Optimal Reach Estimation and Metric Learning

2022/07/13 by Eddie Aamari, Clément Berenfeld, Aamari, Eddie +3
Computer Science · Mathematics · #62C20 #62G05 #68U05 #FOS: Mathematics #Metric Geometry (math.MG) #Morphological variations and asymmetry #Statistics Theory (math.ST) #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.2207.06074

openalex publication_date 2022/07/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 C^k-smooth submanifold of 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.

Related

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• A free lunch: manifolds of positive reach can be smoothed without decreasing the reach
• Convergence of the Reach for a Sequence of Gaussian-Embedded Manifolds
• Computable bounds for the reach and r-convexity of subsets of ℝ^d