Optimal reach estimation and metric learning
Eddie Aamari, Clément Berenfeld, Clément Levrard
The Annals of Statistics · 2023 · Volume 51 · Issue 3
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 Ck-smooth submanifold M 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. 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.
Keywords
62C20, 62G05, 68U05, Geometric inference, manifold learning, metric learning, minimax risk, reach
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