University of Virginia, Statistics Colloquium

Abstract

The problem of identifying geometric structure in heterogeneous, high-dimensional data is a cornerstone of Representation Learning. In this talk, we study this problem from the perspective of Discrete Geometry. We start by reviewing discrete notions of curvature with a focus on Ricci curvature. Then we discuss how curvature characterizations of graphs can be used to improve the efficiency of Graph Neural Networks. Specifically, we propose curvature-based rewiring and encoding approaches and study their impact on the Graph Neural Network’s downstream performance through theoretical and computational analysis. We further discuss applications of discrete Ricci curvature in Manifold Learning, where discrete-to-continuum consistency results allow for characterizing the geometry of a suitable embedding space both locally and in the sense of global curvature bounds. Based on joint work with Lukas Fesser and Nicolás García Trillos.