Coarse Ricci Curvature as a Function on $$\varvec{M\times M}$$

Abstract

We use 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. This function can be used to recover the Ricci tensor on smooth Riemannian manifolds by the formula

Ricci curvature and the manifold learning problem

“…Based on estimates in [2] and calculations similar to the proof of Theorem C, we can prove Theorem D (Non-uniform case). Consider the metric space (Σ, · ) where Σ d ⊂ R N is a smooth closed embedded submanifold.…”

“…In this section we recall (cf. [2]) how Ricci curvature on general metric spaces can be constructed with an operator, in particular the infinitesimal generator of a diffusion semi-group. When the space is a metric measure space, we use a family of operators which are intended to approximate a Laplace operator on the space at scale t. As this definition holds on metric measure spaces constructed from sampling points from a manifold, we can define an empirical or sample version of the Ricci curvature, given a bandwidth parameter t. As mentioned above, this last construction will have an application to the manifold learning problem, namely it will serve to predict the Ricci curvature of an embedded submanifold of R N if one only has a point cloud on the manifold and the distribution of the sample has a smooth positive density.…”

“…It is the goal of this paper, together with [2] and [1], to demonstrate that the above discussed approximation of the rough Laplacian can be continued to approximate Ricci curvature as well. In fact, our approximation method is based on writing sample counterparts of the Ricci curvature.…”

“…More generally, we will show that it is possible to show that one can define sample counterparts of more general objects, for example of the notions of Carré du Champ and iterated Carré du Champ associated to a diffusion semi-group. Our idea for estimating the Carré du Champ (and ultimately the Ricci curvature) from a sample is closely related to the results in [2,1]. For example, in [1] we define a family of coarse Ricci curvatures which depend on a scale parameter t, and show that when taken on a smooth embedded submanifold on Euclidean space, these recover the Ricci curvature as t → 0.…”

“…We will show that as long as we sample points adequately from the submanifold Σ, it is possible to choose a scale t n depending only on the size of the data set (equal to n) to obtain almost sure convergence to the actual Ricci curvature of the submanifold at a given point. We will summarize the results in [2,1] relevant to the present article in Section 3 (for example Theorem 3.1 and Proposition 3.2).…”