Tight bounds for the learning of homotopy \`a la Niyogi, Smale, and Weinberger for subsets of Euclidean spaces and of Riemannian manifolds

TL;DR

This paper extends the theoretical understanding of homotopy learning from samples of manifolds and sets of positive reach, providing explicit bounds and considering both Euclidean and Riemannian settings with bounded curvature.

Contribution

It generalizes previous work by considering sets of positive reach and Riemannian manifolds, introducing new bounds and a Riemannian reach concept.

Findings

Provided explicit bounds on sample proximity ensuring homotopy equivalence.

Extended results to Riemannian manifolds with bounded sectional curvature.

Introduced a new Riemannian reach concept and demonstrated tightness of bounds.

Abstract

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.