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。