Continuum Limits of Ollivier's Ricci Curvature on data clouds: pointwise consistency and global lower bounds

TL;DR

This paper establishes that the discrete Ricci curvature of random geometric graphs converges to the manifold's curvature, enabling curvature estimation and analysis of heat kernel properties in data-driven manifold learning.

Contribution

It provides the first non-asymptotic pointwise consistency results for Ollivier's Ricci curvature on data clouds and links global curvature bounds to manifold properties.

Findings

Discrete Ricci curvature converges to manifold curvature with high probability.

Global lower bounds on manifold Ricci curvature are inherited by the graph.

Results facilitate intrinsic curvature estimation from data samples.

Abstract

Let $M$ denote a low-dimensional manifold embedded in Euclidean space and let ${X}= \{ x_1, \dots, x_n \}$ be a collection of points uniformly sampled from it. We study the relationship between the curvature of a random geometric graph built from ${X}$ and the curvature of the manifold $M$ via continuum limits of Ollivier's discrete Ricci curvature. We prove pointwise, non-asymptotic consistency results and also show that if $M$ has Ricci curvature bounded from below by a positive constant, then the random geometric graph will inherit this global structural property with high probability. We discuss applications of the global discrete curvature bounds to contraction properties of heat kernels on graphs, as well as implications for manifold learning from data clouds. In particular, we show that our consistency results allow for estimating the intrinsic curvature of a manifold by first estimating concrete extrinsic quantities.