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Mathematics > Statistics Theory

arXiv:2207.06074 (math)
[Submitted on 13 Jul 2022]

Title:Optimal Reach Estimation and Metric Learning

Authors:Eddie Aamari, Clément Berenfeld, Clément Levrard
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Abstract: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 $\mathcal{C}^k$-smooth submanifold of $\mathbb{R}^D$, 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.
Subjects: Statistics Theory (math.ST); Metric Geometry (math.MG)
MSC classes: 62G05, 62C20, 68U05
Cite as: arXiv:2207.06074 [math.ST]
  (or arXiv:2207.06074v1 [math.ST] for this version)
  https://doi.org/10.48550/arXiv.2207.06074

Submission history

From: Eddie Aamari [view email]
[v1] Wed, 13 Jul 2022 09:31:40 UTC (534 KB)
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