Title: Ricci Curvature, Ricci Flow, and the Geometry of Learning

Abstract: Geometric structure in data plays a crucial role in machine learning. In this talk, we study this observation through the lens of Ricci curvature and its associated Ricci flow. We start by reviewing a discrete notion of Ricci curvature introduced by Ollivier and the geometric flow that it induces. We further discuss the relationship between discrete Ricci curvature and its continuous counterpart via discrete-to-continuum consistency results, which imply that discrete Ricci curvature can provably characterize the geometry of a data manifold based on a finite sample. This provides a theoretical foundation for several applications of discrete Ricci curvature in machine learning, two of which we discuss in the remainder of this talk. First, we analyze learned feature representations in deep neural networks and show that they transform during training in ways that closely resemble a Ricci flow. Our analysis reveals that nonlinear activations shape class separability and suggests geometry-informed training principles such as early stopping and depth selection. Second, we turn to deep learning on graphs, where we address representational limitations of state of the art graph neural networks through curvature-based data augmentations. We show that augmenting input graphs with geometric information provably increases the representational power of such models and yields performance gains in practise.