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Optimal Reach Estimation and Metric Learning

Aamari, Eddie; Berenfeld, Clément; Levrard, Clément (2022), Optimal Reach Estimation and Metric Learning. https://basepub.dauphine.psl.eu/handle/123456789/23324

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Type
Document de travail / Working paper
Date
2022
Series title
Cahier de recherche CEREMADE, Université Paris Dauphine-PSL
Published in
Paris
Pages
47
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Author(s)
Aamari, Eddie cc
Laboratoire de Probabilités, Statistique et Modélisation [LPSM (UMR_8001)]
Berenfeld, Clément
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Levrard, Clément
Laboratoire de Probabilités, Statistique et Modélisation [LPSM (UMR_8001)]
Abstract (EN)
We study the estimation of the reach, an ubiquitous regularity parameter in manifold estimation and geometric data analysis. Given an i.i.d. sample over an unknown d-dimensional Ck-smooth submanifold of RD , we provide optimal nonasymptotic bounds for the estimation of its reach. We build upon a formulation of the reach in terms of maximal curvature on one hand, and geodesic metric distortion on the other hand. The derived rates are adaptive, with rates depending on whether the reach of M arises from curvature or from a bottleneck structure. In the process, we derive optimal geodesic metric estimation bounds.