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arXiv · 2001.08006

Estimating the reach of a manifold via its convexity defect function

Clément Berenfeld·John Harvey·Marc Hoffmann·Krishnan Shankar
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Abstract

The reach of a submanifold is a crucial regularity parameter for manifold learning and geometric inference from point clouds. This paper relates the reach of a submanifold to its convexity defect function. Using the stability properties of convexity defect functions, along with some new bounds and the recent submanifold estimator of Aamari and Levrard [Ann. Statist. 47 177-204 (2019)], an estimator for the reach is given. A uniform expected loss bound over a C^k model is found. Lower bounds for the minimax rate for estimating the reach over these models are also provided. The estimator almost achieves these rates in the C^3 and C^4 cases, with a gap given by a logarithmic factor.

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Clément Berenfeld, John Harvey, Marc Hoffmann, Krishnan Shankar. 2021-03-26. Estimating the reach of a manifold via its convexity defect function. https://doi.org/10.1007/s00454-021-00290-8

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