Universality in Random Persistent Homology and Scale-Invariant Functionals

TL;DR

This paper establishes a universal limiting distribution for persistence diagrams from geometric filtrations over random point processes, demonstrating scale-invariance and independence from the underlying distribution.

Contribution

It introduces a novel framework for universality in scale-invariant functionals on point processes and extends Morse theory results in random geometric complexes.

Findings

Limiting distribution of persistence ratios is universal across point processes.

Framework for universality in scale-invariant functionals is developed.

New Morse theory results for random geometric complexes are presented.

Abstract

In this paper, we prove a universality result for the limiting distribution of persistence diagrams arising from geometric filtrations over random point processes. Specifically, we consider the distribution of the ratio of persistence values (death/birth), and show that for fixed dimension, homological degree and filtration type (Cech or Vietoris-Rips), the limiting distribution is independent of the underlying point process distribution, i.e., universal. In proving this result, we present a novel general framework for universality in scale-invariant functionals on point processes. Finally, we also provide a number of new results related to Morse theory in random geometric complexes, which may be of an independent interest.