Ricci Curvature and the Manifold Learning Problem

10/13/2014

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by   Antonio G. Ache, et al.

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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.