通过其柔性缺陷功能估计一个方块的覆盖范围 (Estimating the reach of a manifold via its convexity defect function)

from arxiv, 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

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. 47 177-204 (2019)], an estimator for the reach is given. A uniform expected loss bound over a C^k 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 C^3 and C^4 cases, with a gap given by a logarithmic factor.

翻译：从点云中进行多重学习和几何推断的关键常规参数是次元值的伸展。本文将次元的伸展与它的直立性缺陷功能联系起来。使用粘结性缺陷功能的稳定性特性,加上一些新的界限和最近对Aamari和Levrard[Ann. Statist. 47 177-204 (2019年)]的子元值测量器,给出了一个测距的测距符。找到一个统一的预期损失范围在CQ模型上。还提供了估算这些模型的伸展率的下限。估计值几乎达到了C3和C4的这些速率,但有一个对数系数给出了差距。