latent_activation; init=Flux.glorot_uniform)
Constructs and initializes a JointGaussianLogDecoder object for variational autoencoders (VAEs). This function sets up a decoder network that first processes the latent space and then maps it separately to both its mean (µ) and log standard deviation (logσ).
Arguments
  • n_input::Int: Dimensionality of the output data (or the data to be reconstructed).
  • n_latent::Int: Dimensionality of the latent space.
  • decoder_neurons::Vector{<:Int}: Vector of layer sizes for the primary decoder network, not including the input latent layer.
  • decoder_activation::Vector{<:Function}: Activation functions for each primary decoder layer.
  • output_activation::Vector{<:Function}: Activation functions for the mean (µ) and log standard deviation (logσ) layers.
Optional Keyword Arguments
  • init::Function=Flux.glorot_uniform: Initialization function for the network parameters.
Returns
A JointGaussianLogDecoder object with the specified architecture and initialized weights.
Description
This function constructs a JointGaussianLogDecoder object, setting up its primary decoder network based on the provided specifications. The architecture begins with a dense layer mapping from the latent space and goes through a sequence of middle layers if specified. After processing the latent space through the primary decoder, it then maps separately to both its mean (µ) and log standard deviation (logσ).
Example
n_input = 28*28
 n_latent = 64
 decoder_neurons = [128, 256]
@@ -48,7 +48,7 @@
 output_activation = [tanh, identity]
 decoder = JointGaussianLogDecoder(
     n_input, n_latent, decoder_neurons, decoder_activation, latent_activation
-)
Note
Ensure that the lengths of decoderneurons and decoderactivation match.
AutoEncoderToolkit.JointGaussianDecoder — Method
JointGaussianDecoder(n_input, n_latent, decoder_neurons, decoder_activation, 
+)
Note
Ensure that the lengths of decoderneurons and decoderactivation match.
AutoEncoderToolkit.JointGaussianDecoder — Method
JointGaussianDecoder(n_input, n_latent, decoder_neurons, decoder_activation, 
             latent_activation; init=Flux.glorot_uniform)
Constructs and initializes a JointGaussianLogDecoder object for variational autoencoders (VAEs). This function sets up a decoder network that first processes the latent space and then maps it separately to both its mean (µ) and log standard deviation (logσ).
Arguments
  • n_input::Int: Dimensionality of the output data (or the data to be reconstructed).
  • n_latent::Int: Dimensionality of the latent space.
  • decoder_neurons::Vector{<:Int}: Vector of layer sizes for the primary decoder network, not including the input latent layer.
  • decoder_activation::Vector{<:Function}: Activation functions for each primary decoder layer.
  • output_activation::Function: Activation function for the mean (µ) and log standard deviation (logσ) layers.
Optional Keyword Arguments
  • init::Function=Flux.glorot_uniform: Initialization function for the network parameters.
Returns
A JointGaussianDecoder object with the specified architecture and initialized weights.
Description
This function constructs a JointGaussianDecoder object, setting up its primary decoder network based on the provided specifications. The architecture begins with a dense layer mapping from the latent space and goes through a sequence of middle layers if specified. After processing the latent space through the primary decoder, it then maps separately to both its mean (µ) and standard deviation (σ).
Example
n_input = 28*28
 n_latent = 64
 decoder_neurons = [128, 256]
@@ -56,7 +56,7 @@
 output_activation = tanh
 decoder = JointGaussianDecoder(
     n_input, n_latent, decoder_neurons, decoder_activation, output_activation
-)
Note
Ensure that the lengths of decoderneurons and decoderactivation match.
AutoEncoderToolkit.JointGaussianDecoder — Method
JointGaussianDecoder(n_input, n_latent, decoder_neurons, decoder_activation, 
+)
Note
Ensure that the lengths of decoderneurons and decoderactivation match.
AutoEncoderToolkit.JointGaussianDecoder — Method
JointGaussianDecoder(n_input, n_latent, decoder_neurons, decoder_activation, 
             latent_activation; init=Flux.glorot_uniform)
Constructs and initializes a JointGaussianDecoder object for variational autoencoders (VAEs). This function sets up a decoder network that first processes the latent space and then maps it separately to both its mean (µ) and standard deviation (σ).
Arguments
  • n_input::Int: Dimensionality of the output data (or the data to be reconstructed).
  • n_latent::Int: Dimensionality of the latent space.
  • decoder_neurons::Vector{<:Int}: Vector of layer sizes for the primary decoder network, not including the input latent layer.
  • decoder_activation::Vector{<:Function}: Activation functions for each primary decoder layer.
  • output_activation::Function: Activation function for the mean (µ) and standard deviation (σ) layers.
Optional Keyword Arguments
  • init::Function=Flux.glorot_uniform: Initialization function for the network parameters.
Returns
A JointGaussianDecoder object with the specified architecture and initialized weights.
Description
This function constructs a JointGaussianDecoder object, setting up its primary decoder network based on the provided specifications. The architecture begins with a dense layer mapping from the latent space and goes through a sequence of middle layers if specified. After processing the latent space through the primary decoder, it then maps separately to both its mean (µ) and standard deviation (σ).
Example
n_input = 28*28
 n_latent = 64
 decoder_neurons = [128, 256]
@@ -64,7 +64,7 @@
 latent_activation = [tanh, softplus]
 decoder = JointGaussianDecoder(
     n_input, n_latent, decoder_neurons, decoder_activation, latent_activation
-)
Note
Ensure that the lengths of decoderneurons and decoderactivation match.
AutoEncoderToolkit.SplitGaussianLogDecoder — Method
SplitGaussianLogDecoder(n_input, n_latent, µ_neurons, µ_activation, logσ_neurons, 
+)
Note
Ensure that the lengths of decoderneurons and decoderactivation match.
AutoEncoderToolkit.SplitGaussianLogDecoder — Method
SplitGaussianLogDecoder(n_input, n_latent, µ_neurons, µ_activation, logσ_neurons, 
             logσ_activation; init=Flux.glorot_uniform)
Constructs and initializes a SplitGaussianLogDecoder object for variational autoencoders (VAEs). This function sets up two distinct decoder networks, one dedicated for determining the mean (µ) and the other for the log standard deviation (logσ) of the latent space.
Arguments
  • n_input::Int: Dimensionality of the output data (or the data to be reconstructed).
  • n_latent::Int: Dimensionality of the latent space.
  • µ_neurons::Vector{<:Int}: Vector of layer sizes for the µ decoder network, not including the input latent layer.
  • µ_activation::Vector{<:Function}: Activation functions for each µ decoder layer.
  • logσ_neurons::Vector{<:Int}: Vector of layer sizes for the logσ decoder network, not including the input latent layer.
  • logσ_activation::Vector{<:Function}: Activation functions for each logσ decoder layer.
Optional Keyword Arguments
  • init::Function=Flux.glorot_uniform: Initialization function for the network parameters.
Returns
A SplitGaussianLogDecoder object with two distinct networks initialized with the specified architectures and weights.
Description
This function constructs a SplitGaussianLogDecoder object, setting up two separate decoder networks based on the provided specifications. The first network, dedicated to determining the mean (µ), and the second for the log standard deviation (logσ), both begin with a dense layer mapping from the latent space and go through a sequence of middle layers if specified.
Example
n_latent = 64
 µ_neurons = [128, 256]
 µ_activation = [relu, relu]
@@ -72,7 +72,7 @@
 logσ_activation = [relu, relu]
 decoder = SplitGaussianLogDecoder(
     n_latent, µ_neurons, µ_activation, logσ_neurons, logσ_activation
-)
Notes
  • Ensure that the lengths of µneurons with µactivation and logσneurons with logσactivation match respectively.
  • If µneurons[end] or logσneurons[end] do not match n_input, the function automatically changes this number to match the right dimensionality
AutoEncoderToolkit.SplitGaussianDecoder — Method
SplitGaussianDecoder(n_input, n_latent, µ_neurons, µ_activation, logσ_neurons, 
+)
Notes
  • Ensure that the lengths of µneurons with µactivation and logσneurons with logσactivation match respectively.
  • If µneurons[end] or logσneurons[end] do not match n_input, the function automatically changes this number to match the right dimensionality
AutoEncoderToolkit.SplitGaussianDecoder — Method
SplitGaussianDecoder(n_input, n_latent, µ_neurons, µ_activation, logσ_neurons, 
             logσ_activation; init=Flux.glorot_uniform)
Constructs and initializes a SplitGaussianDecoder object for variational autoencoders (VAEs). This function sets up two distinct decoder networks, one dedicated for determining the mean (µ) and the other for the standard deviation (σ) of the latent space.
Arguments
  • n_input::Int: Dimensionality of the output data (or the data to be reconstructed).
  • n_latent::Int: Dimensionality of the latent space.
  • µ_neurons::Vector{<:Int}: Vector of layer sizes for the µ decoder network, not including the input latent layer.
  • µ_activation::Vector{<:Function}: Activation functions for each µ decoder layer.
  • σ_neurons::Vector{<:Int}: Vector of layer sizes for the σ decoder network, not including the input latent layer.
  • σ_activation::Vector{<:Function}: Activation functions for each σ decoder layer.
Optional Keyword Arguments
  • init::Function=Flux.glorot_uniform: Initialization function for the network parameters.
Returns
A SplitGaussianDecoder object with two distinct networks initialized with the specified architectures and weights.
Description
This function constructs a SplitGaussianDecoder object, setting up two separate decoder networks based on the provided specifications. The first network, dedicated to determining the mean (µ), and the second for the standard deviation (σ), both begin with a dense layer mapping from the latent space and go through a sequence of middle layers if specified.
Example
n_latent = 64
 µ_neurons = [128, 256]
 µ_activation = [relu, relu]
@@ -80,7 +80,7 @@
 σ_activation = [relu, relu]
 decoder = SplitGaussianDecoder(
     n_latent, µ_neurons, µ_activation, σ_neurons, σ_activation
-)
Notes
  • Ensure that the lengths of µneurons with µactivation and σneurons with σactivation match respectively.
  • If µneurons[end] or σneurons[end] do not match n_input, the function automatically changes this number to match the right dimensionality
  • Ensure that σ_neurons[end] maps to a positive value. Activation functions such as softplus are needed to guarantee the positivity of the standard deviation.
AutoEncoderToolkit.BernoulliDecoder — Method
    BernoulliDecoder(n_input, n_latent, decoder_neurons, decoder_activation, 
+)
Notes
  • Ensure that the lengths of µneurons with µactivation and σneurons with σactivation match respectively.
  • If µneurons[end] or σneurons[end] do not match n_input, the function automatically changes this number to match the right dimensionality
  • Ensure that σ_neurons[end] maps to a positive value. Activation functions such as softplus are needed to guarantee the positivity of the standard deviation.
AutoEncoderToolkit.BernoulliDecoder — Method
    BernoulliDecoder(n_input, n_latent, decoder_neurons, decoder_activation, 
                             output_activation; init=Flux.glorot_uniform)
Constructs and initializes a BernoulliDecoder object designed for variational autoencoders (VAEs). This function sets up a decoder network that maps from a latent space to an output space.
Arguments
  • n_input::Int: Dimensionality of the output data (or the data to be   reconstructed).
  • n_latent::Int: Dimensionality of the latent space.
  • decoder_neurons::Vector{<:Int}: Vector of layer sizes for the decoder   network, not including the input latent layer and the final output layer.
  • decoder_activation::Vector{<:Function}: Activation functions for each   decoder layer, not including the final output layer.
  • output_activation::Function: Activation function for the final output layer.
Optional Keyword Arguments
  • init::Function=Flux.glorot_uniform: Initialization function for the network   parameters.
Returns
A BernoulliDecoder object with the specified architecture and initialized weights.
Description
This function constructs a BernoulliDecoder object, setting up its decoder network based on the provided specifications. The architecture begins with a dense layer mapping from the latent space, goes through a sequence of middle layers if specified, and finally maps to the output space.
The function ensures that there are appropriate activation functions provided for each layer in the decoder_neurons and checks for potential mismatches in length.
Example
n_input = 28*28
 n_latent = 64
 decoder_neurons = [128, 256]
@@ -92,13 +92,13 @@
     decoder_neurons, 
     decoder_activation, 
     output_activation
-)
Note
Ensure that the lengths of decoderneurons and decoderactivation match, excluding the output layer. Also, the output activation function should return values between 0 and 1, as the decoder models the output data as a Bernoulli distribution. 
AutoEncoderToolkit.CategoricalDecoder — Method
    CategoricalDecoder(
+)
Note
Ensure that the lengths of decoderneurons and decoderactivation match, excluding the output layer. Also, the output activation function should return values between 0 and 1, as the decoder models the output data as a Bernoulli distribution. 
AutoEncoderToolkit.CategoricalDecoder — Method
    CategoricalDecoder(
         size_input, n_latent, decoder_neurons, decoder_activation, 
         output_activation; init=Flux.glorot_uniform
-    )
Constructs and initializes a CategoricalDecoder object designed for variational autoencoders (VAEs). This function sets up a decoder network that maps from a latent space to an output space.
Arguments
  • size_input::AbstractVector{<:Int}: Dimensionality of the output data (or the data to be reconstructed) in the form of a vector where each element represents the size of a dimension.
  • n_latent::Int: Dimensionality of the latent space.
  • decoder_neurons::Vector{<:Int}: Vector of layer sizes for the decoder network, not including the input latent layer and the final output layer.
  • decoder_activation::Vector{<:Function}: Activation functions for each decoder layer, not including the final output layer.
  • output_activation::Function: Activation function for the final output layer.
Optional Keyword Arguments
  • init::Function=Flux.glorot_uniform: Initialization function for the network parameters.
Returns
A CategoricalDecoder object with the specified architecture and initialized weights.
Description
This function constructs a CategoricalDecoder object, setting up its decoder network based on the provided specifications. The architecture begins with a dense layer mapping from the latent space, goes through a sequence of middle layers if specified, and finally maps to the output space.
The function ensures that there are appropriate activation functions provided for each layer in the decoder_neurons and checks for potential mismatches in length.
The output layer uses the identity function as its activation function, and the output is reshaped to match the dimensions specified in size_input. The output_activation function is then applied over the first dimension of the reshaped output.
Note
Ensure that the lengths of decoderneurons and decoderactivation match, excluding the output layer. Also, the output activation function should return values that can be interpreted as probabilities, as the decoder models the output data as a categorical distribution. 
AutoEncoderToolkit.CategoricalDecoder — Method
CategoricalDecoder(
+    )
Constructs and initializes a CategoricalDecoder object designed for variational autoencoders (VAEs). This function sets up a decoder network that maps from a latent space to an output space.
Arguments
  • size_input::AbstractVector{<:Int}: Dimensionality of the output data (or the data to be reconstructed) in the form of a vector where each element represents the size of a dimension.
  • n_latent::Int: Dimensionality of the latent space.
  • decoder_neurons::Vector{<:Int}: Vector of layer sizes for the decoder network, not including the input latent layer and the final output layer.
  • decoder_activation::Vector{<:Function}: Activation functions for each decoder layer, not including the final output layer.
  • output_activation::Function: Activation function for the final output layer.
Optional Keyword Arguments
  • init::Function=Flux.glorot_uniform: Initialization function for the network parameters.
Returns
A CategoricalDecoder object with the specified architecture and initialized weights.
Description
This function constructs a CategoricalDecoder object, setting up its decoder network based on the provided specifications. The architecture begins with a dense layer mapping from the latent space, goes through a sequence of middle layers if specified, and finally maps to the output space.
The function ensures that there are appropriate activation functions provided for each layer in the decoder_neurons and checks for potential mismatches in length.
The output layer uses the identity function as its activation function, and the output is reshaped to match the dimensions specified in size_input. The output_activation function is then applied over the first dimension of the reshaped output.
Note
Ensure that the lengths of decoderneurons and decoderactivation match, excluding the output layer. Also, the output activation function should return values that can be interpreted as probabilities, as the decoder models the output data as a categorical distribution. 
AutoEncoderToolkit.CategoricalDecoder — Method
CategoricalDecoder(
     n_input, n_latent, decoder_neurons, decoder_activation,
     output_activation; init=Flux.glorot_uniform
-)
Constructs and initializes a CategoricalDecoder object designed for variational autoencoders (VAEs). This function sets up a decoder network that maps from a latent space to an output space.
Arguments
  • size_input::AbstractVector{<:Int}: Dimensionality of the output data (or the data to be reconstructed).
  • n_latent::Int: Dimensionality of the latent space.
  • decoder_neurons::Vector{<:Int}: Vector of layer sizes for the decoder network, not including the input latent layer and the final output layer.
  • decoder_activation::Vector{<:Function}: Activation functions for each decoder layer, not including the final output layer.
  • output_activation::Function: Activation function for the final output layer.
Optional Keyword Arguments
  • init::Function=Flux.glorot_uniform: Initialization function for the network parameters.
Returns
A CategoricalDecoder object with the specified architecture and initialized weights.
Description
This function constructs a CategoricalDecoder object, setting up its decoder network based on the provided specifications. The architecture begins with a dense layer mapping from the latent space, goes through a sequence of middle layers if specified, and finally maps to the output space.
The function ensures that there are appropriate activation functions provided for each layer in the decoder_neurons and checks for potential mismatches in length.
Note
Ensure that the lengths of decoderneurons and decoderactivation match, excluding the output layer. Also, the output activation function should return values that can be interpreted as probabilities, as the decoder models the output data as a categorical distribution. 
Probabilistic functions
Given the probability-centered design of AutoEncoderToolkit.jl, each variational encoder and decoder has an associated probabilistic function used when computing the evidence lower bound (ELBO). The following functions are available:
AutoEncoderToolkit.encoder_logposterior — Function
encoder_logposterior(
+)
Constructs and initializes a CategoricalDecoder object designed for variational autoencoders (VAEs). This function sets up a decoder network that maps from a latent space to an output space.
Arguments
  • size_input::AbstractVector{<:Int}: Dimensionality of the output data (or the data to be reconstructed).
  • n_latent::Int: Dimensionality of the latent space.
  • decoder_neurons::Vector{<:Int}: Vector of layer sizes for the decoder network, not including the input latent layer and the final output layer.
  • decoder_activation::Vector{<:Function}: Activation functions for each decoder layer, not including the final output layer.
  • output_activation::Function: Activation function for the final output layer.
Optional Keyword Arguments
  • init::Function=Flux.glorot_uniform: Initialization function for the network parameters.
Returns
A CategoricalDecoder object with the specified architecture and initialized weights.
Description
This function constructs a CategoricalDecoder object, setting up its decoder network based on the provided specifications. The architecture begins with a dense layer mapping from the latent space, goes through a sequence of middle layers if specified, and finally maps to the output space.
The function ensures that there are appropriate activation functions provided for each layer in the decoder_neurons and checks for potential mismatches in length.
Note
Ensure that the lengths of decoderneurons and decoderactivation match, excluding the output layer. Also, the output activation function should return values that can be interpreted as probabilities, as the decoder models the output data as a categorical distribution. 
Probabilistic functions
Given the probability-centered design of AutoEncoderToolkit.jl, each variational encoder and decoder has an associated probabilistic function used when computing the evidence lower bound (ELBO). The following functions are available:
AutoEncoderToolkit.encoder_logposterior — Function
encoder_logposterior(
     z::AbstractVector,
     encoder::AbstractGaussianLogEncoder,
     encoder_output::NamedTuple
@@ -111,10 +111,10 @@
     encoder::AbstractGaussianLogEncoder,
     encoder_output::NamedTuple,
     index::Int
-)
Computes the log-posterior of the latent variable z for a single data point specified by index given the encoder output under a Gaussian distribution with mean and standard deviation given by the encoder.
Arguments
  • z::AbstractVector: The latent variable for which the log-posterior is to be computed. 
  • encoder::AbstractGaussianLogEncoder: The encoder of the VAE, which is not used in the computation of the log-posterior. This argument is only used to know which method to call.
  • encoder_output::NamedTuple: The output of the encoder, which includes the mean and log standard deviation of the Gaussian distribution for multiple data points.
  • index::Int: The index of the data point for which the log-posterior is to be computed.
Returns
  • logposterior::Float32: The computed log-posterior of the latent variable z   for the specified data point given the encoder output.
Description
The function computes the log-posterior of the latent variable z for a single data point specified by index given the encoder output under a Gaussian distribution. The mean and log standard deviation of the Gaussian distribution are extracted from the encoder_output for the specified data point. The standard deviation is then computed by exponentiating the log standard deviation. The log-posterior is computed using the formula for the log-posterior of a Gaussian distribution.
Note
Ensure the dimensions of z match the expected input dimensionality of the encoder. Also, ensure that index is a valid index for the data points in encoder_output.
AutoEncoderToolkit.encoder_kl — Function
encoder_kl(
+)
Computes the log-posterior of the latent variable z for a single data point specified by index given the encoder output under a Gaussian distribution with mean and standard deviation given by the encoder.
Arguments
  • z::AbstractVector: The latent variable for which the log-posterior is to be computed. 
  • encoder::AbstractGaussianLogEncoder: The encoder of the VAE, which is not used in the computation of the log-posterior. This argument is only used to know which method to call.
  • encoder_output::NamedTuple: The output of the encoder, which includes the mean and log standard deviation of the Gaussian distribution for multiple data points.
  • index::Int: The index of the data point for which the log-posterior is to be computed.
Returns
  • logposterior::Float32: The computed log-posterior of the latent variable z   for the specified data point given the encoder output.
Description
The function computes the log-posterior of the latent variable z for a single data point specified by index given the encoder output under a Gaussian distribution. The mean and log standard deviation of the Gaussian distribution are extracted from the encoder_output for the specified data point. The standard deviation is then computed by exponentiating the log standard deviation. The log-posterior is computed using the formula for the log-posterior of a Gaussian distribution.
Note
Ensure the dimensions of z match the expected input dimensionality of the encoder. Also, ensure that index is a valid index for the data points in encoder_output.
AutoEncoderToolkit.encoder_kl — Function
encoder_kl(
     encoder::AbstractGaussianLogEncoder,
     encoder_output::NamedTuple
-)
Calculate the Kullback-Leibler (KL) divergence between the approximate posterior distribution and the prior distribution in a variational autoencoder with a Gaussian encoder.
The KL divergence for a Gaussian encoder with mean encoder_µ and log standard deviation encoder_logσ is computed against a standard Gaussian prior.
Arguments
  • encoder::AbstractGaussianLogEncoder: Encoder network. This argument is not used in the computation of the KL divergence, but is included to allow for multiple encoder types to be used with the same function.
  • encoder_output::NamedTuple: NamedTuple containing all the encoder outputs. It should have fields μ and logσ representing the mean and log standard deviation of the encoder's output.
Returns
  • kl_div::Union{Number, Vector}: The KL divergence for the entire batch of data points. If encoder_µ is a vector, kl_div is a scalar. If encoder_µ is a matrix, kl_div is a vector where each element corresponds to the KL divergence for a batch of data points.
Note
  • It is assumed that the mapping from data space to latent parameters (encoder_µ and encoder_logσ) has been performed prior to calling this function. The encoder argument is provided to indicate the type of decoder network used, but it is not used within the function itself.
AutoEncoderToolkit.spherical_logprior — Function
spherical_logprior(z::AbstractVector, σ::Real=1.0f0)
Computes the log-prior of the latent variable z under a spherical Gaussian distribution with zero mean and standard deviation σ.
Arguments
  • z::AbstractVector: The latent variable for which the log-prior is to be computed.
  • σ::T=1.0f0: The standard deviation of the spherical Gaussian distribution. Defaults to 1.0f0.
Returns
  • logprior::T: The computed log-prior of the latent variable z.
Description
The function computes the log-prior of the latent variable z under a spherical Gaussian distribution with zero mean and standard deviation σ. The log-prior is computed using the formula for the log-prior of a Gaussian distribution.
Note
Ensure the dimension of z matches the expected dimensionality of the latent space.
spherical_logprior(z::AbstractMatrix, σ::Real=1.0f0)
Computes the log-prior of the latent variable z under a spherical Gaussian distribution with zero mean and standard deviation σ.
Arguments
  • z::AbstractMatrix: The latent variable for which the log-prior is to be computed. Each column of z represents a different latent variable.
  • σ::Real=1.0f0: The standard deviation of the spherical Gaussian distribution. Defaults to 1.0f0.
Returns
  • logprior::T: The computed log-prior(s) of the latent variable z.
Description
The function computes the log-prior of the latent variable z under a spherical Gaussian distribution with zero mean and standard deviation σ. The log-prior is computed using the formula for the log-prior of a Gaussian distribution.
Note
Ensure the dimension of z matches the expected dimensionality of the latent space.
Defining custom encoder and decoder types
Note
We will omit all docstrings in the following examples for brevity. However, every struct and function in AutoEncoderToolkit.jl is well-documented.
Let us imagine your particular task requires a custom encoder or decoder type. For example, let's imagine that for a particular application, you need a decoder whose output distribution is Poisson. In other words, the assumption is that each dimension in the input $x_i$ is a sample from a Poisson distribution with mean $\lambda_i$. Thus, on the decoder side, what the decoder return is a vector of these $\lambda$ paraeters. We thus need to define a custom decoder type.
struct PoissonDecoder <: AbstractVariationalDecoder
+)
Calculate the Kullback-Leibler (KL) divergence between the approximate posterior distribution and the prior distribution in a variational autoencoder with a Gaussian encoder.
The KL divergence for a Gaussian encoder with mean encoder_µ and log standard deviation encoder_logσ is computed against a standard Gaussian prior.
Arguments
  • encoder::AbstractGaussianLogEncoder: Encoder network. This argument is not used in the computation of the KL divergence, but is included to allow for multiple encoder types to be used with the same function.
  • encoder_output::NamedTuple: NamedTuple containing all the encoder outputs. It should have fields μ and logσ representing the mean and log standard deviation of the encoder's output.
Returns
  • kl_div::Union{Number, Vector}: The KL divergence for the entire batch of data points. If encoder_µ is a vector, kl_div is a scalar. If encoder_µ is a matrix, kl_div is a vector where each element corresponds to the KL divergence for a batch of data points.
Note
  • It is assumed that the mapping from data space to latent parameters (encoder_µ and encoder_logσ) has been performed prior to calling this function. The encoder argument is provided to indicate the type of decoder network used, but it is not used within the function itself.
AutoEncoderToolkit.spherical_logprior — Function
spherical_logprior(z::AbstractVector, σ::Real=1.0f0)

Computes the log-prior of the latent variable z under a spherical Gaussian distribution with zero mean and standard deviation σ.

Arguments

  • z::AbstractVector: The latent variable for which the log-prior is to be computed.
  • σ::T=1.0f0: The standard deviation of the spherical Gaussian distribution. Defaults to 1.0f0.

Returns

  • logprior::T: The computed log-prior of the latent variable z.

Description

The function computes the log-prior of the latent variable z under a spherical Gaussian distribution with zero mean and standard deviation σ. The log-prior is computed using the formula for the log-prior of a Gaussian distribution.

Note

Ensure the dimension of z matches the expected dimensionality of the latent space.

spherical_logprior(z::AbstractMatrix, σ::Real=1.0f0)

Computes the log-prior of the latent variable z under a spherical Gaussian distribution with zero mean and standard deviation σ.

Arguments

  • z::AbstractMatrix: The latent variable for which the log-prior is to be computed. Each column of z represents a different latent variable.
  • σ::Real=1.0f0: The standard deviation of the spherical Gaussian distribution. Defaults to 1.0f0.

Returns

  • logprior::T: The computed log-prior(s) of the latent variable z.

Description

The function computes the log-prior of the latent variable z under a spherical Gaussian distribution with zero mean and standard deviation σ. The log-prior is computed using the formula for the log-prior of a Gaussian distribution.

Note

Ensure the dimension of z matches the expected dimensionality of the latent space.

Defining custom encoder and decoder types

Note

We will omit all docstrings in the following examples for brevity. However, every struct and function in AutoEncoderToolkit.jl is well-documented.

Let us imagine your particular task requires a custom encoder or decoder type. For example, let's imagine that for a particular application, you need a decoder whose output distribution is Poisson. In other words, the assumption is that each dimension in the input $x_i$ is a sample from a Poisson distribution with mean $\lambda_i$. Thus, on the decoder side, what the decoder return is a vector of these $\lambda$ paraeters. We thus need to define a custom decoder type.

struct PoissonDecoder <: AbstractVariationalDecoder
     decoder::Flux.Chain
 end # struct

With this struct defined, we need to define the forward-pass function for our custom PoissonDecoder. All decoders in AutoEncoderToolkit.jl return a NamedTuple with the corresponding parameters of the distribution that defines them. In this case, the Poisson distribution is defined by a single parameter $\lambda$. Thus, we have a forward-pass of the form

function (decoder::PoissonDecoder)(z::AbstractArray)
     # Run input to decoder network
@@ -136,4 +136,4 @@
         loglikelihood = sum(x .* log.(λ) - λ - loggamma.(x .+ 1))
 
         return loglikelihood
-end # function

where we use the loggamma function from SpecialFunctions.jl to compute the log of the factorial of x_i.

Warning

We only defined the decoder_loglikelihood method for z::AbstractVector. One should also include a method for z::AbstractMatrix used when performing batch training.

With these two functions defined, our PoissonDecoder is ready to be used with any of the different VAE flavors included in AutoEncoderToolkit.jl!

+end # function

where we use the loggamma function from SpecialFunctions.jl to compute the log of the factorial of x_i.

Warning

We only defined the decoder_loglikelihood method for z::AbstractVector. One should also include a method for z::AbstractMatrix used when performing batch training.

With these two functions defined, our PoissonDecoder is ready to be used with any of the different VAE flavors included in AutoEncoderToolkit.jl!

diff --git a/dev/guidelines/index.html b/dev/guidelines/index.html index 034bb86..7580bf3 100644 --- a/dev/guidelines/index.html +++ b/dev/guidelines/index.html @@ -1,2 +1,2 @@ -Community Guidelines · AutoEncoderToolkit

Community Guidelines

Contributing to the Software

For those interested in contributing to AutoEncoderToolkit.jl, please refer to the GitHub repository. The project welcomes contributions to

  • Expand the list of available models.
  • Improve the performance of existing models.
  • Add new features to the toolkit.
  • Improve the documentation.

Reporting Issues or Problems

If you encounter any issues or problems with the software, you can report them directly on the GitHub repository's issues page.

Seeking Support

For support and further inquiries, consider checking the documentation and existing issues on the GitHub repository. If you still do not find the answer, you can open a new issue on the GitHub repository's issues page.

+Community Guidelines · AutoEncoderToolkit

Community Guidelines

Contributing to the Software

For those interested in contributing to AutoEncoderToolkit.jl, please refer to the GitHub repository. The project welcomes contributions to

  • Expand the list of available models.
  • Improve the performance of existing models.
  • Add new features to the toolkit.
  • Improve the documentation.

Reporting Issues or Problems

If you encounter any issues or problems with the software, you can report them directly on the GitHub repository's issues page.

Seeking Support

For support and further inquiries, consider checking the documentation and existing issues on the GitHub repository. If you still do not find the answer, you can open a new issue on the GitHub repository's issues page.

diff --git a/dev/hvae/index.html b/dev/hvae/index.html index a6cd2a8..2ef2463 100644 --- a/dev/hvae/index.html +++ b/dev/hvae/index.html @@ -1,7 +1,7 @@ -HVAE · AutoEncoderToolkit

Hamiltonian Variational Autoencoder

The Hamiltonian Variational Autoencoder (HVAE) is a variant of the Variational autoencoder (VAE) that uses Hamiltonian dynamics to improve the sampling of the latent space representation. HVAE combines ideas from Hamiltonian Monte Carlo, annealed importance sampling, and variational inference to improve the latent space representation of the VAE.

For the implementation of the HVAE in AutoEncoderToolkit.jl, the HVAE struct inherits directly from the VAE struct and adds the necessary functions to compute the Hamiltonian dynamics steps as part of the training protocol. An HVAE object is created by simply passing a VAE object to the constructor. This way, we can use Julias multiple dispatch to extend the functionality of the VAE object without having to redefine the entire structure.

Warning

HVAEs require the computation of nested gradients. This means that the AutoDiff framework must differentiate a function of an already AutoDiff differentiated function. This is known to be problematic for Julia's AutoDiff backends. See details below to understand how to we circumvent this problem.

Reference

Caterini, A. L., Doucet, A. & Sejdinovic, D. Hamiltonian Variational Auto-Encoder. 11 (2018).

HVAE struct

AutoEncoderToolkit.HVAEs.HVAE — Type
struct HVAE{
+HVAE · AutoEncoderToolkit
Hamiltonian Variational Autoencoder
The Hamiltonian Variational Autoencoder (HVAE) is a variant of the Variational autoencoder (VAE) that uses Hamiltonian dynamics to improve the sampling of the latent space representation. HVAE combines ideas from Hamiltonian Monte Carlo, annealed importance sampling, and variational inference to improve the latent space representation of the VAE.
For the implementation of the HVAE in AutoEncoderToolkit.jl, the HVAE struct inherits directly from the VAE struct and adds the necessary functions to compute the Hamiltonian dynamics steps as part of the training protocol. An HVAE object is created by simply passing a VAE object to the constructor. This way, we can use Julias multiple dispatch to extend the functionality of the VAE object without having to redefine the entire structure.
Warning
HVAEs require the computation of nested gradients. This means that the AutoDiff framework must differentiate a function of an already AutoDiff differentiated function. This is known to be problematic for Julia's AutoDiff backends. See details below to understand how to we circumvent this problem.
Reference
Caterini, A. L., Doucet, A. & Sejdinovic, D. Hamiltonian Variational Auto-Encoder. 11 (2018).
HVAE struct
AutoEncoderToolkit.HVAEs.HVAE — Type
struct HVAE{
     V<:VAE{<:AbstractVariationalEncoder,<:AbstractVariationalDecoder}
-} <: AbstractVariationalAutoEncoder
Hamiltonian Variational Autoencoder (HVAE) model defined for Flux.jl.
Fields
  • vae::V: A Variational Autoencoder (VAE) model that forms the basis of the   HVAE. V is a subtype of VAE with a specific AbstractVariationalEncoder   and AbstractVariationalDecoder.
An HVAE is a type of Variational Autoencoder (VAE) that uses Hamiltonian Monte Carlo (HMC) to sample from the posterior distribution in the latent space. The VAE's encoder compresses the input into a low-dimensional probabilistic representation q(z|x). The VAE's decoder tries to reconstruct the original input from a sampled point in the latent space p(x|z). 
The HMC sampling in the latent space allows the HVAE to better capture complex posterior distributions compared to a standard VAE, which assumes a simple Gaussian posterior. This can lead to more accurate reconstructions and better disentanglement of latent variables.
Forward pass
AutoEncoderToolkit.HVAEs.HVAE — Method
(hvae::HVAE{VAE{E,D}})(
+} <: AbstractVariationalAutoEncoder
Hamiltonian Variational Autoencoder (HVAE) model defined for Flux.jl.
Fields
  • vae::V: A Variational Autoencoder (VAE) model that forms the basis of the   HVAE. V is a subtype of VAE with a specific AbstractVariationalEncoder   and AbstractVariationalDecoder.
An HVAE is a type of Variational Autoencoder (VAE) that uses Hamiltonian Monte Carlo (HMC) to sample from the posterior distribution in the latent space. The VAE's encoder compresses the input into a low-dimensional probabilistic representation q(z|x). The VAE's decoder tries to reconstruct the original input from a sampled point in the latent space p(x|z). 
The HMC sampling in the latent space allows the HVAE to better capture complex posterior distributions compared to a standard VAE, which assumes a simple Gaussian posterior. This can lead to more accurate reconstructions and better disentanglement of latent variables.
Forward pass
AutoEncoderToolkit.HVAEs.HVAE — Method
(hvae::HVAE{VAE{E,D}})(
     x::AbstractArray;
     ϵ::Union{<:Number,<:AbstractVector}=Float32(1E-4),
     K::Int=3,
@@ -12,7 +12,7 @@
     ),
     tempering_schedule::Function=quadratic_tempering,
     latent::Bool=false,
-) where {E<:AbstractGaussianLogEncoder,D<:AbstractVariationalDecoder}
Run the Hamiltonian Variational Autoencoder (HVAE) on the given input.
Arguments
  • x::AbstractArray: The input to the HVAE. If Vector, it represents a single data point. If Array, the last dimension must contain each of the data points.
Optional Keyword Arguments
  • ϵ::Union{<:Number,<:AbstractVector}=0.0001: The step size for the leapfrog steps in the HMC part of the HVAE. If it is a scalar, the same step size is used for all dimensions. If it is an array, each element corresponds to the step size for a specific dimension.
  • K::Int=3: The number of leapfrog steps to perform in the Hamiltonian Monte Carlo (HMC) part of the HVAE.
  • βₒ::Number=0.3f0: The initial inverse temperature for the tempering schedule.
  • ∇U_kwargs::NamedTuple: Additional keyword arguments to be passed to the ∇potential_energy function. Default is a NamedTuple with reconstruction_loglikelihood and latent_logprior.
  • tempering_schedule::Function=quadratic_tempering: The function to compute   the tempering schedule in the HVAE.
  • latent::Bool=false: If true, the function returns a NamedTuple containing the outputs of the encoder and decoder, and the final state of the phase space after the leapfrog and tempering steps. If false, the function only returns the output of the decoder.
Returns
If latent=true, the function returns a NamedTuple with the following fields:
  • encoder: The outputs of the encoder.
  • decoder: The output of the decoder.
  • phase_space: The final state of the phase space after the leapfrog and   tempering steps.
If latent=false, the function only returns the output of the decoder.
Description
This function runs the HVAE on the given input. It first passes the input through the encoder to obtain the mean and log standard deviation of the latent space. It then uses the reparameterization trick to sample from the latent space. After that, it performs the leapfrog and tempering steps to refine the sample from the latent space. Finally, it passes the refined sample through the decoder to obtain the output.
Notes
Ensure that the dimensions of x match the input dimensions of the HVAE, and that the dimensions of ϵ match the dimensions of the latent space.
Loss function
AutoEncoderToolkit.HVAEs.loss — Function
loss(
+) where {E<:AbstractGaussianLogEncoder,D<:AbstractVariationalDecoder}
Run the Hamiltonian Variational Autoencoder (HVAE) on the given input.
Arguments
  • x::AbstractArray: The input to the HVAE. If Vector, it represents a single data point. If Array, the last dimension must contain each of the data points.
Optional Keyword Arguments
  • ϵ::Union{<:Number,<:AbstractVector}=0.0001: The step size for the leapfrog steps in the HMC part of the HVAE. If it is a scalar, the same step size is used for all dimensions. If it is an array, each element corresponds to the step size for a specific dimension.
  • K::Int=3: The number of leapfrog steps to perform in the Hamiltonian Monte Carlo (HMC) part of the HVAE.
  • βₒ::Number=0.3f0: The initial inverse temperature for the tempering schedule.
  • ∇U_kwargs::NamedTuple: Additional keyword arguments to be passed to the ∇potential_energy function. Default is a NamedTuple with reconstruction_loglikelihood and latent_logprior.
  • tempering_schedule::Function=quadratic_tempering: The function to compute   the tempering schedule in the HVAE.
  • latent::Bool=false: If true, the function returns a NamedTuple containing the outputs of the encoder and decoder, and the final state of the phase space after the leapfrog and tempering steps. If false, the function only returns the output of the decoder.
Returns
If latent=true, the function returns a NamedTuple with the following fields:
  • encoder: The outputs of the encoder.
  • decoder: The output of the decoder.
  • phase_space: The final state of the phase space after the leapfrog and   tempering steps.
If latent=false, the function only returns the output of the decoder.
Description
This function runs the HVAE on the given input. It first passes the input through the encoder to obtain the mean and log standard deviation of the latent space. It then uses the reparameterization trick to sample from the latent space. After that, it performs the leapfrog and tempering steps to refine the sample from the latent space. Finally, it passes the refined sample through the decoder to obtain the output.
Notes
Ensure that the dimensions of x match the input dimensions of the HVAE, and that the dimensions of ϵ match the dimensions of the latent space.
Loss function
AutoEncoderToolkit.HVAEs.loss — Function
loss(
     hvae::HVAE,
     x::AbstractArray;
     K::Int=3,
@@ -45,7 +45,7 @@
     reg_strength::Float32=1.0f0,
     logp_prefactor::AbstractArray=ones(Float32, 3),
     logq_prefactor::AbstractArray=ones(Float32, 3),
-)
Compute the loss for a Hamiltonian Variational Autoencoder (HVAE).
Arguments
  • hvae::HVAE: The HVAE used to encode the input data and decode the latent space.
  • x_in::AbstractArray: Input data to the HVAE encoder. The last dimension is taken as having each of the samples in a batch.
  • x_out::AbstractArray: The data against which the reconstruction is compared. If Array, the last dimension must contain each of the data points.
Optional Keyword Arguments
  • K::Int: The number of HMC steps (default is 3).
  • ϵ::Union{<:Number,<:AbstractVector}: The step size for the leapfrog integrator (default is 0.001).
  • βₒ::Number: The initial inverse temperature (default is 0.3).
  • ∇U_kwargs::NamedTuple: Additional keyword arguments to be passed to the ∇potential_energy function.
  • tempering_schedule::Function: The tempering schedule function used in the HMC (default is quadratic_tempering).
  • reg_function::Union{Function, Nothing}=nothing: A function that computes the regularization term based on the VAE outputs. This function must take as input the VAE outputs and the keyword arguments provided in reg_kwargs.
  • reg_kwargs::NamedTuple=NamedTuple(): Keyword arguments to pass to the regularization function.
  • reg_strength::Float32=1.0f0: The strength of the regularization term.
  • logp_prefactor::AbstractArray: A 3-element array to scale the log likelihood, log prior of the latent variables, and log prior of the momentum variables. Default is an array of ones.
  • logq_prefactor::AbstractArray: A 3-element array to scale the log posterior of the initial latent variables, log prior of the initial momentum variables, and the tempering Jacobian term. Default is an array of ones.
Returns
  • The computed loss.
Training
AutoEncoderToolkit.HVAEs.train! — Function
train!(
+)
Compute the loss for a Hamiltonian Variational Autoencoder (HVAE).
Arguments
  • hvae::HVAE: The HVAE used to encode the input data and decode the latent space.
  • x_in::AbstractArray: Input data to the HVAE encoder. The last dimension is taken as having each of the samples in a batch.
  • x_out::AbstractArray: The data against which the reconstruction is compared. If Array, the last dimension must contain each of the data points.
Optional Keyword Arguments
  • K::Int: The number of HMC steps (default is 3).
  • ϵ::Union{<:Number,<:AbstractVector}: The step size for the leapfrog integrator (default is 0.001).
  • βₒ::Number: The initial inverse temperature (default is 0.3).
  • ∇U_kwargs::NamedTuple: Additional keyword arguments to be passed to the ∇potential_energy function.
  • tempering_schedule::Function: The tempering schedule function used in the HMC (default is quadratic_tempering).
  • reg_function::Union{Function, Nothing}=nothing: A function that computes the regularization term based on the VAE outputs. This function must take as input the VAE outputs and the keyword arguments provided in reg_kwargs.
  • reg_kwargs::NamedTuple=NamedTuple(): Keyword arguments to pass to the regularization function.
  • reg_strength::Float32=1.0f0: The strength of the regularization term.
  • logp_prefactor::AbstractArray: A 3-element array to scale the log likelihood, log prior of the latent variables, and log prior of the momentum variables. Default is an array of ones.
  • logq_prefactor::AbstractArray: A 3-element array to scale the log posterior of the initial latent variables, log prior of the initial momentum variables, and the tempering Jacobian term. Default is an array of ones.
Returns
  • The computed loss.
Training
AutoEncoderToolkit.HVAEs.train! — Function
train!(
     hvae::HVAE, 
     x::AbstractArray, 
     opt::NamedTuple; 
@@ -67,7 +67,7 @@
     :∇U_kwargs => Dict(
         :adtype => :finite
     )
-)
Note
Although verbose, the nested dictionaries help to keep everything organized. (PRs with better design ideas are welcome!)
The default both for cpu and gpu devices is :finite.
AutoEncoderToolkit.HVAEs.∇potential_energy_finite — Function
∇potential_energy_finite(
+)
Note
Although verbose, the nested dictionaries help to keep everything organized. (PRs with better design ideas are welcome!)
The default both for cpu and gpu devices is :finite.
AutoEncoderToolkit.HVAEs.∇potential_energy_finite — Function
∇potential_energy_finite(
     x::AbstractArray,
     z::AbstractVecOrMat,
     decoder::AbstractVariationalDecoder,
@@ -82,7 +82,7 @@
     reconstruction_loglikelihood::Function=decoder_loglikelihood,
     latent_logprior::Function=spherical_logprior,
     fdtype::Symbol=:central
-)
Compute the gradient of the potential energy of a Hamiltonian Variational Autoencoder (HVAE) with respect to the latent variables z using finite difference method. This function returns the gradient of the potential energy computed for given data x and latent variable z.
Arguments
  • x::AbstractArray: An array representing the input data. The last dimension corresponds to different data points.
  • z::AbstractVecOrMat: A latent variable encoding of the input data. If a matrix, each column corresponds to a different data point.
  • hvae::HVAE: An HVAE model that contains a decoder which maps the latent variables to the data space.
Optional Keyword Arguments
  • reconstruction_loglikelihood::Function=decoder_loglikelihood: A function representing the log-likelihood function used by the decoder. The function must take as first input an array x representing the data, as second input a vector or matrix z representing the latent variable, and as third input a decoder. Default is decoder_loglikelihood.
  • latent_logprior::Function=spherical_logprior: A function representing the log-prior distribution used in the autoencoder. The function must take as single input a vector or matrix z representing the latent variable. Default is spherical_logprior.  
  • fdtype::Symbol=:central: A symbol representing the type of finite difference method to use. Default is :central, but it can also be :forward.
Returns
  • gradient: The computed gradient of the potential energy for the given input x and latent variable z.
AutoEncoderToolkit.HVAEs.∇potential_energy_TaylorDiff — Function
∇potential_energy_TaylorDiff(
+)
Compute the gradient of the potential energy of a Hamiltonian Variational Autoencoder (HVAE) with respect to the latent variables z using finite difference method. This function returns the gradient of the potential energy computed for given data x and latent variable z.
Arguments
  • x::AbstractArray: An array representing the input data. The last dimension corresponds to different data points.
  • z::AbstractVecOrMat: A latent variable encoding of the input data. If a matrix, each column corresponds to a different data point.
  • hvae::HVAE: An HVAE model that contains a decoder which maps the latent variables to the data space.
Optional Keyword Arguments
  • reconstruction_loglikelihood::Function=decoder_loglikelihood: A function representing the log-likelihood function used by the decoder. The function must take as first input an array x representing the data, as second input a vector or matrix z representing the latent variable, and as third input a decoder. Default is decoder_loglikelihood.
  • latent_logprior::Function=spherical_logprior: A function representing the log-prior distribution used in the autoencoder. The function must take as single input a vector or matrix z representing the latent variable. Default is spherical_logprior.  
  • fdtype::Symbol=:central: A symbol representing the type of finite difference method to use. Default is :central, but it can also be :forward.
Returns
  • gradient: The computed gradient of the potential energy for the given input x and latent variable z.
AutoEncoderToolkit.HVAEs.∇potential_energy_TaylorDiff — Function
∇potential_energy_TaylorDiff(
     x::AbstractArray,
     z::AbstractVecOrMat,
     hvae::HVAE;
@@ -94,7 +94,7 @@
     hvae::HVAE;
     reconstruction_loglikelihood::Function=decoder_loglikelihood,
     latent_logprior::Function=spherical_logprior,
-)
Compute the gradient of the potential energy of a Hamiltonian Variational Autoencoder (HVAE) with respect to the latent variables z using Taylor series differentiation. This function returns the gradient of the potential energy computed for given data x and latent variable z.
Arguments
  • x::AbstractArray: An array representing the input data. The last dimension corresponds to different data points.
  • z::AbstractVecOrMat: A latent variable encoding of the input data. If a matrix, each column corresponds to a different data point.
  • hvae::HVAE: An HVAE model that contains a decoder which maps the latent variables to the data space.
Optional Keyword Arguments
  • reconstruction_loglikelihood::Function=decoder_loglikelihood: A function representing the log-likelihood function used by the decoder. The function must take as first input an array x representing the data, as second input a vector or matrix z representing the latent variable, and as third input a decoder. Default is decoder_loglikelihood.
  • latent_logprior::Function=spherical_logprior: A function representing the log-prior distribution used in the autoencoder. The function must take as single input a vector or matrix z representing the latent variable.  Default is spherical_logprior.  
Returns
  • gradient: The computed gradient of the potential energy for the given input   x and latent variable z.
Other Functions
AutoEncoderToolkit.HVAEs.potential_energy — Function
potential_energy(
+)
Compute the gradient of the potential energy of a Hamiltonian Variational Autoencoder (HVAE) with respect to the latent variables z using Taylor series differentiation. This function returns the gradient of the potential energy computed for given data x and latent variable z.
Arguments
  • x::AbstractArray: An array representing the input data. The last dimension corresponds to different data points.
  • z::AbstractVecOrMat: A latent variable encoding of the input data. If a matrix, each column corresponds to a different data point.
  • hvae::HVAE: An HVAE model that contains a decoder which maps the latent variables to the data space.
Optional Keyword Arguments
  • reconstruction_loglikelihood::Function=decoder_loglikelihood: A function representing the log-likelihood function used by the decoder. The function must take as first input an array x representing the data, as second input a vector or matrix z representing the latent variable, and as third input a decoder. Default is decoder_loglikelihood.
  • latent_logprior::Function=spherical_logprior: A function representing the log-prior distribution used in the autoencoder. The function must take as single input a vector or matrix z representing the latent variable.  Default is spherical_logprior.  
Returns
  • gradient: The computed gradient of the potential energy for the given input   x and latent variable z.
Other Functions
AutoEncoderToolkit.HVAEs.potential_energy — Function
potential_energy(
     x::AbstractVector,
     z::AbstractVector,
     decoder::AbstractVariationalDecoder,
@@ -107,7 +107,7 @@
     hvae::HVAE;
     reconstruction_loglikelihood::Function=decoder_loglikelihood,
     latent_logprior::Function=spherical_logprior
-)
Compute the potential energy of a Hamiltonian Variational Autoencoder (HVAE). In the context of Hamiltonian Monte Carlo (HMC), the potential energy is defined as the negative log-posterior. This function computes the potential energy for given data x and latent variable z. It does this by computing the log-likelihood of x under the distribution defined by reconstruction_loglikelihood(x, z, hvae.vae.decoder, decoder_output), and the log-prior of z under the latent_logprior distribution. The potential energy is then computed as:
            U(x, z) = -log p(x | z) - log p(z)
Arguments
  • x::AbstractArray: An array representing the input data. The last dimension corresponds to different data points.
  • z::AbstractVecOrMat: A latent variable encoding of the input data. If a matrix, each column corresponds to a different data point.
  • hvae::HVAE: A Hamiltonian Variational Autoencoder that contains the decoder.
Optional Keyword Arguments
  • reconstruction_loglikelihood::Function=decoder_loglikelihood: A function representing the log-likelihood function used by the decoder. The function must take as first input an array x representing the data, as second input a vector or matrix z representing the latent variable, as third input a decoder, and as fourth input a NamedTuple representing the decoder output. Default is decoder_loglikelihood.
  • latent_logprior::Function=spherical_logprior: A function representing the log-prior distribution used in the autoencoder. The function must take as single input a vector or matrix z representing the latent variable.  Default is spherical_logprior.  
Returns
  • energy: The computed potential energy for the given input x and latent   variable z.
AutoEncoderToolkit.HVAEs.∇potential_energy — Function
∇potential_energy(
+)
Compute the potential energy of a Hamiltonian Variational Autoencoder (HVAE). In the context of Hamiltonian Monte Carlo (HMC), the potential energy is defined as the negative log-posterior. This function computes the potential energy for given data x and latent variable z. It does this by computing the log-likelihood of x under the distribution defined by reconstruction_loglikelihood(x, z, hvae.vae.decoder, decoder_output), and the log-prior of z under the latent_logprior distribution. The potential energy is then computed as:
            U(x, z) = -log p(x | z) - log p(z)
Arguments
  • x::AbstractArray: An array representing the input data. The last dimension corresponds to different data points.
  • z::AbstractVecOrMat: A latent variable encoding of the input data. If a matrix, each column corresponds to a different data point.
  • hvae::HVAE: A Hamiltonian Variational Autoencoder that contains the decoder.
Optional Keyword Arguments
  • reconstruction_loglikelihood::Function=decoder_loglikelihood: A function representing the log-likelihood function used by the decoder. The function must take as first input an array x representing the data, as second input a vector or matrix z representing the latent variable, as third input a decoder, and as fourth input a NamedTuple representing the decoder output. Default is decoder_loglikelihood.
  • latent_logprior::Function=spherical_logprior: A function representing the log-prior distribution used in the autoencoder. The function must take as single input a vector or matrix z representing the latent variable.  Default is spherical_logprior.  
Returns
  • energy: The computed potential energy for the given input x and latent   variable z.
AutoEncoderToolkit.HVAEs.∇potential_energy — Function
∇potential_energy(
     x::AbstractArray,
     z::AbstractVecOrMat,
     decoder::AbstractVariationalDecoder,
@@ -124,7 +124,7 @@
     latent_logprior::Function=spherical_logprior,
     adtype::Union{Symbol,Nothing}=nothing,
     adkwargs::NamedTuple=NamedTuple(),
-)
Compute the gradient of the potential energy of a Hamiltonian Variational Autoencoder (HVAE) with respect to the latent variables z using the specified automatic differentiation method. This function returns the gradient of the potential energy computed for given data x and latent variable z.
Arguments
  • x::AbstractArray: An array representing the input data. The last dimension corresponds to different data points.
  • z::AbstractVecOrMat: A latent variable encoding of the input data. If a matrix, each column corresponds to a different data point.
  • hvae::HVAE: An HVAE model that contains a decoder which maps the latent variables to the data space.
Optional Keyword Arguments
  • reconstruction_loglikelihood::Function=decoder_loglikelihood: A function representing the log-likelihood function used by the decoder. The function must take as first input an array x representing the data, as second input a vector or matrix z representing the latent variable, and as third input a decoder. Default is decoder_loglikelihood.
  • latent_logprior::Function=spherical_logprior: A function representing the log-prior distribution used in the autoencoder. The function must take as single input a vector or matrix z representing the latent variable.  Default is spherical_logprior.  
    • adtype::Symbol=:finite`: The type of automatic differentiation method to
    use. Must be :finite or :TaylorDiff. Default is :finite.
  • adkwargs::NamedTuple=NamedTuple(): Additional keyword arguments to pass to the automatic differentiation method.
Returns
  • gradient: The computed gradient of the potential energy for the given input x and latent variable z.
AutoEncoderToolkit.HVAEs.leapfrog_step — Function
leapfrog_step(
+)
Compute the gradient of the potential energy of a Hamiltonian Variational Autoencoder (HVAE) with respect to the latent variables z using the specified automatic differentiation method. This function returns the gradient of the potential energy computed for given data x and latent variable z.
Arguments
  • x::AbstractArray: An array representing the input data. The last dimension corresponds to different data points.
  • z::AbstractVecOrMat: A latent variable encoding of the input data. If a matrix, each column corresponds to a different data point.
  • hvae::HVAE: An HVAE model that contains a decoder which maps the latent variables to the data space.
Optional Keyword Arguments
  • reconstruction_loglikelihood::Function=decoder_loglikelihood: A function representing the log-likelihood function used by the decoder. The function must take as first input an array x representing the data, as second input a vector or matrix z representing the latent variable, and as third input a decoder. Default is decoder_loglikelihood.
  • latent_logprior::Function=spherical_logprior: A function representing the log-prior distribution used in the autoencoder. The function must take as single input a vector or matrix z representing the latent variable.  Default is spherical_logprior.  
    • adtype::Symbol=:finite`: The type of automatic differentiation method to
    use. Must be :finite or :TaylorDiff. Default is :finite.
  • adkwargs::NamedTuple=NamedTuple(): Additional keyword arguments to pass to the automatic differentiation method.
Returns
  • gradient: The computed gradient of the potential energy for the given input x and latent variable z.
AutoEncoderToolkit.HVAEs.leapfrog_step — Function
leapfrog_step(
     x::AbstractArray,
     z::AbstractVecOrMat,
     ρ::AbstractVecOrMat,
@@ -145,10 +145,10 @@
         reconstruction_loglikelihood=reconstruction_loglikelihood,
         latent_logprior=spherical_logprior,
     )
-)
Perform a full step of the leapfrog integrator for Hamiltonian dynamics.
The leapfrog integrator is a numerical integration scheme used to simulate Hamiltonian dynamics. It consists of three steps:
  1. Half update of the momentum variable: 
     ρ(t + ϵ/2) = ρ(t) - 0.5 * ϵ * ∇z_U(z(t), ρ(t + ϵ/2)).
  2. Full update of the position variable: 
     z(t + ϵ) = z(t) + ϵ * ρ(t + ϵ/2).
  3. Half update of the momentum variable: 
     ρ(t + ϵ) = ρ(t + ϵ/2) - 0.5 * ϵ * ∇z_U(z(t + ϵ), ρ(t + ϵ/2)).
This function performs these three steps in sequence.
Arguments
  • x::AbstractArray: The point in the data space. This does not necessarily need to be a vector. Array inputs are supported. The last dimension is assumed to have each of the data points.
  • z::AbstractVecOrMat: The point in the latent space. If matrix, each column represents a point in the latent space.
  • ρ::AbstractVecOrMat: The momentum. If matrix, each column represents a momentum vector.
  • hvae::HVAE: An HVAE model that contains the decoder.
Optional Keyword Arguments
  • ϵ::Union{<:Number,<:AbstractVector}=Float32(1E-4): The step size. Default is 0.0001.
  • ∇U_kwargs::NamedTuple: The keyword arguments for ∇potential_energy.  Default is a tuple with reconstruction_loglikelihood and latent_logprior.
Returns
A tuple (z̄, ρ̄, decoder_output_z̄) representing the updated position and momentum after performing the full leapfrog step as well as the decoder output of the updated position.
AutoEncoderToolkit.HVAEs.quadratic_tempering — Function
quadratic_tempering(βₒ::AbstractFloat, k::Int, K::Int)
Compute the inverse temperature βₖ at a given stage k of a tempering schedule with K total stages, using a quadratic tempering scheme. 
Tempering is a technique used in sampling algorithms to improve mixing and convergence. It involves running parallel chains of the algorithm at different "temperatures", and swapping states between the chains. The "temperature" of a chain is controlled by an inverse temperature parameter β, which is varied according to a tempering schedule. 
In a quadratic tempering schedule, the inverse temperature βₖ at stage k is computed as the square of the quantity ((1 - 1 / √(βₒ)) * (k / K)^2 + 1 / √(βₒ)), where βₒ is the initial inverse temperature. This schedule starts at βₒ when k = 0, and increases quadratically as k increases, reaching 1 when k = K.
Arguments
  • βₒ::AbstractFloat: The initial inverse temperature.
  • k::Int: The current stage of the tempering schedule.
  • K::Int: The total number of stages in the tempering schedule.
Returns
  • βₖ::AbstractFloat: The inverse temperature at stage k.
AutoEncoderToolkit.HVAEs.null_tempering — Function
    null_tempering(βₒ::T, k::Int, K::Int) where {T<:AbstractFloat}
Return the initial inverse temperature βₒ. This function is used in the context of tempered Hamiltonian Monte Carlo (HMC) methods, where tempering involves running HMC at different "temperatures" to improve mixing and convergence. 
In this case, null_tempering is a simple tempering schedule that does not actually change the temperature—it always returns the initial inverse temperature βₒ. This can be useful as a default or placeholder tempering schedule.
Arguments
  • βₒ::AbstractFloat: The initial inverse temperature. 
  • k::Int: The current step in the tempering schedule. Not used in this   function, but included for compatibility with other tempering schedules.
  • K::Int: The total number of steps in the tempering schedule. Not used in   this function, but included for compatibility with other tempering   schedules.
Returns
  • β::T: The inverse temperature for the current step, which is always βₒ in this case.
Example
βₒ = 0.5
+)
Perform a full step of the leapfrog integrator for Hamiltonian dynamics.
The leapfrog integrator is a numerical integration scheme used to simulate Hamiltonian dynamics. It consists of three steps:
  1. Half update of the momentum variable: 
     ρ(t + ϵ/2) = ρ(t) - 0.5 * ϵ * ∇z_U(z(t), ρ(t + ϵ/2)).
  2. Full update of the position variable: 
     z(t + ϵ) = z(t) + ϵ * ρ(t + ϵ/2).
  3. Half update of the momentum variable: 
     ρ(t + ϵ) = ρ(t + ϵ/2) - 0.5 * ϵ * ∇z_U(z(t + ϵ), ρ(t + ϵ/2)).
This function performs these three steps in sequence.
Arguments
  • x::AbstractArray: The point in the data space. This does not necessarily need to be a vector. Array inputs are supported. The last dimension is assumed to have each of the data points.
  • z::AbstractVecOrMat: The point in the latent space. If matrix, each column represents a point in the latent space.
  • ρ::AbstractVecOrMat: The momentum. If matrix, each column represents a momentum vector.
  • hvae::HVAE: An HVAE model that contains the decoder.
Optional Keyword Arguments
  • ϵ::Union{<:Number,<:AbstractVector}=Float32(1E-4): The step size. Default is 0.0001.
  • ∇U_kwargs::NamedTuple: The keyword arguments for ∇potential_energy.  Default is a tuple with reconstruction_loglikelihood and latent_logprior.
Returns
A tuple (z̄, ρ̄, decoder_output_z̄) representing the updated position and momentum after performing the full leapfrog step as well as the decoder output of the updated position.
AutoEncoderToolkit.HVAEs.quadratic_tempering — Function
quadratic_tempering(βₒ::AbstractFloat, k::Int, K::Int)
Compute the inverse temperature βₖ at a given stage k of a tempering schedule with K total stages, using a quadratic tempering scheme. 
Tempering is a technique used in sampling algorithms to improve mixing and convergence. It involves running parallel chains of the algorithm at different "temperatures", and swapping states between the chains. The "temperature" of a chain is controlled by an inverse temperature parameter β, which is varied according to a tempering schedule. 
In a quadratic tempering schedule, the inverse temperature βₖ at stage k is computed as the square of the quantity ((1 - 1 / √(βₒ)) * (k / K)^2 + 1 / √(βₒ)), where βₒ is the initial inverse temperature. This schedule starts at βₒ when k = 0, and increases quadratically as k increases, reaching 1 when k = K.
Arguments
  • βₒ::AbstractFloat: The initial inverse temperature.
  • k::Int: The current stage of the tempering schedule.
  • K::Int: The total number of stages in the tempering schedule.
Returns
  • βₖ::AbstractFloat: The inverse temperature at stage k.
AutoEncoderToolkit.HVAEs.null_tempering — Function
    null_tempering(βₒ::T, k::Int, K::Int) where {T<:AbstractFloat}
Return the initial inverse temperature βₒ. This function is used in the context of tempered Hamiltonian Monte Carlo (HMC) methods, where tempering involves running HMC at different "temperatures" to improve mixing and convergence. 
In this case, null_tempering is a simple tempering schedule that does not actually change the temperature—it always returns the initial inverse temperature βₒ. This can be useful as a default or placeholder tempering schedule.
Arguments
  • βₒ::AbstractFloat: The initial inverse temperature. 
  • k::Int: The current step in the tempering schedule. Not used in this   function, but included for compatibility with other tempering schedules.
  • K::Int: The total number of steps in the tempering schedule. Not used in   this function, but included for compatibility with other tempering   schedules.
Returns
  • β::T: The inverse temperature for the current step, which is always βₒ in this case.
Example
βₒ = 0.5
 k = 1
 K = 10
-β = null_tempering(βₒ, k, K)  # β will be 0.5
AutoEncoderToolkit.HVAEs.leapfrog_tempering_step — Function
leapfrog_tempering_step(
     x::AbstractArray,
     zₒ::AbstractVecOrMat,
     decoder::AbstractVariationalDecoder,
@@ -173,20 +173,20 @@
         latent_logprior=spherical_logprior,
     ),
     tempering_schedule::Function=quadratic_tempering,
-)
Combines the leapfrog and tempering steps into a single function for the Hamiltonian Variational Autoencoder (HVAE).
Arguments
  • x::AbstractArray: The data to be processed. If Array, the last dimension must be of size 1.
  • zₒ::AbstractVecOrMat: The initial latent variable. 
  • hvae::HVAE: An HVAE model that contains the decoder.
Optional Keyword Arguments
  • ϵ::Union{<:Number,<:AbstractVector}: The step size for the leapfrog steps in the HMC algorithm. This can be a scalar or an array. Default is 0.0001.  
  • K::Int: The number of leapfrog steps to perform in the Hamiltonian Monte Carlo (HMC) algorithm. Default is 3.
  • βₒ::Number: The initial inverse temperature for the tempering schedule. Default is 0.3f0.
  • ∇U_kwargs::NamedTuple: Additional keyword arguments to be passed to the ∇potential_energy function. Default is a NamedTuple with reconstruction_loglikelihood and latent_logprior.
  • tempering_schedule::Function: The function to compute the inverse temperature at each step in the HMC algorithm. Defaults to quadratic_tempering. This function must take three arguments: First, βₒ, an initial inverse temperature, second, k, the current step in the tempering schedule, and third, K, the total number of steps in the tempering schedule.
Returns
  • A NamedTuple with the following keys: 
    • z_init: The initial latent variable. 
    • ρ_init: The initial momentum variable. 
    • z_final: The final latent variable after K leapfrog steps. 
    • ρ_final: The final momentum variable after K leapfrog steps. 
  • The decoder output at the final latent variable is also returned. Note: This is not in the same named tuple as the other outputs, but as a separate output.
Description
The function first samples a random momentum variable γₒ from a standard normal distribution and scales it by the inverse square root of the initial inverse temperature βₒ to obtain the initial momentum variable ρₒ. Then, it performs K leapfrog steps, each followed by a tempering step, to generate a new sample from the latent space.
Note
Ensure the input data x and the initial latent variable zₒ match the expected input dimensionality for the HVAE model.
AutoEncoderToolkit.HVAEs._log_p̄ — Function
_log_p̄(
+)
Combines the leapfrog and tempering steps into a single function for the Hamiltonian Variational Autoencoder (HVAE).
Arguments
  • x::AbstractArray: The data to be processed. If Array, the last dimension must be of size 1.
  • zₒ::AbstractVecOrMat: The initial latent variable. 
  • hvae::HVAE: An HVAE model that contains the decoder.
Optional Keyword Arguments
  • ϵ::Union{<:Number,<:AbstractVector}: The step size for the leapfrog steps in the HMC algorithm. This can be a scalar or an array. Default is 0.0001.  
  • K::Int: The number of leapfrog steps to perform in the Hamiltonian Monte Carlo (HMC) algorithm. Default is 3.
  • βₒ::Number: The initial inverse temperature for the tempering schedule. Default is 0.3f0.
  • ∇U_kwargs::NamedTuple: Additional keyword arguments to be passed to the ∇potential_energy function. Default is a NamedTuple with reconstruction_loglikelihood and latent_logprior.
  • tempering_schedule::Function: The function to compute the inverse temperature at each step in the HMC algorithm. Defaults to quadratic_tempering. This function must take three arguments: First, βₒ, an initial inverse temperature, second, k, the current step in the tempering schedule, and third, K, the total number of steps in the tempering schedule.
Returns
  • A NamedTuple with the following keys: 
    • z_init: The initial latent variable. 
    • ρ_init: The initial momentum variable. 
    • z_final: The final latent variable after K leapfrog steps. 
    • ρ_final: The final momentum variable after K leapfrog steps. 
  • The decoder output at the final latent variable is also returned. Note: This is not in the same named tuple as the other outputs, but as a separate output.
Description
The function first samples a random momentum variable γₒ from a standard normal distribution and scales it by the inverse square root of the initial inverse temperature βₒ to obtain the initial momentum variable ρₒ. Then, it performs K leapfrog steps, each followed by a tempering step, to generate a new sample from the latent space.
Note
Ensure the input data x and the initial latent variable zₒ match the expected input dimensionality for the HVAE model.
AutoEncoderToolkit.HVAEs._log_p̄ — Function
_log_p̄(
     x::AbstractArray,
     hvae::HVAE{VAE{E,D}},
     hvae_outputs::NamedTuple;
     reconstruction_loglikelihood::Function=decoder_loglikelihood,
     logprior::Function=spherical_logprior,
     prefactor::AbstractArray=ones(Float32, 3),
-)
This is an internal function used in hamiltonian_elbo to compute the numerator of the unbiased estimator of the marginal likelihood. The function computes the sum of the log likelihood of the data given the latent variables, the log prior of the latent variables, and the log prior of the momentum variables.
    log p̄ = log p(x | zₖ) + log p(zₖ) + log p(ρₖ)
Arguments
  • x::AbstractArray: The input data. If Array, the last dimension must contain each of the data points.
  • hvae::HVAE{<:VAE{<:AbstractGaussianEncoder,<:AbstractGaussianLogDecoder}}: The Hamiltonian Variational Autoencoder (HVAE) model.
  • hvae_outputs::NamedTuple: The outputs of the HVAE, including the final latent variables zₖ and the final momentum variables ρₖ.
Optional Keyword Arguments
  • reconstruction_loglikelihood::Function: The function to compute the log likelihood of the data given the latent variables. Default is decoder_loglikelihood.
  • logprior::Function: The function to compute the log prior of the latent variables. Default is spherical_logprior.
  • prefactor::AbstractArray: A 3-element array to scale the log likelihood, log prior of the latent variables, and log prior of the momentum variables. Default is an array of ones.
Returns
  • log_p̄::AbstractVector: The first term of the log of the unbiased estimator of the marginal likelihood for each data point.
Note
This is an internal function and should not be called directly. It is used as part of the hamiltonian_elbo function.
AutoEncoderToolkit.HVAEs._log_q̄ — Function
_log_q̄(
+)
This is an internal function used in hamiltonian_elbo to compute the numerator of the unbiased estimator of the marginal likelihood. The function computes the sum of the log likelihood of the data given the latent variables, the log prior of the latent variables, and the log prior of the momentum variables.
    log p̄ = log p(x | zₖ) + log p(zₖ) + log p(ρₖ)
Arguments
  • x::AbstractArray: The input data. If Array, the last dimension must contain each of the data points.
  • hvae::HVAE{<:VAE{<:AbstractGaussianEncoder,<:AbstractGaussianLogDecoder}}: The Hamiltonian Variational Autoencoder (HVAE) model.
  • hvae_outputs::NamedTuple: The outputs of the HVAE, including the final latent variables zₖ and the final momentum variables ρₖ.
Optional Keyword Arguments
  • reconstruction_loglikelihood::Function: The function to compute the log likelihood of the data given the latent variables. Default is decoder_loglikelihood.
  • logprior::Function: The function to compute the log prior of the latent variables. Default is spherical_logprior.
  • prefactor::AbstractArray: A 3-element array to scale the log likelihood, log prior of the latent variables, and log prior of the momentum variables. Default is an array of ones.
Returns
  • log_p̄::AbstractVector: The first term of the log of the unbiased estimator of the marginal likelihood for each data point.
Note
This is an internal function and should not be called directly. It is used as part of the hamiltonian_elbo function.
AutoEncoderToolkit.HVAEs._log_q̄ — Function
_log_q̄(
     hvae::HVAE,
     hvae_outputs::NamedTuple,
     βₒ::Number;
     logprior::Function=spherical_logprior,
     prefactor::AbstractArray=ones(Float32, 3),
-)
This is an internal function used in hamiltonian_elbo to compute the second term of the unbiased estimator of the marginal likelihood. The function computes the sum of the log posterior of the initial latent variables and the log prior of the initial momentum variables, minus a term that depends on the dimensionality of the latent space and the initial temperature.
log q̄ = log q(zₒ | x) + log p(ρₒ | zₒ) - d/2 log(βₒ)
Arguments
  • hvae::HVAE: The Hamiltonian Variational Autoencoder (HVAE) model.
  • hvae_outputs::NamedTuple: The outputs of the HVAE, including the initial latent variables zₒ and the initial momentum variables ρₒ.
  • βₒ::Number: The initial temperature for the tempering steps.
Optional Keyword Arguments
  • logprior::Function: The function to compute the log prior of the momentum variables. Default is spherical_logprior.
  • prefactor::AbstractArray: A 3-element array to scale the log posterior of the initial latent variables, log prior of the initial momentum variables, and the tempering Jacobian term. Default is an array of ones.
Returns
  • log_q̄::Vector: The second term of the log of the unbiased estimator of the marginal likelihood for each data point.
Note
This is an internal function and should not be called directly. It is used as part of the hamiltonian_elbo function.
AutoEncoderToolkit.HVAEs.hamiltonian_elbo — Function
hamiltonian_elbo(
+)
This is an internal function used in hamiltonian_elbo to compute the second term of the unbiased estimator of the marginal likelihood. The function computes the sum of the log posterior of the initial latent variables and the log prior of the initial momentum variables, minus a term that depends on the dimensionality of the latent space and the initial temperature.
log q̄ = log q(zₒ | x) + log p(ρₒ | zₒ) - d/2 log(βₒ)
Arguments
  • hvae::HVAE: The Hamiltonian Variational Autoencoder (HVAE) model.
  • hvae_outputs::NamedTuple: The outputs of the HVAE, including the initial latent variables zₒ and the initial momentum variables ρₒ.
  • βₒ::Number: The initial temperature for the tempering steps.
Optional Keyword Arguments
  • logprior::Function: The function to compute the log prior of the momentum variables. Default is spherical_logprior.
  • prefactor::AbstractArray: A 3-element array to scale the log posterior of the initial latent variables, log prior of the initial momentum variables, and the tempering Jacobian term. Default is an array of ones.
Returns
  • log_q̄::Vector: The second term of the log of the unbiased estimator of the marginal likelihood for each data point.
Note
This is an internal function and should not be called directly. It is used as part of the hamiltonian_elbo function.
AutoEncoderToolkit.HVAEs.hamiltonian_elbo — Function
hamiltonian_elbo(
     hvae::HVAE,
     x::AbstractArray;
     ϵ::Union{<:Number,<:AbstractVector}=Float32(1E-4),
@@ -215,4 +215,4 @@
     return_outputs::Bool=false,
     logp_prefactor::AbstractArray=ones(Float32, 3),
     logq_prefactor::AbstractArray=ones(Float32, 3),
-)
Compute the Hamiltonian Monte Carlo (HMC) estimate of the evidence lower bound (ELBO) for a Hamiltonian Variational Autoencoder (HVAE).
This function takes as input an HVAE and a vector of input data x. It performs K HMC steps with a leapfrog integrator and a tempering schedule to estimate the ELBO. The ELBO is computed as the difference between the log p̄ and log q̄ as
elbo = mean(log p̄ - log q̄),
Arguments
  • hvae::HVAE: The HVAE used to encode the input data and decode the latent space.
  • x_in::AbstractArray: The input data. If Array, the last dimension must contain each of the data points.
  • x_out::AbstractArray: The data against which the reconstruction is compared. If Array, the last dimension must contain each of the data points.
Optional Keyword Arguments
  • ϵ::Union{<:Number,<:AbstractVector}: The step size for the leapfrog integrator (default is 0.01).
  • K::Int: The number of HMC steps (default is 3).
  • βₒ::Number: The initial inverse temperature (default is 0.3).
  • ∇U_kwargs::NamedTuple: Additional keyword arguments to be passed to the ∇potential_energy function. Defaults to a NamedTuple with :reconstruction_loglikelihood set to decoder_loglikelihood and :latent_logprior set to spherical_logprior.
  • tempering_schedule::Function: The tempering schedule function used in the HMC (default is quadratic_tempering).
  • return_outputs::Bool: Whether to return the outputs of the HVAE. Defaults to false. NOTE: This is necessary to avoid computing the forward pass twice when computing the loss function with regularization.
  • logp_prefactor::AbstractArray: A 3-element array to scale the log likelihood, log prior of the latent variables, and log prior of the momentum variables. Default is an array of ones.
  • logq_prefactor::AbstractArray: A 3-element array to scale the log posterior of the initial latent variables, log prior of the initial momentum variables, and the tempering Jacobian term. Default is an array of ones.
Returns
  • elbo::Number: The HMC estimate of the ELBO. If return_outputs is true, also returns the outputs of the HVAE.

+)

Compute the Hamiltonian Monte Carlo (HMC) estimate of the evidence lower bound (ELBO) for a Hamiltonian Variational Autoencoder (HVAE).

This function takes as input an HVAE and a vector of input data x. It performs K HMC steps with a leapfrog integrator and a tempering schedule to estimate the ELBO. The ELBO is computed as the difference between the log p̄ and log q̄ as

elbo = mean(log p̄ - log q̄),

Arguments

  • hvae::HVAE: The HVAE used to encode the input data and decode the latent space.
  • x_in::AbstractArray: The input data. If Array, the last dimension must contain each of the data points.
  • x_out::AbstractArray: The data against which the reconstruction is compared. If Array, the last dimension must contain each of the data points.

Optional Keyword Arguments

  • ϵ::Union{<:Number,<:AbstractVector}: The step size for the leapfrog integrator (default is 0.01).
  • K::Int: The number of HMC steps (default is 3).
  • βₒ::Number: The initial inverse temperature (default is 0.3).
  • ∇U_kwargs::NamedTuple: Additional keyword arguments to be passed to the ∇potential_energy function. Defaults to a NamedTuple with :reconstruction_loglikelihood set to decoder_loglikelihood and :latent_logprior set to spherical_logprior.
  • tempering_schedule::Function: The tempering schedule function used in the HMC (default is quadratic_tempering).
  • return_outputs::Bool: Whether to return the outputs of the HVAE. Defaults to false. NOTE: This is necessary to avoid computing the forward pass twice when computing the loss function with regularization.
  • logp_prefactor::AbstractArray: A 3-element array to scale the log likelihood, log prior of the latent variables, and log prior of the momentum variables. Default is an array of ones.
  • logq_prefactor::AbstractArray: A 3-element array to scale the log posterior of the initial latent variables, log prior of the initial momentum variables, and the tempering Jacobian term. Default is an array of ones.

Returns

  • elbo::Number: The HMC estimate of the ELBO. If return_outputs is true, also returns the outputs of the HVAE.
diff --git a/dev/index.html b/dev/index.html index d4a051e..210704c 100644 --- a/dev/index.html +++ b/dev/index.html @@ -49,4 +49,4 @@ decoder = AutoEncoderToolkit.SimpleGaussianDecoder(deconv_layers) # Re-defining the variational autoencoder -vae = encoder * decoder

Everything else in our training pipeline would remain the same thanks to multiple dispatch.

Furthermore, let's say that we would like to use a different flavor for our variational autoencoder. In particular the InfoVAE (also known as MMD-VAE) includes extra terms in the loss function to maximize mutual information between the latent space and the input data. We can easily take our vae model and convert it into a MMDVAE-type object from the MMDVAEs submodule as follows:

mmdvae = AutoEncoderToolkit.MMDVAEs.MMDVAE(vae)

This is the power of AutoEncoderToolkit.jl and Julia's multiple dispatch!

Implemented Autoencoders

modelmoduledescription
AutoencoderAEsVanilla deterministic autoencoder
Variational AutoencoderVAEsVanilla variational autoencoder
β-VAEVAEsbeta-VAE to weigh the reconstruction vs. KL divergence in ELBO
MMD-VAEsMMDsMaximum-Mean Discrepancy Variational Autoencoders
InfoMax-VAEsInfoMaxVAEsInformation Maximization Variational Autoencoders
Hamiltonian VAEHVAEsHamiltonian Variational Autoencoders
Riemannian Hamiltonian-VAERHVAEsRiemannian-Hamiltonian Variational Autoencoder
Looking for contributors!

If you are interested in contributing to the package to add a new model, please check the GitHub repository. We are always looking to expand the list of available models. And AutoEncoderToolkit.jl's structure should make it relatively easy.

GPU support

AutoEncoderToolkit.jl supports GPU training out of the box for CUDA.jl-compatible GPUs. The CUDA functionality is provided as an extension. Therefore, to train a model on the GPU, simply import CUDA into the current environment, then move the model and data to the GPU. The rest of the training pipeline remains the same.

+vae = encoder * decoder

Everything else in our training pipeline would remain the same thanks to multiple dispatch.

Furthermore, let's say that we would like to use a different flavor for our variational autoencoder. In particular the InfoVAE (also known as MMD-VAE) includes extra terms in the loss function to maximize mutual information between the latent space and the input data. We can easily take our vae model and convert it into a MMDVAE-type object from the MMDVAEs submodule as follows:

mmdvae = AutoEncoderToolkit.MMDVAEs.MMDVAE(vae)

This is the power of AutoEncoderToolkit.jl and Julia's multiple dispatch!

Implemented Autoencoders

modelmoduledescription
AutoencoderAEsVanilla deterministic autoencoder
Variational AutoencoderVAEsVanilla variational autoencoder
β-VAEVAEsbeta-VAE to weigh the reconstruction vs. KL divergence in ELBO
MMD-VAEsMMDsMaximum-Mean Discrepancy Variational Autoencoders
InfoMax-VAEsInfoMaxVAEsInformation Maximization Variational Autoencoders
Hamiltonian VAEHVAEsHamiltonian Variational Autoencoders
Riemannian Hamiltonian-VAERHVAEsRiemannian-Hamiltonian Variational Autoencoder
Looking for contributors!

If you are interested in contributing to the package to add a new model, please check the GitHub repository. We are always looking to expand the list of available models. And AutoEncoderToolkit.jl's structure should make it relatively easy.

GPU support

AutoEncoderToolkit.jl supports GPU training out of the box for CUDA.jl-compatible GPUs. The CUDA functionality is provided as an extension. Therefore, to train a model on the GPU, simply import CUDA into the current environment, then move the model and data to the GPU. The rest of the training pipeline remains the same.

diff --git a/dev/infomaxvae/index.html b/dev/infomaxvae/index.html index 437a14f..3815be5 100644 --- a/dev/infomaxvae/index.html +++ b/dev/infomaxvae/index.html @@ -1,16 +1,16 @@ -InfoMax-VAE · AutoEncoderToolkit

InfoMax VAE

The InfoMax VAE is a variant of the Variational Autoencoder (VAE) that aims to explicitly account for the maximization of mutual information between the latent space representation and the input data. The main difference between the InfoMax VAE and the MMD-VAE (InfoVAE) is that rather than using the Maximum-Mean Discrepancy (MMD) as a measure of the "distance" between the latent space, the InfoMax VAE explicitly models the mutual information between latent representations and data inputs via a separate neural network. The loss function for this separate network then takes the form of a variational lower bound on the mutual information between the latent space and the input data.

Because of the need of this separate network, the InfoMaxVAE struct in AutoEncoderToolkit.jl takes two arguments to construct: the original VAE struct and a network to compute the mutual information. To properly deploy all relevant functions associated with this second network, we also provide a MutualInfoChain struct.

Furthermore, because of the two networks and the way the training algorithm is set up, the loss function for the InfoMax VAE includes two separate loss functions: one for the MutualInfoChain and one for the InfoMaxVAE.

References

Rezaabad, A. L. & Vishwanath, S. Learning Representations by Maximizing Mutual Information in Variational Autoencoders. Preprint at http://arxiv.org/abs/1912.13361 (2020).

MutualInfoChain struct

AutoEncoderToolkit.InfoMaxVAEs.MutualInfoChain — Type
MutualInfoChain

A MutualInfoChain is used to compute the variational mutual information when training an InfoMaxVAE. The chain is composed of a series of layers that must end with a single output: the mutual information between the latent variables and the input data.

Arguments

  • data::Union{Flux.Dense,Flux.Chain}: The data layer of the MutualInfoChain. This layer is used to input the data.
  • latent::Union{Flux.Dense,Flux.Chain}: The latent layer of the MutualInfoChain. This layer is used to input the latent variables.
  • mlp::Flux.Chain: A multi-layer perceptron (MLP) that is used to compute the mutual information between the inputs and the latent representations. The MLP takes as input the latent variables and outputs a scalar representing the estimated variational mutual information.

Citation

Rezaabad, A. L. & Vishwanath, S. Learning Representations by Maximizing Mutual Information in Variational Autoencoders. in 2020 IEEE International Symposium on Information Theory (ISIT) 2729–2734 (IEEE, 2020). doi:10.1109/ISIT44484.2020.9174424.

Note

If the input data is not a flat array, make sure to include a flattening layer within data.

InfoMaxVAE struct

AutoEncoderToolkit.InfoMaxVAEs.InfoMaxVAE — Type
`InfoMaxVAE <: AbstractVariationalAutoEncoder`

struct encapsulating an InfoMax variational autoencoder (InfoMaxVAE), an architecture designed to enhance the VAE framework by maximizing mutual information between the inputs and the latent representations, as per the methods described by Rezaabad and Vishwanath (2020).

The model aims to learn representations that preserve mutual information with the input data, arguably capturing more meaningful factors of variation.

Fields

  • vae::VAE: The core variational autoencoder, consisting of an encoder that maps input data into a latent space representation, and a decoder that attempts to reconstruct the input from the latent representation.
  • mi::MutualInfoChain: A multi-layer perceptron (MLP) that estimates the mutual information between the input data and the latent representations.

Usage

The InfoMaxVAE struct is utilized in a similar manner to a standard VAE, with the added capability of mutual information maximization as part of the training process. This involves an additional loss term that considers the output of the mi network to encourage latent representations that are informative about the input data.

Example

# Assuming definitions for `encoder`, `decoder`, and `mi` are provided:
+InfoMax-VAE · AutoEncoderToolkit
InfoMax VAE
The InfoMax VAE is a variant of the Variational Autoencoder (VAE) that aims to explicitly account for the maximization of mutual information between the latent space representation and the input data. The main difference between the InfoMax VAE and the MMD-VAE (InfoVAE) is that rather than using the Maximum-Mean Discrepancy (MMD) as a measure of the "distance" between the latent space, the InfoMax VAE explicitly models the mutual information between latent representations and data inputs via a separate neural network. The loss function for this separate network then takes the form of a variational lower bound on the mutual information between the latent space and the input data.
Because of the need of this separate network, the InfoMaxVAE struct in AutoEncoderToolkit.jl takes two arguments to construct: the original VAE struct and a network to compute the mutual information. To properly deploy all relevant functions associated with this second network, we also provide a MutualInfoChain struct.
Furthermore, because of the two networks and the way the training algorithm is set up, the loss function for the InfoMax VAE includes two separate loss functions: one for the MutualInfoChain and one for the InfoMaxVAE.
References
Rezaabad, A. L. & Vishwanath, S. Learning Representations by Maximizing Mutual Information in Variational Autoencoders. Preprint at http://arxiv.org/abs/1912.13361 (2020).
MutualInfoChain struct
AutoEncoderToolkit.InfoMaxVAEs.MutualInfoChain — Type
MutualInfoChain
A MutualInfoChain is used to compute the variational mutual information when training an InfoMaxVAE. The chain is composed of a series of layers that must end with a single output: the mutual information between the latent variables and the input data.
Arguments
  • data::Union{Flux.Dense,Flux.Chain}: The data layer of the MutualInfoChain. This layer is used to input the data.
  • latent::Union{Flux.Dense,Flux.Chain}: The latent layer of the MutualInfoChain. This layer is used to input the latent variables.
  • mlp::Flux.Chain: A multi-layer perceptron (MLP) that is used to compute the mutual information between the inputs and the latent representations. The MLP takes as input the latent variables and outputs a scalar representing the estimated variational mutual information.
Citation
Rezaabad, A. L. & Vishwanath, S. Learning Representations by Maximizing Mutual Information in Variational Autoencoders. in 2020 IEEE International Symposium on Information Theory (ISIT) 2729–2734 (IEEE, 2020). doi:10.1109/ISIT44484.2020.9174424.
Note
If the input data is not a flat array, make sure to include a flattening layer within data.
InfoMaxVAE struct
AutoEncoderToolkit.InfoMaxVAEs.InfoMaxVAE — Type
`InfoMaxVAE <: AbstractVariationalAutoEncoder`
struct encapsulating an InfoMax variational autoencoder (InfoMaxVAE), an architecture designed to enhance the VAE framework by maximizing mutual information between the inputs and the latent representations, as per the methods described by Rezaabad and Vishwanath (2020).
The model aims to learn representations that preserve mutual information with the input data, arguably capturing more meaningful factors of variation.
Fields
  • vae::VAE: The core variational autoencoder, consisting of an encoder that maps input data into a latent space representation, and a decoder that attempts to reconstruct the input from the latent representation.
  • mi::MutualInfoChain: A multi-layer perceptron (MLP) that estimates the mutual information between the input data and the latent representations.
Usage
The InfoMaxVAE struct is utilized in a similar manner to a standard VAE, with the added capability of mutual information maximization as part of the training process. This involves an additional loss term that considers the output of the mi network to encourage latent representations that are informative about the input data.
Example
# Assuming definitions for `encoder`, `decoder`, and `mi` are provided:
 info_max_vae = InfoMaxVAE(VAE(encoder, decoder), mi)
 
 # During training, one would maximize both the variational lower bound and the 
-# mutual information estimate provided by `mlp`.
Citation
Rezaabad, A. L. & Vishwanath, S. Learning Representations by Maximizing Mutual Information in Variational Autoencoders. in 2020 IEEE International Symposium on Information Theory (ISIT) 2729–2734 (IEEE, 2020). doi:10.1109/ISIT44484.2020.9174424.
Forward pass
Mutual Information Network
AutoEncoderToolkit.InfoMaxVAEs.MutualInfoChain — Method
(mi::MutualInfoChain)(x::AbstractArray, z::AbstractVecOrMat)
Forward pass function for the MutualInfoChain, which applies the MLP to an input x.
Arguments
  • x::AbstractArray: The input array to be processed. The last dimension represents each data sample.
  • z::AbstractVecOrMat: The latent representation of the input data. The last dimension represents each data sample.
Returns
  • The result of applying the MutualInfoChain to the input data and the latent representation simultaneously.
Description
This function applies the MLP (Multilayer Perceptron) of a MutualInfoChain instance to an input array. The MLP is a type of neural network used in the MutualInfoChain for processing the input data.
InfoMax VAE
AutoEncoderToolkit.InfoMaxVAEs.InfoMaxVAE — Method
(vae::InfoMaxVAE)(x::AbstractArray; latent::Bool=false)
Processes the input data x through an InfoMaxVAE, which consists of an encoder, a decoder, and a multi-layer perceptron (MLP) to estimate variational mutual information.
Arguments
  • x::AbstractArray: The data to be decoded. If array, the last dimension contains each data sample. 
Optional Keyword Arguments
  • latent::Bool: If true, returns a dictionary with latent variables and mutual information estimations along with the reconstruction. Defaults to false.
  • seed::Union{Nothing,Int}: Optional argument. The seed for the random number generator used for shuffling the latent codes. If not provided, a random seed will be used.
Returns
  • If latent=false: The decoder output as a NamedTuple.
  • If latent=true: A NamedTuple with the :vae field that contains the outputs of the VAE, and the :mi field that contains the estimate of the variational mutual information. Note that this estimate requires shuffling the latent codes between data samples. Therefore, it is only meaningful for batch data cases.
Description
This function first encodes the input x . It then samples from this distribution using the reparametrization trick. The sampled latent vectors are then decoded, and the MutualInfoChain is used to estimate the mutual information.
Note
Ensure the input data x matches the expected input dimensionality for the encoder in the InfoMaxVAE.
[Loss functions]
Mutual Information Network
AutoEncoderToolkit.InfoMaxVAEs.miloss — Function
miloss(
+# mutual information estimate provided by `mlp`.
Citation
Rezaabad, A. L. & Vishwanath, S. Learning Representations by Maximizing Mutual Information in Variational Autoencoders. in 2020 IEEE International Symposium on Information Theory (ISIT) 2729–2734 (IEEE, 2020). doi:10.1109/ISIT44484.2020.9174424.
Forward pass
Mutual Information Network
AutoEncoderToolkit.InfoMaxVAEs.MutualInfoChain — Method
(mi::MutualInfoChain)(x::AbstractArray, z::AbstractVecOrMat)
Forward pass function for the MutualInfoChain, which applies the MLP to an input x.
Arguments
  • x::AbstractArray: The input array to be processed. The last dimension represents each data sample.
  • z::AbstractVecOrMat: The latent representation of the input data. The last dimension represents each data sample.
Returns
  • The result of applying the MutualInfoChain to the input data and the latent representation simultaneously.
Description
This function applies the MLP (Multilayer Perceptron) of a MutualInfoChain instance to an input array. The MLP is a type of neural network used in the MutualInfoChain for processing the input data.
InfoMax VAE
AutoEncoderToolkit.InfoMaxVAEs.InfoMaxVAE — Method
(vae::InfoMaxVAE)(x::AbstractArray; latent::Bool=false)
Processes the input data x through an InfoMaxVAE, which consists of an encoder, a decoder, and a multi-layer perceptron (MLP) to estimate variational mutual information.
Arguments
  • x::AbstractArray: The data to be decoded. If array, the last dimension contains each data sample. 
Optional Keyword Arguments
  • latent::Bool: If true, returns a dictionary with latent variables and mutual information estimations along with the reconstruction. Defaults to false.
  • seed::Union{Nothing,Int}: Optional argument. The seed for the random number generator used for shuffling the latent codes. If not provided, a random seed will be used.
Returns
  • If latent=false: The decoder output as a NamedTuple.
  • If latent=true: A NamedTuple with the :vae field that contains the outputs of the VAE, and the :mi field that contains the estimate of the variational mutual information. Note that this estimate requires shuffling the latent codes between data samples. Therefore, it is only meaningful for batch data cases.
Description
This function first encodes the input x . It then samples from this distribution using the reparametrization trick. The sampled latent vectors are then decoded, and the MutualInfoChain is used to estimate the mutual information.
Note
Ensure the input data x matches the expected input dimensionality for the encoder in the InfoMaxVAE.
[Loss functions]
Mutual Information Network
AutoEncoderToolkit.InfoMaxVAEs.miloss — Function
miloss(
     vae::VAE,
     mi::MutualInfoChain,
     x::AbstractArray;
     regularization::Union{Function,Nothing}=nothing,
     reg_strength::Float32=1.0f0,
     seed::Union{Nothing,Int}=nothing
-)
Calculates the loss for training the MutualInfoChain in the InfoMaxVAE algorithm to estimate mutual information between the input x and the latent representation z. The loss function is based on a variational approximation of mutual information, using the MutualInfoChain's output g(x, z). The variational mutual information is then calculated as the difference between the MutualInfoChain's output for the true x and latent z, and the exponentiated average of the MLP's output for x and the shuffled latent z_shuffle, adjusted for the regularization term if provided.
Arguments
  • vae::VAE: The variational autoencoder.
  • mi::MutualInfoChain: The MutualInfoChain used for estimating mutual information.
  • x::AbstractArray: The input vector for the VAE.
Optional Keyword Arguments
  • regularization::Union{Function, Nothing}=nothing: A regularization function applied to the MLP's output.
  • reg_strength::Float32=1.0f0: The strength of the regularization term.
  • seed::Union{Nothing,Int}=nothing: The seed for the random number generator used for shuffling the latent codes. If not provided, a random seed will be used.
Returns
  • Float32: The computed loss, representing negative variational mutual information, adjusted by the regularization term.
Description
The function computes the loss as follows:
loss = -sum(I(x; z)) + sum(exp(I(x; z̃) - 1)) + regstrength * regterm
where I(x; z) is the MLP's output representing an estimation of mutual information for true x and latent z, and z̃ represents shuffled latent variables, meaning, the latent codes are randomly swap between data points.
The function is used to separately train the MLP to estimate mutual information, which is a component of the larger InfoMaxVAE model.
Notes
  • This function takes the vae and mi instances of an InfoMaxVAE model as separate arguments to be able to compute a gradient only with respect to the mi parameters.
  • Ensure that the dimensionality of the input data x aligns with the encoder's   expected input in the VAE.
  • InfoMaxVAEs fully depend on batch training as the estimation of mutual   information depends on shuffling the latent codes. This method works for   large enough batches (≥ 64 samples).
InfoMax VAE
AutoEncoderToolkit.InfoMaxVAEs.infomaxloss — Function
infomaxloss(
+)
Calculates the loss for training the MutualInfoChain in the InfoMaxVAE algorithm to estimate mutual information between the input x and the latent representation z. The loss function is based on a variational approximation of mutual information, using the MutualInfoChain's output g(x, z). The variational mutual information is then calculated as the difference between the MutualInfoChain's output for the true x and latent z, and the exponentiated average of the MLP's output for x and the shuffled latent z_shuffle, adjusted for the regularization term if provided.
Arguments
  • vae::VAE: The variational autoencoder.
  • mi::MutualInfoChain: The MutualInfoChain used for estimating mutual information.
  • x::AbstractArray: The input vector for the VAE.
Optional Keyword Arguments
  • regularization::Union{Function, Nothing}=nothing: A regularization function applied to the MLP's output.
  • reg_strength::Float32=1.0f0: The strength of the regularization term.
  • seed::Union{Nothing,Int}=nothing: The seed for the random number generator used for shuffling the latent codes. If not provided, a random seed will be used.
Returns
  • Float32: The computed loss, representing negative variational mutual information, adjusted by the regularization term.
Description
The function computes the loss as follows:
loss = -sum(I(x; z)) + sum(exp(I(x; z̃) - 1)) + regstrength * regterm
where I(x; z) is the MLP's output representing an estimation of mutual information for true x and latent z, and z̃ represents shuffled latent variables, meaning, the latent codes are randomly swap between data points.
The function is used to separately train the MLP to estimate mutual information, which is a component of the larger InfoMaxVAE model.
Notes
  • This function takes the vae and mi instances of an InfoMaxVAE model as separate arguments to be able to compute a gradient only with respect to the mi parameters.
  • Ensure that the dimensionality of the input data x aligns with the encoder's   expected input in the VAE.
  • InfoMaxVAEs fully depend on batch training as the estimation of mutual   information depends on shuffling the latent codes. This method works for   large enough batches (≥ 64 samples).
InfoMax VAE
AutoEncoderToolkit.InfoMaxVAEs.infomaxloss — Function
infomaxloss(
     vae::VAE,
     mi::MutualInfoChain,
     x::AbstractArray;
@@ -35,7 +35,7 @@
     regularization::Union{Function,Nothing}=nothing,
     reg_strength::Float32=1.0f0,
     seed::Union{Nothing,Int}=nothing
-)
Computes the loss for an InfoMax variational autoencoder (VAE) with mutual information constraints, by averaging over n_samples latent space samples.
The loss function combines the reconstruction loss with the Kullback-Leibler (KL) divergence, the variational mutual information between input and latent representations, and possibly a regularization term, defined as:
loss = -⟨log p(x|z)⟩ + β × Dₖₗ[qᵩ(z|x) || p(z)] - α × I(x;z) + regstrength × regterm
Where:
  • ⟨log p(x|z)⟩ is the expected log likelihood of the probabilistic decoder. -
Dₖₗ[qᵩ(z|x) || p(z)] is the KL divergence between the approximated encoder and the prior over the latent space.
  • I(x;z) is the variational mutual information between the inputs x and the latent variables z.
Arguments
  • vae::VAE: A VAE model with encoder and decoder networks.
  • mi::MutualInfoChain: A MutualInfoChain instance used to estimate mutual information term.
  • x_in::AbstractArray: Input matrix. The last dimension represents each data sample.
  • x_out::AbstractArray: Output matrix against wich reconstructions are compared. The last dimension represents each data sample.
Optional Keyword Arguments
  • β::Float32=1.0f0: Weighting factor for the KL-divergence term, used for annealing.
  • α::Float32=1.0f0: Weighting factor for the mutual information term.
  • n_samples::Int=1: The number of samples to draw from the latent space when computing the loss.
  • reconstruction_loglikelihood::Function=decoder_loglikelihood: A function that computes the log likelihood of the decoder's output.
  • kl_divergence::Function=encoder_kl: A function that computes the KL divergence between the encoder's output and the prior.
  • regularization::Union{Function, Nothing}=nothing: A function that computes the regularization term based on the VAE outputs. Should return a Float32.
  • reg_strength::Float32=1.0f0: The strength of the regularization term.
  • seed::Union{Nothing,Int}: The seed for the random number generator used for shuffling the latent codes. If not provided, a random seed will be used.
Returns
  • Float32: The computed average loss value for the input x and its reconstructed counterparts over n_samples samples, including possible regularization terms and the mutual information constraint.
Note
  • This function takes the vae and mi instances of an InfoMaxVAE model as separate arguments to be able to compute a gradient only with respect to the vae parameters.
  • Ensure that the input data x match the expected input dimensionality for the encoder in the VAE.
  • InfoMaxVAEs fully depend on batch training as the estimation of mutual information depends on shuffling the latent codes. This method works for large enough batches (≥ 64 samples).
Training
AutoEncoderToolkit.InfoMaxVAEs.train! — Function
    train!(
+)
Computes the loss for an InfoMax variational autoencoder (VAE) with mutual information constraints, by averaging over n_samples latent space samples.
The loss function combines the reconstruction loss with the Kullback-Leibler (KL) divergence, the variational mutual information between input and latent representations, and possibly a regularization term, defined as:
loss = -⟨log p(x|z)⟩ + β × Dₖₗ[qᵩ(z|x) || p(z)] - α × I(x;z) + regstrength × regterm
Where:
  • ⟨log p(x|z)⟩ is the expected log likelihood of the probabilistic decoder. -
Dₖₗ[qᵩ(z|x) || p(z)] is the KL divergence between the approximated encoder and the prior over the latent space.
  • I(x;z) is the variational mutual information between the inputs x and the latent variables z.
Arguments
  • vae::VAE: A VAE model with encoder and decoder networks.
  • mi::MutualInfoChain: A MutualInfoChain instance used to estimate mutual information term.
  • x_in::AbstractArray: Input matrix. The last dimension represents each data sample.
  • x_out::AbstractArray: Output matrix against wich reconstructions are compared. The last dimension represents each data sample.
Optional Keyword Arguments
  • β::Float32=1.0f0: Weighting factor for the KL-divergence term, used for annealing.
  • α::Float32=1.0f0: Weighting factor for the mutual information term.
  • n_samples::Int=1: The number of samples to draw from the latent space when computing the loss.
  • reconstruction_loglikelihood::Function=decoder_loglikelihood: A function that computes the log likelihood of the decoder's output.
  • kl_divergence::Function=encoder_kl: A function that computes the KL divergence between the encoder's output and the prior.
  • regularization::Union{Function, Nothing}=nothing: A function that computes the regularization term based on the VAE outputs. Should return a Float32.
  • reg_strength::Float32=1.0f0: The strength of the regularization term.
  • seed::Union{Nothing,Int}: The seed for the random number generator used for shuffling the latent codes. If not provided, a random seed will be used.
Returns
  • Float32: The computed average loss value for the input x and its reconstructed counterparts over n_samples samples, including possible regularization terms and the mutual information constraint.
Note
  • This function takes the vae and mi instances of an InfoMaxVAE model as separate arguments to be able to compute a gradient only with respect to the vae parameters.
  • Ensure that the input data x match the expected input dimensionality for the encoder in the VAE.
  • InfoMaxVAEs fully depend on batch training as the estimation of mutual information depends on shuffling the latent codes. This method works for large enough batches (≥ 64 samples).
Training
AutoEncoderToolkit.InfoMaxVAEs.train! — Function
    train!(
         infomaxvae, x, opt; 
         infomaxloss_function=infomaxloss,
         infomaxloss_kwargs, 
@@ -53,15 +53,15 @@
         loss_return::Bool=false,
         verbose::Bool=false
     )
Customized training function to update parameters of an InfoMax variational autoencoder (VAE) given a loss function of the specified form.
The InfoMax VAE loss function can be defined as:
loss_infoMax = argmin -⟨log p(x|z)⟩ + β Dₖₗ(qᵩ(z) || p(z)) -
-                                 α [⟨g(x, z)⟩ - ⟨exp(g(x, z) - 1)⟩],
where ⟨log p(x|z)⟩ is the expected log likelihood of the probabilistic decoder, Dₖₗ[qᵩ(z) || p(z)] is the KL divergence between the approximated encoder distribution and the prior over the latent space, and g(x, z) is the output of the MutualInfoChain estimating the mutual information between the input data and the latent representation.
This function simultaneously optimizes two neural networks: the VAE itself and a multi-layer perceptron MutualInfoChain used to compute the mutual information between input and latent variables.
Arguments
  • infomaxvae::InfoMaxVAE: Struct containing the elements of an InfoMax VAE.
  • x::AbstractArray: Matrix containing the data on which to evaluate the loss function. Each column represents a single data point.
  • opt::NamedTuple: State of the optimizer for updating parameters. Typically initialized using Flux.Optimisers.update!.
Optional Keyword arguments
  • infomaxloss_function::Function: The loss function to be used during training for the VAE, defaulting to infomaxloss.
  • infomaxloss_kwargs::NamedTuple: Additional keyword arguments to be passed to the VAE loss function.
  • miloss_function::Function: The loss function to be used during training for the MutualInfoChain computing the variational free energy, defaulting to miloss.
  • miloss_kwargs::NamedTuple: Additional keyword arguments to be passed to the MutualInfoChain loss function.
  • loss_return::Bool: If true, the function returns the loss values for the VAE and MLP. Defaults to false.
Description
Performs one step of gradient descent on the InfoMaxVAE loss function to jointly train the VAE and MutualInfoChain. The VAE parameters are updated to minimize the InfoMaxVAE loss, while the MutualInfoChain parameters are updated to maximize the estimated mutual information. The function allows for customization of loss hyperparameters during training.
Notes
  • Ensure that the dimensionality of the input data x aligns with the encoder's expected input in the VAE.
  • InfoMaxVAEs fully depend on batch training as the estimation of mutual information depends on shuffling the latent codes. This method works best for large enough batches (≥ 64 samples).
Other Functions
AutoEncoderToolkit.InfoMaxVAEs.shuffle_latent — Function
shuffle_latent(z::AbstractMatrix, seed::Int=Random.seed!())
Shuffle the elements of the second dimension of a matrix representing latent space points.
Arguments
  • z::AbstractMatrix: A matrix representing latent codes. Each column corresponds to a single latent code.
Optional Keyword Arguments
  • seed::Union{Nothing, Int}: Optional argument. The seed for the random number generator. If not provided, a random seed will be used.
Returns
  • AbstractMatrix: A new matrix with the second dimension shuffled.
AutoEncoderToolkit.InfoMaxVAEs.variational_mutual_info — Function
variational_mutual_info(mi, x, z, z_shuffle)
Compute a variational approximation of the mutual information between the input x and the latent code z using a MutualInfoChain. Note that this estimate requires shuffling the latent codes between data samples. Therefore, it only applies to batch data cases. A single sample will not provide a meaningful estimate.
Arguments
  • mi::MutualInfoChain: A MutualInfoChain instance used to estimate mutual information.
  • x::AbstractArray: Array of input data. The last dimension represents each data sample.
  • z::AbstractMatrix: Matrix of corresponding latent representations of the input data.
  • z_shuffle::AbstractMatrix: Matrix of latent representations where the second dimension has been shuffled.
Returns
  • Float32: An approximation of the mutual information between the input data and its corresponding latent representation.
References
Rezaabad, A. L. & Vishwanath, S. Learning Representations by Maximizing Mutual Information in Variational Autoencoders. Preprint at http://arxiv.org/abs/1912.13361 (2020).
variational_mutual_info(infomaxvae, x, z, z_shuffle)
Compute a variational approximation of the mutual information between the input x and the latent code z using an InfoMaxVAE instance. Note that this estimate requires shuffling the latent codes between data samples. Therefore, it only applies to batch data cases. A single sample will not provide a meaningful estimate.
Arguments
  • infomaxvae::InfoMaxVAE: An InfoMaxVAE instance used to estimate mutual information.
  • x::AbstractArray: Array of input data. The last dimension represents each data sample.
  • z::AbstractMatrix: Matrix of corresponding latent representations of the input data.
  • z_shuffle::AbstractMatrix: Matrix of latent representations where the second dimension has been shuffled.
Returns
  • Float32: An approximation of the mutual information between the input data and its corresponding latent representation.
References
Rezaabad, A. L. & Vishwanath, S. Learning Representations by Maximizing Mutual Information in Variational Autoencoders. Preprint at http://arxiv.org/abs/1912.13361 (2020).
variational_mutual_info(
+                                 α [⟨g(x, z)⟩ - ⟨exp(g(x, z) - 1)⟩],
where ⟨log p(x|z)⟩ is the expected log likelihood of the probabilistic decoder, Dₖₗ[qᵩ(z) || p(z)] is the KL divergence between the approximated encoder distribution and the prior over the latent space, and g(x, z) is the output of the MutualInfoChain estimating the mutual information between the input data and the latent representation.
This function simultaneously optimizes two neural networks: the VAE itself and a multi-layer perceptron MutualInfoChain used to compute the mutual information between input and latent variables.
Arguments
  • infomaxvae::InfoMaxVAE: Struct containing the elements of an InfoMax VAE.
  • x::AbstractArray: Matrix containing the data on which to evaluate the loss function. Each column represents a single data point.
  • opt::NamedTuple: State of the optimizer for updating parameters. Typically initialized using Flux.Optimisers.update!.
Optional Keyword arguments
  • infomaxloss_function::Function: The loss function to be used during training for the VAE, defaulting to infomaxloss.
  • infomaxloss_kwargs::NamedTuple: Additional keyword arguments to be passed to the VAE loss function.
  • miloss_function::Function: The loss function to be used during training for the MutualInfoChain computing the variational free energy, defaulting to miloss.
  • miloss_kwargs::NamedTuple: Additional keyword arguments to be passed to the MutualInfoChain loss function.
  • loss_return::Bool: If true, the function returns the loss values for the VAE and MLP. Defaults to false.
Description
Performs one step of gradient descent on the InfoMaxVAE loss function to jointly train the VAE and MutualInfoChain. The VAE parameters are updated to minimize the InfoMaxVAE loss, while the MutualInfoChain parameters are updated to maximize the estimated mutual information. The function allows for customization of loss hyperparameters during training.
Notes
  • Ensure that the dimensionality of the input data x aligns with the encoder's expected input in the VAE.
  • InfoMaxVAEs fully depend on batch training as the estimation of mutual information depends on shuffling the latent codes. This method works best for large enough batches (≥ 64 samples).
Other Functions
AutoEncoderToolkit.InfoMaxVAEs.shuffle_latent — Function
shuffle_latent(z::AbstractMatrix, seed::Int=Random.seed!())
Shuffle the elements of the second dimension of a matrix representing latent space points.
Arguments
  • z::AbstractMatrix: A matrix representing latent codes. Each column corresponds to a single latent code.
Optional Keyword Arguments
  • seed::Union{Nothing, Int}: Optional argument. The seed for the random number generator. If not provided, a random seed will be used.
Returns
  • AbstractMatrix: A new matrix with the second dimension shuffled.
AutoEncoderToolkit.InfoMaxVAEs.variational_mutual_info — Function
variational_mutual_info(mi, x, z, z_shuffle)
Compute a variational approximation of the mutual information between the input x and the latent code z using a MutualInfoChain. Note that this estimate requires shuffling the latent codes between data samples. Therefore, it only applies to batch data cases. A single sample will not provide a meaningful estimate.
Arguments
  • mi::MutualInfoChain: A MutualInfoChain instance used to estimate mutual information.
  • x::AbstractArray: Array of input data. The last dimension represents each data sample.
  • z::AbstractMatrix: Matrix of corresponding latent representations of the input data.
  • z_shuffle::AbstractMatrix: Matrix of latent representations where the second dimension has been shuffled.
Returns
  • Float32: An approximation of the mutual information between the input data and its corresponding latent representation.
References
Rezaabad, A. L. & Vishwanath, S. Learning Representations by Maximizing Mutual Information in Variational Autoencoders. Preprint at http://arxiv.org/abs/1912.13361 (2020).
variational_mutual_info(infomaxvae, x, z, z_shuffle)
Compute a variational approximation of the mutual information between the input x and the latent code z using an InfoMaxVAE instance. Note that this estimate requires shuffling the latent codes between data samples. Therefore, it only applies to batch data cases. A single sample will not provide a meaningful estimate.
Arguments
  • infomaxvae::InfoMaxVAE: An InfoMaxVAE instance used to estimate mutual information.
  • x::AbstractArray: Array of input data. The last dimension represents each data sample.
  • z::AbstractMatrix: Matrix of corresponding latent representations of the input data.
  • z_shuffle::AbstractMatrix: Matrix of latent representations where the second dimension has been shuffled.
Returns
  • Float32: An approximation of the mutual information between the input data and its corresponding latent representation.
References
Rezaabad, A. L. & Vishwanath, S. Learning Representations by Maximizing Mutual Information in Variational Autoencoders. Preprint at http://arxiv.org/abs/1912.13361 (2020).
variational_mutual_info(
     infomaxvae::InfoMaxVAE,
     x::AbstractArray;
     seed::Union{Nothing,Int}=nothing
-)
Compute a variational approximation of the mutual information between the input x and the latent code z using an InfoMaxVAE instance. This function also shuffles the latent codes between data samples to provide a meaningful estimate even for a single data sample.
Arguments
  • infomaxvae::InfoMaxVAE: An InfoMaxVAE instance used to estimate mutual information.
  • x::AbstractArray: Array of input data. The last dimension represents each data sample.
Optional Keyword Arguments
  • seed::Union{Nothing,Int}: Optional argument. The seed for the random number generator used for shuffling the latent codes. If not provided, a random seed will be used.
Returns
  • Float32: An approximation of the mutual information between the input data and its corresponding latent representation.
References
Rezaabad, A. L. & Vishwanath, S. Learning Representations by Maximizing Mutual Information in Variational Autoencoders. Preprint at http://arxiv.org/abs/1912.13361 (2020).
Default initializations
AutoEncoderToolkit.jl provides default initializations for the MutualInfoChain. Although it gives the user less flexibility, it can be useful for quick prototyping.
AutoEncoderToolkit.InfoMaxVAEs.MutualInfoChain — Method
MutualInfoChain(
+)
Compute a variational approximation of the mutual information between the input x and the latent code z using an InfoMaxVAE instance. This function also shuffles the latent codes between data samples to provide a meaningful estimate even for a single data sample.
Arguments
  • infomaxvae::InfoMaxVAE: An InfoMaxVAE instance used to estimate mutual information.
  • x::AbstractArray: Array of input data. The last dimension represents each data sample.
Optional Keyword Arguments
  • seed::Union{Nothing,Int}: Optional argument. The seed for the random number generator used for shuffling the latent codes. If not provided, a random seed will be used.
Returns
  • Float32: An approximation of the mutual information between the input data and its corresponding latent representation.
References
Rezaabad, A. L. & Vishwanath, S. Learning Representations by Maximizing Mutual Information in Variational Autoencoders. Preprint at http://arxiv.org/abs/1912.13361 (2020).
Default initializations
AutoEncoderToolkit.jl provides default initializations for the MutualInfoChain. Although it gives the user less flexibility, it can be useful for quick prototyping.
AutoEncoderToolkit.InfoMaxVAEs.MutualInfoChain — Method
MutualInfoChain(
     size_input::Union{Int,Vector{<:Int}},
     n_latent::Int,
     mlp_neurons::Vector{<:Int},
     mlp_activations::Vector{<:Function},
     output_activation::Function;
     init::Function = Flux.glorot_uniform
-)
Constructs a default MutualInfoChain. 
Arguments
  • n_input::Int: Number of input features to the MutualInfoChain.
  • n_latent::Int: The dimensionality of the latent space.
  • mlp_neurons::Vector{<:Int}: A vector of integers where each element represents the number of neurons in the corresponding hidden layer of the MLP.
  • mlp_activations::Vector{<:Function}: A vector of activation functions to be used in the hidden layers. Length must match that of mlp_neurons.
  • output_activation::Function: Activation function for the output neuron of the MLP.
Optional Keyword Arguments
  • init::Function: Initialization function for the weights of all layers in the MutualInfoChain. Defaults to Flux.glorot_uniform.
Returns
  • MutualInfoChain: A MutualInfoChain instance with the specified MLP architecture.
Notes
The function will throw an error if the number of provided activation functions does not match the number of layers specified in mlp_neurons.

+)

Constructs a default MutualInfoChain.

Arguments

  • n_input::Int: Number of input features to the MutualInfoChain.
  • n_latent::Int: The dimensionality of the latent space.
  • mlp_neurons::Vector{<:Int}: A vector of integers where each element represents the number of neurons in the corresponding hidden layer of the MLP.
  • mlp_activations::Vector{<:Function}: A vector of activation functions to be used in the hidden layers. Length must match that of mlp_neurons.
  • output_activation::Function: Activation function for the output neuron of the MLP.

Optional Keyword Arguments

  • init::Function: Initialization function for the weights of all layers in the MutualInfoChain. Defaults to Flux.glorot_uniform.

Returns

  • MutualInfoChain: A MutualInfoChain instance with the specified MLP architecture.

Notes

The function will throw an error if the number of provided activation functions does not match the number of layers specified in mlp_neurons.

diff --git a/dev/layers/index.html b/dev/layers/index.html index 832ae58..883a0f0 100644 --- a/dev/layers/index.html +++ b/dev/layers/index.html @@ -1,15 +1,15 @@ -Custom Layers · AutoEncoderToolkit

Custom Layers

AutoEncoderToolkit.jl provides a set of commonly-used custom layers for building autoencoders. These layers need to be explicitly defined if you want to save a train model and load it later. For example, if the input to the encoder is an image in format HWC (height, width, channel), somewhere in the encoder there must be a function that flattens its input to a vector for the mapping to the latent space to be possible. If you were to define this with a simple function, the libraries to save the the model such as JLD2 or BSON would not work with these anonymous function. This is why we provide this set of custom layers that play along these libraries.

Reshape

AutoEncoderToolkit.Reshape — Type
Reshape(shape)

A custom layer for Flux that reshapes its input to a specified shape.

This layer is useful when you need to change the dimensions of your data within a Flux model. Unlike the built-in reshape operation in Julia, this custom layer can be saved and loaded using packages such as BSON or JLD2.

Arguments

  • shape: The target shape. This can be any tuple of integers and colons. Colons are used to indicate dimensions whose size should be inferred such that the total number of elements remains the same.

Examples

julia> r = Reshape(10, :)
+Custom Layers · AutoEncoderToolkit
Custom Layers
AutoEncoderToolkit.jl provides a set of commonly-used custom layers for building autoencoders. These layers need to be explicitly defined if you want to save a train model and load it later. For example, if the input to the encoder is an image in format HWC (height, width, channel), somewhere in the encoder there must be a function that flattens its input to a vector for the mapping to the latent space to be possible. If you were to define this with a simple function, the libraries to save the the model such as JLD2 or BSON would not work with these anonymous function. This is why we provide this set of custom layers that play along these libraries.
Reshape
AutoEncoderToolkit.Reshape — Type
Reshape(shape)
A custom layer for Flux that reshapes its input to a specified shape.
This layer is useful when you need to change the dimensions of your data within a Flux model. Unlike the built-in reshape operation in Julia, this custom layer can be saved and loaded using packages such as BSON or JLD2.
Arguments
  • shape: The target shape. This can be any tuple of integers and colons. Colons are used to indicate dimensions whose size should be inferred such that the total number of elements remains the same.
Examples
julia> r = Reshape(10, :)
 Reshape((10, :))
 
 julia> r(rand(5, 2))
-10×1 Matrix{Float64}:
Note
When saving and loading the model, make sure to include Reshape in the list of layers to be processed by BSON or JLD2.
AutoEncoderToolkit.Reshape — Method
Reshape(args...)
Constructor for the Reshape struct that takes variable arguments.
This function allows us to create a Reshape instance with any shape.
Arguments
  • args...: Variable arguments representing the dimensions of the target shape.
Returns
  • A Reshape instance with the target shape set to the provided dimensions.
Examples
julia> r = Reshape(10, :)
+10×1 Matrix{Float64}:
Note
When saving and loading the model, make sure to include Reshape in the list of layers to be processed by BSON or JLD2.
AutoEncoderToolkit.Reshape — Method
Reshape(args...)
Constructor for the Reshape struct that takes variable arguments.
This function allows us to create a Reshape instance with any shape.
Arguments
  • args...: Variable arguments representing the dimensions of the target shape.
Returns
  • A Reshape instance with the target shape set to the provided dimensions.
Examples
julia> r = Reshape(10, :)
 Reshape((10, :))
(r::Reshape)(x)
This function is called during the forward pass of the model. It reshapes the input x to the target shape stored in the Reshape instance r.
Arguments
  • r::Reshape: An instance of the Reshape struct.
  • x: The input to be reshaped.
Returns
  • The reshaped input.
Examples
julia> r = Reshape(10, :)
 Reshape((10, :))
 
 julia> r(rand(5, 2))
 10×1 Matrix{Float64}:
- ...
Flatten
AutoEncoderToolkit.Flatten — Type
Flatten()
A custom layer for Flux that flattens its input into a 1D vector.
This layer is useful when you need to change the dimensions of your data within a Flux model. Unlike the built-in flatten operation in Julia, this custom layer can be saved and loaded by packages such as BSON and JLD2.
Examples
julia> f = Flatten()
+ ...
Flatten
AutoEncoderToolkit.Flatten — Type
Flatten()
A custom layer for Flux that flattens its input into a 1D vector.
This layer is useful when you need to change the dimensions of your data within a Flux model. Unlike the built-in flatten operation in Julia, this custom layer can be saved and loaded by packages such as BSON and JLD2.
Examples
julia> f = Flatten()
 
 julia> f(rand(5, 2))
-10-element Vector{Float64}:
Note
When saving and loading the model, make sure to include Flatten in the list of layers to be processed by BSON or JLD2.
AutoEncoderToolkit.Flatten — Method
(f::Flatten)(x)
This function is called during the forward pass of the model. It flattens the input x into a 1D vector.
Arguments
  • f::Flatten: An instance of the Flatten struct.
  • x: The input to be flattened.
Returns
The flattened input.
ActivationOverDims
AutoEncoderToolkit.ActivationOverDims — Type
ActivationOverDims(σ::Function, dims::Int)
A custom layer for Flux that applies an activation function over specified dimensions.
This layer is useful when you need to apply an activation function over specific dimensions of your data within a Flux model. Unlike the built-in activation functions in Julia, this custom layer can be saved and loaded using the BSON or JLD2 package.
Arguments
  • σ::Function: The activation function to be applied.
  • dims: The dimensions over which the activation function should be applied.
Note
When saving and loading the model, make sure to include ActivationOverDims in the list of layers to be processed by BSON or JLD2.
AutoEncoderToolkit.ActivationOverDims — Method
(σ::ActivationOverDims)(x)
This function is called during the forward pass of the model. It applies the activation function σ.σ over the dimensions σ.dims of the input x.
Arguments
  • σ::ActivationOverDims: An instance of the ActivationOverDims struct.
  • x: The input to which the activation function should be applied.
Returns
  • The input x with the activation function applied over the specified dimensions.
Note
This custom layer can be saved and loaded using the BSON package. When saving and loading the model, make sure to include ActivationOverDims in the list of layers to be processed by BSON or JLD2.

+10-element Vector{Float64}:

Note

When saving and loading the model, make sure to include Flatten in the list of layers to be processed by BSON or JLD2.

AutoEncoderToolkit.Flatten — Method
(f::Flatten)(x)

This function is called during the forward pass of the model. It flattens the input x into a 1D vector.

Arguments

  • f::Flatten: An instance of the Flatten struct.
  • x: The input to be flattened.

Returns

The flattened input.

ActivationOverDims

AutoEncoderToolkit.ActivationOverDims — Type
ActivationOverDims(σ::Function, dims::Int)

A custom layer for Flux that applies an activation function over specified dimensions.

This layer is useful when you need to apply an activation function over specific dimensions of your data within a Flux model. Unlike the built-in activation functions in Julia, this custom layer can be saved and loaded using the BSON or JLD2 package.

Arguments

  • σ::Function: The activation function to be applied.
  • dims: The dimensions over which the activation function should be applied.

Note

When saving and loading the model, make sure to include ActivationOverDims in the list of layers to be processed by BSON or JLD2.

AutoEncoderToolkit.ActivationOverDims — Method
(σ::ActivationOverDims)(x)

This function is called during the forward pass of the model. It applies the activation function σ.σ over the dimensions σ.dims of the input x.

Arguments

  • σ::ActivationOverDims: An instance of the ActivationOverDims struct.
  • x: The input to which the activation function should be applied.

Returns

  • The input x with the activation function applied over the specified dimensions.

Note

This custom layer can be saved and loaded using the BSON package. When saving and loading the model, make sure to include ActivationOverDims in the list of layers to be processed by BSON or JLD2.

diff --git a/dev/mmdvae/index.html b/dev/mmdvae/index.html index 4670ca4..eef1866 100644 --- a/dev/mmdvae/index.html +++ b/dev/mmdvae/index.html @@ -1,7 +1,7 @@ -MMD-VAE (InfoVAE) · AutoEncoderToolkit

MMD-VAE (InfoVAE)

The Maximum-Mean Discrepancy Variational Autoencoder (MMD-VAE) is a variant of the Variational Autoencoder (VAE) that adds an extra term to the evidence lower bound (ELBO) that aims to maximize the mutual information between the latent space representation and the input data. In particular, the MMD-VAE uses the Maximum-Mean Discrepancy (MMD) as a measure of the "distance" between the latent space distribution and the input data distribution.

For the implementation of the MMD-VAE in AutoEncoderToolkit.jl, the MMDVAE struct inherits directly from the VAE struct and adds the necessary functions to compute the extra terms in the loss function. An MMDVAE object is created by simply passing a VAE object to the constructor. This way, we can use Julias multiple dispatch to extend the functionality of the VAE object without having to redefine the entire structure.

Reference

Maximum-Mean Discrepancy Variational Autoencoders Zhao, S., Song, J. & Ermon, S. InfoVAE: Information Maximizing Variational Autoencoders. Preprint at http://arxiv.org/abs/1706.02262 (2018).

MMDVAE struct

AutoEncoderToolkit.MMDVAEs.MMDVAE — Type
`MMDVAE{
+MMD-VAE (InfoVAE) · AutoEncoderToolkit
MMD-VAE (InfoVAE)
The Maximum-Mean Discrepancy Variational Autoencoder (MMD-VAE) is a variant of the Variational Autoencoder (VAE) that adds an extra term to the evidence lower bound (ELBO) that aims to maximize the mutual information between the latent space representation and the input data. In particular, the MMD-VAE uses the Maximum-Mean Discrepancy (MMD) as a measure of the "distance" between the latent space distribution and the input data distribution.
For the implementation of the MMD-VAE in AutoEncoderToolkit.jl, the MMDVAE struct inherits directly from the VAE struct and adds the necessary functions to compute the extra terms in the loss function. An MMDVAE object is created by simply passing a VAE object to the constructor. This way, we can use Julias multiple dispatch to extend the functionality of the VAE object without having to redefine the entire structure.
Reference
Maximum-Mean Discrepancy Variational Autoencoders Zhao, S., Song, J. & Ermon, S. InfoVAE: Information Maximizing Variational Autoencoders. Preprint at http://arxiv.org/abs/1706.02262 (2018).
MMDVAE struct
AutoEncoderToolkit.MMDVAEs.MMDVAE — Type
`MMDVAE{
     V<:VAE{<:AbstractVariationalEncoder,<:AbstractVariationalDecoder}
-    } <: AbstractVariationalAutoEncoder`
A struct representing a Maximum-Mean Discrepancy Variational Autoencoder (MMD-VAE).
Fields
  • vae::V: A Variational Autoencoder (VAE) that forms the basis of the MMD-VAE. The VAE should be composed of an AbstractVariationalEncoder and an AbstractVariationalDecoder.
Description
The MMDVAE struct is a subtype of AbstractVariationalAutoEncoder and represents a specific type of VAE known as an MMD-VAE. The MMD-VAE modifies the standard VAE by replacing the KL-divergence term in the loss function with a Maximum-Mean Discrepancy (MMD) term, which measures the distance between the aggregated posterior of the latent codes and the prior. This can help to alleviate the issue of posterior collapse, where the aggregated posterior fails to cover significant parts of the prior, commonly seen in VAEs.
Citation
Maximum-Mean Discrepancy Variational Autoencoders. Zhao, S., Song, J. & Ermon, S. InfoVAE: Information Maximizing Variational Autoencoders. Preprint at http://arxiv.org/abs/1706.02262 (2018).
Forward pass
AutoEncoderToolkit.MMDVAEs.MMDVAE — Method
(mmdvae::MMDVAE)(x::AbstractArray; latent::Bool=false)
Defines the forward pass for the Maximum-Mean Discrepancy Variational Autoencoder (MMD-VAE).
Arguments
  • x::AbstractArray: Input data.
Optional Keyword Arguments
  • latent::Bool: Whether to return the latent variables along with the decoder output. If true, the function returns a tuple containing the encoder outputs, the latent sample, and the decoder outputs. If false, the function only returns the decoder outputs. Defaults to false.  
Returns
  • If latent is true, returns a NamedTuple containing:
    • encoder: The outputs of the encoder.
    • z: The latent sample.
    • decoder: The outputs of the decoder.
  • If latent is false, returns the outputs of the decoder.
Loss function
AutoEncoderToolkit.MMDVAEs.loss — Function
loss(mmdvae::MMDVAE, x::AbstractArray; σ::Number=1.0f0, λ::Number=1.0f0, α::Number=0.0f0, n_latent_samples::Int=50, kernel::Function=gaussian_kernel, kernel_kwargs::NamedTuple=NamedTuple(), reconstruction_loglikelihood::Function=decoder_loglikelihood, kl_divergence::Function=encoder_kl)
Loss function for the Maximum-Mean Discrepancy variational autoencoder (MMD-VAE). The loss function is defined as:
loss = -⟨log p(x|z)⟩ + (1 - α) * Dₖₗ(qᵩ(z | x) || p(z)) + (λ + α - 1) * MMD-D(qᵩ(z) || p(z)),
Arguments
  • mmdvae::MMDVAE: Struct containing the elements of the MMD-VAE.
  • x::AbstractArray: Input data.
Optional Arguments
  • λ::Number=1.0f0: Hyperparameter that emphasizes the importance of the KL divergence between qᵩ(z) and π(z) during training.
  • α::Number=0.0f0: Hyperparameter that emphasizes the importance of the Mutual Information term during optimization.
  • n_latent_samples::Int=50: Number of samples to take from the latent space prior π(z) when computing the MMD divergence.
  • kernel::Function=gaussian_kernel: Kernel used to compute the divergence. Default is the Gaussian Kernel.
  • kernel_kwargs::NamedTuple=NamedTuple(): Additional keyword arguments to be passed to the kernel function.
  • reconstruction_loglikelihood::Function=decoder_loglikelihood: Function that computes the log likelihood of the reconstructed input.
  • kl_divergence::Function=encoder_kl: Function that computes the Kullback-Leibler divergence between the encoder distribution and the prior.
Returns
  • Single value defining the loss function for entry x when compared with reconstructed output x̂.
Description
This function calculates the loss for the MMD-VAE. It computes the log likelihood of the reconstructed input, the MMD divergence between the encoder distribution and the prior, and the Kullback-Leibler divergence between the approximate decoder and the prior. These quantities are combined according to the formula above to compute the loss.
loss(
+    } <: AbstractVariationalAutoEncoder`
A struct representing a Maximum-Mean Discrepancy Variational Autoencoder (MMD-VAE).
Fields
  • vae::V: A Variational Autoencoder (VAE) that forms the basis of the MMD-VAE. The VAE should be composed of an AbstractVariationalEncoder and an AbstractVariationalDecoder.
Description
The MMDVAE struct is a subtype of AbstractVariationalAutoEncoder and represents a specific type of VAE known as an MMD-VAE. The MMD-VAE modifies the standard VAE by replacing the KL-divergence term in the loss function with a Maximum-Mean Discrepancy (MMD) term, which measures the distance between the aggregated posterior of the latent codes and the prior. This can help to alleviate the issue of posterior collapse, where the aggregated posterior fails to cover significant parts of the prior, commonly seen in VAEs.
Citation
Maximum-Mean Discrepancy Variational Autoencoders. Zhao, S., Song, J. & Ermon, S. InfoVAE: Information Maximizing Variational Autoencoders. Preprint at http://arxiv.org/abs/1706.02262 (2018).
Forward pass
AutoEncoderToolkit.MMDVAEs.MMDVAE — Method
(mmdvae::MMDVAE)(x::AbstractArray; latent::Bool=false)
Defines the forward pass for the Maximum-Mean Discrepancy Variational Autoencoder (MMD-VAE).
Arguments
  • x::AbstractArray: Input data.
Optional Keyword Arguments
  • latent::Bool: Whether to return the latent variables along with the decoder output. If true, the function returns a tuple containing the encoder outputs, the latent sample, and the decoder outputs. If false, the function only returns the decoder outputs. Defaults to false.  
Returns
  • If latent is true, returns a NamedTuple containing:
    • encoder: The outputs of the encoder.
    • z: The latent sample.
    • decoder: The outputs of the decoder.
  • If latent is false, returns the outputs of the decoder.
Loss function
AutoEncoderToolkit.MMDVAEs.loss — Function
loss(mmdvae::MMDVAE, x::AbstractArray; σ::Number=1.0f0, λ::Number=1.0f0, α::Number=0.0f0, n_latent_samples::Int=50, kernel::Function=gaussian_kernel, kernel_kwargs::NamedTuple=NamedTuple(), reconstruction_loglikelihood::Function=decoder_loglikelihood, kl_divergence::Function=encoder_kl)
Loss function for the Maximum-Mean Discrepancy variational autoencoder (MMD-VAE). The loss function is defined as:
loss = -⟨log p(x|z)⟩ + (1 - α) * Dₖₗ(qᵩ(z | x) || p(z)) + (λ + α - 1) * MMD-D(qᵩ(z) || p(z)),
Arguments
  • mmdvae::MMDVAE: Struct containing the elements of the MMD-VAE.
  • x::AbstractArray: Input data.
Optional Arguments
  • λ::Number=1.0f0: Hyperparameter that emphasizes the importance of the KL divergence between qᵩ(z) and π(z) during training.
  • α::Number=0.0f0: Hyperparameter that emphasizes the importance of the Mutual Information term during optimization.
  • n_latent_samples::Int=50: Number of samples to take from the latent space prior π(z) when computing the MMD divergence.
  • kernel::Function=gaussian_kernel: Kernel used to compute the divergence. Default is the Gaussian Kernel.
  • kernel_kwargs::NamedTuple=NamedTuple(): Additional keyword arguments to be passed to the kernel function.
  • reconstruction_loglikelihood::Function=decoder_loglikelihood: Function that computes the log likelihood of the reconstructed input.
  • kl_divergence::Function=encoder_kl: Function that computes the Kullback-Leibler divergence between the encoder distribution and the prior.
Returns
  • Single value defining the loss function for entry x when compared with reconstructed output x̂.
Description
This function calculates the loss for the MMD-VAE. It computes the log likelihood of the reconstructed input, the MMD divergence between the encoder distribution and the prior, and the Kullback-Leibler divergence between the approximate decoder and the prior. These quantities are combined according to the formula above to compute the loss.
loss(
     mmdvae::MMDVAE, x_in::AbstractArray, x_out::AbstractArray; 
     λ::Number=1.0f0, α::Number=0.0f0, 
     n_latent_samples::Int=50, 
@@ -9,15 +9,15 @@
     kernel_kwargs::NamedTuple=NamedTuple(), 
     reconstruction_loglikelihood::Function=decoder_loglikelihood, 
     kl_divergence::Function=encoder_kl
-)
Loss function for the Maximum-Mean Discrepancy variational autoencoder (MMD-VAE). The loss function is defined as:
loss = -⟨log p(x|z)⟩ + (1 - α) * Dₖₗ(qᵩ(z | x) || p(z)) + (λ + α - 1) * MMD-D(qᵩ(z) || p(z)),
Arguments
  • mmdvae::MMDVAE: Struct containing the elements of the MMD-VAE.
  • x_in::AbstractArray: Input data.
  • x_out::AbstractArray: Data against which to compare the reconstructed output.
Optional Arguments
  • λ::Number=1.0f0: Hyperparameter that emphasizes the importance of the KL divergence between qᵩ(z) and π(z) during training.
  • α::Number=0.0f0: Hyperparameter that emphasizes the importance of the Mutual Information term during optimization.
  • n_latent_samples::Int=50: Number of samples to take from the latent space prior π(z) when computing the MMD divergence.
  • kernel::Function=gaussian_kernel: Kernel used to compute the divergence. Default is the Gaussian Kernel.
  • kernel_kwargs::NamedTuple=NamedTuple(): Additional keyword arguments to be passed to the kernel function.
  • reconstruction_loglikelihood::Function=decoder_loglikelihood: Function that computes the log likelihood of the reconstructed input.
  • kl_divergence::Function=encoder_kl: Function that computes the Kullback-Leibler divergence between the encoder distribution and the prior.
Returns
  • Single value defining the loss function for entry x when compared with reconstructed output x̂.
Description
This function calculates the loss for the MMD-VAE. It computes the log likelihood of the reconstructed input, the MMD divergence between the encoder distribution and the prior, and the Kullback-Leibler divergence between the approximate decoder and the prior. These quantities are combined according to the formula above to compute the loss.
Training
AutoEncoderToolkit.MMDVAEs.train! — Function
train!(mmdvae, x, opt; loss_function, loss_kwargs, verbose, loss_return)
Customized training function to update parameters of a variational autoencoder given a specified loss function.
Arguments
  • mmdvae::MMDVAE: A struct containing the elements of a Maximum-Mean Discrepancy Variational Autoencoder (MMD-VAE).
  • x::AbstractArray: Data on which to evaluate the loss function. The last dimension is taken as having each of the samples in a batch.
  • opt::NamedTuple: State of the optimizer for updating parameters. Typically initialized using Flux.Train.setup.
Optional Keyword Arguments
  • loss_function::Function=loss: The loss function used for training. It should accept the MMDVAE model, data x, and keyword arguments in that order.
  • loss_kwargs::NamedTuple=NamedTuple(): Arguments for the loss function. These might include parameters like α, or β, depending on the specific loss function in use.
  • verbose::Bool=false: If true, the loss value will be printed during training.
  • loss_return::Bool=false: If true, the loss value will be returned after training.
Description
Trains the MMDVAE by:
  1. Computing the gradient of the loss w.r.t the MMDVAE parameters.
  2. Updating the MMDVAE parameters using the optimizer.
train!(mmdvae, x_in, x_out, opt; loss_function, loss_kwargs, verbose, loss_return)
Customized training function to update parameters of a variational autoencoder given a specified loss function.
Arguments
  • mmdvae::MMDVAE: A struct containing the elements of a Maximum-Mean Discrepancy Variational Autoencoder (MMD-VAE).
  • x_in::AbstractArray: Data on which to evaluate the loss function. The last dimension is taken as having each of the samples in a batch.
  • x_out::AbstractArray: Data against which to compare the reconstructed output.
  • opt::NamedTuple: State of the optimizer for updating parameters. Typically initialized using Flux.Train.setup.
Optional Keyword Arguments
  • loss_function::Function=loss: The loss function used for training. It should accept the MMDVAE model, data x, and keyword arguments in that order.
  • loss_kwargs::NamedTuple=NamedTuple(): Arguments for the loss function. These might include parameters like α, or β, depending on the specific loss function in use.
  • verbose::Bool=false: If true, the loss value will be printed during training.
  • loss_return::Bool=false: If true, the loss value will be returned after training.
Description
Trains the MMDVAE by:
  1. Computing the gradient of the loss w.r.t the MMDVAE parameters.
  2. Updating the MMDVAE parameters using the optimizer.
Other Functions
AutoEncoderToolkit.MMDVAEs.gaussian_kernel — Function
gaussian_kernel(
+)
Loss function for the Maximum-Mean Discrepancy variational autoencoder (MMD-VAE). The loss function is defined as:
loss = -⟨log p(x|z)⟩ + (1 - α) * Dₖₗ(qᵩ(z | x) || p(z)) + (λ + α - 1) * MMD-D(qᵩ(z) || p(z)),
Arguments
  • mmdvae::MMDVAE: Struct containing the elements of the MMD-VAE.
  • x_in::AbstractArray: Input data.
  • x_out::AbstractArray: Data against which to compare the reconstructed output.
Optional Arguments
  • λ::Number=1.0f0: Hyperparameter that emphasizes the importance of the KL divergence between qᵩ(z) and π(z) during training.
  • α::Number=0.0f0: Hyperparameter that emphasizes the importance of the Mutual Information term during optimization.
  • n_latent_samples::Int=50: Number of samples to take from the latent space prior π(z) when computing the MMD divergence.
  • kernel::Function=gaussian_kernel: Kernel used to compute the divergence. Default is the Gaussian Kernel.
  • kernel_kwargs::NamedTuple=NamedTuple(): Additional keyword arguments to be passed to the kernel function.
  • reconstruction_loglikelihood::Function=decoder_loglikelihood: Function that computes the log likelihood of the reconstructed input.
  • kl_divergence::Function=encoder_kl: Function that computes the Kullback-Leibler divergence between the encoder distribution and the prior.
Returns
  • Single value defining the loss function for entry x when compared with reconstructed output x̂.
Description
This function calculates the loss for the MMD-VAE. It computes the log likelihood of the reconstructed input, the MMD divergence between the encoder distribution and the prior, and the Kullback-Leibler divergence between the approximate decoder and the prior. These quantities are combined according to the formula above to compute the loss.
Training
AutoEncoderToolkit.MMDVAEs.train! — Function
train!(mmdvae, x, opt; loss_function, loss_kwargs, verbose, loss_return)
Customized training function to update parameters of a variational autoencoder given a specified loss function.
Arguments
  • mmdvae::MMDVAE: A struct containing the elements of a Maximum-Mean Discrepancy Variational Autoencoder (MMD-VAE).
  • x::AbstractArray: Data on which to evaluate the loss function. The last dimension is taken as having each of the samples in a batch.
  • opt::NamedTuple: State of the optimizer for updating parameters. Typically initialized using Flux.Train.setup.
Optional Keyword Arguments
  • loss_function::Function=loss: The loss function used for training. It should accept the MMDVAE model, data x, and keyword arguments in that order.
  • loss_kwargs::NamedTuple=NamedTuple(): Arguments for the loss function. These might include parameters like α, or β, depending on the specific loss function in use.
  • verbose::Bool=false: If true, the loss value will be printed during training.
  • loss_return::Bool=false: If true, the loss value will be returned after training.
Description
Trains the MMDVAE by:
  1. Computing the gradient of the loss w.r.t the MMDVAE parameters.
  2. Updating the MMDVAE parameters using the optimizer.
train!(mmdvae, x_in, x_out, opt; loss_function, loss_kwargs, verbose, loss_return)
Customized training function to update parameters of a variational autoencoder given a specified loss function.
Arguments
  • mmdvae::MMDVAE: A struct containing the elements of a Maximum-Mean Discrepancy Variational Autoencoder (MMD-VAE).
  • x_in::AbstractArray: Data on which to evaluate the loss function. The last dimension is taken as having each of the samples in a batch.
  • x_out::AbstractArray: Data against which to compare the reconstructed output.
  • opt::NamedTuple: State of the optimizer for updating parameters. Typically initialized using Flux.Train.setup.
Optional Keyword Arguments
  • loss_function::Function=loss: The loss function used for training. It should accept the MMDVAE model, data x, and keyword arguments in that order.
  • loss_kwargs::NamedTuple=NamedTuple(): Arguments for the loss function. These might include parameters like α, or β, depending on the specific loss function in use.
  • verbose::Bool=false: If true, the loss value will be printed during training.
  • loss_return::Bool=false: If true, the loss value will be returned after training.
Description
Trains the MMDVAE by:
  1. Computing the gradient of the loss w.r.t the MMDVAE parameters.
  2. Updating the MMDVAE parameters using the optimizer.
Other Functions
AutoEncoderToolkit.MMDVAEs.gaussian_kernel — Function
gaussian_kernel(
     x::AbstractArray, y::AbstractArray; ρ::Float32=1.0f0, dims::Int=2
-)
Function to compute the Gaussian Kernel between two arrays x and y, defined as 
    k(x, y) = exp(-||x - y ||² / ρ²)
Arguments
  • x::AbstractArray: First input array for the kernel.
  • y::AbstractArray: Second input array for the kernel.  
Optional Keyword Arguments
  • ρ=1.0f0: Kernel amplitude hyperparameter. Larger ρ gives a smoother kernel.
  • dims::Int=2: Number of dimensions to compute pairwise distances over.
Returns
  • k::AbstractArray: Kernel matrix where each element is computed as 
Theory
The Gaussian kernel measures the similarity between two points x and y. It is widely used in many machine learning algorithms. This implementation computes the squared Euclidean distance between all pairs of rows in x and y, scales the distance by ρ² and takes the exponential.
AutoEncoderToolkit.MMDVAEs.mmd_div — Function
mmd_div(
+)
Function to compute the Gaussian Kernel between two arrays x and y, defined as 
    k(x, y) = exp(-||x - y ||² / ρ²)
Arguments
  • x::AbstractArray: First input array for the kernel.
  • y::AbstractArray: Second input array for the kernel.  
Optional Keyword Arguments
  • ρ=1.0f0: Kernel amplitude hyperparameter. Larger ρ gives a smoother kernel.
  • dims::Int=2: Number of dimensions to compute pairwise distances over.
Returns
  • k::AbstractArray: Kernel matrix where each element is computed as 
Theory
The Gaussian kernel measures the similarity between two points x and y. It is widely used in many machine learning algorithms. This implementation computes the squared Euclidean distance between all pairs of rows in x and y, scales the distance by ρ² and takes the exponential.
AutoEncoderToolkit.MMDVAEs.mmd_div — Function
mmd_div(
     x::AbstractArray, y::AbstractArray; 
     kernel::Function=gaussian_kernel, 
     kernel_kwargs::NamedTuple=NamedTuple()
-)
Compute the Maximum Mean Discrepancy (MMD) divergence between two arrays x and y.
Arguments
  • x::AbstractArray: First input array.
  • y::AbstractArray: Second input array.
Keyword Arguments
  • kernel::Function=gaussian_kernel: Kernel function to use. Default is the Gaussian kernel.
  • kernel_kwargs::NamedTuple=NamedTuple(): Additional keyword arguments to be passed to the kernel function.
Returns
  • mmd::Number: MMD divergence value. 
Theory
MMD measures the difference between two distributions based on embeddings in a Reproducing Kernel Hilbert Space (RKHS). It is widely used for two-sample tests.
This function implements MMD as:
MMD(x, y) = mean(k(x, x)) - 2 * mean(k(x, y)) + mean(k(y, y))
where k is a positive definite kernel (e.g., Gaussian).
AutoEncoderToolkit.MMDVAEs.logP_mmd_ratio — Function
logP_mmd_ratio(
+)
Compute the Maximum Mean Discrepancy (MMD) divergence between two arrays x and y.
Arguments
  • x::AbstractArray: First input array.
  • y::AbstractArray: Second input array.
Keyword Arguments
  • kernel::Function=gaussian_kernel: Kernel function to use. Default is the Gaussian kernel.
  • kernel_kwargs::NamedTuple=NamedTuple(): Additional keyword arguments to be passed to the kernel function.
Returns
  • mmd::Number: MMD divergence value. 
Theory
MMD measures the difference between two distributions based on embeddings in a Reproducing Kernel Hilbert Space (RKHS). It is widely used for two-sample tests.
This function implements MMD as:
MMD(x, y) = mean(k(x, x)) - 2 * mean(k(x, y)) + mean(k(y, y))
where k is a positive definite kernel (e.g., Gaussian).
AutoEncoderToolkit.MMDVAEs.logP_mmd_ratio — Function
logP_mmd_ratio(
     mmdvae::MMDVAE, x::AbstractArray; 
     n_latent_samples::Int=100, kernel=gaussian_kernel, 
     kernel_kwargs::NamedTuple=NamedTuple(), 
     reconstruction_loglikelihood::Function=decoder_loglikelihood
-)
Function to compute the absolute ratio between the log likelihood ⟨log p(x|z)⟩ and the MMD divergence MMD-D(qᵩ(z|x)||p(z)).
Arguments
  • mmdvae::MMDVAE: Struct containing the elements of the MMD-VAE.
  • x::AbstractArray: Data to train the MMD-VAE.
Optional Keyword Arguments
  • n_latent_samples::Int=100: Number of samples to take from the latent space prior p(z) when computing the MMD divergence.
  • kernel=gaussian_kernel: Kernel used to compute the divergence.  Default is the Gaussian Kernel.
  • kernel_kwargs::NamedTuple=NamedTuple(): Tuple containing arguments for the Kernel function.
  • reconstruction_loglikelihood::Function=decoder_loglikelihood: Function that computes the log likelihood of the reconstructed input.
Returns
abs(⟨log p(x|z)⟩ / MMD-D(qᵩ(z|x)||p(z)))
Description
This function calculates:
  1. The log likelihood ⟨log p(x|z)⟩ of x under the MMD-VAE decoder, averaged over
all samples. 2. The MMD divergence between the encoder distribution q(z|x) and prior p(z). 
The absolute ratio of these two quantities is returned.
Note
This ratio is useful for setting the Lagrangian multiplier λ in training MMD-VAEs.

+)

Function to compute the absolute ratio between the log likelihood ⟨log p(x|z)⟩ and the MMD divergence MMD-D(qᵩ(z|x)||p(z)).

Arguments

  • mmdvae::MMDVAE: Struct containing the elements of the MMD-VAE.
  • x::AbstractArray: Data to train the MMD-VAE.

Optional Keyword Arguments

  • n_latent_samples::Int=100: Number of samples to take from the latent space prior p(z) when computing the MMD divergence.
  • kernel=gaussian_kernel: Kernel used to compute the divergence. Default is the Gaussian Kernel.
  • kernel_kwargs::NamedTuple=NamedTuple(): Tuple containing arguments for the Kernel function.
  • reconstruction_loglikelihood::Function=decoder_loglikelihood: Function that computes the log likelihood of the reconstructed input.

Returns

abs(⟨log p(x|z)⟩ / MMD-D(qᵩ(z|x)||p(z)))

Description

This function calculates:

  1. The log likelihood ⟨log p(x|z)⟩ of x under the MMD-VAE decoder, averaged over

all samples. 2. The MMD divergence between the encoder distribution q(z|x) and prior p(z).

The absolute ratio of these two quantities is returned.

Note

This ratio is useful for setting the Lagrangian multiplier λ in training MMD-VAEs.

diff --git a/dev/quickstart/index.html b/dev/quickstart/index.html index 54ad645..6dfc614 100644 --- a/dev/quickstart/index.html +++ b/dev/quickstart/index.html @@ -388,4 +388,4 @@ # Save model output nng_ex[:, :, (epoch÷n_save)+1] = nng(t_array) end # if -end # for

Now that we have trained the network, we can visualize the path between the initial and final points in the latent space. The color code in the following plot matches the epoch at which the path was computed.

+end # for

Now that we have trained the network, we can visualize the path between the initial and final points in the latent space. The color code in the following plot matches the epoch at which the path was computed.

diff --git a/dev/rhvae/index.html b/dev/rhvae/index.html index 0d3711d..94b86b1 100644 --- a/dev/rhvae/index.html +++ b/dev/rhvae/index.html @@ -1,12 +1,12 @@ -RHVAE · AutoEncoderToolkit

Riemannian Hamiltonian Variational Autoencoder

The Riemannian Hamiltonian Variational Autoencoder (RHVAE) is a variant of the Hamiltonian Variational Autoencoder (HVAE) that uses concepts from Riemannian geometry to improve the sampling of the latent space representation. As the HVAE, the RHVAE uses Hamiltonian dynamics to improve the sampling of the latent. However, the RHVAE accounts for the geometry of the latent space by learning a Riemannian metric tensor that is used to compute the kinetic energy of the dynamical system. This allows the RHVAE to sample the latent space more evenly while learning the curvature of the latent space.

For the implementation of the RHVAE in AutoEncoderToolkit.jl, the RHVAE requires two arguments to construct: the original VAE as well as a separate neural network used to compute the metric tensor. To facilitate the dispatch of the necessary functions associated with this second network, we also provide a MetricChain struct.

Warning

RHVAEs require the computation of nested gradients. This means that the AutoDiff framework must differentiate a function of an already AutoDiff differentiated function. This is known to be problematic for Julia's AutoDiff backends. See details below to understand how to we circumvent this problem.

Reference

Chadebec, C., Mantoux, C. & Allassonnière, S. Geometry-Aware Hamiltonian Variational Auto-Encoder. Preprint at http://arxiv.org/abs/2010.11518 (2020).

MetricChain struct

AutoEncoderToolkit.RHVAEs.MetricChain — Type
MetricChain <: AbstractMetricChain

A MetricChain is used to compute the Riemannian metric tensor in the latent space of a Riemannian Hamiltonian Variational AutoEncoder (RHVAE).

Fields

  • mlp::Flux.Chain: A multi-layer perceptron (MLP) consisting of the hidden layers. The inputs are first run through this MLP.
  • diag::Flux.Dense: A dense layer that computes the diagonal elements of a lower-triangular matrix. The output of the mlp is fed into this layer.
  • lower::Flux.Dense: A dense layer that computes the off-diagonal elements of the lower-triangular matrix. The output of the mlp is also fed into this layer.

The outputs of diag and lower are used to construct a lower-triangular matrix used to compute the Riemannian metric tensor in latent space.

Note

If the dimension of the latent space is n, the number of neurons in the output layer of diag must be n, and the number of neurons in the output layer of lower must be n * (n - 1) ÷ 2.

Example

mlp = Flux.Chain(Dense(10, 10, relu), Dense(10, 10, relu))
+RHVAE · AutoEncoderToolkit
Riemannian Hamiltonian Variational Autoencoder
The Riemannian Hamiltonian Variational Autoencoder (RHVAE) is a variant of the Hamiltonian Variational Autoencoder (HVAE) that uses concepts from Riemannian geometry to improve the sampling of the latent space representation. As the HVAE, the RHVAE uses Hamiltonian dynamics to improve the sampling of the latent. However, the RHVAE accounts for the geometry of the latent space by learning a Riemannian metric tensor that is used to compute the kinetic energy of the dynamical system. This allows the RHVAE to sample the latent space more evenly while learning the curvature of the latent space.
For the implementation of the RHVAE in AutoEncoderToolkit.jl, the RHVAE requires two arguments to construct: the original VAE as well as a separate neural network used to compute the metric tensor. To facilitate the dispatch of the necessary functions associated with this second network, we also provide a MetricChain struct.
Warning
RHVAEs require the computation of nested gradients. This means that the AutoDiff framework must differentiate a function of an already AutoDiff differentiated function. This is known to be problematic for Julia's AutoDiff backends. See details below to understand how to we circumvent this problem.
Reference
Chadebec, C., Mantoux, C. & Allassonnière, S. Geometry-Aware Hamiltonian Variational Auto-Encoder. Preprint at http://arxiv.org/abs/2010.11518 (2020).
MetricChain struct
AutoEncoderToolkit.RHVAEs.MetricChain — Type
MetricChain <: AbstractMetricChain
A MetricChain is used to compute the Riemannian metric tensor in the latent space of a Riemannian Hamiltonian Variational AutoEncoder (RHVAE).
Fields
  • mlp::Flux.Chain: A multi-layer perceptron (MLP) consisting of the hidden layers. The inputs are first run through this MLP.
  • diag::Flux.Dense: A dense layer that computes the diagonal elements of a lower-triangular matrix. The output of the mlp is fed into this layer.
  • lower::Flux.Dense: A dense layer that computes the off-diagonal elements of the lower-triangular matrix. The output of the mlp is also fed into this layer.
The outputs of diag and lower are used to construct a lower-triangular matrix used to compute the Riemannian metric tensor in latent space.
Note
If the dimension of the latent space is n, the number of neurons in the output layer of diag must be n, and the number of neurons in the output layer of lower must be n * (n - 1) ÷ 2.
Example
mlp = Flux.Chain(Dense(10, 10, relu), Dense(10, 10, relu))
 diag = Flux.Dense(10, 5)
 lower = Flux.Dense(10, 15)
-metric_chain = MetricChain(mlp, diag, lower)
RHVAE struct
RHVAE struct
AutoEncoderToolkit.RHVAEs.RHVAE — Type
RHVAE{
     V<:VAE{<:AbstractVariationalEncoder,<:AbstractVariationalDecoder}
-} <: AbstractVariationalAutoEncoder
A Riemannian Hamiltonian Variational AutoEncoder (RHVAE) as described in Chadebec, C., Mantoux, C. & Allassonnière, S. Geometry-Aware Hamiltonian Variational Auto-Encoder. Preprint at http://arxiv.org/abs/2010.11518 (2020).
The RHVAE is a type of Variational AutoEncoder (VAE) that incorporates a Riemannian metric in the latent space. This metric is computed by a MetricChain, which is a struct that contains a multi-layer perceptron (MLP) and two dense layers for computing the elements of a lower-triangular matrix.
The inverse metric is computed as follows:
G⁻¹(z) = ∑ᵢ₌₁ⁿ Lψᵢ Lψᵢᵀ exp(-‖z - cᵢ‖₂² / T²) + λIₗ
where L_ψᵢ is computed by the MetricChain, T is the temperature, λ is a regularization factor, and each column of centroids are the cᵢ.
Fields
  • vae::V: The underlying VAE, where V is a subtype of VAE with an AbstractVariationalEncoder and an AbstractVariationalDecoder.
  • metric_chain::MetricChain: The MetricChain that computes the Riemannian metric in the latent space.
  • centroids_data::AbstractArray: An array where the last dimension represents a data point xᵢ from which the centroids cᵢ are computed by passing them through the encoder.
  • centroids_latent::AbstractMatrix: A matrix where each column represents a centroid cᵢ in the inverse metric computation.
  • L::AbstractArray{<:Number, 3}: A 3D array where each slice represents a Lψᵢ matrix. Lψᵢ can intuitively be seen as the triangular matrix in the Cholesky decomposition of G⁻¹(centroids_latentᵢ) up to a regularization factor.
  • M::AbstractArray{<:Number, 3}: A 3D array where each slice represents a Lψᵢ Lψᵢᵀ.
  • T::Number: The temperature parameter in the inverse metric computation.  
  • λ::Number: The regularization factor in the inverse metric computation.
Forward pass
Metric Network
AutoEncoderToolkit.RHVAEs.MetricChain — Method
(m::MetricChain)(x::AbstractArray; matrix::Bool=false)
Perform a forward pass through the MetricChain.
Arguments
  • x::AbstractArray: The input data to be processed. 
  • matrix::Bool=false: A boolean flag indicating whether to return the result as a lower triangular matrix (if true) or as a tuple of diagonal and lower off-diagonal elements (if false). Defaults to false.
Returns
  • If matrix is true, returns a lower triangular matrix constructed from the outputs of the diag and lower components of the MetricChain.
  • If matrix is false, returns a NamedTuple with two elements: diag, the output of the diag component of the MetricChain, and lower, the output of the lower component of the MetricChain.
Example
m = MetricChain(...)
+} <: AbstractVariationalAutoEncoder
A Riemannian Hamiltonian Variational AutoEncoder (RHVAE) as described in Chadebec, C., Mantoux, C. & Allassonnière, S. Geometry-Aware Hamiltonian Variational Auto-Encoder. Preprint at http://arxiv.org/abs/2010.11518 (2020).
The RHVAE is a type of Variational AutoEncoder (VAE) that incorporates a Riemannian metric in the latent space. This metric is computed by a MetricChain, which is a struct that contains a multi-layer perceptron (MLP) and two dense layers for computing the elements of a lower-triangular matrix.
The inverse metric is computed as follows:
G⁻¹(z) = ∑ᵢ₌₁ⁿ Lψᵢ Lψᵢᵀ exp(-‖z - cᵢ‖₂² / T²) + λIₗ
where L_ψᵢ is computed by the MetricChain, T is the temperature, λ is a regularization factor, and each column of centroids are the cᵢ.
Fields
  • vae::V: The underlying VAE, where V is a subtype of VAE with an AbstractVariationalEncoder and an AbstractVariationalDecoder.
  • metric_chain::MetricChain: The MetricChain that computes the Riemannian metric in the latent space.
  • centroids_data::AbstractArray: An array where the last dimension represents a data point xᵢ from which the centroids cᵢ are computed by passing them through the encoder.
  • centroids_latent::AbstractMatrix: A matrix where each column represents a centroid cᵢ in the inverse metric computation.
  • L::AbstractArray{<:Number, 3}: A 3D array where each slice represents a Lψᵢ matrix. Lψᵢ can intuitively be seen as the triangular matrix in the Cholesky decomposition of G⁻¹(centroids_latentᵢ) up to a regularization factor.
  • M::AbstractArray{<:Number, 3}: A 3D array where each slice represents a Lψᵢ Lψᵢᵀ.
  • T::Number: The temperature parameter in the inverse metric computation.  
  • λ::Number: The regularization factor in the inverse metric computation.
Forward pass
Metric Network
AutoEncoderToolkit.RHVAEs.MetricChain — Method
(m::MetricChain)(x::AbstractArray; matrix::Bool=false)
Perform a forward pass through the MetricChain.
Arguments
  • x::AbstractArray: The input data to be processed. 
  • matrix::Bool=false: A boolean flag indicating whether to return the result as a lower triangular matrix (if true) or as a tuple of diagonal and lower off-diagonal elements (if false). Defaults to false.
Returns
  • If matrix is true, returns a lower triangular matrix constructed from the outputs of the diag and lower components of the MetricChain.
  • If matrix is false, returns a NamedTuple with two elements: diag, the output of the diag component of the MetricChain, and lower, the output of the lower component of the MetricChain.
Example
m = MetricChain(...)
 x = rand(Float32, 100, 10)
-m(x, matrix=true)  # Returns a lower triangular matrix
RHVAE
RHVAE
AutoEncoderToolkit.RHVAEs.RHVAE — Method
(rhvae::RHVAE{VAE{E,D}})(
     x::AbstractArray;
     ϵ::Union{<:Number,<:AbstractVector}=Float32(1E-4),
     K::Int=3,
@@ -20,7 +20,7 @@
     ),
     tempering_schedule::Function=quadratic_tempering,
     latent::Bool=false,
-) where where {E<:AbstractGaussianLogEncoder,D<:AbstractVariationalDecoder}
Run the Riemannian Hamiltonian Variational Autoencoder (RHVAE) on the given input.
Arguments
  • x::AbstractArray: The input to the RHVAE. If it is a vector, it represents a single data point. If Array, the last dimension must contain each of the data points.
Optional Keyword Arguments
  • K::Int=3: The number of leapfrog steps to perform in the Hamiltonian Monte Carlo (HMC) part of the RHVAE.
  • ϵ::Union{<:Number,<:AbstractVector}=0.01f0: The step size for the leapfrog steps in the HMC part of the RHVAE. If it is a scalar, the same step size is used for all dimensions. If it is an array, each element corresponds to the step size for a specific dimension.
  • βₒ::Number=0.3f0: The initial inverse temperature for the tempering schedule.
  • steps::Int: The number of fixed-point iterations to perform. Default is 3.
  • ∇H::Function=∇hamiltonian_finite: The function to compute the gradient of the Hamiltonian in the HMC part of the RHVAE.
  • ∇H_kwargs::NamedTuple: Additional keyword arguments to be passed to the ∇hamiltonian function. Default is a NamedTuple with reconstruction_loglikelihood, position_logprior, and momentum_logprior.  
  • G_inv::Function=G_inv: The function to compute the inverse of the Riemannian metric tensor.
  • tempering_schedule::Function=quadratic_tempering: The function to compute the tempering schedule in the RHVAE.
  • latent::Bool=false: If true, the function returns a NamedTuple containing the outputs of the encoder and decoder, and the final state of the phase space after the leapfrog and tempering steps. If false, the function only returns the output of the decoder.
Returns
If latent=true, the function returns a NamedTuple with the following fields:
  • encoder: The outputs of the encoder.
  • decoder: The output of the decoder.
  • phase_space: The final state of the phase space after the leapfrog and   tempering steps.
If latent=false, the function only returns the output of the decoder.
Description
This function runs the RHVAE on the given input. It first passes the input through the encoder to obtain the mean and log standard deviation of the latent space. It then uses the reparameterization trick to sample from the latent space. After that, it performs the leapfrog and tempering steps to refine the sample from the latent space. Finally, it passes the refined sample through the decoder to obtain the output.
Notes
Ensure that the dimensions of x match the input dimensions of the RHVAE, and that the dimensions of ϵ match the dimensions of the latent space.
Loss function
AutoEncoderToolkit.RHVAEs.loss — Function
loss(
+) where where {E<:AbstractGaussianLogEncoder,D<:AbstractVariationalDecoder}
Run the Riemannian Hamiltonian Variational Autoencoder (RHVAE) on the given input.
Arguments
  • x::AbstractArray: The input to the RHVAE. If it is a vector, it represents a single data point. If Array, the last dimension must contain each of the data points.
Optional Keyword Arguments
  • K::Int=3: The number of leapfrog steps to perform in the Hamiltonian Monte Carlo (HMC) part of the RHVAE.
  • ϵ::Union{<:Number,<:AbstractVector}=0.01f0: The step size for the leapfrog steps in the HMC part of the RHVAE. If it is a scalar, the same step size is used for all dimensions. If it is an array, each element corresponds to the step size for a specific dimension.
  • βₒ::Number=0.3f0: The initial inverse temperature for the tempering schedule.
  • steps::Int: The number of fixed-point iterations to perform. Default is 3.
  • ∇H::Function=∇hamiltonian_finite: The function to compute the gradient of the Hamiltonian in the HMC part of the RHVAE.
  • ∇H_kwargs::NamedTuple: Additional keyword arguments to be passed to the ∇hamiltonian function. Default is a NamedTuple with reconstruction_loglikelihood, position_logprior, and momentum_logprior.  
  • G_inv::Function=G_inv: The function to compute the inverse of the Riemannian metric tensor.
  • tempering_schedule::Function=quadratic_tempering: The function to compute the tempering schedule in the RHVAE.
  • latent::Bool=false: If true, the function returns a NamedTuple containing the outputs of the encoder and decoder, and the final state of the phase space after the leapfrog and tempering steps. If false, the function only returns the output of the decoder.
Returns
If latent=true, the function returns a NamedTuple with the following fields:
  • encoder: The outputs of the encoder.
  • decoder: The output of the decoder.
  • phase_space: The final state of the phase space after the leapfrog and   tempering steps.
If latent=false, the function only returns the output of the decoder.
Description
This function runs the RHVAE on the given input. It first passes the input through the encoder to obtain the mean and log standard deviation of the latent space. It then uses the reparameterization trick to sample from the latent space. After that, it performs the leapfrog and tempering steps to refine the sample from the latent space. Finally, it passes the refined sample through the decoder to obtain the output.
Notes
Ensure that the dimensions of x match the input dimensions of the RHVAE, and that the dimensions of ϵ match the dimensions of the latent space.
Loss function
AutoEncoderToolkit.RHVAEs.loss — Function
loss(
     rhvae::RHVAE,
     x::AbstractArray;
     K::Int=3,
@@ -59,7 +59,7 @@
     reg_strength::Number=1.0f0,
     logp_prefactor::AbstractArray=ones(Float32, 3),
     logq_prefactor::AbstractArray=ones(Float32, 3),
-)
Compute the loss for a Riemannian Hamiltonian Variational Autoencoder (RHVAE).
Arguments
  • rhvae::RHVAE: The RHVAE used to encode the input data and decode the latent space.
  • x_in::AbstractArray: Input data to the RHVAE encoder. The last dimension is taken as having each of the samples in a batch.
  • x_out::AbstractArray: Target data to compute the reconstruction error. The last dimension is taken as having each of the samples in a batch.
Optional Keyword Arguments
  • K::Int: The number of HMC steps (default is 3).
  • ϵ::Union{<:Number,<:AbstractVector}: The step size for the leapfrog integrator (default is 0.001).
  • βₒ::Number: The initial inverse temperature (default is 0.3).
  • steps::Int: The number of steps in the leapfrog integrator (default is 3).
  • ∇H_kwargs::NamedTuple: Additional keyword arguments to be passed to the ∇hamiltonian function.
  • G_inv::Function: The function to compute the inverse of the Riemannian metric tensor (default is G_inv).
  • tempering_schedule::Function: The tempering schedule function used in the HMC (default is quadratic_tempering).
  • reg_function::Union{Function, Nothing}=nothing: A function that computes the regularization term based on the VAE outputs. This function must take as input the VAE outputs and the keyword arguments provided in reg_kwargs.
  • reg_kwargs::NamedTuple=NamedTuple(): Keyword arguments to pass to the regularization function.
  • reg_strength::Number=1.0f0: The strength of the regularization term.
  • logp_prefactor::AbstractArray: A 3-element array to scale the log likelihood, log prior of the latent variables, and log prior of the momentum variables. Default is an array of ones.
  • logq_prefactor::AbstractArray: A 3-element array to scale the log posterior of the initial latent variables, log prior of the initial momentum variables, and the tempering Jacobian term. Default is an array of ones.
Returns
  • The computed loss.
Training
AutoEncoderToolkit.RHVAEs.train! — Function
train!(
+)
Compute the loss for a Riemannian Hamiltonian Variational Autoencoder (RHVAE).
Arguments
  • rhvae::RHVAE: The RHVAE used to encode the input data and decode the latent space.
  • x_in::AbstractArray: Input data to the RHVAE encoder. The last dimension is taken as having each of the samples in a batch.
  • x_out::AbstractArray: Target data to compute the reconstruction error. The last dimension is taken as having each of the samples in a batch.
Optional Keyword Arguments
  • K::Int: The number of HMC steps (default is 3).
  • ϵ::Union{<:Number,<:AbstractVector}: The step size for the leapfrog integrator (default is 0.001).
  • βₒ::Number: The initial inverse temperature (default is 0.3).
  • steps::Int: The number of steps in the leapfrog integrator (default is 3).
  • ∇H_kwargs::NamedTuple: Additional keyword arguments to be passed to the ∇hamiltonian function.
  • G_inv::Function: The function to compute the inverse of the Riemannian metric tensor (default is G_inv).
  • tempering_schedule::Function: The tempering schedule function used in the HMC (default is quadratic_tempering).
  • reg_function::Union{Function, Nothing}=nothing: A function that computes the regularization term based on the VAE outputs. This function must take as input the VAE outputs and the keyword arguments provided in reg_kwargs.
  • reg_kwargs::NamedTuple=NamedTuple(): Keyword arguments to pass to the regularization function.
  • reg_strength::Number=1.0f0: The strength of the regularization term.
  • logp_prefactor::AbstractArray: A 3-element array to scale the log likelihood, log prior of the latent variables, and log prior of the momentum variables. Default is an array of ones.
  • logq_prefactor::AbstractArray: A 3-element array to scale the log posterior of the initial latent variables, log prior of the initial momentum variables, and the tempering Jacobian term. Default is an array of ones.
Returns
  • The computed loss.
Training
AutoEncoderToolkit.RHVAEs.train! — Function
train!(
     rhvae::RHVAE, 
     x::AbstractArray, 
     opt::NamedTuple; 
@@ -81,7 +81,7 @@
     :∇H_kwargs => Dict(
         :adtype => :finite
     )
-)
Note
Although verbose, the nested dictionaries help to keep everything organized. (PRs with better design ideas are welcome!)
The default both for cpu and gpu devices is :finite.
AutoEncoderToolkit.RHVAEs.∇hamiltonian_finite — Function
∇hamiltonian_finite(
+)
Note
Although verbose, the nested dictionaries help to keep everything organized. (PRs with better design ideas are welcome!)
The default both for cpu and gpu devices is :finite.
AutoEncoderToolkit.RHVAEs.∇hamiltonian_finite — Function
∇hamiltonian_finite(
     x::AbstractArray,
     z::AbstractVecOrMat,
     ρ::AbstractVecOrMat,
@@ -105,7 +105,7 @@
     momentum_logprior::Function=riemannian_logprior,
     G_inv::Function=G_inv,
     fdtype::Symbol=:central,
-)
Compute the gradient of the Hamiltonian with respect to a given variable using a naive finite difference method.
This function takes a point x in the data space, a point z in the latent space, a momentum ρ, an instance of RHVAE, and a variable var (:z or :ρ), and computes the gradient of the Hamiltonian with respect to var using a simple finite differences method. The computation is based on the log-likelihood of the decoder, the log-prior of the latent space, and the inverse of the metric tensor G at the point z.
The Hamiltonian is computed as follows:
Hₓ(z, ρ) = Uₓ(z) + κ(ρ),
where Uₓ(z) is the potential energy, and κ(ρ) is the kinetic energy. The potential energy is defined as follows:
Uₓ(z) = -log p(x|z) - log p(z),
where p(x|z) is the log-likelihood of the decoder and p(z) is the log-prior in latent space. The kinetic energy is defined as follows:
κ(ρ) = 0.5 * log((2π)ᴰ det G(z)) + 0.5 * ρᵀ G⁻¹ ρ
where D is the dimension of the latent space, and G(z) is the metric tensor at the point z.
Arguments
  • x::AbstractArray: The point in the data space. This does not necessarily need to be a vector. Array inputs are supported. The last dimension is assumed to have each of the data points.
  • z::AbstractVecOrMat: The point in the latent space. If matrix, each column represents a point in the latent space.
  • ρ::AbstractVecOrMat: The momentum. If matrux, each column represents a momentum vector.
  • rhvae::RHVAE: An instance of the RHVAE model.
  • var::Symbol: The variable with respect to which the gradient is computed. Must be :z or :ρ.
Optional Keyword Arguments
  • reconstruction_loglikelihood::Function: The function to compute the log-likelihood of the decoder reconstruction. Default is decoder_loglikelihood. This function must take as input the decoder, the point x in the data space, and the decoder_output.
  • position_logprior::Function: The function to compute the log-prior of the latent space position. Default is spherical_logprior. This function must take as input the point z in the latent space.
  • momentum_logprior::Function: The function to compute the log-prior of the momentum. Default is riemannian_logprior. This function must take as input the momentum ρ and the inverse of the Riemannian metric tensor G⁻¹.
  • G_inv::Function: The function to compute the inverse of the Riemannian metric tensor. Default is G_inv. This function must take as input the point z in the latent space and the rhvae instance.
  • fdtype::Symbol=:central: The type of finite difference method to use. Must be :central or :forward. Default is :central.
Returns
A vector representing the gradient of the Hamiltonian at the point (z, ρ) with respect to variable var.
Note
The inverse of the Riemannian metric tensor G⁻¹, the log determinant of the metric tensor, and the output of the decoder are computed internally in this function. The user does not need to provide these as inputs.
AutoEncoderToolkit.RHVAEs.∇hamiltonian_TaylorDiff — Function
∇hamiltonian_TaylorDiff(
+)
Compute the gradient of the Hamiltonian with respect to a given variable using a naive finite difference method.
This function takes a point x in the data space, a point z in the latent space, a momentum ρ, an instance of RHVAE, and a variable var (:z or :ρ), and computes the gradient of the Hamiltonian with respect to var using a simple finite differences method. The computation is based on the log-likelihood of the decoder, the log-prior of the latent space, and the inverse of the metric tensor G at the point z.
The Hamiltonian is computed as follows:
Hₓ(z, ρ) = Uₓ(z) + κ(ρ),
where Uₓ(z) is the potential energy, and κ(ρ) is the kinetic energy. The potential energy is defined as follows:
Uₓ(z) = -log p(x|z) - log p(z),
where p(x|z) is the log-likelihood of the decoder and p(z) is the log-prior in latent space. The kinetic energy is defined as follows:
κ(ρ) = 0.5 * log((2π)ᴰ det G(z)) + 0.5 * ρᵀ G⁻¹ ρ
where D is the dimension of the latent space, and G(z) is the metric tensor at the point z.
Arguments
  • x::AbstractArray: The point in the data space. This does not necessarily need to be a vector. Array inputs are supported. The last dimension is assumed to have each of the data points.
  • z::AbstractVecOrMat: The point in the latent space. If matrix, each column represents a point in the latent space.
  • ρ::AbstractVecOrMat: The momentum. If matrux, each column represents a momentum vector.
  • rhvae::RHVAE: An instance of the RHVAE model.
  • var::Symbol: The variable with respect to which the gradient is computed. Must be :z or :ρ.
Optional Keyword Arguments
  • reconstruction_loglikelihood::Function: The function to compute the log-likelihood of the decoder reconstruction. Default is decoder_loglikelihood. This function must take as input the decoder, the point x in the data space, and the decoder_output.
  • position_logprior::Function: The function to compute the log-prior of the latent space position. Default is spherical_logprior. This function must take as input the point z in the latent space.
  • momentum_logprior::Function: The function to compute the log-prior of the momentum. Default is riemannian_logprior. This function must take as input the momentum ρ and the inverse of the Riemannian metric tensor G⁻¹.
  • G_inv::Function: The function to compute the inverse of the Riemannian metric tensor. Default is G_inv. This function must take as input the point z in the latent space and the rhvae instance.
  • fdtype::Symbol=:central: The type of finite difference method to use. Must be :central or :forward. Default is :central.
Returns
A vector representing the gradient of the Hamiltonian at the point (z, ρ) with respect to variable var.
Note
The inverse of the Riemannian metric tensor G⁻¹, the log determinant of the metric tensor, and the output of the decoder are computed internally in this function. The user does not need to provide these as inputs.
AutoEncoderToolkit.RHVAEs.∇hamiltonian_TaylorDiff — Function
∇hamiltonian_TaylorDiff(
     x::AbstractArray,
     z::AbstractVector,
     ρ::AbstractVector,
@@ -127,7 +127,7 @@
     position_logprior::Function=spherical_logprior,
     momentum_logprior::Function=riemannian_logprior,
     G_inv::Function=G_inv,
-)
Compute the gradient of the Hamiltonian with respect to a given variable using the TaylorDiff.jl automatic differentiation library.
This function takes a point x in the data space, a point z in the latent space, a momentum ρ, an instance of RHVAE, and a variable var (:z or :ρ), and computes the gradient of the Hamiltonian with respect to var using TaylorDiff.jl.
The Hamiltonian is computed as follows:
Hₓ(z, ρ) = Uₓ(z) + κ(ρ),
where Uₓ(z) is the potential energy, and κ(ρ) is the kinetic energy. The potential energy is defined as follows:
Uₓ(z) = -log p(x|z) - log p(z),
where p(x|z) is the log-likelihood of the decoder and p(z) is the log-prior in latent space. The kinetic energy is defined as follows:
κ(ρ) = 0.5 * log((2π)ᴰ det G(z)) + 0.5 * ρᵀ G⁻¹ ρ
where D is the dimension of the latent space, and G(z) is the metric tensor at the point z.
Arguments
  • x::AbstractArray: The point in the data space. This does not necessarily need to be a vector. Array inputs are supported. The last dimension is assumed to have each of the data points.
  • z::AbstractVecOrMat: The point in the latent space. If matrix, each column represents a point in the latent space.
  • ρ::AbstractVecOrMat: The momentum. If matrix, each column represents a momentum vector.
  • rhvae::RHVAE: An instance of the RHVAE model.
  • var::Symbol: The variable with respect to which the gradient is computed. Must be :z or :ρ.
Optional Keyword Arguments
  • reconstruction_loglikelihood::Function: The function to compute the log-likelihood of the decoder reconstruction. Default is decoder_loglikelihood. This function must take as input the decoder, the point x in the data space, and the decoder_output.
  • position_logprior::Function: The function to compute the log-prior of the latent space position. Default is spherical_logprior. This function must take as input the point z in the latent space.
  • momentum_logprior::Function: The function to compute the log-prior of the momentum. Default is riemannian_logprior. This function must take as input the momentum ρ and the inverse of the Riemannian metric tensor G⁻¹.
  • G_inv::Function: The function to compute the inverse of the Riemannian metric tensor. Default is G_inv. This function must take as input the point z in the latent space and the rhvae instance.
Returns
A matrix representing the gradient of the Hamiltonian at the point (z, ρ) with respect to variable var.
AutoEncoderToolkit.RHVAEs.∇hamiltonian_ForwardDiff — Function
∇hamiltonian_ForwardDiff(
+)
Compute the gradient of the Hamiltonian with respect to a given variable using the TaylorDiff.jl automatic differentiation library.
This function takes a point x in the data space, a point z in the latent space, a momentum ρ, an instance of RHVAE, and a variable var (:z or :ρ), and computes the gradient of the Hamiltonian with respect to var using TaylorDiff.jl.
The Hamiltonian is computed as follows:
Hₓ(z, ρ) = Uₓ(z) + κ(ρ),
where Uₓ(z) is the potential energy, and κ(ρ) is the kinetic energy. The potential energy is defined as follows:
Uₓ(z) = -log p(x|z) - log p(z),
where p(x|z) is the log-likelihood of the decoder and p(z) is the log-prior in latent space. The kinetic energy is defined as follows:
κ(ρ) = 0.5 * log((2π)ᴰ det G(z)) + 0.5 * ρᵀ G⁻¹ ρ
where D is the dimension of the latent space, and G(z) is the metric tensor at the point z.
Arguments
  • x::AbstractArray: The point in the data space. This does not necessarily need to be a vector. Array inputs are supported. The last dimension is assumed to have each of the data points.
  • z::AbstractVecOrMat: The point in the latent space. If matrix, each column represents a point in the latent space.
  • ρ::AbstractVecOrMat: The momentum. If matrix, each column represents a momentum vector.
  • rhvae::RHVAE: An instance of the RHVAE model.
  • var::Symbol: The variable with respect to which the gradient is computed. Must be :z or :ρ.
Optional Keyword Arguments
  • reconstruction_loglikelihood::Function: The function to compute the log-likelihood of the decoder reconstruction. Default is decoder_loglikelihood. This function must take as input the decoder, the point x in the data space, and the decoder_output.
  • position_logprior::Function: The function to compute the log-prior of the latent space position. Default is spherical_logprior. This function must take as input the point z in the latent space.
  • momentum_logprior::Function: The function to compute the log-prior of the momentum. Default is riemannian_logprior. This function must take as input the momentum ρ and the inverse of the Riemannian metric tensor G⁻¹.
  • G_inv::Function: The function to compute the inverse of the Riemannian metric tensor. Default is G_inv. This function must take as input the point z in the latent space and the rhvae instance.
Returns
A matrix representing the gradient of the Hamiltonian at the point (z, ρ) with respect to variable var.
AutoEncoderToolkit.RHVAEs.∇hamiltonian_ForwardDiff — Function
∇hamiltonian_ForwardDiff(
     x::AbstractArray,
     z::AbstractVector,
     ρ::AbstractVector,
@@ -161,16 +161,16 @@
     position_logprior::Function=spherical_logprior,
     momentum_logprior::Function=riemannian_logprior,
     G_inv::Function=G_inv,
-)
Compute the gradient of the Hamiltonian with respect to a given variable using the ForwardDiff.jl automatic differentiation library.
This function takes a point x in the data space, a point z in the latent space, a momentum ρ, an instance of RHVAE, and a variable var (:z or :ρ), and computes the gradient of the Hamiltonian with respect to var using ForwardDiff.jl.
The Hamiltonian is computed as follows:
Hₓ(z, ρ) = Uₓ(z) + κ(ρ),
where Uₓ(z) is the potential energy, and κ(ρ) is the kinetic energy. The potential energy is defined as follows:
Uₓ(z) = -log p(x|z) - log p(z),
where p(x|z) is the log-likelihood of the decoder and p(z) is the log-prior in latent space. The kinetic energy is defined as follows:
κ(ρ) = 0.5 * log((2π)ᴰ det G(z)) + 0.5 * ρᵀ G⁻¹ ρ
where D is the dimension of the latent space, and G(z) is the metric tensor at the point z.
Arguments
  • x::AbstractArray: The point in the data space. This does not necessarily need to be a vector. Array inputs are supported. The last dimension is assumed to have each of the data points.
  • z::AbstractVecOrMat: The point in the latent space. If matrix, each column represents a point in the latent space.
  • ρ::AbstractVecOrMat: The momentum. If matrix, each column represents a momentum vector.
  • rhvae::RHVAE: An instance of the RHVAE model.
  • var::Symbol: The variable with respect to which the gradient is computed. Must be :z or :ρ.
Optional Keyword Arguments
  • reconstruction_loglikelihood::Function: The function to compute the log-likelihood of the decoder reconstruction. Default is decoder_loglikelihood. This function must take as input the decoder, the point x in the data space, and the decoder_output.
  • position_logprior::Function: The function to compute the log-prior of the latent space position. Default is spherical_logprior. This function must take as input the point z in the latent space.
  • momentum_logprior::Function: The function to compute the log-prior of the momentum. Default is riemannian_logprior. This function must take as input the momentum ρ and the inverse of the Riemannian metric tensor G⁻¹.
  • G_inv::Function: The function to compute the inverse of the Riemannian metric tensor. Default is G_inv. This function must take as input the point z in the latent space and the rhvae instance.
Returns
A matrix representing the gradient of the Hamiltonian at the point (z, ρ) with respect to variable var.
Note
ForwardDiff.jl is not composable with Zygote.jl. Thus, for backpropagation using this function one should use ReverseDiff.jl.
Other Functions
AutoEncoderToolkit.RHVAEs.update_metric — Function
update_metric(
+)
Compute the gradient of the Hamiltonian with respect to a given variable using the ForwardDiff.jl automatic differentiation library.
This function takes a point x in the data space, a point z in the latent space, a momentum ρ, an instance of RHVAE, and a variable var (:z or :ρ), and computes the gradient of the Hamiltonian with respect to var using ForwardDiff.jl.
The Hamiltonian is computed as follows:
Hₓ(z, ρ) = Uₓ(z) + κ(ρ),
where Uₓ(z) is the potential energy, and κ(ρ) is the kinetic energy. The potential energy is defined as follows:
Uₓ(z) = -log p(x|z) - log p(z),
where p(x|z) is the log-likelihood of the decoder and p(z) is the log-prior in latent space. The kinetic energy is defined as follows:
κ(ρ) = 0.5 * log((2π)ᴰ det G(z)) + 0.5 * ρᵀ G⁻¹ ρ
where D is the dimension of the latent space, and G(z) is the metric tensor at the point z.
Arguments
  • x::AbstractArray: The point in the data space. This does not necessarily need to be a vector. Array inputs are supported. The last dimension is assumed to have each of the data points.
  • z::AbstractVecOrMat: The point in the latent space. If matrix, each column represents a point in the latent space.
  • ρ::AbstractVecOrMat: The momentum. If matrix, each column represents a momentum vector.
  • rhvae::RHVAE: An instance of the RHVAE model.
  • var::Symbol: The variable with respect to which the gradient is computed. Must be :z or :ρ.
Optional Keyword Arguments
  • reconstruction_loglikelihood::Function: The function to compute the log-likelihood of the decoder reconstruction. Default is decoder_loglikelihood. This function must take as input the decoder, the point x in the data space, and the decoder_output.
  • position_logprior::Function: The function to compute the log-prior of the latent space position. Default is spherical_logprior. This function must take as input the point z in the latent space.
  • momentum_logprior::Function: The function to compute the log-prior of the momentum. Default is riemannian_logprior. This function must take as input the momentum ρ and the inverse of the Riemannian metric tensor G⁻¹.
  • G_inv::Function: The function to compute the inverse of the Riemannian metric tensor. Default is G_inv. This function must take as input the point z in the latent space and the rhvae instance.
Returns
A matrix representing the gradient of the Hamiltonian at the point (z, ρ) with respect to variable var.
Note
ForwardDiff.jl is not composable with Zygote.jl. Thus, for backpropagation using this function one should use ReverseDiff.jl.
Other Functions
AutoEncoderToolkit.RHVAEs.update_metric — Function
update_metric(
     rhvae::RHVAE{<:VAE{<:AbstractGaussianEncoder,<:AbstractVariationalDecoder}}
-)
Compute the centroids_latent and M field of a RHVAE instance without modifying the instance. This method is used when needing to backpropagate through the RHVAE during training.
Arguments
  • rhvae::RHVAE{<:VAE{<:AbstractGaussianEncoder,<:AbstractVariationalDecoder}}: The RHVAE instance to be updated.
Returns
  • NamedTuple with the following fields:
    • centroids_latent::Matrix: A matrix where each column represents a centroid cᵢ in the inverse metric computation.
    • L::Array{<:Number, 3}: A 3D array where each slice represents a L_ψᵢ matrix.
    • M::Array{<:Number, 3}: A 3D array where each slice represents a Lψᵢ Lψᵢᵀ.
AutoEncoderToolkit.RHVAEs.update_metric! — Function
update_metric!(
+)
Compute the centroids_latent and M field of a RHVAE instance without modifying the instance. This method is used when needing to backpropagate through the RHVAE during training.
Arguments
  • rhvae::RHVAE{<:VAE{<:AbstractGaussianEncoder,<:AbstractVariationalDecoder}}: The RHVAE instance to be updated.
Returns
  • NamedTuple with the following fields:
    • centroids_latent::Matrix: A matrix where each column represents a centroid cᵢ in the inverse metric computation.
    • L::Array{<:Number, 3}: A 3D array where each slice represents a L_ψᵢ matrix.
    • M::Array{<:Number, 3}: A 3D array where each slice represents a Lψᵢ Lψᵢᵀ.
AutoEncoderToolkit.RHVAEs.update_metric! — Function
update_metric!(
     rhvae::RHVAE{<:VAE{<:AbstractGaussianEncoder,<:AbstractVariationalDecoder}},
     params::NamedTuple
 )
Update the centroids_latent and M fields of a RHVAE instance in place.
This function takes a RHVAE instance and a named tuple params containing the new values for centroids_latent and M. It updates the centroids_latent, L, and M fields of the RHVAE instance with the provided values.
Arguments
  • rhvae::RHVAE{<:VAE{<:AbstractGaussianEncoder,<:AbstractVariationalDecoder}}: The RHVAE instance to update.
  • params::NamedTuple: A named tuple containing the new values for centroids_latent and M. Must have the keys :centroids_latent, :L, and :M.
Returns
Nothing. The RHVAE instance is updated in place.
update_metric!(
     rhvae::RHVAE{
         <:VAE{<:AbstractGaussianEncoder,<:AbstractVariationalDecoder}
     }
-)
Update the centroids_latent, and M fields of a RHVAE instance in place.
This function takes a RHVAE instance as input and modifies its centroids_latent and M fields. The centroids_latent field is updated by running the centroids_data through the encoder of the underlying VAE and extracting the mean (µ) of the resulting Gaussian distribution. The M field is updated by running each column of the centroids_data through the metric_chain and concatenating the results along the third dimension, then each slice is updated by multiplying each slice of L by its transpose and concating the results along the third dimension.
Arguments
  • rhvae::RHVAE{<:VAE{<:AbstractGaussianEncoder,<:AbstractVariationalDecoder}}: The RHVAE instance to be updated.
Notes
This function modifies the RHVAE instance in place, so it does not return anything. The changes are made directly to the centroids_latent, L, and M fields of the input RHVAE instance.
AutoEncoderToolkit.RHVAEs.G_inv — Function
G_inv(
+)
Update the centroids_latent, and M fields of a RHVAE instance in place.
This function takes a RHVAE instance as input and modifies its centroids_latent and M fields. The centroids_latent field is updated by running the centroids_data through the encoder of the underlying VAE and extracting the mean (µ) of the resulting Gaussian distribution. The M field is updated by running each column of the centroids_data through the metric_chain and concatenating the results along the third dimension, then each slice is updated by multiplying each slice of L by its transpose and concating the results along the third dimension.
Arguments
  • rhvae::RHVAE{<:VAE{<:AbstractGaussianEncoder,<:AbstractVariationalDecoder}}: The RHVAE instance to be updated.
Notes
This function modifies the RHVAE instance in place, so it does not return anything. The changes are made directly to the centroids_latent, L, and M fields of the input RHVAE instance.
AutoEncoderToolkit.RHVAEs.G_inv — Function
G_inv(
     z::AbstractVecOrMat,
     centroids_latent::AbstractMatrix,
     M::AbstractArray{<:Number,3},
@@ -179,10 +179,10 @@
 )
Compute the inverse of the metric tensor G for a given point in the latent space.
This function takes a point z in the latent space, the centroids_latent of the RHVAE instance, a 3D array M representing the metric tensor, a temperature T, and a regularization factor λ, and computes the inverse of the metric tensor G at that point. The computation is based on the centroids and the temperature, as well as a regularization term. The inverse metric is computed as follows:
G⁻¹(z) = ∑ᵢ₌₁ⁿ Lψᵢ Lψᵢᵀ exp(-‖z - cᵢ‖₂² / T²) + λIₗ,
where Lψᵢ is computed by the MetricChain, T is the temperature, λ is a regularization factor, and each column of `centroidslatent` are the cᵢ.
Arguments
  • z::AbstractVecOrMat: The point in the latent space. If a matrix, each column represents a point in the latent space.
  • centroids_latent::AbstractMatrix: The centroids in the latent space.
  • M::AbstractArray{<:Number,3}: The 3D array containing the symmetric matrices used to compute the inverse metric tensor.
  • T::N: The temperature.
  • λ::N: The regularization factor.
Returns
A matrix or 3D array representing the inverse of the metric tensor G at the point z. If a 3D array, each slice represents the inverse metric tensor at a different point in the latent space.
Notes
The computation involves the squared Euclidean distance between z and each centroid, the exponential of the negative of these distances divided by the square of the temperature, and a regularization term proportional to the identity matrix. The result is a matrix of the same size as the latent space.
GPU support
This function supports CPU and GPU arrays.
G_inv( 
     z::AbstractVecOrMat,
     metric_param::Union{RHVAE,NamedTuple},
-)
Compute the inverse of the metric tensor G for a given point in the latent space.
This function takes a RHVAE instance and a point z in the latent space, and computes the inverse of the metric tensor G at that point. The computation is based on the centroids and the temperature of the RHVAE instance, as well as a regularization term. The inverse metric is computed as follows:
G⁻¹(z) = ∑ᵢ₌₁ⁿ Lψᵢ Lψᵢᵀ exp(-‖z - cᵢ‖₂² / T²) + λIₗ,
where Lψᵢ is computed by the MetricChain, T is the temperature, λ is a regularization factor, and each column of `centroidslatent` are the cᵢ.
Arguments
  • z::AbstractVecOrMat: The point in the latent space. If a matrix, each column represents a point in the latent space.
  • metric_param::Union{RHVAE,NamedTuple}: Either an RHVAE instance or a named tuple containing the fields centroids_latent, M, T, and λ.
Returns
A matrix representing the inverse of the metric tensor G at the point z.
Notes
The computation involves the squared Euclidean distance between z and each centroid of the RHVAE instance, the exponential of the negative of these distances divided by the square of the temperature, and a regularization term proportional to the identity matrix. The result is a matrix of the same size as the latent space.
AutoEncoderToolkit.RHVAEs.metric_tensor — Function
metric_tensor(
+)
Compute the inverse of the metric tensor G for a given point in the latent space.
This function takes a RHVAE instance and a point z in the latent space, and computes the inverse of the metric tensor G at that point. The computation is based on the centroids and the temperature of the RHVAE instance, as well as a regularization term. The inverse metric is computed as follows:
G⁻¹(z) = ∑ᵢ₌₁ⁿ Lψᵢ Lψᵢᵀ exp(-‖z - cᵢ‖₂² / T²) + λIₗ,
where Lψᵢ is computed by the MetricChain, T is the temperature, λ is a regularization factor, and each column of `centroidslatent` are the cᵢ.
Arguments
  • z::AbstractVecOrMat: The point in the latent space. If a matrix, each column represents a point in the latent space.
  • metric_param::Union{RHVAE,NamedTuple}: Either an RHVAE instance or a named tuple containing the fields centroids_latent, M, T, and λ.
Returns
A matrix representing the inverse of the metric tensor G at the point z.
Notes
The computation involves the squared Euclidean distance between z and each centroid of the RHVAE instance, the exponential of the negative of these distances divided by the square of the temperature, and a regularization term proportional to the identity matrix. The result is a matrix of the same size as the latent space.
AutoEncoderToolkit.RHVAEs.metric_tensor — Function
metric_tensor(
     z::AbstractVecOrMat,
     metric_param::Union{RHVAE,NamedTuple},
-)
Compute the metric tensor G for a given point in the latent space. This function is a wrapper that determines the type of the input z and calls the appropriate specialized function _metric_tensor to perform the actual computation.
This function takes a RHVAE instance or a named tuple containing the fields centroids_latent, M, T, and λ, and a point z in the latent space, and computes the metric tensor G at that point. The computation is based on the inverse of the metric tensor G, which is computed by the G_inv function.
Arguments
  • z::AbstractVecOrMat: The point in the latent space. If a matrix, each column represents a point in the latent space.
  • metric_param::Union{RHVAE,NamedTuple}: Either an RHVAE instance or a named tuple containing the fields centroids_latent, M, T, and λ.
Returns
A matrix representing the metric tensor G at the point z.
Notes
The computation involves the inverse of the metric tensor G at the point z. The result is a matrix of the same size as the latent space.
GPU Support
This function supports CPU and GPU arrays.
AutoEncoderToolkit.RHVAEs.riemannian_logprior — Function
riemannian_logprior(
+)
Compute the metric tensor G for a given point in the latent space. This function is a wrapper that determines the type of the input z and calls the appropriate specialized function _metric_tensor to perform the actual computation.
This function takes a RHVAE instance or a named tuple containing the fields centroids_latent, M, T, and λ, and a point z in the latent space, and computes the metric tensor G at that point. The computation is based on the inverse of the metric tensor G, which is computed by the G_inv function.
Arguments
  • z::AbstractVecOrMat: The point in the latent space. If a matrix, each column represents a point in the latent space.
  • metric_param::Union{RHVAE,NamedTuple}: Either an RHVAE instance or a named tuple containing the fields centroids_latent, M, T, and λ.
Returns
A matrix representing the metric tensor G at the point z.
Notes
The computation involves the inverse of the metric tensor G at the point z. The result is a matrix of the same size as the latent space.
GPU Support
This function supports CPU and GPU arrays.
AutoEncoderToolkit.RHVAEs.riemannian_logprior — Function
riemannian_logprior(
     ρ::AbstractVector,
     G⁻¹::AbstractMatrix,
     logdetG::Number;
@@ -190,7 +190,7 @@
     ρ::AbstractVector,
     G⁻¹::AbstractMatrix,
     logdetG::Number,
-)
CPU AbstractMatrix version of the riemannian_logprior function.
AutoEncoderToolkit.RHVAEs.hamiltonian — Function
hamiltonian(
     x::AbstractArray,
     z::AbstractVecOrMat,
     ρ::AbstractVecOrMat,
@@ -210,7 +210,7 @@
     position_logprior::Function=spherical_logprior,
     momentum_logprior::Function=riemannian_logprior,
     G_inv::Function=G_inv,
-)
Compute the Hamiltonian for a given point in the latent space and a given momentum.
This function takes a point x in the data space, a point z in the latent space, a momentum ρ, and an instance of RHVAE. It computes the inverse of the Riemannian metric tensor G⁻¹ and the output of the decoder internally, and then computes the Hamiltonian. The computation is based on the log-likelihood of the decoder, the log-prior of the latent space, and the inverse of the metric tensor G at the point z.
The Hamiltonian is computed as follows:
Hₓ(z, ρ) = Uₓ(z) + κ(ρ),
where Uₓ(z) is the potential energy, and κ(ρ) is the kinetic energy. The potential energy is defined as follows:
Uₓ(z) = -log p(x|z) - log p(z),
where p(x|z) is the log-likelihood of the decoder and p(z) is the log-prior in latent space. The kinetic energy is defined as follows:
κ(ρ) = -log p(ρ),
where p(ρ) is the log-prior of the momentum.
Arguments
  • x::AbstractArray: The point in the data space. This does not necessarily need to be a vector. Array inputs are supported, but the last dimension of the array should be of size 1.
  • z::AbstractVector: The point in the latent space.
  • ρ::AbstractVector: The momentum.
  • rhvae::RHVAE: An instance of the RHVAE model.
Optional Keyword Arguments
  • reconstruction_loglikelihood::Function: The function to compute the log-likelihood of the decoder reconstruction. Default is decoder_loglikelihood. This function must take as input the decoder, the point x in the data space, and the decoder_output.
  • position_logprior::Function: The function to compute the log-prior of the latent space position. Default is spherical_logprior. This function must take as input the point z in the latent space.
  • momentum_logprior::Function: The function to compute the log-prior of the momentum. Default is riemannian_logprior. This function must take as input the momentum ρ and the inverse of the Riemannian metric tensor G⁻¹.
  • G_inv::Function: The function to compute the inverse of the Riemannian metric tensor. Default is G_inv. This function must take as input the point z in the latent space and the rhvae instance.
Returns
A scalar representing the Hamiltonian at the point z with the momentum ρ.
Note
The inverse of the Riemannian metric tensor G⁻¹, the log determinant of the metric tensor, and the output of the decoder are computed internally in this function. The user does not need to provide these as inputs.
AutoEncoderToolkit.RHVAEs.∇hamiltonian — Function
∇hamiltonian(
+)
Compute the Hamiltonian for a given point in the latent space and a given momentum.
This function takes a point x in the data space, a point z in the latent space, a momentum ρ, and an instance of RHVAE. It computes the inverse of the Riemannian metric tensor G⁻¹ and the output of the decoder internally, and then computes the Hamiltonian. The computation is based on the log-likelihood of the decoder, the log-prior of the latent space, and the inverse of the metric tensor G at the point z.
The Hamiltonian is computed as follows:
Hₓ(z, ρ) = Uₓ(z) + κ(ρ),
where Uₓ(z) is the potential energy, and κ(ρ) is the kinetic energy. The potential energy is defined as follows:
Uₓ(z) = -log p(x|z) - log p(z),
where p(x|z) is the log-likelihood of the decoder and p(z) is the log-prior in latent space. The kinetic energy is defined as follows:
κ(ρ) = -log p(ρ),
where p(ρ) is the log-prior of the momentum.
Arguments
  • x::AbstractArray: The point in the data space. This does not necessarily need to be a vector. Array inputs are supported, but the last dimension of the array should be of size 1.
  • z::AbstractVector: The point in the latent space.
  • ρ::AbstractVector: The momentum.
  • rhvae::RHVAE: An instance of the RHVAE model.
Optional Keyword Arguments
  • reconstruction_loglikelihood::Function: The function to compute the log-likelihood of the decoder reconstruction. Default is decoder_loglikelihood. This function must take as input the decoder, the point x in the data space, and the decoder_output.
  • position_logprior::Function: The function to compute the log-prior of the latent space position. Default is spherical_logprior. This function must take as input the point z in the latent space.
  • momentum_logprior::Function: The function to compute the log-prior of the momentum. Default is riemannian_logprior. This function must take as input the momentum ρ and the inverse of the Riemannian metric tensor G⁻¹.
  • G_inv::Function: The function to compute the inverse of the Riemannian metric tensor. Default is G_inv. This function must take as input the point z in the latent space and the rhvae instance.
Returns
A scalar representing the Hamiltonian at the point z with the momentum ρ.
Note
The inverse of the Riemannian metric tensor G⁻¹, the log determinant of the metric tensor, and the output of the decoder are computed internally in this function. The user does not need to provide these as inputs.
AutoEncoderToolkit.RHVAEs.∇hamiltonian — Function
∇hamiltonian(
     x::AbstractArray,
     z::AbstractVecOrMat,
     ρ::AbstractVecOrMat,
@@ -236,7 +236,7 @@
     G_inv::Function=G_inv,
     adtype::Symbol=:TaylorDiff,
     adkwargs::NamedTuple=NamedTuple(),
-)
Compute the gradient of the Hamiltonian with respect to a given variable using a specified automatic differentiation method.
This function takes a point x in the data space, a point z in the latent space, a momentum ρ, an instance of RHVAE, and a variable var (:z or :ρ), and computes the gradient of the Hamiltonian with respect to var using the specified automatic differentiation method. The computation is based on the log-likelihood of the decoder, the log-prior of the latent space, and G_inv.
Arguments
  • x::AbstractArray: The point in the data space. This does not necessarily need to be a vector. Array inputs are supported. The last dimension is assumed to have each of the data points.
  • z::AbstractVecOrMat: The point in the latent space. If matrix, each column represents a point in the latent space.
  • ρ::AbstractVecOrMat: The momentum. If matrix, each column represents a momentum vector.
  • rhvae::RHVAE: An instance of the RHVAE model.
  • var::Symbol: The variable with respect to which the gradient is computed. Must be :z or :ρ.
Optional Keyword Arguments
  • reconstruction_loglikelihood::Function: The function to compute the log-likelihood of the decoder reconstruction. Default is decoder_loglikelihood. This function must take as input the decoder, the point x in the data space, and the decoder_output.
  • position_logprior::Function: The function to compute the log-prior of the latent space position. Default is spherical_logprior. This function must take as input the point z in the latent space.
  • momentum_logprior::Function: The function to compute the log-prior of the momentum. Default is riemannian_logprior. This function must take as input the momentum ρ and G_inv.
  • G_inv::Function: The function to compute the inverse of the Riemannian metric tensor.  Default is G_inv.
  • adtype::Symbol=:finite: The type of automatic differentiation method to use. Must be:finite,:ForwardDiff, or:TaylorDiff. Default is:finite`.
  • adkwargs::NamedTuple=NamedTuple(): Additional keyword arguments to pass to the automatic differentiation method.
Returns
A vector representing the gradient of the Hamiltonian at the point (z, ρ) with respect to variable var.
AutoEncoderToolkit.RHVAEs._leapfrog_first_step — Function
_leapfrog_first_step(
+)
Compute the gradient of the Hamiltonian with respect to a given variable using a specified automatic differentiation method.
This function takes a point x in the data space, a point z in the latent space, a momentum ρ, an instance of RHVAE, and a variable var (:z or :ρ), and computes the gradient of the Hamiltonian with respect to var using the specified automatic differentiation method. The computation is based on the log-likelihood of the decoder, the log-prior of the latent space, and G_inv.
Arguments
  • x::AbstractArray: The point in the data space. This does not necessarily need to be a vector. Array inputs are supported. The last dimension is assumed to have each of the data points.
  • z::AbstractVecOrMat: The point in the latent space. If matrix, each column represents a point in the latent space.
  • ρ::AbstractVecOrMat: The momentum. If matrix, each column represents a momentum vector.
  • rhvae::RHVAE: An instance of the RHVAE model.
  • var::Symbol: The variable with respect to which the gradient is computed. Must be :z or :ρ.
Optional Keyword Arguments
  • reconstruction_loglikelihood::Function: The function to compute the log-likelihood of the decoder reconstruction. Default is decoder_loglikelihood. This function must take as input the decoder, the point x in the data space, and the decoder_output.
  • position_logprior::Function: The function to compute the log-prior of the latent space position. Default is spherical_logprior. This function must take as input the point z in the latent space.
  • momentum_logprior::Function: The function to compute the log-prior of the momentum. Default is riemannian_logprior. This function must take as input the momentum ρ and G_inv.
  • G_inv::Function: The function to compute the inverse of the Riemannian metric tensor.  Default is G_inv.
  • adtype::Symbol=:finite: The type of automatic differentiation method to use. Must be:finite,:ForwardDiff, or:TaylorDiff. Default is:finite`.
  • adkwargs::NamedTuple=NamedTuple(): Additional keyword arguments to pass to the automatic differentiation method.
Returns
A vector representing the gradient of the Hamiltonian at the point (z, ρ) with respect to variable var.
AutoEncoderToolkit.RHVAEs._leapfrog_first_step — Function
_leapfrog_first_step(
     x::AbstractArray,
     z::AbstractVecOrMat,
     ρ::AbstractVecOrMat,
@@ -264,7 +264,7 @@
         momentum_logprior=riemannian_logprior,
     ),
     G_inv::Function=G_inv,
-)
Perform the first step of the generalized leapfrog integrator for Hamiltonian dynamics, defined as
ρ(t + ϵ/2) = ρ(t) - 0.5 * ϵ * ∇z_H(z(t), ρ(t + ϵ/2)).
This function is part of the generalized leapfrog integrator used in Hamiltonian dynamics. Unlike the standard leapfrog integrator, the generalized leapfrog integrator is implicit, which means it requires the use of fixed-point iterations to be solved.
The function takes a RHVAE instance, a point x in the data space, a point z in the latent space, a momentum ρ, a step size ϵ, and optionally the number of fixed-point iterations to perform (steps), a function to compute the gradient of the Hamiltonian (∇H), and a set of keyword arguments for ∇H (∇H_kwargs).
The function performs the following update for steps times:
ρ̃ = ρ̃ - 0.5 * ϵ * ∇hamiltonian(rhvae, x, z, ρ̃, :z; ∇H_kwargs...)
where ∇H is the gradient of the Hamiltonian with respect to the position variables z. The result is returned as ρ̃.
Arguments
  • x::AbstractArray: The point in the data space. This does not necessarily need to be a vector. Array inputs are supported. The last dimension is assumed to have each of the data points.
  • z::AbstractVecOrMat: The point in the latent space. If matrix, each column represents a point in the latent space.
  • ρ::AbstractVecOrMat: The momentum. If matrux, each column represents a momentum vector.
  • rhvae::RHVAE: The RHVAE instance.
Optional Keyword Arguments
  • ϵ::Union{<:Number,<:AbstractVector}=0.01f0: The leapfrog step size. Default is 0.01f0.
  • steps::Int=3: The number of fixed-point iterations to perform. Default is 3.
  • ∇H_kwargs::NamedTuple: The keyword arguments for ∇hamiltonian. Default is a tuple with reconstruction_loglikelihood, position_logprior, and momentum_logprior.
  • G_inv::Function: The function to compute the inverse of the Riemannian metric tensor. Default is G_inv.
Returns
A vector representing the updated momentum after performing the first step of the generalized leapfrog integrator.
AutoEncoderToolkit.RHVAEs._leapfrog_second_step — Function
_leapfrog_second_step(
+)
Perform the first step of the generalized leapfrog integrator for Hamiltonian dynamics, defined as
ρ(t + ϵ/2) = ρ(t) - 0.5 * ϵ * ∇z_H(z(t), ρ(t + ϵ/2)).
This function is part of the generalized leapfrog integrator used in Hamiltonian dynamics. Unlike the standard leapfrog integrator, the generalized leapfrog integrator is implicit, which means it requires the use of fixed-point iterations to be solved.
The function takes a RHVAE instance, a point x in the data space, a point z in the latent space, a momentum ρ, a step size ϵ, and optionally the number of fixed-point iterations to perform (steps), a function to compute the gradient of the Hamiltonian (∇H), and a set of keyword arguments for ∇H (∇H_kwargs).
The function performs the following update for steps times:
ρ̃ = ρ̃ - 0.5 * ϵ * ∇hamiltonian(rhvae, x, z, ρ̃, :z; ∇H_kwargs...)
where ∇H is the gradient of the Hamiltonian with respect to the position variables z. The result is returned as ρ̃.
Arguments
  • x::AbstractArray: The point in the data space. This does not necessarily need to be a vector. Array inputs are supported. The last dimension is assumed to have each of the data points.
  • z::AbstractVecOrMat: The point in the latent space. If matrix, each column represents a point in the latent space.
  • ρ::AbstractVecOrMat: The momentum. If matrux, each column represents a momentum vector.
  • rhvae::RHVAE: The RHVAE instance.
Optional Keyword Arguments
  • ϵ::Union{<:Number,<:AbstractVector}=0.01f0: The leapfrog step size. Default is 0.01f0.
  • steps::Int=3: The number of fixed-point iterations to perform. Default is 3.
  • ∇H_kwargs::NamedTuple: The keyword arguments for ∇hamiltonian. Default is a tuple with reconstruction_loglikelihood, position_logprior, and momentum_logprior.
  • G_inv::Function: The function to compute the inverse of the Riemannian metric tensor. Default is G_inv.
Returns
A vector representing the updated momentum after performing the first step of the generalized leapfrog integrator.
AutoEncoderToolkit.RHVAEs._leapfrog_second_step — Function
_leapfrog_second_step(
     x::AbstractArray,
     z::AbstractVecOrMat,
     ρ::AbstractVecOrMat,
@@ -292,7 +292,7 @@
             momentum_logprior=riemannian_logprior,
     ),
     G_inv::Function=G_inv,
-)
Perform the second step of the generalized leapfrog integrator for Hamiltonian dynamics, defined as
z(t + ϵ) = z(t) + 0.5 * ϵ * [∇ρH(z(t), ρ(t+ϵ/2)) + ∇ρH(z(t + ϵ), ρ(t+ϵ/2))].
This function is part of the generalized leapfrog integrator used in Hamiltonian dynamics. Unlike the standard leapfrog integrator, the generalized leapfrog integrator is implicit, which means it requires the use of fixed-point iterations to be solved.
The function takes a RHVAE instance, a point x in the data space, a point z in the latent space, a momentum ρ, a step size ϵ, and optionally the number of fixed-point iterations to perform (steps), a function to compute the gradient of the Hamiltonian (∇H), and a set of keyword arguments for ∇H (∇H_kwargs).
The function performs the following update for steps times:
z̄ = z̄ + 0.5 * ϵ * ( ∇hamiltonian(rhvae, x, z̄, ρ, :ρ; ∇Hkwargs...) +         ∇hamiltonian(rhvae, x, z, ρ, :ρ; ∇Hkwargs...) )
where ∇H is the gradient of the Hamiltonian with respect to the momentum variables ρ. The result is returned as z̄.
Arguments
  • x::AbstractArray: The point in the data space. This does not necessarily need to be a vector. Array inputs are supported. The last dimension is assumed to have each of the data points.
  • z::AbstractVecOrMat: The point in the latent space. If matrix, each column represents a point in the latent space.
  • ρ::AbstractVecOrMat: The momentum. If matrux, each column represents a momentum vector.
  • rhvae::RHVAE: The RHVAE instance.
Optional Keyword Arguments
  • ϵ::Union{<:Number,<:AbstractVector}=0.01f0: The leapfrog step size. Default is 0.01f0.
  • steps::Int=3: The number of fixed-point iterations to perform. Default is 3. Typically, 3 iterations are sufficient.
  • ∇H_kwargs::NamedTuple: The keyword arguments for ∇hamiltonian. Default is a tuple with reconstruction_loglikelihood, position_logprior, and momentum_logprior.
  • G_inv::Function: The function to compute the inverse of the Riemannian metric tensor. Default is G_inv.
Returns
A vector representing the updated position after performing the second step of the generalized leapfrog integrator.
AutoEncoderToolkit.RHVAEs._leapfrog_third_step — Function
_leapfrog_third_step(
+)
Perform the second step of the generalized leapfrog integrator for Hamiltonian dynamics, defined as
z(t + ϵ) = z(t) + 0.5 * ϵ * [∇ρH(z(t), ρ(t+ϵ/2)) + ∇ρH(z(t + ϵ), ρ(t+ϵ/2))].
This function is part of the generalized leapfrog integrator used in Hamiltonian dynamics. Unlike the standard leapfrog integrator, the generalized leapfrog integrator is implicit, which means it requires the use of fixed-point iterations to be solved.
The function takes a RHVAE instance, a point x in the data space, a point z in the latent space, a momentum ρ, a step size ϵ, and optionally the number of fixed-point iterations to perform (steps), a function to compute the gradient of the Hamiltonian (∇H), and a set of keyword arguments for ∇H (∇H_kwargs).
The function performs the following update for steps times:
z̄ = z̄ + 0.5 * ϵ * ( ∇hamiltonian(rhvae, x, z̄, ρ, :ρ; ∇Hkwargs...) +         ∇hamiltonian(rhvae, x, z, ρ, :ρ; ∇Hkwargs...) )
where ∇H is the gradient of the Hamiltonian with respect to the momentum variables ρ. The result is returned as z̄.
Arguments
  • x::AbstractArray: The point in the data space. This does not necessarily need to be a vector. Array inputs are supported. The last dimension is assumed to have each of the data points.
  • z::AbstractVecOrMat: The point in the latent space. If matrix, each column represents a point in the latent space.
  • ρ::AbstractVecOrMat: The momentum. If matrux, each column represents a momentum vector.
  • rhvae::RHVAE: The RHVAE instance.
Optional Keyword Arguments
  • ϵ::Union{<:Number,<:AbstractVector}=0.01f0: The leapfrog step size. Default is 0.01f0.
  • steps::Int=3: The number of fixed-point iterations to perform. Default is 3. Typically, 3 iterations are sufficient.
  • ∇H_kwargs::NamedTuple: The keyword arguments for ∇hamiltonian. Default is a tuple with reconstruction_loglikelihood, position_logprior, and momentum_logprior.
  • G_inv::Function: The function to compute the inverse of the Riemannian metric tensor. Default is G_inv.
Returns
A vector representing the updated position after performing the second step of the generalized leapfrog integrator.
AutoEncoderToolkit.RHVAEs._leapfrog_third_step — Function
_leapfrog_third_step(
     x::AbstractArray,
     z::AbstractVecOrMat,
     ρ::AbstractVecOrMat,
@@ -319,7 +319,7 @@
             momentum_logprior=riemannian_logprior,
     ),
     G_inv::Function=G_inv,
-)
Perform the third step of the generalized leapfrog integrator for Hamiltonian dynamics, defined as
ρ(t + ϵ) = ρ(t + ϵ/2) - 0.5 * ϵ * ∇z_H(z(t + ϵ), ρ(t + ϵ/2)).
This function is part of the generalized leapfrog integrator used in Hamiltonian dynamics. Unlike the standard leapfrog integrator, the generalized leapfrog integrator is implicit, which means it requires the use of fixed-point iterations to be solved.
The function takes a RHVAE instance, a point x in the data space, a point z in the latent space, a momentum ρ, a step size ϵ, the number of fixed-point iterations to perform (steps), a function to compute the gradient of the Hamiltonian (∇H), and a set of keyword arguments for ∇H (∇H_kwargs).
The function performs the following update:
ρ̃ = ρ - 0.5 * ϵ * ∇hamiltonian(rhvae, x, z, ρ, :z; ∇H_kwargs...)
where ∇H is the gradient of the Hamiltonian with respect to the position variables z. The result is returned as ρ̃.
Arguments
  • x::AbstractArray: The point in the data space. This does not necessarily need to be a vector. Array inputs are supported. The last dimension is assumed to have each of the data points.
  • z::AbstractVecOrMat: The point in the latent space. If matrix, each column represents a point in the latent space.
  • ρ::AbstractVecOrMat: The momentum. If matrux, each column represents a momentum vector.
  • rhvae::RHVAE: The RHVAE instance.
Optional Keyword Arguments
  • ϵ::Union{<:Number,<:AbstractVector}: The leapfrog step size. Default is 0.01f0.
  • steps::Int=3: The number of fixed-point iterations to perform. Default is 3.
  • ∇H_kwargs::NamedTuple: The keyword arguments for ∇hamiltonian. Default is a tuple with reconstruction_loglikelihood, position_logprior, and momentum_logprior.
  • G_inv::Function: The function to compute the inverse of the Riemannian metric tensor. Default is G_inv.
Returns
A vector representing the updated momentum after performing the third step of the generalized leapfrog integrator.
AutoEncoderToolkit.RHVAEs.general_leapfrog_step — Function
general_leapfrog_step(
+)
Perform the third step of the generalized leapfrog integrator for Hamiltonian dynamics, defined as
ρ(t + ϵ) = ρ(t + ϵ/2) - 0.5 * ϵ * ∇z_H(z(t + ϵ), ρ(t + ϵ/2)).
This function is part of the generalized leapfrog integrator used in Hamiltonian dynamics. Unlike the standard leapfrog integrator, the generalized leapfrog integrator is implicit, which means it requires the use of fixed-point iterations to be solved.
The function takes a RHVAE instance, a point x in the data space, a point z in the latent space, a momentum ρ, a step size ϵ, the number of fixed-point iterations to perform (steps), a function to compute the gradient of the Hamiltonian (∇H), and a set of keyword arguments for ∇H (∇H_kwargs).
The function performs the following update:
ρ̃ = ρ - 0.5 * ϵ * ∇hamiltonian(rhvae, x, z, ρ, :z; ∇H_kwargs...)
where ∇H is the gradient of the Hamiltonian with respect to the position variables z. The result is returned as ρ̃.
Arguments
  • x::AbstractArray: The point in the data space. This does not necessarily need to be a vector. Array inputs are supported. The last dimension is assumed to have each of the data points.
  • z::AbstractVecOrMat: The point in the latent space. If matrix, each column represents a point in the latent space.
  • ρ::AbstractVecOrMat: The momentum. If matrux, each column represents a momentum vector.
  • rhvae::RHVAE: The RHVAE instance.
Optional Keyword Arguments
  • ϵ::Union{<:Number,<:AbstractVector}: The leapfrog step size. Default is 0.01f0.
  • steps::Int=3: The number of fixed-point iterations to perform. Default is 3.
  • ∇H_kwargs::NamedTuple: The keyword arguments for ∇hamiltonian. Default is a tuple with reconstruction_loglikelihood, position_logprior, and momentum_logprior.
  • G_inv::Function: The function to compute the inverse of the Riemannian metric tensor. Default is G_inv.
Returns
A vector representing the updated momentum after performing the third step of the generalized leapfrog integrator.
AutoEncoderToolkit.RHVAEs.general_leapfrog_step — Function
general_leapfrog_step(
     x::AbstractArray,
     z::AbstractVecOrMat,
     ρ::AbstractVecOrMat,
@@ -349,7 +349,7 @@
             momentum_logprior=riemannian_logprior,
             G_inv=G_inv,
     ),
-)
Perform a full step of the generalized leapfrog integrator for Hamiltonian dynamics.
The leapfrog integrator is a numerical integration scheme used to simulate Hamiltonian dynamics. It consists of three steps:
  1. Half update of the momentum variable: ρ(t + ϵ/2) = ρ(t) - 0.5 * ϵ *  ∇z_H(z(t), ρ(t + ϵ/2)).
  2. Full update of the position variable: z(t + ϵ) = z(t) + 0.5 * ϵ * [∇ρ_H(z(t),
ρ(t+ϵ/2)) + ∇ρ_H(z(t + ϵ), ρ(t+ϵ/2))].
  1. Half update of the momentum variable: ρ(t + ϵ) = ρ(t + ϵ/2) - 0.5 * ϵ *  ∇z_H(z(t + ϵ), ρ(t + ϵ/2)).
This function performs these three steps in sequence, using the _leapfrog_first_step and _leapfrog_second_step helper functions.
Arguments
  • x::AbstractArray: The point in the data space. This does not necessarily need to be a vector. Array inputs are supported. The last dimension is assumed to have each of the data points.
  • z::AbstractVecOrMat: The point in the latent space. If matrix, each column represents a point in the latent space.
  • ρ::AbstractVecOrMat: The momentum. If matrux, each column represents a momentum vector.
  • rhvae::RHVAE: The RHVAE instance.
Optional Keyword Arguments
  • ϵ::Union{<:Number,<:AbstractVector}=0.01f0: The leapfrog step size. Default is 0.01f0.
  • steps::Int=3: The number of fixed-point iterations to perform. Default is 3. Typically, 3 iterations are sufficient.
  • ∇H_kwargs::NamedTuple: The keyword arguments for ∇hamiltonian. Default is a tuple with decoder_loglikelihood, position_logprior, and momentum_logprior
  • G_inv::Function: The function to compute the inverse of the Riemannian metric tensor. Default is G_inv.
    A tuple (z̄, ρ̄, Ḡ⁻¹, logdetḠ, decoder_update) representing the updated position, momentum, the inverse of the updated Riemannian metric tensor, the log of the determinant of the metric tensor, and the updated decoder outputs after performing the full leapfrog step.
AutoEncoderToolkit.RHVAEs.general_leapfrog_tempering_step — Function
general_leapfrog_tempering_step(
+)
Perform a full step of the generalized leapfrog integrator for Hamiltonian dynamics.
The leapfrog integrator is a numerical integration scheme used to simulate Hamiltonian dynamics. It consists of three steps:
  1. Half update of the momentum variable: ρ(t + ϵ/2) = ρ(t) - 0.5 * ϵ *  ∇z_H(z(t), ρ(t + ϵ/2)).
  2. Full update of the position variable: z(t + ϵ) = z(t) + 0.5 * ϵ * [∇ρ_H(z(t),
ρ(t+ϵ/2)) + ∇ρ_H(z(t + ϵ), ρ(t+ϵ/2))].
  1. Half update of the momentum variable: ρ(t + ϵ) = ρ(t + ϵ/2) - 0.5 * ϵ *  ∇z_H(z(t + ϵ), ρ(t + ϵ/2)).
This function performs these three steps in sequence, using the _leapfrog_first_step and _leapfrog_second_step helper functions.
Arguments
  • x::AbstractArray: The point in the data space. This does not necessarily need to be a vector. Array inputs are supported. The last dimension is assumed to have each of the data points.
  • z::AbstractVecOrMat: The point in the latent space. If matrix, each column represents a point in the latent space.
  • ρ::AbstractVecOrMat: The momentum. If matrux, each column represents a momentum vector.
  • rhvae::RHVAE: The RHVAE instance.
Optional Keyword Arguments
  • ϵ::Union{<:Number,<:AbstractVector}=0.01f0: The leapfrog step size. Default is 0.01f0.
  • steps::Int=3: The number of fixed-point iterations to perform. Default is 3. Typically, 3 iterations are sufficient.
  • ∇H_kwargs::NamedTuple: The keyword arguments for ∇hamiltonian. Default is a tuple with decoder_loglikelihood, position_logprior, and momentum_logprior
  • G_inv::Function: The function to compute the inverse of the Riemannian metric tensor. Default is G_inv.
    A tuple (z̄, ρ̄, Ḡ⁻¹, logdetḠ, decoder_update) representing the updated position, momentum, the inverse of the updated Riemannian metric tensor, the log of the determinant of the metric tensor, and the updated decoder outputs after performing the full leapfrog step.
AutoEncoderToolkit.RHVAEs.general_leapfrog_tempering_step — Function
general_leapfrog_tempering_step(
     x::AbstractArray,
     zₒ::AbstractVecOrMat,
     Gₒ⁻¹::AbstractArray,
@@ -383,7 +383,7 @@
     ),
     G_inv::Function=G_inv,
     tempering_schedule::Function=quadratic_tempering,
-)
Combines the leapfrog and tempering steps into a single function for the Riemannian Hamiltonian Variational Autoencoder (RHVAE).
Arguments
  • x::AbstractArray: The data to be processed. If Array, the last dimension must be of size 1.
  • zₒ::AbstractVecOrMat: The initial latent variable. 
Optional Keyword Arguments
  • ϵ::Union{<:Number,<:AbstractVector}: The step size for the leapfrog steps in the HMC algorithm. This can be a scalar or an array. Default is 0.01f0.  
  • K::Int: The number of leapfrog steps to perform in the Hamiltonian Monte Carlo (HMC) algorithm. Default is 3.
  • βₒ::Number: The initial inverse temperature for the tempering schedule. Default is 0.3f0.
  • steps::Int: The number of fixed-point iterations to perform. Default is 3.
  • ∇H_kwargs::NamedTuple: Additional keyword arguments to be passed to the ∇hamiltonian function. Default is a NamedTuple with reconstruction_loglikelihood, position_logprior, and momentum_logprior.
  • tempering_schedule::Function: The function to compute the inverse temperature at each step in the HMC algorithm. Defaults to quadratic_tempering. This function must take three arguments: First, βₒ, an initial inverse temperature, second, k, the current step in the tempering schedule, and third, K, the total number of steps in the tempering schedule.
Returns
  • A NamedTuple with the following keys: 
    • z_init: The initial latent variable. 
    • ρ_init: The initial momentum variable. 
    • Ginv_init: The initial inverse of the Riemannian metric tensor. 
    • z_final: The final latent variable after K leapfrog steps. 
    • ρ_final: The final momentum variable after K leapfrog steps. 
    • Ginv_final: The final inverse of the Riemannian metric tensor after K leapfrog steps.
  • The decoder output at the final latent variable is also returned. Note: This is not in the same named tuple as the other outputs, but as a separate output.
Description
The function first samples a random momentum variable γₒ from a standard normal distribution and scales it by the inverse square root of the initial inverse temperature βₒ to obtain the initial momentum variable ρₒ. Then, it performs K leapfrog steps, each followed by a tempering step, to generate a new sample from the latent space.
Note
Ensure the input data x and the initial latent variable zₒ match the expected input dimensionality for the RHVAE model.
AutoEncoderToolkit.RHVAEs._log_p̄ — Function
_log_p̄(
+)
Combines the leapfrog and tempering steps into a single function for the Riemannian Hamiltonian Variational Autoencoder (RHVAE).
Arguments
  • x::AbstractArray: The data to be processed. If Array, the last dimension must be of size 1.
  • zₒ::AbstractVecOrMat: The initial latent variable. 
Optional Keyword Arguments
  • ϵ::Union{<:Number,<:AbstractVector}: The step size for the leapfrog steps in the HMC algorithm. This can be a scalar or an array. Default is 0.01f0.  
  • K::Int: The number of leapfrog steps to perform in the Hamiltonian Monte Carlo (HMC) algorithm. Default is 3.
  • βₒ::Number: The initial inverse temperature for the tempering schedule. Default is 0.3f0.
  • steps::Int: The number of fixed-point iterations to perform. Default is 3.
  • ∇H_kwargs::NamedTuple: Additional keyword arguments to be passed to the ∇hamiltonian function. Default is a NamedTuple with reconstruction_loglikelihood, position_logprior, and momentum_logprior.
  • tempering_schedule::Function: The function to compute the inverse temperature at each step in the HMC algorithm. Defaults to quadratic_tempering. This function must take three arguments: First, βₒ, an initial inverse temperature, second, k, the current step in the tempering schedule, and third, K, the total number of steps in the tempering schedule.
Returns
  • A NamedTuple with the following keys: 
    • z_init: The initial latent variable. 
    • ρ_init: The initial momentum variable. 
    • Ginv_init: The initial inverse of the Riemannian metric tensor. 
    • z_final: The final latent variable after K leapfrog steps. 
    • ρ_final: The final momentum variable after K leapfrog steps. 
    • Ginv_final: The final inverse of the Riemannian metric tensor after K leapfrog steps.
  • The decoder output at the final latent variable is also returned. Note: This is not in the same named tuple as the other outputs, but as a separate output.
Description
The function first samples a random momentum variable γₒ from a standard normal distribution and scales it by the inverse square root of the initial inverse temperature βₒ to obtain the initial momentum variable ρₒ. Then, it performs K leapfrog steps, each followed by a tempering step, to generate a new sample from the latent space.
Note
Ensure the input data x and the initial latent variable zₒ match the expected input dimensionality for the RHVAE model.
AutoEncoderToolkit.RHVAEs._log_p̄ — Function
_log_p̄(
     x::AbstractArray,
     rhvae::RHVAE{VAE{E,D}},
     rhvae_outputs::NamedTuple;
@@ -391,13 +391,13 @@
     position_logprior::Function=spherical_logprior,
     momentum_logprior::Function=riemannian_logprior,
     prefactor::AbstractArray=ones(Float32, 3),
-)
This is an internal function used in riemannian_hamiltonian_elbo to compute the numerator of the unbiased estimator of the marginal likelihood. The function computes the sum of the log likelihood of the data given the latent variables, the log prior of the latent variables, and the log prior of the momentum variables.
log p̄ = log p(x | zₖ) + log p(zₖ) + log p(ρₖ(zₖ))
Arguments
  • x::AbstractArray: The input data. If Array, the last dimension must contain each of the data points.
  • rhvae::RHVAE{<:VAE{<:AbstractGaussianEncoder,<:AbstractGaussianLogDecoder}}: The Riemannian Hamiltonian Variational Autoencoder (RHVAE) model.
  • rhvae_outputs::NamedTuple: The outputs of the RHVAE, including the final latent variables zₖ and the final momentum variables ρₖ.
Optional Keyword Arguments
  • reconstruction_loglikelihood::Function: The function to compute the log likelihood of the data given the latent variables. Default is decoder_loglikelihood.
  • position_logprior::Function: The function to compute the log prior of the latent variables. Default is spherical_logprior.
  • momentum_logprior::Function: The function to compute the log prior of the momentum variables. Default is riemannian_logprior.
  • prefactor::AbstractArray: A 3-element array to scale the log likelihood, log prior of the latent variables, and log prior of the momentum variables. Default is an array of ones.
Returns
  • log_p̄::AbstractVector: The first term of the log of the unbiased estimator of the marginal likelihood for each data point.
Note
This is an internal function and should not be called directly. It is used as part of the riemannian_hamiltonian_elbo function.
AutoEncoderToolkit.RHVAEs._log_q̄ — Function
_log_q̄(
+)
This is an internal function used in riemannian_hamiltonian_elbo to compute the numerator of the unbiased estimator of the marginal likelihood. The function computes the sum of the log likelihood of the data given the latent variables, the log prior of the latent variables, and the log prior of the momentum variables.
log p̄ = log p(x | zₖ) + log p(zₖ) + log p(ρₖ(zₖ))
Arguments
  • x::AbstractArray: The input data. If Array, the last dimension must contain each of the data points.
  • rhvae::RHVAE{<:VAE{<:AbstractGaussianEncoder,<:AbstractGaussianLogDecoder}}: The Riemannian Hamiltonian Variational Autoencoder (RHVAE) model.
  • rhvae_outputs::NamedTuple: The outputs of the RHVAE, including the final latent variables zₖ and the final momentum variables ρₖ.
Optional Keyword Arguments
  • reconstruction_loglikelihood::Function: The function to compute the log likelihood of the data given the latent variables. Default is decoder_loglikelihood.
  • position_logprior::Function: The function to compute the log prior of the latent variables. Default is spherical_logprior.
  • momentum_logprior::Function: The function to compute the log prior of the momentum variables. Default is riemannian_logprior.
  • prefactor::AbstractArray: A 3-element array to scale the log likelihood, log prior of the latent variables, and log prior of the momentum variables. Default is an array of ones.
Returns
  • log_p̄::AbstractVector: The first term of the log of the unbiased estimator of the marginal likelihood for each data point.
Note
This is an internal function and should not be called directly. It is used as part of the riemannian_hamiltonian_elbo function.
AutoEncoderToolkit.RHVAEs._log_q̄ — Function
_log_q̄(
     rhvae::RHVAE,
     rhvae_outputs::NamedTuple,
     βₒ::Number;
     momentum_logprior::Function=riemannian_logprior,
     prefactor::AbstractArray=ones(Float32, 3),
-)
This is an internal function used in riemannian_hamiltonian_elbo to compute the second term of the unbiased estimator of the marginal likelihood. The function computes the sum of the log posterior of the initial latent variables and the log prior of the initial momentum variables, minus a term that depends on the dimensionality of the latent space and the initial temperature.
    log q̄ = log q(zₒ) + log p(ρₒ) - d/2 log(βₒ)
Arguments
  • rhvae::RHVAE: The Riemannian Hamiltonian Variational Autoencoder (RHVAE) model.
  • rhvae_outputs::NamedTuple: The outputs of the RHVAE, including the initial latent variables zₒ and the initial momentum variables ρₒ.
  • βₒ::Number: The initial temperature for the tempering steps.
Optional Keyword Arguments
  • momentum_logprior::Function: The function to compute the log prior of the momentum variables. Default is riemannian_logprior.
  • prefactor::AbstractArray: A 3-element array to scale the log posterior of the initial latent variables, log prior of the initial momentum variables, and the tempering Jacobian term. Default is an array of ones.
Returns
  • log_q̄::Vector: The second term of the log of the unbiased estimator of the   marginal likelihood for each data point.
Note
This is an internal function and should not be called directly. It is used as part of the riemannian_hamiltonian_elbo function.
AutoEncoderToolkit.RHVAEs.riemannian_hamiltonian_elbo — Function
riemannian_hamiltonian_elbo(
+)
This is an internal function used in riemannian_hamiltonian_elbo to compute the second term of the unbiased estimator of the marginal likelihood. The function computes the sum of the log posterior of the initial latent variables and the log prior of the initial momentum variables, minus a term that depends on the dimensionality of the latent space and the initial temperature.
    log q̄ = log q(zₒ) + log p(ρₒ) - d/2 log(βₒ)
Arguments
  • rhvae::RHVAE: The Riemannian Hamiltonian Variational Autoencoder (RHVAE) model.
  • rhvae_outputs::NamedTuple: The outputs of the RHVAE, including the initial latent variables zₒ and the initial momentum variables ρₒ.
  • βₒ::Number: The initial temperature for the tempering steps.
Optional Keyword Arguments
  • momentum_logprior::Function: The function to compute the log prior of the momentum variables. Default is riemannian_logprior.
  • prefactor::AbstractArray: A 3-element array to scale the log posterior of the initial latent variables, log prior of the initial momentum variables, and the tempering Jacobian term. Default is an array of ones.
Returns
  • log_q̄::Vector: The second term of the log of the unbiased estimator of the   marginal likelihood for each data point.
Note
This is an internal function and should not be called directly. It is used as part of the riemannian_hamiltonian_elbo function.
AutoEncoderToolkit.RHVAEs.riemannian_hamiltonian_elbo — Function
riemannian_hamiltonian_elbo(
     rhvae::RHVAE,
     metric_param::NamedTuple,
     x::AbstractArray;
@@ -469,17 +469,17 @@
     return_outputs::Bool=false,
     logp_prefactor::AbstractArray=ones(Float32, 3),
     logq_prefactor::AbstractArray=ones(Float32, 3),
-)
Compute the Riemannian Hamiltonian Monte Carlo (RHMC) estimate of the evidence lower bound (ELBO) for a Riemannian Hamiltonian Variational Autoencoder (RHVAE).
This function takes as input an RHVAE, a NamedTuple of metric parameters, and a vector of input data x. It performs K RHMC steps with a leapfrog integrator and a tempering schedule to estimate the ELBO. The ELBO is computed as the difference between the log p̄ and log q̄ as
elbo = mean(log p̄ - log q̄).
Arguments
  • rhvae::RHVAE: The RHVAE used to encode the input data and decode the latent space.
  • x_in::AbstractArray: Input data to the RHVAE encoder. The last dimension is taken as having each of the samples in a batch.
  • x_out::AbstractArray: Target data to compute the reconstruction error. The last dimension is taken as having each of the samples in a batch.
Optional Keyword Arguments
  • ∇H_kwargs::NamedTuple: Additional keyword arguments to be passed to the ∇hamiltonian function. Defaults to a NamedTuple with :decoder_loglikelihood set to decoder_loglikelihood, :position_logprior set to spherical_logprior, :momentum_logprior set to riemannian_logprior, and :G_inv set to G_inv.
  • K::Int: The number of RHMC steps (default is 3).
  • ϵ::Union{<:Number,<:AbstractVector}: The step size for the leapfrog integrator (default is 0.001).
  • βₒ::Number: The initial inverse temperature (default is 0.3).
  • steps::Int: The number of leapfrog steps (default is 3).
  • G_inv::Function: The function to compute the inverse of the Riemannian metric tensor (default is G_inv).
  • tempering_schedule::Function: The tempering schedule function used in the RHMC (default is quadratic_tempering).
  • return_outputs::Bool: Whether to return the outputs of the RHVAE. Defaults to false. NOTE: This is necessary to avoid computing the forward pass twice when computing the loss function with regularization.
  • logp_prefactor::AbstractArray: A 3-element array to scale the log likelihood, log prior of the latent variables, and log prior of the momentum variables. Default is an array of ones.
  • logq_prefactor::AbstractArray: A 3-element array to scale the log posterior of the initial latent variables, log prior of the initial momentum variables, and the tempering Jacobian term. Default is an array of ones.
Returns
  • elbo::Number: The RHMC estimate of the ELBO. If return_outputs is true, also returns the outputs of the RHVAE.
Default initializations
AutoEncoderToolkit.jl provides default initializations for both the metric tensor network and the RHVAE. Although less flexible than defining your own initial networks, these can serve as a good starting point for your experiments.
AutoEncoderToolkit.RHVAEs.MetricChain — Method
MetricChain(
+)
Compute the Riemannian Hamiltonian Monte Carlo (RHMC) estimate of the evidence lower bound (ELBO) for a Riemannian Hamiltonian Variational Autoencoder (RHVAE).
This function takes as input an RHVAE, a NamedTuple of metric parameters, and a vector of input data x. It performs K RHMC steps with a leapfrog integrator and a tempering schedule to estimate the ELBO. The ELBO is computed as the difference between the log p̄ and log q̄ as
elbo = mean(log p̄ - log q̄).
Arguments
  • rhvae::RHVAE: The RHVAE used to encode the input data and decode the latent space.
  • x_in::AbstractArray: Input data to the RHVAE encoder. The last dimension is taken as having each of the samples in a batch.
  • x_out::AbstractArray: Target data to compute the reconstruction error. The last dimension is taken as having each of the samples in a batch.
Optional Keyword Arguments
  • ∇H_kwargs::NamedTuple: Additional keyword arguments to be passed to the ∇hamiltonian function. Defaults to a NamedTuple with :decoder_loglikelihood set to decoder_loglikelihood, :position_logprior set to spherical_logprior, :momentum_logprior set to riemannian_logprior, and :G_inv set to G_inv.
  • K::Int: The number of RHMC steps (default is 3).
  • ϵ::Union{<:Number,<:AbstractVector}: The step size for the leapfrog integrator (default is 0.001).
  • βₒ::Number: The initial inverse temperature (default is 0.3).
  • steps::Int: The number of leapfrog steps (default is 3).
  • G_inv::Function: The function to compute the inverse of the Riemannian metric tensor (default is G_inv).
  • tempering_schedule::Function: The tempering schedule function used in the RHMC (default is quadratic_tempering).
  • return_outputs::Bool: Whether to return the outputs of the RHVAE. Defaults to false. NOTE: This is necessary to avoid computing the forward pass twice when computing the loss function with regularization.
  • logp_prefactor::AbstractArray: A 3-element array to scale the log likelihood, log prior of the latent variables, and log prior of the momentum variables. Default is an array of ones.
  • logq_prefactor::AbstractArray: A 3-element array to scale the log posterior of the initial latent variables, log prior of the initial momentum variables, and the tempering Jacobian term. Default is an array of ones.
Returns
  • elbo::Number: The RHMC estimate of the ELBO. If return_outputs is true, also returns the outputs of the RHVAE.
Default initializations
AutoEncoderToolkit.jl provides default initializations for both the metric tensor network and the RHVAE. Although less flexible than defining your own initial networks, these can serve as a good starting point for your experiments.
AutoEncoderToolkit.RHVAEs.MetricChain — Method
MetricChain(
     n_input::Int,
     n_latent::Int,
     metric_neurons::Vector{<:Int},
     metric_activation::Vector{<:Function},
     output_activation::Function;
     init::Function=Flux.glorot_uniform
-) -> MetricChain
Construct a MetricChain for computing the Riemannian metric tensor in the latent space.
Arguments
  • n_input::Int: The number of input features.
  • n_latent::Int: The dimension of the latent space.
  • metric_neurons::Vector{<:Int}: The number of neurons in each hidden layer of the MLP.
  • metric_activation::Vector{<:Function}: The activation function for each hidden layer of the MLP.
  • output_activation::Function: The activation function for the output layer.
  • init::Function: The initialization function for the weights in the layers (default is Flux.glorot_uniform).
Returns
  • MetricChain: A MetricChain object that includes the MLP, and two dense layers for computing the elements of a lower-triangular matrix used to compute the Riemannian metric tensor in latent space.
AutoEncoderToolkit.RHVAEs.RHVAE — Method
RHVAE(
+) -> MetricChain
Construct a MetricChain for computing the Riemannian metric tensor in the latent space.
Arguments
  • n_input::Int: The number of input features.
  • n_latent::Int: The dimension of the latent space.
  • metric_neurons::Vector{<:Int}: The number of neurons in each hidden layer of the MLP.
  • metric_activation::Vector{<:Function}: The activation function for each hidden layer of the MLP.
  • output_activation::Function: The activation function for the output layer.
  • init::Function: The initialization function for the weights in the layers (default is Flux.glorot_uniform).
Returns
  • MetricChain: A MetricChain object that includes the MLP, and two dense layers for computing the elements of a lower-triangular matrix used to compute the Riemannian metric tensor in latent space.
AutoEncoderToolkit.RHVAEs.RHVAE — Method
RHVAE(
     vae::VAE, 
     metric_chain::MetricChain, 
     centroids_data::AbstractArray, 
     T::Number, 
     λ::Number
-)
Construct a Riemannian Hamiltonian Variational Autoencoder (RHVAE) from a standard VAE and a metric chain.
Arguments
  • vae::VAE: A standard Variational Autoencoder (VAE) model.
  • metric_chain::MetricChain: A chain of metrics to be used for the Riemannian Hamiltonian Monte Carlo (RHMC) sampler.
  • centroids_data::AbstractArray: An array of data centroids. Each column represents a centroid. N is a subtype of Number.
  • T::N: The temperature parameter for the inverse metric tensor. N is a subtype of Number.
  • λ::N: The regularization parameter for the inverse metric tensor. N is a subtype of Number.
Returns
  • A new RHVAE object.
Description
The constructor initializes the latent centroids and the metric tensor M to their default values. The latent centroids are initialized to a zero matrix of the same size as centroids_data, and M is initialized to a 3D array of identity matrices, one for each centroid.

+)

Construct a Riemannian Hamiltonian Variational Autoencoder (RHVAE) from a standard VAE and a metric chain.

Arguments

  • vae::VAE: A standard Variational Autoencoder (VAE) model.
  • metric_chain::MetricChain: A chain of metrics to be used for the Riemannian Hamiltonian Monte Carlo (RHMC) sampler.
  • centroids_data::AbstractArray: An array of data centroids. Each column represents a centroid. N is a subtype of Number.
  • T::N: The temperature parameter for the inverse metric tensor. N is a subtype of Number.
  • λ::N: The regularization parameter for the inverse metric tensor. N is a subtype of Number.

Returns

  • A new RHVAE object.

Description

The constructor initializes the latent centroids and the metric tensor M to their default values. The latent centroids are initialized to a zero matrix of the same size as centroids_data, and M is initialized to a 3D array of identity matrices, one for each centroid.

diff --git a/dev/utils/index.html b/dev/utils/index.html index 66c9a8d..90ed5a2 100644 --- a/dev/utils/index.html +++ b/dev/utils/index.html @@ -1,5 +1,5 @@ -Utilities · AutoEncoderToolkit

Utils

AutoEncoderToolkit.jl offers a series of utility functions for different tasks.

Training Utilities

AutoEncoderToolkit.utils.step_scheduler — Function
`step_scheduler(epoch, epoch_change, learning_rates)`

Simple function to define different learning rates at specified epochs.

Arguments

  • epoch::Int: Epoch at which to define learning rate.
  • epoch_change::Vector{<:Int}: Number of epochs at which to change learning rate. It must include the initial learning rate!
  • learning_rates::Vector{<:AbstractFloat}: Learning rate value for the epoch range. Must be the same length as epoch_change

Returns

  • η::AbstractFloat: Learning rate for the current epoch.
AutoEncoderToolkit.utils.cycle_anneal — Function
cycle_anneal(
+Utilities · AutoEncoderToolkit
Utils
AutoEncoderToolkit.jl offers a series of utility functions for different tasks. 
Training Utilities
AutoEncoderToolkit.utils.step_scheduler — Function
`step_scheduler(epoch, epoch_change, learning_rates)`
Simple function to define different learning rates at specified epochs.
Arguments
  • epoch::Int: Epoch at which to define learning rate.
  • epoch_change::Vector{<:Int}: Number of epochs at which to change learning rate. It must include the initial learning rate!
  • learning_rates::Vector{<:AbstractFloat}: Learning rate value for the epoch range. Must be the same length as epoch_change
Returns
  • η::AbstractFloat: Learning rate for the current epoch.
AutoEncoderToolkit.utils.cycle_anneal — Function
cycle_anneal(
     epoch::Int, 
     n_epoch::Int, 
     n_cycles::Int; 
@@ -7,9 +7,9 @@
     βmax::Number=1.0f0, 
     βmin::Number=0.0f0, 
     T::Type=Float32
-)
Function that computes the value of the annealing parameter β for a variational autoencoder as a function of the epoch number according to the cyclical annealing strategy.
Arguments
  • epoch::Int: Epoch on which to evaluate the value of the annealing parameter.
  • n_epoch::Int: Number of epochs that will be run to train the VAE.
  • n_cycles::Int: Number of annealing cycles to be fit within the number of epochs.
Optional Arguments
  • frac::AbstractFloat= 0.5f0: Fraction of the cycle in which the annealing parameter β will increase from the minimum to the maximum value.
  • βmax::Number=1.0f0: Maximum value that the annealing parameter can reach.
  • βmin::Number=0.0f0: Minimum value that the annealing parameter can reach.
  • T::Type=Float32: The type of the output. The function will convert the output to this type.
Returns
  • β::T: Value of the annealing parameter.
Citation
Fu, H. et al. Cyclical Annealing Schedule: A Simple Approach to Mitigating KL Vanishing. Preprint at http://arxiv.org/abs/1903.10145 (2019).
AutoEncoderToolkit.utils.locality_sampler — Function
locality_sampler(data, dist_tree, n_primary, n_secondary, k_neighbors; index=false)
Algorithm to generate mini-batches based on spatial locality as determined by a pre-constructed nearest neighbors tree.
Arguments
  • data::AbstractArray: An array containing the data points. The data points can be of any dimension.
  • dist_tree::NearestNeighbors.NNTree: NearestNeighbors.jl tree used to determine the distance between data points.
  • n_primary::Int: Number of primary points to sample.
  • n_secondary::Int: Number of secondary points to sample from the neighbors of each primary point.
  • k_neighbors::Int: Number of nearest neighbors from which to potentially sample the secondary points.
Optional Keyword Arguments
  • index::Bool: If true, returns the indices of the selected samples. If false, returns the data corresponding to the indexes. Defaults to false.
Returns
  • If index is true, returns sample_idx::Vector{Int64}: Indices of data points to include in the mini-batch.
  • If index is false, returns sample_data::AbstractArray: The data points to include in the mini-batch.
Description
This sampling algorithm consists of three steps:
  1. For each datapoint, determine the k_neighbors nearest neighbors using the dist_tree.
  2. Uniformly sample n_primary points without replacement from all data points.
  3. For each primary point, sample n_secondary points without replacement from its k_neighbors nearest neighbors.
Examples
# Pre-constructed NearestNeighbors.jl tree
+)
Function that computes the value of the annealing parameter β for a variational autoencoder as a function of the epoch number according to the cyclical annealing strategy.
Arguments
  • epoch::Int: Epoch on which to evaluate the value of the annealing parameter.
  • n_epoch::Int: Number of epochs that will be run to train the VAE.
  • n_cycles::Int: Number of annealing cycles to be fit within the number of epochs.
Optional Arguments
  • frac::AbstractFloat= 0.5f0: Fraction of the cycle in which the annealing parameter β will increase from the minimum to the maximum value.
  • βmax::Number=1.0f0: Maximum value that the annealing parameter can reach.
  • βmin::Number=0.0f0: Minimum value that the annealing parameter can reach.
  • T::Type=Float32: The type of the output. The function will convert the output to this type.
Returns
  • β::T: Value of the annealing parameter.
Citation
Fu, H. et al. Cyclical Annealing Schedule: A Simple Approach to Mitigating KL Vanishing. Preprint at http://arxiv.org/abs/1903.10145 (2019).
AutoEncoderToolkit.utils.locality_sampler — Function
locality_sampler(data, dist_tree, n_primary, n_secondary, k_neighbors; index=false)
Algorithm to generate mini-batches based on spatial locality as determined by a pre-constructed nearest neighbors tree.
Arguments
  • data::AbstractArray: An array containing the data points. The data points can be of any dimension.
  • dist_tree::NearestNeighbors.NNTree: NearestNeighbors.jl tree used to determine the distance between data points.
  • n_primary::Int: Number of primary points to sample.
  • n_secondary::Int: Number of secondary points to sample from the neighbors of each primary point.
  • k_neighbors::Int: Number of nearest neighbors from which to potentially sample the secondary points.
Optional Keyword Arguments
  • index::Bool: If true, returns the indices of the selected samples. If false, returns the data corresponding to the indexes. Defaults to false.
Returns
  • If index is true, returns sample_idx::Vector{Int64}: Indices of data points to include in the mini-batch.
  • If index is false, returns sample_data::AbstractArray: The data points to include in the mini-batch.
Description
This sampling algorithm consists of three steps:
  1. For each datapoint, determine the k_neighbors nearest neighbors using the dist_tree.
  2. Uniformly sample n_primary points without replacement from all data points.
  3. For each primary point, sample n_secondary points without replacement from its k_neighbors nearest neighbors.
Examples
# Pre-constructed NearestNeighbors.jl tree
 dist_tree = NearestNeighbors.KDTree(data, metric)
-sample_indices = locality_sampler(data, dist_tree, 10, 5, 50)
Citation
Skafte, N., Jø rgensen, M. & Hauberg, S. ren. Reliable training and estimation of variance networks. in Advances in Neural Information Processing Systems vol. 32 (Curran Associates, Inc., 2019).
Centroid Finding Utilities
Some VAE models, such as the RHVAE, require clustering of the data. Specifically RHVAE can take a fixed subset of the training data as a reference for the computation of the metric tensor. The following functions can be used to define this reference subset to be used as centroids for the metric tensor computation.
AutoEncoderToolkit.utils.centroids_kmeans — Function
centroids_kmeans(
+sample_indices = locality_sampler(data, dist_tree, 10, 5, 50)
Citation
Skafte, N., Jø rgensen, M. & Hauberg, S. ren. Reliable training and estimation of variance networks. in Advances in Neural Information Processing Systems vol. 32 (Curran Associates, Inc., 2019).
Centroid Finding Utilities
Some VAE models, such as the RHVAE, require clustering of the data. Specifically RHVAE can take a fixed subset of the training data as a reference for the computation of the metric tensor. The following functions can be used to define this reference subset to be used as centroids for the metric tensor computation.
AutoEncoderToolkit.utils.centroids_kmeans — Function
centroids_kmeans(
     x::AbstractMatrix, 
     n_centroids::Int; 
     assign::Bool=false
@@ -20,7 +20,7 @@
     reshape_centroids::Bool=true, 
     assign::Bool=false
 )
Perform k-means clustering on the input and return the centers. This function can be used to down-sample the number of points used when computing the metric tensor in training a Riemannian Hamiltonian Variational Autoencoder (RHVAE).
The input data is flattened into a matrix before performing k-means clustering. This is done because k-means operates on a set of data points in a vector space and cannot handle multi-dimensional arrays. Flattening the input ensures that the k-means algorithm can process the data correctly.
By default, the output centroids are reshaped back to the original input shape. This is controlled by the reshape_centroids argument.
Arguments
  • x::AbstractArray: The input data. It can be a multi-dimensional array where the last dimension represents individual samples.
  • n_centroids::Int: The number of centroids to compute.
Optional Keyword Arguments
  • reshape_centroids::Bool=true: If true, reshape the output centroids back to the original input shape.
  • assign::Bool=false: If true, also return the assignments of each point to a centroid.
Returns
  • If assign is false, returns a matrix where each column is a centroid.
  • If assign is true, returns a tuple where the first element is the matrix of centroids and the second element is a vector of assignments.
Examples
data = rand(100, 10)
-centroids = centroids_kmeans(data, 5)
AutoEncoderToolkit.utils.centroids_kmedoids — Function
    centroids_kmedoids(
         x::AbstractMatrix, n_centroids::Int; assign::Bool=false
     )
Perform k-medoids clustering on the input and return the centers. This function can be used to down-sample the number of points used when computing the metric tensor in training a Riemannian Hamiltonian Variational Autoencoder (RHVAE).
Arguments
  • x::AbstractMatrix: The input data. Rows represent individual samples.
  • n_centroids::Int: The number of centroids to compute.
  • dist::Distances.PreMetric=Distances.Euclidean(): The distance metric to use when computing the pairwise distance matrix.
Optional Keyword Arguments
  • assign::Bool=false: If true, also return the assignments of each point to a centroid.
Returns
  • If assign is false, returns a matrix where each column is a centroid.
  • If assign is true, returns a tuple where the first element is the matrix of centroids and the second element is a vector of assignments.
Examples
data = rand(100, 10)
 centroids = centroids_kmedoids(data, 5)
centroids_kmedoids(
@@ -29,7 +29,7 @@
     dist::Distances.PreMetric=Distances.Euclidean();
     assign::Bool=false
 )
Perform k-medoids clustering on the input and return the centers. This function can be used to down-sample the number of points used when computing the metric tensor in training a Riemannian Hamiltonian Variational Autoencoder (RHVAE).
Arguments
  • x::AbstractArray: The input data. The last dimension of x should contain each of the samples that should be clustered.
  • n_centroids::Int: The number of centroids to compute.
  • dist::Distances.PreMetric=Distances.Euclidean(): The distance metric to use for the clustering. Defaults to Euclidean distance.
Optional Keyword Arguments
  • assign::Bool=false: If true, also return the assignments of each point to a centroid.
Returns
  • If assign is false, returns an array where each column is a centroid.
  • If assign is true, returns a tuple where the first element is the array of centroids and the second element is a vector of assignments.
Examples
data = rand(10, 100)
-centroids = centroids_kmedoids(data, 5)
Other Utilities
AutoEncoderToolkit.utils.storage_type — Function
storage_type(A::AbstractArray)
Determine the storage type of an array.
This function recursively checks the parent of the array until it finds the base storage type. This is useful for determining whether an array or its subarrays are stored on the CPU or GPU.
Arguments
  • A::AbstractArray: The array whose storage type is to be determined.
Returns
The type of the array that is the base storage of A.
AutoEncoderToolkit.utils.vec_to_ltri — Function
    vec_to_ltri(diag::AbstractVecOrMat, lower::AbstractVecOrMat)
Convert two one-dimensional vectors or matrices into a lower triangular matrix or a 3D tensor.
Arguments
  • diag::AbstractVecOrMat: The input vector or matrix to be converted into the diagonal of the matrix. If it's a matrix, each column is considered as a separate vector.
  • lower::AbstractVecOrMat: The input vector or matrix to be converted into the lower triangular part of the matrix. The length of this vector or the number of rows in this matrix should be a triangular number (i.e., the sum of the first n natural numbers for some n). If it's a matrix, each column is considered the lower part of a separate lower triangular matrix.
Returns
  • A lower triangular matrix or a 3D tensor where each slice is a lower triangular matrix constructed from diag and lower.
Description
This function constructs a lower triangular matrix or a 3D tensor from two input vectors or matrices, diag and lower. The diag vector or matrix provides the diagonal elements of the matrix, while the lower vector or matrix provides the elements below the diagonal. The function uses a comprehension to construct the matrix or tensor, with the lower_index function calculating the appropriate index in the lower vector or matrix for each element below the diagonal.
GPU Support
The function supports both CPU and GPU arrays. For GPU arrays, the data is first transferred to the CPU, the lower triangular matrix or tensor is constructed, and then it is transferred back to the GPU.
Other Utilities
AutoEncoderToolkit.utils.storage_type — Function
storage_type(A::AbstractArray)
Determine the storage type of an array.
This function recursively checks the parent of the array until it finds the base storage type. This is useful for determining whether an array or its subarrays are stored on the CPU or GPU.
Arguments
  • A::AbstractArray: The array whose storage type is to be determined.
Returns
The type of the array that is the base storage of A.
AutoEncoderToolkit.utils.vec_to_ltri — Function
    vec_to_ltri(diag::AbstractVecOrMat, lower::AbstractVecOrMat)
Convert two one-dimensional vectors or matrices into a lower triangular matrix or a 3D tensor.
Arguments
  • diag::AbstractVecOrMat: The input vector or matrix to be converted into the diagonal of the matrix. If it's a matrix, each column is considered as a separate vector.
  • lower::AbstractVecOrMat: The input vector or matrix to be converted into the lower triangular part of the matrix. The length of this vector or the number of rows in this matrix should be a triangular number (i.e., the sum of the first n natural numbers for some n). If it's a matrix, each column is considered the lower part of a separate lower triangular matrix.
Returns
  • A lower triangular matrix or a 3D tensor where each slice is a lower triangular matrix constructed from diag and lower.
Description
This function constructs a lower triangular matrix or a 3D tensor from two input vectors or matrices, diag and lower. The diag vector or matrix provides the diagonal elements of the matrix, while the lower vector or matrix provides the elements below the diagonal. The function uses a comprehension to construct the matrix or tensor, with the lower_index function calculating the appropriate index in the lower vector or matrix for each element below the diagonal.
GPU Support
The function supports both CPU and GPU arrays. For GPU arrays, the data is first transferred to the CPU, the lower triangular matrix or tensor is constructed, and then it is transferred back to the GPU.
AutoEncoderToolkit.utils.vec_mat_vec_batched — Function
vec_mat_vec_batched(
     v::AbstractVector, 
     M::AbstractMatrix, 
     w::AbstractVector
@@ -45,11 +45,11 @@
     v::AbstractMatrix{T}, 
     M::AbstractArray{S,3}, 
     w::AbstractMatrix{T}
-) where {T<:TaylorDiff.TaylorScalar{Float32,2},S<:Number}
Compute the batched product of vectors and matrices in the form v̲ᵀ M̲̲ w̲ for a specific type of matrices and vectors.
This function takes two matrices v and w of type TaylorDiff.TaylorScalar{Float32,2}, and a 3D array M of type Number, and computes the batched product v̲ M̲̲ w̲. The computation is performed by first extracting each slice of M and each column of w, then performing the vector-matrix multiplication for each pair of slices, and finally computing the element-wise multiplication of the resulting matrix with v and summing over the dimensions.
Arguments
  • v::AbstractMatrix{T}: A d×n matrix, where d is the dimension of the vectors and n is the number of vectors. T is a subtype of TaylorDiff.TaylorScalar{Float32,2}.
  • M::AbstractArray{S,3}: A d×d×n array, where d is the dimension of the matrices and n is the number of matrices. S is a subtype of Number.
  • w::AbstractMatrix{T}: A d×n matrix, where d is the dimension of the vectors and n is the number of vectors. T is a subtype of TaylorDiff.TaylorScalar{Float32,2}.
Returns
An n dimensional array where each element is the result of the product v̲ M̲̲ w̲ for the corresponding vectors and matrix.
Notes
This function uses the eachslice and eachcol functions to extract the slices of M and the columns of w, respectively. It then uses a list comprehension to perform the vector-matrix multiplication for each pair of slices, and finally computes the element-wise multiplication of the resulting matrix with v and sums over the dimensions to obtain the final result.
AutoEncoderToolkit.utils.slogdet — Function
slogdet(A::AbstractArray{T}; check::Bool=false) where {T<:Number}
Compute the log determinant of a positive-definite matrix A or a 3D array of such matrices.
Arguments
  • A::AbstractArray{T}: A positive-definite matrix or a 3D array of positive-definite matrices whose log determinant is to be computed.  
  • check::Bool=false: A flag that determines whether to check if the input matrix A is positive-definite. Defaults to false due to numerical instability.
Returns
  • The log determinant of A. If A is a 3D array, returns a 1D array of log determinants, one for each slice along the third dimension of A.
Description
This function computes the log determinant of a positive-definite matrix A or a 3D array of such matrices. It first computes the Cholesky decomposition of A, and then calculates the log determinant as twice the sum of the log of the diagonal elements of the lower triangular matrix from the Cholesky decomposition.
Conditions
The input matrix A must be a positive-definite matrix, i.e., it must be symmetric and all its eigenvalues must be positive. If check is set to true, the function will throw an error if A is not positive-definite.
GPU Support
The function supports both CPU and GPU arrays. 
AutoEncoderToolkit.utils.sample_MvNormalCanon — Function
sample_MvNormalCanon(Σ⁻¹::AbstractArray{T}) where {T<:Number}
Draw a random sample from a multivariate normal distribution in canonical form.
Arguments
  • Σ⁻¹::AbstractArray{T}: The precision matrix (inverse of the covariance matrix) of the multivariate normal distribution. This can be a 2D array (matrix) or a 3D array.
Returns
  • A random sample drawn from the multivariate normal distribution specified by the input precision matrix. If Σ⁻¹ is a 3D array, returns a 2D array of samples, one for each slice along the third dimension of Σ⁻¹.
Description
This function draws a random sample from a multivariate normal distribution specified by a precision matrix Σ⁻¹. The precision matrix can be a 2D array (matrix) or a 3D array. If Σ⁻¹ is a 3D array, the function draws a sample for each slice along the third dimension of Σ⁻¹.
The function first inverts the precision matrix to obtain the covariance matrix, then performs a Cholesky decomposition of the covariance matrix. It then draws a sample from a standard normal distribution and multiplies it by the lower triangular matrix from the Cholesky decomposition to obtain the final sample.
GPU Support
The function supports both CPU and GPU arrays.
AutoEncoderToolkit.utils.unit_vector — Function
unit_vector(x::AbstractVector, i::Int)
Create a unit vector of the same length as x with the i-th element set to 1.
Arguments
  • x::AbstractVector: The vector whose length is used to determine the dimension of the unit vector.
  • i::Int: The index of the element to be set to 1.
Returns
  • A unit vector of type eltype(x) and length equal to x with the i-th element set to 1.
Description
This function creates a unit vector of the same length as x with the i-th element set to 1. All other elements are set to 0.
Note
This function is marked with the @ignore_derivatives macro from the ChainRulesCore package, which means that all AutoDiff backends will ignore any call to this function when computing gradients.
unit_vector(x::AbstractMatrix, i::Int)
Create a unit vector of the same length as the number of rows in x with the i-th element set to 1.
Arguments
  • x::AbstractMatrix: The matrix whose number of rows is used to determine the dimension of the unit vector.
  • i::Int: The index of the element to be set to 1.
Returns
  • A unit vector of type eltype(x) and length equal to the number of rows in x with the i-th element set to 1.
Description
This function creates a unit vector of the same length as the number of rows in x with the i-th element set to 1. All other elements are set to 0. 
AutoEncoderToolkit.utils.finite_difference_gradient — Function
finite_difference_gradient(
+) where {T<:TaylorDiff.TaylorScalar{Float32,2},S<:Number}
Compute the batched product of vectors and matrices in the form v̲ᵀ M̲̲ w̲ for a specific type of matrices and vectors.
This function takes two matrices v and w of type TaylorDiff.TaylorScalar{Float32,2}, and a 3D array M of type Number, and computes the batched product v̲ M̲̲ w̲. The computation is performed by first extracting each slice of M and each column of w, then performing the vector-matrix multiplication for each pair of slices, and finally computing the element-wise multiplication of the resulting matrix with v and summing over the dimensions.
Arguments
  • v::AbstractMatrix{T}: A d×n matrix, where d is the dimension of the vectors and n is the number of vectors. T is a subtype of TaylorDiff.TaylorScalar{Float32,2}.
  • M::AbstractArray{S,3}: A d×d×n array, where d is the dimension of the matrices and n is the number of matrices. S is a subtype of Number.
  • w::AbstractMatrix{T}: A d×n matrix, where d is the dimension of the vectors and n is the number of vectors. T is a subtype of TaylorDiff.TaylorScalar{Float32,2}.
Returns
An n dimensional array where each element is the result of the product v̲ M̲̲ w̲ for the corresponding vectors and matrix.
Notes
This function uses the eachslice and eachcol functions to extract the slices of M and the columns of w, respectively. It then uses a list comprehension to perform the vector-matrix multiplication for each pair of slices, and finally computes the element-wise multiplication of the resulting matrix with v and sums over the dimensions to obtain the final result.
AutoEncoderToolkit.utils.slogdet — Function
slogdet(A::AbstractArray{T}; check::Bool=false) where {T<:Number}
Compute the log determinant of a positive-definite matrix A or a 3D array of such matrices.
Arguments
  • A::AbstractArray{T}: A positive-definite matrix or a 3D array of positive-definite matrices whose log determinant is to be computed.  
  • check::Bool=false: A flag that determines whether to check if the input matrix A is positive-definite. Defaults to false due to numerical instability.
Returns
  • The log determinant of A. If A is a 3D array, returns a 1D array of log determinants, one for each slice along the third dimension of A.
Description
This function computes the log determinant of a positive-definite matrix A or a 3D array of such matrices. It first computes the Cholesky decomposition of A, and then calculates the log determinant as twice the sum of the log of the diagonal elements of the lower triangular matrix from the Cholesky decomposition.
Conditions
The input matrix A must be a positive-definite matrix, i.e., it must be symmetric and all its eigenvalues must be positive. If check is set to true, the function will throw an error if A is not positive-definite.
GPU Support
The function supports both CPU and GPU arrays. 
AutoEncoderToolkit.utils.sample_MvNormalCanon — Function
sample_MvNormalCanon(Σ⁻¹::AbstractArray{T}) where {T<:Number}
Draw a random sample from a multivariate normal distribution in canonical form.
Arguments
  • Σ⁻¹::AbstractArray{T}: The precision matrix (inverse of the covariance matrix) of the multivariate normal distribution. This can be a 2D array (matrix) or a 3D array.
Returns
  • A random sample drawn from the multivariate normal distribution specified by the input precision matrix. If Σ⁻¹ is a 3D array, returns a 2D array of samples, one for each slice along the third dimension of Σ⁻¹.
Description
This function draws a random sample from a multivariate normal distribution specified by a precision matrix Σ⁻¹. The precision matrix can be a 2D array (matrix) or a 3D array. If Σ⁻¹ is a 3D array, the function draws a sample for each slice along the third dimension of Σ⁻¹.
The function first inverts the precision matrix to obtain the covariance matrix, then performs a Cholesky decomposition of the covariance matrix. It then draws a sample from a standard normal distribution and multiplies it by the lower triangular matrix from the Cholesky decomposition to obtain the final sample.
GPU Support
The function supports both CPU and GPU arrays.
AutoEncoderToolkit.utils.unit_vector — Function
unit_vector(x::AbstractVector, i::Int)
Create a unit vector of the same length as x with the i-th element set to 1.
Arguments
  • x::AbstractVector: The vector whose length is used to determine the dimension of the unit vector.
  • i::Int: The index of the element to be set to 1.
Returns
  • A unit vector of type eltype(x) and length equal to x with the i-th element set to 1.
Description
This function creates a unit vector of the same length as x with the i-th element set to 1. All other elements are set to 0.
Note
This function is marked with the @ignore_derivatives macro from the ChainRulesCore package, which means that all AutoDiff backends will ignore any call to this function when computing gradients.
unit_vector(x::AbstractMatrix, i::Int)
Create a unit vector of the same length as the number of rows in x with the i-th element set to 1.
Arguments
  • x::AbstractMatrix: The matrix whose number of rows is used to determine the dimension of the unit vector.
  • i::Int: The index of the element to be set to 1.
Returns
  • A unit vector of type eltype(x) and length equal to the number of rows in x with the i-th element set to 1.
Description
This function creates a unit vector of the same length as the number of rows in x with the i-th element set to 1. All other elements are set to 0. 
AutoEncoderToolkit.utils.finite_difference_gradient — Function
finite_difference_gradient(
     f::Function,
     x::AbstractVecOrMat;
     fdtype::Symbol=:central
-)
Compute the finite difference gradient of a function f at a point x.
Arguments
  • f::Function: The function for which the gradient is to be computed. This function must return a scalar value.
  • x::AbstractVecOrMat: The point at which the gradient is to be computed. Can be a vector or a matrix. If a matrix, each column represents a point where the function f is to be evaluated and the derivative computed.
Optional Keyword Arguments
  • fdtype::Symbol=:central: The finite difference type. It can be either :forward or :central. Defaults to :central.
Returns
  • A vector or a matrix representing the gradient of f at x, depending on the input type of x.
Description
This function computes the finite difference gradient of a function f at a point x. The gradient is a vector or a matrix where the i-th element is the partial derivative of f with respect to the i-th element of x.
The partial derivatives are computed using the forward or central difference formula, depending on the fdtype argument:
  • Forward difference formula: ∂f/∂xᵢ ≈ [f(x + ε * eᵢ) - f(x)] / ε
  • Central difference formula: ∂f/∂xᵢ ≈ [f(x + ε * eᵢ) - f(x - ε * eᵢ)] / 2ε
where ε is the step size and eᵢ is the i-th unit vector.
GPU Support
This function supports both CPU and GPU arrays.
AutoEncoderToolkit.utils.taylordiff_gradient — Function
    taylordiff_gradient(
+)
Compute the finite difference gradient of a function f at a point x.
Arguments
  • f::Function: The function for which the gradient is to be computed. This function must return a scalar value.
  • x::AbstractVecOrMat: The point at which the gradient is to be computed. Can be a vector or a matrix. If a matrix, each column represents a point where the function f is to be evaluated and the derivative computed.
Optional Keyword Arguments
  • fdtype::Symbol=:central: The finite difference type. It can be either :forward or :central. Defaults to :central.
Returns
  • A vector or a matrix representing the gradient of f at x, depending on the input type of x.
Description
This function computes the finite difference gradient of a function f at a point x. The gradient is a vector or a matrix where the i-th element is the partial derivative of f with respect to the i-th element of x.
The partial derivatives are computed using the forward or central difference formula, depending on the fdtype argument:
  • Forward difference formula: ∂f/∂xᵢ ≈ [f(x + ε * eᵢ) - f(x)] / ε
  • Central difference formula: ∂f/∂xᵢ ≈ [f(x + ε * eᵢ) - f(x - ε * eᵢ)] / 2ε
where ε is the step size and eᵢ is the i-th unit vector.
GPU Support
This function supports both CPU and GPU arrays.
AutoEncoderToolkit.utils.taylordiff_gradient — Function
    taylordiff_gradient(
             f::Function,
             x::AbstractVecOrMat
-    )
Compute the gradient of a function f at a point x using Taylor series differentiation.
Arguments
  • f::Function: The function for which the gradient is to be computed. This must be a scalar function.
  • x::AbstractVecOrMat: The point at which the gradient is to be computed. Can be a vector or a matrix. If a matrix, each column represents a point where the function f is to be evaluated and the derivative computed.
Returns
  • A vector or a matrix representing the gradient of f at x, depending on the input type of x.
Description
This function computes the gradient of a function f at a point x using Taylor series differentiation. The gradient is a vector or a matrix where the i-th element or column is the partial derivative of f with respect to the i-th element of x.
The partial derivatives are computed using the TaylorDiff.derivative function.
GPU Support
This function currently only supports CPU arrays.

+    )

Compute the gradient of a function f at a point x using Taylor series differentiation.

Arguments

  • f::Function: The function for which the gradient is to be computed. This must be a scalar function.
  • x::AbstractVecOrMat: The point at which the gradient is to be computed. Can be a vector or a matrix. If a matrix, each column represents a point where the function f is to be evaluated and the derivative computed.

Returns

  • A vector or a matrix representing the gradient of f at x, depending on the input type of x.

Description

This function computes the gradient of a function f at a point x using Taylor series differentiation. The gradient is a vector or a matrix where the i-th element or column is the partial derivative of f with respect to the i-th element of x.

The partial derivatives are computed using the TaylorDiff.derivative function.

GPU Support

This function currently only supports CPU arrays.

diff --git a/dev/vae/index.html b/dev/vae/index.html index 7fffb50..bb4cc4b 100644 --- a/dev/vae/index.html +++ b/dev/vae/index.html @@ -1,5 +1,5 @@ -VAE / β-VAE · AutoEncoderToolkit

β-Variational Autoencoder

Variational Autoencoders, first introduced by Kingma and Welling in 2014, are a type of generative model that learns to encode high-dimensional data into a low-dimensional latent space. The main idea behind VAEs is to learn a probabilistic mapping (via variational inference) from the input data to the latent space, which allows for the generation of new data points by sampling from the latent space.

Their counterpart, the β-VAE, introduced by Higgins et al. in 2017, is a variant of the original VAE that includes a hyperparameter β that controls the relative importance of the reconstruction loss and the KL divergence term in the loss function. By adjusting β, the user can control the trade-off between the reconstruction quality and the disentanglement of the latent space.

In terms of implementation, the VAE struct in AutoEncoderToolkit.jl is a simple feedforward network composed of variational encoder and decoder parts. This means that the encoder has a log-posterior function and a KL divergence function associated with it, while the decoder has a log-likehood function associated with it.

References

VAE

Kingma, D. P. & Welling, M. Auto-Encoding Variational Bayes. Preprint at http://arxiv.org/abs/1312.6114 (2014).

β-VAE

Higgins, I. et al. β-VAE: LEARNING BASIC VISUAL CONCEPTS WITH A CONSTRAINED VARIATIONAL FRAMEWORK. (2017).

VAE struct

AutoEncoderToolkit.VAEs.VAE — Type

struct VAE{E<:AbstractVariationalEncoder, D<:AbstractVariationalDecoder}

Variational autoencoder (VAE) model defined for Flux.jl

Fields

  • encoder::E: Neural network that encodes the input into the latent space. E is a subtype of AbstractVariationalEncoder.
  • decoder::D: Neural network that decodes the latent representation back to the original input space. D is a subtype of AbstractVariationalDecoder.

A VAE consists of an encoder and decoder network with a bottleneck latent space in between. The encoder compresses the input into a low-dimensional probabilistic representation q(z|x). The decoder tries to reconstruct the original input from a sampled point in the latent space p(x|z).

Forward pass

AutoEncoderToolkit.VAEs.VAE — Method
    (vae::VAE)(x::AbstractArray; latent::Bool=false)

Perform the forward pass of a Variational Autoencoder (VAE).

This function takes as input a VAE and a vector or matrix of input data x. It first runs the input through the encoder to obtain the mean and log standard deviation of the latent variables. It then uses the reparameterization trick to sample from the latent distribution. Finally, it runs the latent sample through the decoder to obtain the output.

Arguments

  • vae::VAE: The VAE used to encode the input data and decode the latent space.
  • x::AbstractArray: The input data. If array, the last dimension contains each of the samples in a batch.

Optional Keyword Arguments

  • latent::Bool: Whether to return the latent variables along with the decoder output. If true, the function returns a tuple containing the encoder outputs, the latent sample, and the decoder outputs. If false, the function only returns the decoder outputs. Defaults to false.

Returns

  • If latent is true, returns a tuple containing:
    • encoder: The outputs of the encoder.
    • z: The latent sample.
    • decoder: The outputs of the decoder.
  • If latent is false, returns the outputs of the decoder.

Example

# Define a VAE
+VAE / β-VAE · AutoEncoderToolkit
β-Variational Autoencoder
Variational Autoencoders, first introduced by Kingma and Welling in 2014, are a type of generative model that learns to encode high-dimensional data into a low-dimensional latent space. The main idea behind VAEs is to learn a probabilistic mapping (via variational inference) from the input data to the latent space, which allows for the generation of new data points by sampling from the latent space.
Their counterpart, the β-VAE, introduced by Higgins et al. in 2017, is a variant of the original VAE that includes a hyperparameter β that controls the relative importance of the reconstruction loss and the KL divergence term in the loss function. By adjusting β, the user can control the trade-off between the reconstruction quality and the disentanglement of the latent space.
In terms of implementation, the VAE struct in AutoEncoderToolkit.jl is a simple feedforward network composed of variational encoder and decoder parts. This means that the encoder has a log-posterior function and a KL divergence function associated with it, while the decoder has a log-likehood function associated with it.
References
VAE
Kingma, D. P. & Welling, M. Auto-Encoding Variational Bayes. Preprint at http://arxiv.org/abs/1312.6114 (2014).
β-VAE
Higgins, I. et al. β-VAE: LEARNING BASIC VISUAL CONCEPTS WITH A CONSTRAINED VARIATIONAL FRAMEWORK. (2017).
VAE struct
AutoEncoderToolkit.VAEs.VAE — Type
struct VAE{E<:AbstractVariationalEncoder, D<:AbstractVariationalDecoder}
Variational autoencoder (VAE) model defined for Flux.jl
Fields
  • encoder::E: Neural network that encodes the input into the latent space. E is a subtype of AbstractVariationalEncoder.
  • decoder::D: Neural network that decodes the latent representation back to the original input space. D is a subtype of AbstractVariationalDecoder.
A VAE consists of an encoder and decoder network with a bottleneck latent space in between. The encoder compresses the input into a low-dimensional probabilistic representation q(z|x). The decoder tries to reconstruct the original input from a sampled point in the latent space p(x|z). 
Forward pass
AutoEncoderToolkit.VAEs.VAE — Method
    (vae::VAE)(x::AbstractArray; latent::Bool=false)
Perform the forward pass of a Variational Autoencoder (VAE).
This function takes as input a VAE and a vector or matrix of input data x. It first runs the input through the encoder to obtain the mean and log standard deviation of the latent variables. It then uses the reparameterization trick to sample from the latent distribution. Finally, it runs the latent sample through the decoder to obtain the output.
Arguments
  • vae::VAE: The VAE used to encode the input data and decode the latent space.
  • x::AbstractArray: The input data. If array, the last dimension contains each of the samples in a batch.
Optional Keyword Arguments
  • latent::Bool: Whether to return the latent variables along with the decoder output. If true, the function returns a tuple containing the encoder outputs, the latent sample, and the decoder outputs. If false, the function only returns the decoder outputs. Defaults to false.  
Returns
  • If latent is true, returns a tuple containing:
    • encoder: The outputs of the encoder.
    • z: The latent sample.
    • decoder: The outputs of the decoder.
  • If latent is false, returns the outputs of the decoder.
Example
# Define a VAE
 vae = VAE(
     encoder=Flux.Chain(Flux.Dense(784, 400, relu), Flux.Dense(400, 20)),
     decoder=Flux.Chain(Flux.Dense(20, 400, relu), Flux.Dense(400, 784))
@@ -9,7 +9,7 @@
 x = rand(Float32, 784)
 
 # Perform the forward pass
-outputs = vae(x, latent=true)
Loss function
Loss function
AutoEncoderToolkit.VAEs.loss — Function
loss(
     vae::VAE,
     x::AbstractArray;
     β::Number=1.0f0,
@@ -28,10 +28,10 @@
     reg_function::Union{Function,Nothing}=nothing,
     reg_kwargs::NamedTuple=NamedTuple(),
     reg_strength::Number=1.0f0
-)
Computes the loss for the variational autoencoder (VAE).
The loss function combines the reconstruction loss with the Kullback-Leibler (KL) divergence and possibly a regularization term, defined as:
loss = -⟨logπ(xout|z)⟩ + β × Dₖₗ[qᵩ(z|xin) || π(z)] + regstrength × regterm
Where:
  • π(xout|z) is a probabilistic decoder: π(xout|z) = N(f(z), σ² I̲̲)) - f(z) is
the function defining the mean of the decoder π(xout|z) - qᵩ(z|xin) is the approximated encoder: qᵩ(z|xin) = N(g(xin), h(x_in))
  • g(xin) and h(xin) define the mean and covariance of the encoder respectively.
Arguments
  • vae::VAE: A VAE model with encoder and decoder networks.
  • x_in::AbstractArray: Input data to the VAE encoder. The last dimension is taken as having each of the samples in a batch.
  • x_out::AbstractArray: Target data to compute the reconstruction error. The last dimension is taken as having each of the samples in a batch.
Optional Keyword Arguments
  • β::Number=1.0f0: Weighting factor for the KL-divergence term, used for annealing.
  • reconstruction_loglikelihood::Function=decoder_loglikelihood: A function that computes the reconstruction log likelihood.
  • kl_divergence::Function=encoder_kl: A function that computes the Kullback-Leibler divergence.
  • reg_function::Union{Function, Nothing}=nothing: A function that computes the regularization term based on the VAE outputs. Should return a Float32. This function must take as input the VAE outputs and the keyword arguments provided in reg_kwargs.
  • reg_kwargs::NamedTuple=NamedTuple(): Keyword arguments to pass to the regularization function.
  • reg_strength::Number=1.0f0: The strength of the regularization term.
Returns
  • T: The computed average loss value for the input x_in and its reconstructed counterparts x_out, including possible regularization terms.
Note
  • Ensure that the input data x_in and x_out match the expected input dimensionality for the encoder in the VAE.
Note
The loss function includes the β optional argument that can turn a vanilla VAE into a β-VAE by changing the default value of β from 1.0 to any other value.
Training
AutoEncoderToolkit.VAEs.train! — Function
train!(vae, x, opt; loss_function, loss_kwargs, verbose, loss_return)
Customized training function to update parameters of a variational autoencoder given a specified loss function.
Arguments
  • vae::VAE: A struct containing the elements of a variational autoencoder.
  • x::AbstractArray: Data on which to evaluate the loss function. The last dimension is taken as having each of the samples in a batch.
  • opt::NamedTuple: State of the optimizer for updating parameters. Typically initialized using Flux.Train.setup.
Optional Keyword Arguments
  • loss_function::Function=loss: The loss function used for training. It should accept the VAE model, data x, and keyword arguments in that order.
  • loss_kwargs::NamedTuple=NamedTuple(): Arguments for the loss function. These might include parameters like σ, or β, depending on the specific loss function in use.
  • verbose::Bool=false: If true, the loss value will be printed during training.
  • loss_return::Bool=false: If true, the loss value will be returned after training.
Description
Trains the VAE by:
  1. Computing the gradient of the loss w.r.t the VAE parameters.
  2. Updating the VAE parameters using the optimizer.
    `train!(
+)
Computes the loss for the variational autoencoder (VAE).
The loss function combines the reconstruction loss with the Kullback-Leibler (KL) divergence and possibly a regularization term, defined as:
loss = -⟨logπ(xout|z)⟩ + β × Dₖₗ[qᵩ(z|xin) || π(z)] + regstrength × regterm
Where:
  • π(xout|z) is a probabilistic decoder: π(xout|z) = N(f(z), σ² I̲̲)) - f(z) is
the function defining the mean of the decoder π(xout|z) - qᵩ(z|xin) is the approximated encoder: qᵩ(z|xin) = N(g(xin), h(x_in))
  • g(xin) and h(xin) define the mean and covariance of the encoder respectively.
Arguments
  • vae::VAE: A VAE model with encoder and decoder networks.
  • x_in::AbstractArray: Input data to the VAE encoder. The last dimension is taken as having each of the samples in a batch.
  • x_out::AbstractArray: Target data to compute the reconstruction error. The last dimension is taken as having each of the samples in a batch.
Optional Keyword Arguments
  • β::Number=1.0f0: Weighting factor for the KL-divergence term, used for annealing.
  • reconstruction_loglikelihood::Function=decoder_loglikelihood: A function that computes the reconstruction log likelihood.
  • kl_divergence::Function=encoder_kl: A function that computes the Kullback-Leibler divergence.
  • reg_function::Union{Function, Nothing}=nothing: A function that computes the regularization term based on the VAE outputs. Should return a Float32. This function must take as input the VAE outputs and the keyword arguments provided in reg_kwargs.
  • reg_kwargs::NamedTuple=NamedTuple(): Keyword arguments to pass to the regularization function.
  • reg_strength::Number=1.0f0: The strength of the regularization term.
Returns
  • T: The computed average loss value for the input x_in and its reconstructed counterparts x_out, including possible regularization terms.
Note
  • Ensure that the input data x_in and x_out match the expected input dimensionality for the encoder in the VAE.
Note
The loss function includes the β optional argument that can turn a vanilla VAE into a β-VAE by changing the default value of β from 1.0 to any other value.
Training
AutoEncoderToolkit.VAEs.train! — Function
train!(vae, x, opt; loss_function, loss_kwargs, verbose, loss_return)
Customized training function to update parameters of a variational autoencoder given a specified loss function.
Arguments
  • vae::VAE: A struct containing the elements of a variational autoencoder.
  • x::AbstractArray: Data on which to evaluate the loss function. The last dimension is taken as having each of the samples in a batch.
  • opt::NamedTuple: State of the optimizer for updating parameters. Typically initialized using Flux.Train.setup.
Optional Keyword Arguments
  • loss_function::Function=loss: The loss function used for training. It should accept the VAE model, data x, and keyword arguments in that order.
  • loss_kwargs::NamedTuple=NamedTuple(): Arguments for the loss function. These might include parameters like σ, or β, depending on the specific loss function in use.
  • verbose::Bool=false: If true, the loss value will be printed during training.
  • loss_return::Bool=false: If true, the loss value will be returned after training.
Description
Trains the VAE by:
  1. Computing the gradient of the loss w.r.t the VAE parameters.
  2. Updating the VAE parameters using the optimizer.
    `train!(
         vae, x_in, x_out, opt; 
         loss_function, loss_kwargs, verbose, loss_return
     )`
Customized training function to update parameters of a variational autoencoder given a loss function.
Arguments
  • vae::VAE: A struct containing the elements of a variational autoencoder.
  • x_in::AbstractArray: Input data for the loss function. Represents an individual sample. The last dimension is taken as having each of the samples in a batch.
  • x_out::AbstractArray: Target output data for the loss function. Represents the corresponding output for the x_in sample. The last dimension is taken as having each of the samples in a batch.
  • opt::NamedTuple: State of the optimizer for updating parameters. Typically   initialized using Flux.Optimisers.update!.
Optional Keyword Arguments
  • loss_function::Function=loss: The loss function used for training. It should accept the VAE model, data x_in, x_out, and keyword arguments in that order.  
  • loss_kwargs::NamedTuple=NamedTuple(): Arguments for the loss function. These might include parameters like σ, or β, depending on the specific loss function in use.
  • verbose::Bool=false: Whether to print the loss value after each training step.
  • loss_return::Bool=false: Whether to return the loss value after each training step.
Description
Trains the VAE by:
  1. Computing the gradient of the loss w.r.t the VAE parameters.
  2. Updating the VAE parameters using the optimizer.
Examples
opt = Flux.setup(Optax.adam(1e-3), vae)
 for (x_in, x_out) in dataloader
         train!(vae, x_in, x_out, opt) 
-end

+end