Vietoris$\unicode{x2013}$Rips Complexes of Metric Spaces Near a Metric Graph

TL;DR

This paper provides a method to choose scale parameters for Vietoris–Rips complexes to recover the topology of metric graphs and related spaces from point cloud data, extending previous qualitative results to quantitative guidelines.

Contribution

It introduces a way to quantitatively select scale parameters for Vietoris–Rips complexes to recover metric graph topology from data with small Gromov–Hausdorff distance.

Findings

Derived a description of the scale parameter based on the convexity radius.

Extended the analysis to Euclidean subsets near embedded metric graphs.

Provided a method to choose parameters ensuring homotopy equivalence.

Abstract

For a sufficiently small scale $\beta>0$, the Vietoris$\unicode{x2013}$Rips complex $\mathcal{R}_\beta(S)$ of a metric space $S$ with a small Gromov$\unicode{x2013}$Hausdorff distance to a closed Riemannian manifold $M$ has been already known to recover $M$ up to homotopy type. While the qualitative result is remarkable and generalizes naturally to the recovery of spaces beyond Riemannian manifolds$\unicode{x2014}$such as geodesic metric spaces with a positive convexity radius$\unicode{x2014}$the generality comes at a cost. Although the scale parameter $\beta$ is known to depend only on the geometric properties of the geodesic space, how to quantitatively choose such a $\beta$ for a given geodesic space is still elusive. In this work, we focus on the topological recovery of a special type of geodesic space, called a metric graph. For an abstract metric graph $\mathcal{G}$ and a (sample) metric space $S$ with a small Gromov$\unicode{x2013}$Hausdorff distance to it, we provide a description of $\beta$ based on the convexity radius of $\mathcal{G}$ in order for $\mathcal{R}_\beta(S)$ to be homotopy equivalent to $\mathcal{G}$. Our investigation also extends to the study of the Vietoris$\unicode{x2013}$Rips complexes of a Euclidean subset $S\subset\mathbb{R}^d$ with a small Hausdorff distance to an embedded metric graph $\mathcal{G}\subset\mathbb{R}^d$. From the pairwise Euclidean distances of points of $S$, we introduce a family (parametrized by $\varepsilon$) of path-based Vietoris$\unicode{x2013}$Rips complexes $\mathcal{R}^\varepsilon_\beta(S)$ for a scale $\beta>0$. Based on the convexity radius and distortion of the embedding of $\mathcal{G}$, we show how to choose a suitable parameter $\varepsilon$ and a scale $\beta$ such that $\mathcal{R}^\varepsilon_\beta(S)$ is homotopy equivalent to $\mathcal{G}$.