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Estimating the reach of a manifold via its convexity defect function

Berenfeld, Clément; Harvey, John; Hoffmann, Marc; Krishnan, Shankar (2021), Estimating the reach of a manifold via its convexity defect function, Discrete and Computational Geometry, p. 1-35. 10.1007/s00454-021-00290-8

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Type
Article accepté pour publication ou publié
Date
2021
Journal name
Discrete and Computational Geometry
Publisher
Springer
Pages
1-35
Publication identifier
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Author(s)
Berenfeld, Clément
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Harvey, John
Swansea University
Hoffmann, Marc
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Krishnan, Shankar
National Science Foundation
Abstract (EN)
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.
Subjects / Keywords
Point clouds; Manifold reconstruction; Minimax estimation; Convexity defect function; Reach