Estimating the Reach of a Manifold via its Convexity Defect Function

Berenfeld, Clément; Harvey, John; Hoffmann, Marc; Shankar, Krishnan

doi:10.1007/s00454-021-00290-8

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Estimating the Reach of a Manifold via its Convexity Defect Function

Clément Berenfeld, John Harvey

, Marc Hoffmann, Krishnan Shankar

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

Swansea University Author: John Harvey

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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
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URI:	https://cronfa.swan.ac.uk/Record/cronfa56481
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spelling	2022-03-16T11:29:11.0755106 v2 56481 2021-03-22 Estimating the Reach of a Manifold via its Convexity Defect Function 1a837434ec48367a7ffb596d04690bfd 0000-0001-9211-0060 John Harvey John Harvey true false 2021-03-22 SMA 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. Journal Article Discrete & Computational Geometry 67 2 403 438 Springer Science and Business Media LLC 0179-5376 1432-0444 Point clouds, Manifold reconstruction, Minimax estimation, Convexity defect function, Reach 1 3 2022 2022-03-01 10.1007/s00454-021-00290-8 COLLEGE NANME Mathematics COLLEGE CODE SMA Swansea University SU Library paid the OA fee (TA Institutional Deal) EPSRC, Daphne Jackson Fellowship; U.S. National Science Foundation; 2022-03-16T11:29:11.0755106 2021-03-22T10:28:45.2555046 Faculty of Science and Engineering School of Mathematics and Computer Science - Mathematics Clément Berenfeld 1 John Harvey 0000-0001-9211-0060 2 Marc Hoffmann 3 Krishnan Shankar 4 56481__20205__a3947fc76423429289ca389a08b83c88.pdf 56481.VOR.Berenfeld2021.pdf 2021-06-21T13:02:43.2539569 Output 857804 application/pdf Version of Record true This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. true eng http://creativecommons.org/licenses/by/4.0/
title	Estimating the Reach of a Manifold via its Convexity Defect Function
spellingShingle	Estimating the Reach of a Manifold via its Convexity Defect Function John Harvey
title_short	Estimating the Reach of a Manifold via its Convexity Defect Function
title_full	Estimating the Reach of a Manifold via its Convexity Defect Function
title_fullStr	Estimating the Reach of a Manifold via its Convexity Defect Function
title_full_unstemmed	Estimating the Reach of a Manifold via its Convexity Defect Function
title_sort	Estimating the Reach of a Manifold via its Convexity Defect Function
author_id_str_mv	1a837434ec48367a7ffb596d04690bfd
author_id_fullname_str_mv	1a837434ec48367a7ffb596d04690bfd_***_John Harvey
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container_title	Discrete & Computational Geometry
container_volume	67
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publishDate	2022
institution	Swansea University
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publisher	Springer Science and Business Media LLC
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description	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.
published_date	2022-03-01T04:11:28Z
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Estimating the Reach of a Manifold via its Convexity Defect Function

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