论文标题
最佳范围估计和度量学习
Optimal Reach Estimation and Metric Learning
论文作者
论文摘要
我们研究了覆盖范围的估计，这是一种普遍存在的规则性参数和几何数据分析。给定I.I.D.在未知的$ d $ -D $ \ MATHCAL {C}^k $ -smooth submanifold of $ \ Mathbb {r}^d $中的样本，我们为估计其覆盖范围提供了最佳的非互闭性界限。一方面，我们基于最大曲率的覆盖范围，另一方面是测量指标失真。派生的价格是适应性的，费率取决于$ m $的覆盖率是曲率引起的还是瓶颈结构。在此过程中，我们得出了最佳的测量指标估计界限。
We study the estimation of the reach, an ubiquitous regularity parameter in manifold estimation and geometric data analysis. Given an i.i.d. sample over an unknown $d$-dimensional $\mathcal{C}^k$-smooth submanifold of $\mathbb{R}^D$, we provide optimal nonasymptotic bounds for the estimation of its reach. We build upon a formulation of the reach in terms of maximal curvature on one hand, and geodesic metric distortion on the other hand. The derived rates are adaptive, with rates depending on whether the reach of $M$ arises from curvature or from a bottleneck structure. In the process, we derive optimal geodesic metric estimation bounds.
