Simplicial Hausdorff Distance for Topological Data Analysis

TL;DR

The paper addresses the problem of comparing simplicial complexes by a distance that integrates geometric proximity and topological structure. It introduces the simplicial Hausdorff distance $\delta((X,f),(Y,g))$, defined via directed distances with $\vec{d}\left((X,f),(Y,g)\right)= \max_k \max_{\sigma \in X,\dim \sigma=k} \min_{\tau \in Y,\dim \tau=k} \max_{v\in\sigma} \min_{w\in\tau} d(f(v),g(w))$ and uses $d(\cdot,\cdot)$ as Euclidean; it extends to filtrations via $\hat{\delta}$. The work proves that these distances are extended metrics, analyzes polynomial-time complexity bounds $O(p^{2D+2})$ (with $p=\max\{n,m\}$ and $D=\max\{|X|,|Y|\}$) and tighter $O(p^6)$ when dimensions are bounded by $2$, and studies the impact of non-injective measurement functions (leading to pseudometrics) and Vietoris–Rips collapses. It also discusses practical considerations, including an R implementation compatible with the TDA package and directions for stability analysis and integration with topological learning workflows. Together, these results provide a geometry–topology aware tool for comparing simplicial complexes in TDA pipelines and facilitating robust downstream analyses.

Abstract

Many practical applications in topological data analysis arise from data in the form of point clouds, which then yield simplicial complexes. The combinatorial structure of simplicial complexes captures the topological relationships between the elements of the complex. In addition to the combinatorial structure, simplicial complexes possess a geometric realization that provides a concrete way to visualize the complex and understand its geometric properties. This work presents an amended Hausdorff distance as an extended metric that integrates geometric proximity with the topological features of simplicial complexes. We also present a version of the simplicial Hausdorff metric for filtered complexes and show results on its computational complexity. In addition, we discuss concerns about the monotonicity of the measurement functions involved in the setup of the simplicial complexes.

Figures (3)

• Figure 1: An embedding of a $2$-simplex is possible only for $\mathbb{R}^d$ with $d\geq 2$.
• Figure 2: Can we tell that $X$ and $X'$ are different?
• Figure 3: Illustration of $\epsilon$-closeness: each orange ball has radius $\frac{\epsilon}{2}$. In the figure on the left, the $1$-simplex $wt$ is $\epsilon$-close to $uv$ because $d(u,w)$ and $d(v,t) < \epsilon$; in the figure on the right, the $1$-simplex $uv$ is $\epsilon$-close to $wt$ because $d(u,t)$ and $d(v,t) < \epsilon$; but $wt$ is not $\epsilon$-close to $uv$ because $d(w,u)> \epsilon$ and $d(w,v) > \epsilon$. The $\epsilon$-closeness relation is not symmetric.

Theorems & Definitions (45)

• Definition 2.1: $\epsilon$-closeness
• Definition 2.2: Simplicial Hausdorff distance
• Proposition 2.3
• proof
• Corollary 2.4
• Definition 2.5: Inclusion of members of $\mathbb{X}^d$
• Definition 2.6
• Lemma 2.7
• proof
• Proposition 2.8