Poisson approximation of large-lifetime cycles

TL;DR

This paper investigates the distribution of large-lifetime topological features in point clouds modeled as Poisson processes, establishing Poisson convergence results for their centers and lifetimes in various regimes.

Contribution

It provides the first rigorous Poisson approximation results for large-lifetime cycles in topological data analysis of Poisson point clouds.

Findings

Poisson convergence of cycle centers on the 2D torus.

Joint Poisson convergence of centers, lifetimes, and deathtimes in higher dimensions.

Analysis of large-lifetime cycles in different connectivity regimes.

Abstract

In topological data analysis, the notions of persistent homology, birthtime, lifetime, and deathtime are used to assign and capture relevant cycles (i.e., topological features) of a point cloud, such as loops and cavities. In particular, cycles with a large lifetime are of special interest. In this paper, we study such large-lifetime cycles when the point cloud is modeled as a Poisson point process. First, we consider the case with no bound on the deathtime, where we establish Poisson convergence of the centers of large-lifetime features on the 2-dimensional flat torus. Afterwards, by imposing a bound on the deathtime, we enter a sparse connectivity regime, and we prove joint Poisson convergence of the centers, lifetimes, and deathtimes in dimensions $d \geq 2$ under suitable model conditions.