Ricci Curvature and the Manifold Learning Problem

Antonio G. Ache; Micah W. Warren

Ricci Curvature and the Manifold Learning Problem

Differential Geometry 2018-03-23 v5 Machine Learning Metric Geometry Machine Learning

Authors: Antonio G. Ache , Micah W. Warren

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.

Keywords

ricci curvature and riemannian geometry riemannian geometry differential geometry

Cite

@article{arxiv.1410.3351,
  title  = {Ricci Curvature and the Manifold Learning Problem},
  author = {Antonio G. Ache and Micah W. Warren},
  journal= {arXiv preprint arXiv:1410.3351},
  year   = {2018}
}

Comments

47 pages

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