# Optimal Reach Estimation and Metric Learning

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.
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Preprints, Working Papers, ...
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https://hal.archives-ouvertes.fr/hal-03722236
Contributor : Eddie Aamari Connect in order to contact the contributor
Submitted on : Wednesday, July 13, 2022 - 11:39:39 AM
Last modification on : Friday, August 5, 2022 - 3:00:08 PM

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Optimal Reach Estimation and M...
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• HAL Id : hal-03722236, version 1

### Citation

Eddie Aamari, Clément Berenfeld, Clément Levrard. Optimal Reach Estimation and Metric Learning. 2022. ⟨hal-03722236⟩

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