A point-cloud curvature estimator with sample-size bounds is proposed, but the bounds rely on incorrect probability estimates and the surface estimator is unproven.
Ricci Curvature and the Manifold Learning Problem
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abstract
Consider a sample of $n$ points taken i.i.d from a submanifold $\Sigma$ of Euclidean space. We show that there is a way to estimate the Ricci curvature of $\Sigma$ with respect to the induced metric from the sample. Our method is grounded in the notions of Carr\'e du Champ for diffusion semi-groups, the theory of Empirical processes and local Principal Component Analysis.
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How many points in a point cloud is sufficient for accurate estimation of the curvature
A point-cloud curvature estimator with sample-size bounds is proposed, but the bounds rely on incorrect probability estimates and the surface estimator is unproven.