The self-similar evolution of stationary point processes via persistent homology

TL;DR

This paper investigates the topological features of point cloud data modeled by point processes using persistent homology, introducing measures on persistence diagrams and analyzing their self-similar scaling properties.

Contribution

It introduces a probabilistic framework for persistent homology, including measures on diagrams and a self-similar scaling analysis, with a key packing relation between scaling exponents.

Findings

Established a measure-theoretic approach to persistence diagrams in a probabilistic setting

Derived a self-similar scaling law for a family of persistence diagrams

Proved a packing relation linking the scaling exponents

Abstract

Persistent homology provides a robust methodology to infer topological structures from point cloud data. Here we explore the persistent homology of point clouds embedded into a probabilistic setting, exploiting the theory of point processes. We introduce measures on the space of persistence diagrams and the self-similar scaling of a one-parameter family of these. As the main result we prove a packing relation between the occurring scaling exponents.