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 -smooth submanifold M of , 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. 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.
Acknowledgments
The authors would like to thank heartily Chez Adel for its unconditional warmth and creative atmosphere and Vincent Divol for helpful discussions.
Citation
Eddie Aamari. Clément Berenfeld. Clément Levrard. "Optimal reach estimation and metric learning." Ann. Statist. 51 (3) 1086 - 1108, June 2023. https://doi.org/10.1214/23-AOS2281
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