Estimating the Reach of a Manifold via its Convexity Defect Function

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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Autores principales: Berenfeld, Clément, Harvey, John, Hoffmann, Marc, Shankar, Krishnan
Formato: Online Artículo Texto
Lenguaje:English
Publicado: Springer US 2021
Materias:
Article
Acceso en línea:https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8827210/
https://www.ncbi.nlm.nih.gov/pubmed/35221405
http://dx.doi.org/10.1007/s00454-021-00290-8
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author Berenfeld, Clément
Harvey, John
Hoffmann, Marc
Shankar, Krishnan
author_facet Berenfeld, Clément
Harvey, John
Hoffmann, Marc
Shankar, Krishnan
author_sort Berenfeld, Clément
collection PubMed
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(1), 177–204 (2019)), an estimator for the reach is given. A uniform expected loss bound over a [Formula: see text] 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 [Formula: see text] and [Formula: see text] cases, with a gap given by a logarithmic factor.
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spelling pubmed-88272102022-02-23 Estimating the Reach of a Manifold via its Convexity Defect Function Berenfeld, Clément Harvey, John Hoffmann, Marc Shankar, Krishnan Discrete Comput Geom Article 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 [Formula: see text] 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 [Formula: see text] and [Formula: see text] cases, with a gap given by a logarithmic factor. Springer US 2021-06-14 2022 /pmc/articles/PMC8827210/ /pubmed/35221405 http://dx.doi.org/10.1007/s00454-021-00290-8 Text en © The Author(s) 2021 https://creativecommons.org/licenses/by/4.0/Open AccessThis 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. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/ (https://creativecommons.org/licenses/by/4.0/) .
spellingShingle Article
Berenfeld, Clément
Harvey, John
Hoffmann, Marc
Shankar, Krishnan
Estimating the Reach of a Manifold via its Convexity Defect Function
title 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_short Estimating the Reach of a Manifold via its Convexity Defect Function
title_sort estimating the reach of a manifold via its convexity defect function
topic Article
url https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8827210/
https://www.ncbi.nlm.nih.gov/pubmed/35221405
http://dx.doi.org/10.1007/s00454-021-00290-8
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