Ricci Curvature and the Manifold Learning Problem

Antonio G. Ache; Micah W. Warren

Ricci Curvature and the Manifold Learning Problem

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arxiv 1410.3351 v5 pith:DIZ325F3 submitted 2014-10-13 math.DG cs.LGmath.MGstat.ML

Antonio G. Ache , Micah W. Warren This is my paper · ORCID

classification math.DGcs.LGmath.MGstat.ML

keywords curvaturericcisamplesigmaanalysiscarrchampcomponent

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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
math.DG 2025-06 reject novelty 4.0 of 10

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.

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