Geometric Bounds for Persistence

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 ${\mathrm{\ell }}^{\mathrm{\infty }}$-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.