Coarse Ricci curvature and the manifold learning problem

We consider the framework used by Bakry and Emery in their work on logarithmic Sobolev inequalities to define a notion of coarse Ricci curvature on smooth metric measure spaces alternative to the notion proposed by Y. Ollivier. We discuss applications of our construction to the manifold learning problem, specifically to the statistical problem of estimating the Ricci curvature of a submanifold of Euclidean space from a point cloud assuming that the sample has smooth density with definite lower bounds. More generally we are able to approximate a 1-parameter family of Ricci curvatures that include the Bakry-Emery Ricci curvature.