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

TL;DR

This work advances homotopy learning by deriving tight, explicit bounds on two sampling quality measures \\varepsilon\\ and \\delta\\ for faithful topological recovery from samples of sets with positive reach in both Euclidean spaces and Riemannian manifolds. It develops a geometric framework that guarantees the existence of a radius \\(r\\) such that the thickening \\(P^{\\boxplus r}\\) deformation-retracts to the underlying set, with precise interval bounds for Euclidean and curvature-dependent bounds for Riemannian settings. The authors establish the optimality of these bounds via constructive counterexamples in multiple dimensions and curvature regimes, and extend the analysis to the cut-locus–based reach suitable for Riemannian manifolds with bounded sectional curvature. Collectively, the results tighten and generalize previous work by Niyogi–Smale–Weinberger, providing rigorous guidance for homotopy inference from noisy samples in both flat and curved ambient spaces. The work also connects to manifold learning and stratification, and suggests avenues for applying these bounds to practical topological reconstruction tasks under noise and curvature constraints.

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 $δ$ -- between $P$ and $\mathcal{S}$ are bounded. We provide explicit bounds in terms of $\varepsilon$ and $ δ$, 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 $δ$ for both cases (submanifolds as well as sets of positive reach), exhibiting the tightness by an explicit construction.

Figures (24)

• Figure 1: Left: A fish shaped set ${\mathcal{S}}$ of positive reach (in blue). Its medial axis (in purple) is at a positive distance. For $0 \leq i \leq 3$, we also represent the normal cone of $p_i$ with respect to ${\mathcal{S}}$ (after an intersection with a small disk and a translation to $p_i$). The normal cone of the point $p_2$ is $p_2$ itself. Right: The set ${\mathcal{S}}$ with a sample $P$ and a thickening of $P$. We see that the thickening has the same homotopy type as ${\mathcal{S}}$.
• Figure 2: The blue-gray region bounded by the blue dashed curve represents the set of pairs $(\varepsilon,\delta)$ for which there exists a radius $r$ such that the union of balls of radius $r$ centred at $P$ captures the homotopy type of a set of positive reach $\mathcal{R}\xspace = 1$. The equivalent region for a manifold of reach $\mathcal{R}\xspace = 1$ is depicted in yellow and is a superset of the previous one. The two regions coincide above the diagonal $\delta = \varepsilon$. The bounds for the Euclidean setting are indicated on top, for an ambient manifold with positive curvature bound (${\Lambda_{\ell}} =+2$) in the middle, and for an ambient manifold with negative curvature bound (${\Lambda_{\ell}} =-2$) bottom. In the top picture, the black points indicate the bounds that were known to Niyogi, Smale, and Weinberger.
• Figure 3: A pictorial overview of the proof. The pink shaded region represents a part of the set ${\mathcal{S}}$, the union of balls $P^{\boxplus r}$ is coloured orange. The thickened blue segment shows those points of the segment $L$ that lie a distance less than $\alpha$ from ${\mathcal{S}}$. Per assumption, this segment is contained in the union of balls $P^{\boxplus r}$.
• Figure 5: A pictorial explanation of why $P^{\boxplus r}$ never has the homotopy type of the set ${\mathcal{S}}$. We only depict three annuli in the sequence of $A_i$s. The set ${\mathcal{S}}$ is in blue, the sample $P$ in red, and the thickening of $P$ in pink. The black circles indicate the location of the two isolated sample points of $P$ associated to each annulus.
• Figure 6: The (half of the) torus $T_i$ depicted in blue; the sample --- the set $C_i$ and the points $p_i$ and $\tilde{p}_i$ --- in red. In black we indicate the circle $C'_{i}$ on which the points $p_i$ and $\tilde{p}_i$ lie. The closest point projection of this circle onto $\mathcal{M}$ is indicated in blue.

Theorems & Definitions (87)

• Remark 1
• Theorem 2
• Remark 3
• Proposition 3
• Remark 3
• Proposition 3
• Proposition 3
• Proposition 3
• Remark 4
• Remark 5