HS2026: 63132 Geometry and Topology in Machine Learning

This course will provide an overview of geometric and topological methods in machine learning, with a particular focus on deep learning, that is, learning methods based on neural networks. The goal of the course is to introduce some of the most well-known geometric and topological perspectives on data, such as graph theory, spectral methods, or persistent homology; and to explore their use within deep learning systems. The introduction of geometry and topology in deep learning has typically taken one of two complemetary approaches: either observational or interventional. The former seeks to leverage geometry and topology to better understand how deep learning systems behave, whereas the latter aims to incorporate geometry and topology within neural networks, to allow them to learn from geometric and topological domains (graphs, simplicial complexes, cell complexes, hypergraphs, etc.) or to guide their learning injecting geometric or topological inductive biases. We will cover plenty of examples of both cases throughout the course. This course will be taught in English, with a theoretical and a practical component. In the theory sessions, different geometric and topological tools for data analysis will be introduced, and their use in deep learning will be explored through examples drawn from current research. In the practical sessions, we will explore, through concrete examples again, how to implement these geometric and topological tools using Python. Assessment will take the form of a collaborative project developed during the second half of the course, and an individual oral exam at the end of it.