Geometric methods in data science:
Manifold learning and Diffusion maps

Jeffrey Giansiracusa - email: jeffrey.giansiracusa@durham.ac.uk

Description

In this project we will learn about some classic methods that allow us to study the geometric aspects of data sets. This project will bring together various interesting pure mathematical ideas (diffusion processes and Markov chains, some analysis and spectral theory, geometry of manifolds) with ideas and challenges from data science.  

A data set is often a set of points in a high dimensional Euclidean space.  Visualising points in say R^20 is impossible for human brains, but if we could find a way to project down to 2 or 3 dimensions then we could visualise the data easily.  Dimensional reduction is about finding a good way of doing this.  In 2006 Coifman and Lafon introduced diffusion maps for this problem.  It is a nonlinear dimension reduction algorithm that is computationally efficient.  

In this project we'll learn about how diffusion maps work, and then look at variations, applications, and other methods in the area of manifold learning, such as ISOMAP and Hessian eigenmaps.  The general question here is: 'What can we learn about a manifold if we only have points sampled from it?'  For those who are interested in coding, we might possibly get to do some work with Python and try these methods on some data sets.

Prerequisites

No required prerequisites, but if you enjoyed any of these modules below, then this project could build on what you learned there nicely.

• MATH2581 Data science and statistical computing

• MATH2707 Markov Chains

• MATH3431 Machine Learning

Resources

https://en.wikipedia.org/wiki/Diffusion_map

Ronald R. Coifman and Stéphane Lafon, Diffusion maps, Applied and Computational Harmonic Analysis, Volume 21, Issue 1, 2006, Pages 5-30
https://doi.org/10.1016/j.acha.2006.04.006

Marina Meilă and Hanyu Zhang, Manifold learning: what, how, and why, https://arxiv.org/abs/2311.03757

David L. Donoho and Carrie Grimes, Hessian eigenmaps: Locally linear embedding techniques for high-dimensional data, https://www.pnas.org/doi/10.1073/pnas.1031596100