OPTIMAL REACH ESTIMATION AND METRIC LEARNING

• 成果类型: Article
• 署名作者: Aamari, Eddie; Berenfeld, Clement; Levrard, Clement
• 署名单位: Sorbonne Universite; Universite Paris Cite; University of Potsdam
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/23-AOS2281
• 发表日期: 2023
• 页码: 1086-1108
• 关键词: manifold estimation; rates; CONVERGENCE

摘要

We study the estimation of the reach, an ubiquitous regularity parameter in manifold estimation and geometric data analysis. Given an i.i.d. sample vide 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.