Mathematics of Data

This lecture gives an introduction to how tools from algebraic geometry and topology can be used to tackle problems in data analysis and statistics.

The first half of the course will focus on what is known as Algebraic Statistics: see e.g., Pistone et al. (2001), Pachter and Sturmfels (2005), and Sullivant (2017). To set the scene, the first two lectures will illustrate some classical statistical theory such as linear models and Bayesian vs frequentist approaches to model selection on small-scale examples. We continue by treating topics such as exponential families and (Gaussian) graphical models and show how their properties can be naturally described using the language of (toric) ideals and varieties. We show that questions of equivalence for conditional independence models can be solved using polytopes.

The second half of the course will start with an intuitive introduction to topology, including homotopy equivalent spaces, homology groups, and homotopy groups. We will then move to the realm of data analysis: given only a dataset, i.e. a finite sampling from a space, what can we say about the space's shape (which may be reflective of patterns within the data)? To study the shape we consider different ways of building geometric objects (simplicial complexes) on point clouds and studying their properties. The main technique we cover is persistent homology; we describe its theoretical underpinnings and discuss examples of how it has been used on real-life data. Lastly, we explain how another popular method called 'mapper' works and show some of its applications.

Date and time info
Wednesday 13:30 - 14:30

Keywords
Algebraic Statistics, Topological Data Analysis

Prerequisites
Undergraduate degree in Mathematics, linear algebra, and probability theory.Undergraduate degree in Mathematics, linear algebra, and probability theory.

Audience
MSc students, PhD students, Postdocs

Language
English

Remarks and notes
Almost all lectures will be self-contained.