From a8731536d2898d0205c1e3d13e11b0642d6b57b5 Mon Sep 17 00:00:00 2001 From: marlin-codes Date: Sat, 21 Dec 2024 17:05:47 -0500 Subject: [PATCH 1/7] update --- .../hypformer/assets/exp1_large_scale.png | Bin 0 -> 229071 bytes .../hypformer/assets/exp2_medium_scale.png | Bin 0 -> 243764 bytes .../hypformer/assets/exp3_image_text.png | Bin 0 -> 228415 bytes .../hypformer/assets/framework2024.jpg | Bin 0 -> 346305 bytes app/projects/hypformer/assets/gpucost.png | Bin 0 -> 106416 bytes app/projects/hypformer/page.mdx | 162 ++++++++++++++++++ 6 files changed, 162 insertions(+) create mode 100644 app/projects/hypformer/assets/exp1_large_scale.png create mode 100644 app/projects/hypformer/assets/exp2_medium_scale.png create mode 100644 app/projects/hypformer/assets/exp3_image_text.png create mode 100644 app/projects/hypformer/assets/framework2024.jpg create mode 100644 app/projects/hypformer/assets/gpucost.png create mode 100644 app/projects/hypformer/page.mdx diff --git 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Introduction +Hyperbolic geometry have shown significant potential in modeling complex structured data, particularly those with underlying tree-like and hierarchical structures. Despite the impressive performance of various hyperbolic neural networks across numerous domains, research on adapting the Transformer to hyperbolic space remains limited. Previous attempts have mainly focused on modifying self-attention modules in the Transformer. However, these efforts have fallen short of developing a complete hyperbolic Transformer. This stems primarily from: (i) the absence of well-defined modules in hyperbolic space, including linear transformation layers, LayerNorm layers, activation functions, dropout operations, etc. (ii) the quadratic time complexity of the existing hyperbolic self-attention module w.r.t the number of input tokens, which hinders its scalability. To address these challenges, we propose, Hypformer, a novel hyperbolic Transformer based on the Lorentz model of hyperbolic geometry. In Hypformer, we introduce two foundational blocks that define the essential modules of the Transformer in hyperbolic space. Furthermore, we develop a linear self-attention mechanism in hyperbolic space, enabling hyperbolic Transformer to process billion-scale graph data and long-sequence inputs for the first time. Our experimental results confirm the effectiveness and efficiency of Hypformer across various datasets, demonstrating its potential as an effective and scalable solution for large-scale data representation and large models. + + +--- + +## 2. Method + +We propose constructing the fundamental linear transformation blocks entirely in hyperbolic space. In contrast to existing Lorentz-based transformations, our approach brings two key innovations: +1. Allowing the curvature to be adaptive (i.e., changeable) while still preserving relative ordering, and +2. Decoupling the normalization term from the transformation itself. + +--- + +### 2.1 Two Basic Blocks + +**(1) Hyperbolic Linear Transformation via Pseudo Lorentz Boost with Adaptive Curvature** + +To capture linear transformations in hyperbolic space, we introduce: + +$$ +\operatorname{HTC}(\mathbf{X}; f_t, \mathbf{W},\kappa_1, \kappa_2) := +\left( +\underbrace{\sqrt{\frac{\kappa_1}{\kappa_2}\| f_t(\mathbf{X}; \mathbf{W}) \|_{2,rw}^2 - \frac{1}{\kappa_2}}}_{\text{time-like dimension}}, +\underbrace{\sqrt{\frac{\kappa_1}{\kappa_2}} f_t(\mathbf{X}; \mathbf{W})}_{\text{space-like dimension}} +\right), +$$ + +Here, $\|\mathbf{A}\|_{2,rw}^2$ denotes the row-wise squared norm, and $f_t$ is a standard linear operation (e.g., $f_t(\mathbf{X};\mathbf{W}) = \mathbf{XW}$). The “time-like dimension” helps maintain consistency with hyperbolic geometry by preserving the Lorentz norm constraint, while the “space-like dimension” carries the actual transformed features. + +**(2) Hyperbolic Non-Linear Transformation via Pseudo Lorentz Rotation with Adaptive Curvature** + +Many operations in neural networks are non-linear (e.g., ReLU, Dropout, and normalizations). We model these using: + +$$ +\mathrm{HRC}(\mathbf{x};f_r, \kappa_1, \kappa_2) := +\left( +\underbrace{\sqrt{\frac{\kappa_1}{\kappa_2} \|f_r(\mathbf{X}_{[1:]})\|_{2,rw}^2 - \frac{1}{\kappa_2}}}_{\text{time-like dimension}}, +\underbrace{\sqrt{\frac{\kappa_1}{\kappa_2}} f_r(\mathbf{X}_{[1:]})}_{\text{space-like dimension}} +\right). +$$ + +Here, $f_r$ indicates a general non-linear function (e.g., Dropout, ReLU, LayerNorm, or BatchNorm). Because non-linear steps often occur within the same hyperbolic space (and do not transform the time-like part), we integrate them directly into the “space-like dimension,” using the Lorentz constraint to maintain geometric consistency. + +--- + +### 2.2 Linear Self-attention + +Given $N$ input tokens, each with dimension $d$, we represent the features as $\mathbf{X} \in \mathbb{L}^{N \times d, \kappa_1}$ in the Lorentz model. We then apply transformation matrices $\mathbf{W}^Q, \mathbf{W}^K, \mathbf{W}^V \in \mathbb{R}^{(d+1)\times d'}$ to obtain the query, key, and value: + +$$ +\begin{aligned} +\mathcal{Q} &= \mathrm{HTC}(\mathbf{X}; f_t, \mathbf{W}^Q, \kappa_1, \kappa_2), \\ +\mathcal{K} &= \mathrm{HTC}(\mathbf{X}; f_t, \mathbf{W}^K, \kappa_1, \kappa_2), \\ +\mathcal{V} &= \mathrm{HTC}(\mathbf{X}; f_t, \mathbf{W}^V, \kappa_1, \kappa_2). +\end{aligned} +$$ + +As a result, $\mathcal{Q}$, $\mathcal{K}$, and $\mathcal{V}$ reside in the hyperbolic space $\mathbb{L}^{N\times d', \kappa_2}$. + +Next, we compute pairwise similarities for self-attention. Since the weighted-sum operation itself does not affect the time-like dimension, we adopt the HRC framework for these space-like operations. Specifically, we extract the space-like dimensions: + +$$ +\mathcal{Q}_s, \mathcal{K}_s, \mathcal{V}_s = \phi(\mathcal{Q}_{[1:]}), \phi(\mathcal{K}_{[1:]}), \phi(\mathcal{V}_{[1:]}) +$$ + +to compute linear self-attention. We switch the multiplication order from $(\mathcal{Q}^T \mathcal{K}) \mathcal{V}$ to $\mathcal{Q} (\mathcal{K}^T \mathcal{V})$, thus achieving linear time complexity in the sequence length. In hyperbolic space, we adjust the time-like dimension after the weighted sum: + +$$ +\mathcal{Z}_{s} = \frac{\mathcal{Q}_s(\mathcal{K}_s^T \mathcal{V}_s)}{\mathcal{Q}_s(\mathcal{K}_s^T \mathbf{1})}. +$$ + +Finally, we map the result back via an HRC operation to satisfy the Lorentz constraint: + +$$ +\mathcal{\tilde{Z}}_{s} = \mathrm{HRC}((\mathbf{0}, \mathcal{\tilde{Z}}_{s})). +$$ + +--- + +### 2.3 LayerNorm, Dropout, Activation, and Concatenation + +Fundamental operations such as LayerNorm, Dropout, Activation, and Concatenation are crucial in Transformers. Since these tasks are also performed in the same hyperbolic reference system and do not require modifying the time-like dimension, we use HRC: + +$$ +\begin{aligned} + \mathrm{HypLayerNorm}(\mathbf{X}) &= \mathrm{HRC}(\mathbf{X}, f_{LayerNorm}), \\ + \mathrm{HypBatchNorm}(\mathbf{X}) &= \mathrm{HRC}(\mathbf{X}, f_{BatchNorm}), \\ + \mathrm{HypDropout}(\mathbf{X}) &= \mathrm{HRC}(\mathbf{X}, f_{Dropout}), \\ + \mathrm{HypActivation}(\mathbf{X}) &= \mathrm{HRC}(\mathbf{X}, f_{\sigma}), \\ + \mathrm{HypConcatenation}(\mathbf{X}) &= \mathrm{HRC}((\mathbf{X}_i, \mathbf{X}_j), f_{concatenation}), \\ +\end{aligned} +$$ + +where $f_{Dropout}$, $f_{LayerNorm}$, $f_{BatchNorm}$, $f_{\sigma}$, and $f_{concatenation}$ correspond to their Euclidean counterparts (Dropout, LayerNorm, BatchNorm, Activation, and Concatenation, respectively), repurposed to operate in hyperbolic space. + +By defining these operations through HRC (and HTC for linear transformations), we maintain the geometric constraints crucial for hyperbolic representations, allowing curvature-aware processing at each stage of the network. + +--- + +### 2.4Framework + +Framework of Hypformer. Input data (text, images, graphs) are projected onto the Lorentz model, then transformed via HTC. The result passes through the hyperbolic linear attention block with positional encoding, followed by a Feedforward layer (built by HTC) and LayerNorm (built by HRC). This serves as an encoder which can optionally incorporate a GNN. +For classification tasks in this study, the decoder is the fully connected layer. Dropout, activation, and residual connections are omitted for brevity. + +![framework](./assets/framework2024.jpg) + + + +## 3.Experiments +### 3.1 Experiments on Large-scale Graphs + +We first evaluate Hypformer on diverse large-scale graphs for node classification, with node counts ranging from millions to billions, including ogbn-arxiv, ogbn-protein, and Papers100M. +![title|scale=0.5](./assets/exp1_large_scale.png) + + Hypformer consistently outperforms other models across various large-scale graph datasets, demonstrating substantial improvements. It is worth noting that models, such as GraphFormer, GraphTrans, and GraphGPS, HAN, HNN++ and F-HNN, have difficulty operating effectively on large-scale graph data. +In addition, our method significantly outperforms the recent approaches such as, SGFormer and NodeFormer across all tested scenarios, highlighting its superior effectiveness. Importantly, Hypformer exhibits robust scalability, maintaining its performance advantage even on the largest dataset, ogbn-papers100M, where previous Transformer-based models have encountered limitations. + +### 3.2 Experiments on Medium/Small-scale Graphs +To complement our large-scale evaluations, we assessed Hypformer on small- and medium-scale graph datasets. This additional testing allows for a more comprehensive comparison against current state-of-the-art models, including GNNs, graph transformers, and hyperbolic approaches that may not scale effectively to larger datasets. By expanding our evaluation scope, we aim to isolate Hypformer's effectiveness in graph learning from its scalability advantages. + +![title|scale=0.5](./assets/exp2_medium_scale.png) + +Our findings suggest that the proposed method significantly surpasses both standard GNNs and hyperbolic GNN models by a substantial margin. +Importantly, the method exhibits effectiveness not only in scenarios with hyperbolic datasets (like Disease, Airport) but also in situations with non-hyperbolic dataset (like Cora, CiteSeer and PubMed). + +### 3.3 Comparisons on Text and Vision Datasets +Additionally, we apply our model to semi-supervised image and text classification tasks on the Mini-ImageNet and 20News-Groups datasets. We also construct a graph using k-NN (based on input node features) to utilize graph model. These experiments are conducted closely in Nodeformer. + +![title|scale=0.8](./assets/exp3_image_text.png) + +Hypformer outperforms in seven out of eight cases. In contrast, the performance of competing baselines models varying significantly with different k values, while our method demonstrates greater stability. + +### 3.4 Further Analysis + +**Scalability** +We conducted additional tests on the model’s scalability regarding the number of nodes in a single batch. The Amazon2M dataset was used, and we randomly selected a subset of nodes, with the number of nodes varying from 10K to 200K. We made a comparison between softmax attention defined by Equation (3) and linear attention, keeping all other parameters the same. As depicted in Figure 5, the memory usage of the proposed method exhibits a linear increase with the size of the graph. When the node count exceeds 40K, the softmax attention experiences an out-of-memory (OOM) issue. However, the proposed method continues to function effectively, resulting in a 10X reduction in GPU cost. + +![title|scale=0.5](./assets/gpucost.png) + + +## 4.Conclusion + +In this work, we introduce a efficient hyperbolic Transformer, Hypformer. This method operates directly and fully on hyperbolic representations and employs a linear attention mechanism, enabling it to be both scalable and effective. +Furthermore, this study introduces two basic blocks, HTC and HRC, which are foundational in constructing hyperbolic models. Nonetheless, the research presented is an initial exploration and numerous challenges warrant further investigation. These include the initial determination of a curvature that better reflects the data geometry, the setting of curvature at different levels for Hypformer, and the design of effective decoders for different downstream tasks. We plan to address these issues in our future work. From 09059278d5156552cb160dd689898a8308c75af3 Mon Sep 17 00:00:00 2001 From: marlin-codes Date: Sun, 22 Dec 2024 15:26:29 -0500 Subject: [PATCH 2/7] add page value --- config/publications.ts | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/config/publications.ts b/config/publications.ts index 24892e7..13dea62 100644 --- a/config/publications.ts +++ b/config/publications.ts @@ -94,7 +94,7 @@ export const publications = [ authors: "Menglin Yang, Harshit Verma, Delvin Ce Zhang, Jiahong Liu, Irwin King, Rex Ying", venue: "KDD 2024", - page: null, + page: "hypformer", paper: "https://arxiv.org/abs/2407.01290", code: "https://github.com/Graph-and-Geometric-Learning/hyperbolic-transformer/", tags: [Tag.MultiModalFoundationModel], From 83f234bece1e3c9ab0f0a80a62010e4e3cd77157 Mon Sep 17 00:00:00 2001 From: Boltzmachine Date: Sun, 22 Dec 2024 13:59:27 -0500 Subject: [PATCH 3/7] merge badges into utils --- app/projects/hybrid/page.mdx | 3 +-- app/projects/mage/page.mdx | 3 +-- components/utils.tsx | 1 + 3 files changed, 3 insertions(+), 4 deletions(-) diff --git a/app/projects/hybrid/page.mdx b/app/projects/hybrid/page.mdx index 22f6a33..c5704a3 100644 --- a/app/projects/hybrid/page.mdx +++ b/app/projects/hybrid/page.mdx @@ -1,5 +1,4 @@ -import { Authors } from '@/components/utils' -import { Badges } from '@/components/badges' +import { Authors, Badges } from '@/components/utils' # Learning High-Order Relationships of Brain Regions diff --git a/app/projects/mage/page.mdx b/app/projects/mage/page.mdx index fc93abb..077d06a 100644 --- a/app/projects/mage/page.mdx +++ b/app/projects/mage/page.mdx @@ -1,5 +1,4 @@ -import { Authors } from '@/components/utils' -import { Badges } from '@/components/badges' +import { Authors, Badges } from '@/components/utils' # Explaining Graph Neural Networks via Structure-aware Interaction Index diff --git a/components/utils.tsx b/components/utils.tsx index a0ea306..d593dcf 100644 --- a/components/utils.tsx +++ b/components/utils.tsx @@ -1,3 +1,4 @@ +export { Badges } from './badges'; export interface Author { name: string; affiliation: string; From c97e3b56e05488da94825234ec9b7676cf87b97d Mon Sep 17 00:00:00 2001 From: Boltzmachine <56542320+Boltzmachine@users.noreply.github.com> Date: Sun, 22 Dec 2024 14:02:40 -0500 Subject: [PATCH 4/7] Update README.md --- README.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/README.md b/README.md index 65e99d3..3798a74 100644 --- a/README.md +++ b/README.md @@ -49,7 +49,7 @@ To set up a project page, we support [MDX](https://mdxjs.com/), which is a super 3. Update the `page` key of your publication in `config/publications.ts` with the name of your folder. -Find an example in `app/projects/mage`. +Find an example in `app/projects/hybrid`. ### GGLab-flavored Markdown We extend the markdown syntax to better styling our website. Here are some special usages, From 97333b41f5a1040d4cee65e98c2815d6d4c217eb Mon Sep 17 00:00:00 2001 From: marlin-codes Date: Sun, 22 Dec 2024 15:55:23 -0500 Subject: [PATCH 5/7] add page value and merge import --- app/projects/hypformer/page.mdx | 3 +-- 1 file changed, 1 insertion(+), 2 deletions(-) diff --git a/app/projects/hypformer/page.mdx b/app/projects/hypformer/page.mdx index a6512e7..c5c5c54 100644 --- a/app/projects/hypformer/page.mdx +++ b/app/projects/hypformer/page.mdx @@ -1,5 +1,4 @@ -import { Authors } from '@/components/utils' -import { Badges } from '@/components/badges' +import { Authors, Badges } from '@/components/utils' # Hypformer: Exploring Efficient Transformer Fully in Hyperbolic Space From 77c6c4fef85a3110ac625ffbdf27a8884c3fdc0a Mon Sep 17 00:00:00 2001 From: marlin-codes Date: Sun, 22 Dec 2024 16:04:01 -0500 Subject: [PATCH 6/7] add figure title --- app/projects/hypformer/page.mdx | 10 +++++----- 1 file changed, 5 insertions(+), 5 deletions(-) diff --git a/app/projects/hypformer/page.mdx b/app/projects/hypformer/page.mdx index c5c5c54..3608b00 100644 --- a/app/projects/hypformer/page.mdx +++ b/app/projects/hypformer/page.mdx @@ -119,7 +119,7 @@ By defining these operations through HRC (and HTC for linear transformations), w Framework of Hypformer. Input data (text, images, graphs) are projected onto the Lorentz model, then transformed via HTC. The result passes through the hyperbolic linear attention block with positional encoding, followed by a Feedforward layer (built by HTC) and LayerNorm (built by HRC). This serves as an encoder which can optionally incorporate a GNN. For classification tasks in this study, the decoder is the fully connected layer. Dropout, activation, and residual connections are omitted for brevity. -![framework](./assets/framework2024.jpg) +![Framework](./assets/framework2024.jpg) @@ -127,7 +127,7 @@ For classification tasks in this study, the decoder is the fully connected layer ### 3.1 Experiments on Large-scale Graphs We first evaluate Hypformer on diverse large-scale graphs for node classification, with node counts ranging from millions to billions, including ogbn-arxiv, ogbn-protein, and Papers100M. -![title|scale=0.5](./assets/exp1_large_scale.png) +![Experiments on Large-scale Graphs|scale=0.5](./assets/exp1_large_scale.png) Hypformer consistently outperforms other models across various large-scale graph datasets, demonstrating substantial improvements. It is worth noting that models, such as GraphFormer, GraphTrans, and GraphGPS, HAN, HNN++ and F-HNN, have difficulty operating effectively on large-scale graph data. In addition, our method significantly outperforms the recent approaches such as, SGFormer and NodeFormer across all tested scenarios, highlighting its superior effectiveness. Importantly, Hypformer exhibits robust scalability, maintaining its performance advantage even on the largest dataset, ogbn-papers100M, where previous Transformer-based models have encountered limitations. @@ -135,7 +135,7 @@ In addition, our method significantly outperforms the recent approaches such as, ### 3.2 Experiments on Medium/Small-scale Graphs To complement our large-scale evaluations, we assessed Hypformer on small- and medium-scale graph datasets. This additional testing allows for a more comprehensive comparison against current state-of-the-art models, including GNNs, graph transformers, and hyperbolic approaches that may not scale effectively to larger datasets. By expanding our evaluation scope, we aim to isolate Hypformer's effectiveness in graph learning from its scalability advantages. -![title|scale=0.5](./assets/exp2_medium_scale.png) +![Experiments on Medium/Small-scale Graphs|scale=0.5](./assets/exp2_medium_scale.png) Our findings suggest that the proposed method significantly surpasses both standard GNNs and hyperbolic GNN models by a substantial margin. Importantly, the method exhibits effectiveness not only in scenarios with hyperbolic datasets (like Disease, Airport) but also in situations with non-hyperbolic dataset (like Cora, CiteSeer and PubMed). @@ -143,7 +143,7 @@ Importantly, the method exhibits effectiveness not only in scenarios with hyperb ### 3.3 Comparisons on Text and Vision Datasets Additionally, we apply our model to semi-supervised image and text classification tasks on the Mini-ImageNet and 20News-Groups datasets. We also construct a graph using k-NN (based on input node features) to utilize graph model. These experiments are conducted closely in Nodeformer. -![title|scale=0.8](./assets/exp3_image_text.png) +![Comparisons on Text and Vision Datasets|scale=0.8](./assets/exp3_image_text.png) Hypformer outperforms in seven out of eight cases. In contrast, the performance of competing baselines models varying significantly with different k values, while our method demonstrates greater stability. @@ -152,7 +152,7 @@ Hypformer outperforms in seven out of eight cases. In contrast, the performance **Scalability** We conducted additional tests on the model’s scalability regarding the number of nodes in a single batch. The Amazon2M dataset was used, and we randomly selected a subset of nodes, with the number of nodes varying from 10K to 200K. We made a comparison between softmax attention defined by Equation (3) and linear attention, keeping all other parameters the same. As depicted in Figure 5, the memory usage of the proposed method exhibits a linear increase with the size of the graph. When the node count exceeds 40K, the softmax attention experiences an out-of-memory (OOM) issue. However, the proposed method continues to function effectively, resulting in a 10X reduction in GPU cost. -![title|scale=0.5](./assets/gpucost.png) +![Scalability|scale=0.5](./assets/gpucost.png) ## 4.Conclusion From b3331bc0dc4c4008db1850d23524a600a1e74d5f Mon Sep 17 00:00:00 2001 From: marlin-codes Date: Sun, 22 Dec 2024 16:11:08 -0500 Subject: [PATCH 7/7] update hypformer --- app/projects/hypformer/page.mdx | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/app/projects/hypformer/page.mdx b/app/projects/hypformer/page.mdx index 3608b00..fff0664 100644 --- a/app/projects/hypformer/page.mdx +++ b/app/projects/hypformer/page.mdx @@ -114,7 +114,7 @@ By defining these operations through HRC (and HTC for linear transformations), w --- -### 2.4Framework +### 2.4 Framework Framework of Hypformer. Input data (text, images, graphs) are projected onto the Lorentz model, then transformed via HTC. The result passes through the hyperbolic linear attention block with positional encoding, followed by a Feedforward layer (built by HTC) and LayerNorm (built by HRC). This serves as an encoder which can optionally incorporate a GNN. For classification tasks in this study, the decoder is the fully connected layer. Dropout, activation, and residual connections are omitted for brevity. @@ -123,7 +123,7 @@ For classification tasks in this study, the decoder is the fully connected layer -## 3.Experiments +## 3. Experiments ### 3.1 Experiments on Large-scale Graphs We first evaluate Hypformer on diverse large-scale graphs for node classification, with node counts ranging from millions to billions, including ogbn-arxiv, ogbn-protein, and Papers100M. @@ -155,7 +155,7 @@ We conducted additional tests on the model’s scalability regarding the number ![Scalability|scale=0.5](./assets/gpucost.png) -## 4.Conclusion +## 4. Conclusion In this work, we introduce a efficient hyperbolic Transformer, Hypformer. This method operates directly and fully on hyperbolic representations and employs a linear attention mechanism, enabling it to be both scalable and effective. Furthermore, this study introduces two basic blocks, HTC and HRC, which are foundational in constructing hyperbolic models. Nonetheless, the research presented is an initial exploration and numerous challenges warrant further investigation. These include the initial determination of a curvature that better reflects the data geometry, the setting of curvature at different levels for Hypformer, and the design of effective decoders for different downstream tasks. We plan to address these issues in our future work.