Ricci Curvature and the Manifold Learning Problem

Consider a sample of $n$ points taken i.i.d from a submanifold $\Sigma$ of Euclidean space. We show that there is a way to estimate the Ricci curvature of $\Sigma$ with respect to the induced metric from the sample. Our method is grounded in the notions of Carr\'e du Champ for diffusion semi-groups, the theory of Empirical processes and local Principal Component Analysis.