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：本文扩展并强化了Niyogi、Smale与Weinberger在基于底层空间样本学习同伦类型方面的开创性工作。在其研究中，Niyogi、Smale与Weinberger研究了嵌入$\mathbb{R}^d$中具有正可达性的$C^2$流形样本。我们从以下方面拓展了他们的结果：在第一部分中，我们同时考虑了具有正可达性的流形（比$C^2$流形更一般的设定）以及嵌入$\mathbb{R}^d$中具有正可达性的集合。此类集合$\mathcal{S}$的样本$P$无需直接位于其上，而是假定$P$与$\mathcal{S}$之间的两个单侧豪斯多夫距离$\varepsilon$和$\delta$有界。我们给出了关于$\varepsilon$和$\delta$的显式界，确保存在参数$r$，使得以样本$P$为中心、半径$r$的球并集可形变收缩至$\mathcal{S}$。在第二部分中，我们研究了更广泛设定下的同伦学习——探讨嵌入到具有有界截面曲率的黎曼流形中的正可达性集合和子流形。为此，我们借鉴割轨迹概念引入了黎曼设定下的新版本可达性。同样地，我们为两种情形（子流形与正可达性集合）提供了$\varepsilon$和$\delta$的紧界，并通过显式构造证明了其紧性。