Ricci curvature and the manifold learning problem

Abstract

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