Extremal life times of persistent loops and holes

TL;DR

This paper investigates the extreme values of the lifetimes of topological features like loops and holes in persistent homology, providing scaling laws, Poisson approximations, and comparisons between different complexes.

Contribution

It offers new insights into the asymptotic behavior of feature lifetimes in persistent homology, including scaling laws and probabilistic approximations for extreme values.

Findings

Scaling laws for minimal and maximal feature lifetimes.

Poisson approximation for large and small lifetime features.

Differences between Čech and Vietoris-Rips complexes.

Abstract

Persistent homology captures the appearances and disappearances of topological features such as loops and holes when growing disks centered at a Poisson point process. We study extreme values for the life times of features dying in bounded components and with birth resp. death time bounded away from the threshold for continuum percolation. First, we describe the scaling of the minimal life times for general feature dimensions, and of the maximal life times for holes in the \v{C}ech complex. Then, we proceed to a more refined analysis and establish Poisson approximation for large life times of holes and for small life times of loops. Finally, we also study the scaling of minimal life times in the Vietoris-Rips setting and point to a surprising difference to the \v{C}ech complex.