通过凸性缺陷函数估计流形的可达半径
统计理论
2022-04-27 v4 计算几何
微分几何
统计理论
摘要
子流形的可达半径（reach）是流形学习与从点云进行几何推断的关键正则性参数。本文将子流形的可达半径与其凸性缺陷函数相联系。利用凸性缺陷函数的稳定性性质，结合一些新界以及 Aamari 与 Levrard [Ann. Statist. 47 177-204 (2019)] 最近的子流形估计量，给出了一种可达半径的估计量。得到了在 C^k 模型上的一致期望损失界。还给出了在这些模型上估计可达半径的极小极大率下界。该估计量在 C^3 与 C^4 情形下几乎达到这些率，差距由一个对数因子给出。
引用
@article{arxiv.2001.08006,
title = {Estimating the reach of a manifold via its convexity defect function},
author = {Clément Berenfeld and John Harvey and Marc Hoffmann and Krishnan Shankar},
journal= {arXiv preprint arXiv:2001.08006},
year = {2022}
}
备注
35 pages, 4 figures. Various minor changes in v2 to correct minor errors and/or improve clarity. Thanks to excellent work by peer reviewers, in v3 an error in Lemma 4.9 was rectified, Section 4.2 was substantially revised and other minor changes made throughout and the manuscript was accepted for publication by Discrete & Computational Geometry. Extremely minor changes in v4