Machine Learning is at the core of contemporary AI research and applications. This module develops a foundation for the mathematical theory underpinning key ML methods which are necessary for their understanding and analysis. The module covers six units: Linear Algebra, Geometry, Calculus, Optimisation, Probability and Statistics. All to establish a comprehensive setting to strenghen the student’s understanding of widely used ML models and methods.

Learning Outcomes

On the successful completion of the module, you will be able to:

• Recognize the role of mathematics in the construction of ML models and their assumptions;
• Understand the key mathematical concepts behind ML methods;
• Identify concepts in ML as particular instances of general mathematical theory;
• Formulate subtasks of a learning problem as mathematical objective

Content

This module will cover the following topics:

• Linear algebra: Linear systems, matrices, decompositions, vector spaces and linear independence.
• Geometry: Inner products, norms, projections, distances and angles.
• Calculus: Limits, derivatives (scalar, vector and partial), chain rule, backpropagation, Taylor series, integration and the fundamental theorem of Calculus.
• Optimisation: Convex optimisation, constrained optimisation and Lagrange multipliers, stochastic gradient descent.
• Probability: Probability spaces, discrete and continuous random variables, sum/product/Bayes rules, Change-of-Variable theorem, and the Gaussian distribution.
• Statistics: Learning form data, empirical risk minimisation, bias-variance trade off, the statistical model, maximum likelihood, Bayesian inference, posterior computation.

Reading List

[D20] Marc Peter Deisenroth, A. Aldo Faisal, and Cheng Soon Ong, Mathematics for Machine Learning. Cambridge University Press, 2020.

[F24] Francis Bach, Learning Theory from First Principles. MIT Press, 2024.

[M22] Kevin P. Murphy, Probabilistic Machine Learning: An Introduction. MIT Press, 2022.

[B06] Christopher M. Bishop, Pattern Recognition and Machine Learning. Springer, 2006.