Optimal rates of convergence for persistence diagrams in Topological Data Analysis

TL;DR

This paper investigates the statistical properties of persistent homology in topological data analysis, establishing convergence rates for persistence diagrams in general metric spaces and demonstrating their utility as statistical tools.

Contribution

It introduces a statistical framework for persistent homology in metric spaces and derives optimal convergence rates for persistence diagrams.

Findings

Persistence diagrams have favorable convergence properties as statistical summaries.

Numerical experiments validate the theoretical convergence rates.

Persistent homology can be effectively integrated into statistical analysis in metric spaces.

Abstract

Computational topology has recently known an important development toward data analysis, giving birth to the field of topological data analysis. Topological persistence, or persistent homology, appears as a fundamental tool in this field. In this paper, we study topological persistence in general metric spaces, with a statistical approach. We show that the use of persistent homology can be naturally considered in general statistical frameworks and persistence diagrams can be used as statistics with interesting convergence properties. Some numerical experiments are performed in various contexts to illustrate our results.