论文标题
通过其凸缺损函数估算歧管的范围
Estimating the reach of a manifold via its convexity defect function
论文作者
论文摘要
亚曼佛的范围是一种至关重要的规律性参数，用于流动学习和从点云的几何推断。本文将亚曼叶的触及范围与其凸缺损函数联系起来。使用凸缺损函数的稳定性，以及一些新的界限以及最近的Aamari和Levrard的Submanifold估计器[Ann。统计学家。 47 177-204（2019）]，给出了覆盖范围的估计器。在C^K模型上发现了统一的预期损耗。还提供了最小值速率的下限，以估算这些模型的覆盖范围。估计器在C^3和C^4的情况下几乎达到了这些速率，并具有对数因子给出的空白。
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.
