Introductory description

N/A

Module web page

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

• Preliminaries on Random Variables (limit theorems, classical inequalities, Gaussian models, Monte Carlo)
• Basic Information theory (entropy; Kull-Back Leibler information divergence)
• Concentrations of Sums of Independent Random Variables
• Random Vectors in High Dimensions
• Random Matrices
• Concentration with Dependency structures
• Deviations of Random Matrices and Geometric Consequences
• Graphical models and deep learning

Learning outcomes

By the end of the module, students should be able to:

• Understand the concentration of measure problem in high dimensions
• Be able to distinguish three basic concentration inequalities
• Be able to distinguish between concentration for independent families as well as for various dependency structures
• Understanding of basic concentrations of the norm
• Familiar with random matrices (main properties)
• Be familiar with some application of graphical models
• Be able to understand basic variational problems

Indicative reading list

We won't follow a particular book and will provide lecture notes. The course is based on the following three books where the majority is taken from [1]:
[1] Roman Vershynin, High-Dimensional Probability: An Introduction with Applications in Data Science, Cambridge Series in Statistical and Probabilistic Mathematics, (2018).
[2] Kevin P. Murphy, Machine Learning - A Probabilistic Perspective, MIT Press (2012).
[3] Simon Rogers and Mark Girolami, A first course in Machine Learning, CRC Press (2017).
[4] Alex Kulesza and Ben Taskar, Determinantal point processes for machine learning, Lecture Notes (2013).

Subject specific skills

• Understanding that the concentration of measure problem
  requires analytical expertise as well as some basic probability
• Be able to distinguish three basic concentration inequalities
• Be able to distinguish between concentration for independent families as well as for various dependency structures
• Understanding of basic concentrations of the norm
• Familiar with random matrices (main properties)
• Be familiar with some application of graphical models

Transferable skills

Students will acquire key reasoning and problem solving skills which will empower them to address new problems with confidence.