Estimating the reach of a manifold via its convexity defect function

Berenfeld, Clément, Harvey, John ORCID: https://orcid.org/0000-0001-9211-0060, Hoffmann, Marc and Shankar, Krishnan 2022. Estimating the reach of a manifold via its convexity defect function. Discrete and Computational Geometry 67 , 403–438. 10.1007/s00454-021-00290-8

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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(1), 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.

Item Type:	Article
Date Type:	Publication
Status:	Published
Schools:	Mathematics
Additional Information:	This article is licensed under a Creative Commons Attribution 4.0 International License
Publisher:	Springer
ISSN:	0179-5376
Date of First Compliant Deposit:	5 July 2022
Date of Acceptance:	14 February 2021
Last Modified:	15 May 2023 23:23
URI:	https://orca.cardiff.ac.uk/id/eprint/150993

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