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With recent computational advances, our ability to create novel machine learning models is far outpacing our capabilities for understanding them. However, focusing on the foundational principles of geometry and topology promises additional insights into their capabilities. In this talk, geared towards a general audience of machine learning enthusiasts, I will highlight the role of geometry and topology in machine learning research. I will give particular attention to (i) recent advances in graph learning, where the creation of novel curvature-based concepts is helping uncover the limitations of message passing, and (ii) unsupervised representation learning, where geometrical--topological information can be used to uncover singularities in the data.

00:02:03 Topology
00:04:56 Persistent Homology
00:08:58 Persistent Homology in Machine Learning
00:11:43 Finding Singularities with Persistent Homology
00:22:50 Introduction to Curvature
00:32:38 Using Curvature for Graph Generative Model Evaluation
00:40:18 Discussion

Papers discussed:

https://proceedings.mlr.press/v202/vo...
https://arxiv.org/abs/2301.12906

(This was a talk at the 2023 Departmental Workshop “Computer Science Meets Mathematics”, University of Barcelona. Thanks to Carles Casacuberta and Rubén Ballester for the invitation)