Geometric Bounds for Persistence

TL;DR

This work develops a geometric lens for persistent homology by tying lifespans of topological features to metric-geometry invariants such as filling radii and various widths (Urysohn, Alexandrov, Kolmogorov) and to the novel notions of treewidth and Überspread. It shows how Čech and Vietoris–Rips lifespans can be bounded by cores and their geometric complexity, and introduces extinction times to bound death across all degrees. The results provide both upper bounds on lifespans through width-based constructions and lower bounds via cores, while also offering practical implications for estimating geometric quantities from PH and potentially accelerating computations through extinction bounds. Overall, the paper bridges applied topology and metric geometry, offering quantitative tools to interpret persistence outputs and to connect them with intrinsic geometric size concepts. $d_\omega - b_\omega$ serves as the central quantity bound by invariants like $UW_{k-1}$, $AW_{k-1}$, $KW_k$, TW_k, and Überspread, establishing a cohesive framework for analyzing the size and significance of topological features in data.

Abstract

In this paper, we offer a new perspective on persistent homology by integrating key concepts from metric geometry. For a given compact subset $\mathcal{X}$ of a Banach space $\mathbf{Y}$, we analyze the topological features arising in the family $\mathcal{N}_\bullet(\mathcal{X} \subset \mathbf{Y})$ of nested neighborhoods of $\mathcal{X}$ in $\mathbf{Y}$ and provide several geometric bounds on their persistence (lifespans). We begin by examining the lifespans of these homology classes in terms of their filling radii in $\mathbf{Y}$, establishing connections between these lifespans and fundamental invariants in metric geometry, such as the Urysohn width. We then derive bounds on these lifespans by considering the $\ell^\infty$-principal components of $\mathcal{X}$, also known as Kolmogorov widths. Additionally, we introduce and investigate the concept of extinction time of a metric space $\mathcal{X}$: the critical threshold beyond which no homological features persist in any degree. We propose methods for estimating the Čech and Vietoris-Rips extinction times of $\mathcal{X}$ by relating $\mathcal{X}$ to its convex hull and to its tight span, respectively.

Figures (4)

• Figure 1: A shape with small $1$-treewidth but considerable Alexandrov $0$-width and Kolmogorov $1$-width.
• Figure 2: For the surface $S$, while $\mathop{\mathrm{AW}}\nolimits_1(S \subset \mathbb{R}^3)$ is small, the lifespan of red curve $\alpha\in \widetilde{\mathrm{H}}_1(S)$ is large.
• Figure 3: The bound $d-b\leq 2\,\ddot{u}$ in \ref{['thm:uberspread']} is sharp; here $\mathbf{Y}=\mathbb{R}^2$ and $\ddot{u} = \mathop{\mathrm{{\ddot{u}}-spread}}\nolimits(\mathcal{X}\subset \mathbb{R}^2)$.
• Figure 4: The space $\widehat{\mathrm{E}}$ from \ref{['rmk:ellipsoid-handle']} for the case $N=3$ and $k=1$.

Theorems & Definitions (123)

• Corollary 1.0: VR Lifespans via Urysohn Width
• Corollary 1.0: Čech Lifespans via Alexandrov Width
• Corollary 1.0: Čech Lifespans via Treewidth
• Corollary 1.0: Čech Lifespans via $\ell^\infty$-Variance
• Theorem 1.1: Čech Lifespans via Überspread
• Theorem 1.1: Bounding Čech Extinction
• Definition 2.1: Vietoris--Rips Complexes
• Definition 2.2: Čech Complexes
• Definition 2.3: Filtration
• Example 2.4: Neighborhod Filtrations