In this article we extend and strengthen the seminal work by Niyogi, Smale, and Weinberger on the learning of the homotopy type from a sample of an underlying space. In their work, Niyogi, Smale, and Weinberger studied samples of $C^2$ manifolds with positive reach embedded in $\mathbb{R}^d$. We extend their results in the following ways: In the first part of our paper we consider both manifolds of positive reach -- a more general setting than $C^2$ manifolds -- and sets of positive reach embedded in $\mathbb{R}^d$. The sample $P$ of such a set $\mathcal{S}$ does not have to lie directly on it. Instead, we assume that the two one-sided Hausdorff distances -- $\varepsilon$ and $\delta$ -- between $P$ and $\mathcal{S}$ are bounded. We provide explicit bounds in terms of $\varepsilon$ and $ \delta$, that guarantee that there exists a parameter $r$ such that the union of balls of radius $r$ centred at the sample $P$ deformation-retracts to $\mathcal{S}$. In the second part of our paper we study homotopy learning in a significantly more general setting -- we investigate sets of positive reach and submanifolds of positive reach embedded in a \emph{Riemannian manifold with bounded sectional curvature}. To this end we introduce a new version of the reach in the Riemannian setting inspired by the cut locus. Yet again, we provide tight bounds on $\varepsilon$ and $\delta$ for both cases (submanifolds as well as sets of positive reach), exhibiting the tightness by an explicit construction.

翻译：本文拓展并强化了尼约基（Niyogi）、斯梅尔（Smale）与温伯格（Weinberger）关于从底层空间样本学习同伦型的开创性工作。在其研究中，三位学者考察了嵌入于$\mathbb{R}^d$中具有正触及半径的$C^2$流形样本。我们通过以下方式推广其结论：论文第一部分同时考虑正触及半径流形（比$C^2$流形更一般的设定）和嵌入$\mathbb{R}^d$的正触及半径集合。此类集合$\mathcal{S}$的样本$P$无需直接位于其上，而仅需假设$P$与$\mathcal{S}$之间的两个单向豪斯多夫距离$\varepsilon$和$\delta$有界。我们给出以$\varepsilon$和$\delta$表达的显式界，保证存在参数$r$使得以样本$P$为球心、半径为$r$的球并集可形变收缩至$\mathcal{S}$。论文第二部分在更普适的框架下研究同伦学习——考虑嵌入于具有有界截面曲率的黎曼流形中的正触及半径集合及其子流形。为此，我们借鉴割轨迹引入黎曼设定下触及半径的新定义。针对子流形与正触及半径集合两种情形，我们再次给出$\varepsilon$和$\delta$的紧界，并通过显式构造证明其紧性。