Mathematics for Machine Learning - Code Templates

1. Linear Algebra

Identifying Special Matrices A function that will test if a 4x4 matrix is singular

1. convert a matrix to echelon form,
2. Then test if this fails by leaving zeros that can't be removed on the leading diagonal

Gram-Schmidt process A function to perform the Gram-schmidt procedure, which takes a list of vectors and forms an orthonormal basis from this set

Reflecting Bear A function that will produce a transformation matrix for reflecting vectors in an arbitrarily angled mirror

Page Rank In this notebook, you'll build on your knowledge of eigenvectors and eigenvalues by exploring the PageRank algorithm. The notebook is in two parts:

• the first is a worksheet to get you up to speed with how the algorithm works - here we will look at a micro-internet with fewer than 10 websites and see what it does and what can go wrong.
• the second is an assessment which will test your application of eigentheory to this problem by writing code and calculating the page rank of a large network representing a sub-section of the internet.

2. Multivariate Calculus

The Sandpit Builds an understanding of what the Jacobian is, and how this can be used to find the minimum of a function

Backpropagation Trains a neural network to draw a curve by implementing backpropagation by the chain rule to calculate Jacobians of the cost function

Gradient descent in a sandpit Examines strategies for steepest descent and how effective they are at finding the supervisor's mobile phone

Fitting the distribution of heights data Implements the steepest descent algorithm on the least squares fitting problem for modelling the distribution of heights in a population with a Gaussian

3. PCA

Mean/covariance of a dataset and effect of a linear transformation Computes means and (co)variances of data sets and the effect of linear transformations of the data sets on the mean and covariance

Inner products and angles Computes distances and angles between images and compare images from the MNIST dataset

Orthogonal projections

• Implements orthogonal projection which projects data onto lower-dimensional subspaces
• Then applies this to the problem of "eigenfaces" which will help you understand the application of orthogonal projection in the real world

Principal Components Analysis (PCA)

1. Implement the main steps of PCA,
2. Apply it to an image data set.
3. Implement PCA for high-dimensional dataset and perform a brief analysis of computation time.