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.