通过流形的凸性缺陷函数估计其可达性。

Estimating the Reach of a Manifold via its Convexity Defect Function.

作者信息

Berenfeld Clément, Harvey John, Hoffmann Marc, Shankar Krishnan

机构信息

Université Paris-Dauphine PSL, CEREMADE, Place du Maréchal de Lattre de Tassigny, 75016 Paris, France.

Department of Mathematics, Swansea University, Fabian Way, Swansea, SA1 8EN UK.

出版信息

Discrete Comput Geom. 2022;67(2):403-438. doi: 10.1007/s00454-021-00290-8. Epub 2021 Jun 14.

DOI:10.1007/s00454-021-00290-8

PMID:35221405

原文链接:https://pmc.ncbi.nlm.nih.gov/articles/PMC8827210/

Abstract

The reach of a submanifold is a crucial regularity parameter for manifold learning and geometric inference from point clouds. This paper relates the reach of a submanifold to its convexity defect function. Using the stability properties of convexity defect functions, along with some new bounds and the recent submanifold estimator of Aamari and Levrard (Ann. Statist. (1), 177-204 (2019)), an estimator for the reach is given. A uniform expected loss bound over a model is found. Lower bounds for the minimax rate for estimating the reach over these models are also provided. The estimator almost achieves these rates in the and cases, with a gap given by a logarithmic factor.

摘要

子流形的到达范围是用于从点云进行流形学习和几何推理的关键正则性参数。本文将子流形的到达范围与其凸性缺陷函数联系起来。利用凸性缺陷函数的稳定性性质，结合一些新的界以及Aamari和Levrard最近的子流形估计器（《统计学年鉴》(1)，第177 - 204页(2019年)），给出了到达范围的一个估计器。找到了在一个模型上的一致期望损失界。还提供了在这些模型上估计到达范围的极小极大率的下界。在某些情况下，该估计器几乎达到了这些率，相差一个对数因子。