Optimal reach estimation and metric learning
研究了流形估计中一个常用正则参数——可达距离的估计问题，基于曲率和测地度量失真给出了最优非渐近界，并推导了自适应速率和最优测地度量估计界。
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 Ck-smooth submanifold M of RD, 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. 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.