Title: Coarse and Approximate Ricci Curvatures for points clouds
Speaker: Micah Warren (University of Oregon)
Abstract: We consider the problem of recovering the Ricci curvature on a submanifold of Euclidean space from a sample of points and their extrinsic distances. The idea was to approximate and iterate the Laplacian, using the Carre du Champ of Bakry and Emery together with ideas of Belkin and Niyogi. This led us to the study of Coarse Ricci curvature, which is a function on pairs of points. Following this, a notion of Coarse Ricci curvature can be defined on a large class of spaces. Unfortunately, this turns out to be something that does not work well when considering extrinsic distance functions. Instead, we consider approximate Ricci curvature, which is an approximation of the classical Ricci tensor based on the same approach, applied to the approximate tangent spaces obtained using PCA. We will discuss the pros and cons of both approaches. This is joint work with Antonio Ache.