Basic Metric Geometry of the Bottleneck Distance

TL;DR

This work analyzes the metric geometry of spaces of persistence diagrams endowed with the bottleneck distance, viewed as functors $\mathcal{D}_p(X,A)$ on metric pairs. It provides precise criteria for metrizability, completeness, separability, and geodesicity in terms of the underlying quotient $X/A$ and the geometry of $X$, including a sharp discreteness condition for metrizability and a quotient-completeness criterion for all $p\in[1,\infty]$. The separability result identifies a necessary and sufficient annulus-boundedness condition, while the geodesicity results require lifting geodesics from $X/A$ to $X$ and use ultrafilter techniques, with a notable counterexample showing geodesicity can fail in general. These findings extend the understanding of persistence-diagram spaces beyond the standard Euclidean setting and connect to prior work on generalized diagram spaces and interleaving-type metrics, with implications for stability and geometry in persistent homology.

Abstract

Given a metric pair $(X,A)$, i.e. a metric space $X$ and a distinguished closed set $A \subset X$, one may construct in a functorial way a pointed pseudometric space $\mathcal{D}_\infty(X,A)$ of persistence diagrams equipped with the bottleneck distance. We investigate the basic metric properties of the spaces $\mathcal{D}_\infty(X,A)$ and obtain characterizations of their metrizability, completeness, separability, and geodesicity.

Figures (3)

• Figure 1: The proof of Lemma \ref{['lem:X.complete.then.Dinfty.complete']}: construction of the sequence $(x_n)_{n=0}^\infty = ((\widehat{\sigma}_n)_m)_{n=0}^\infty$ of points in $X$ for a fixed $m \in \mathbb{N}$.
• Figure 2: The proof of Proposition \ref{['prop:Di-geodesic']}: construction for a point $y \in \widetilde{\tau} \setminus \widetilde{\tau}_c$. Here $d = d_\infty(\sigma,\tau)$, and the solid arrows represent the bijection $\widetilde{\delta}_N\colon \widetilde{\sigma} \to \widetilde{\sigma}'$.
• Figure 3: An example of the situation in the proof of Lemma \ref{['lem:Di-geodesic-=>']}; here $d = d(x,a_0)$, the black points represent a diagram $\xi_{x,y}(t)$ for some $t \in [0,\frac{1}{4}]$, and the arrows a 'nearly optimal' bijection $\gamma\colon \{\!\{ x \}\!\} \to \xi_{x,y}(t)$.

Theorems & Definitions (38)

• Theorem 1: Metrizability
• Theorem 2: Completeness
• Corollary 3
• Theorem 4: Separability
• Theorem 5: Geodesicity
• Corollary 6
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Proposition 2.4