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Estimating the Reach of a Manifold via its Convexity Defect Function

Clément Berenfeld, John Harvey Orcid Logo, Marc Hoffmann, Krishnan Shankar

Discrete & Computational Geometry, Volume: 67, Issue: 2, Pages: 403 - 438

Swansea University Author: John Harvey Orcid Logo

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DOI (Published version): 10.1007/s00454-021-00290-8

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 t...

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Published in: Discrete & Computational Geometry
ISSN: 0179-5376 1432-0444
Published: Springer Science and Business Media LLC 2022
Online Access: Check full text

URI: https://cronfa.swan.ac.uk/Record/cronfa56481
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.
Keywords: Point clouds, Manifold reconstruction, Minimax estimation, Convexity defect function, Reach
College: Faculty of Science and Engineering
Funders: EPSRC, Daphne Jackson Fellowship; U.S. National Science Foundation;
Issue: 2
Start Page: 403
End Page: 438

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