Curvature Estimation for Point Clouds, Graphs, and Finite Metric Spaces

Real-world data sets frequently turn out to have low-dimensional manifold structure that can be studied using ideas from differential geometry. In particular, the curvature of a manifold is an important invariant that characterizes the extent to which the manifold deviates from being flat. I’ll first discuss our recently-introduced estimator for the scalar curvature of a data set presented as a finite metric space (e.g., a distance matrix, a point cloud, or a graph with the shortest-path metric). Our estimator depends only on the metric structure of the data (not on an embedding in Euclidean space), and it converges to the ground-truth scalar curvature as the number of points increases. In the second part of the talk, we’ll review Ollivier-Ricci curvature—an edge-based definition of discrete curvature for graphs—and show how one can use it to define scalar curvature at the vertices of a graph. Both projects are joint work with Andrew Blumberg.