Ricci Curvature and the Manifold Learning Problem
This addresses the manifold learning problem for researchers in differential geometry and machine learning, but appears incremental as it applies existing mathematical tools to a specific estimation task.
The paper tackles the problem of estimating the Ricci curvature of a submanifold from a sample of points, showing that it is possible to do so using a method based on Carré du Champ, empirical processes, and local PCA.
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.