Bayesian estimation of topological features of persistence diagrams

TL;DR

This paper introduces a Bayesian clustering approach to estimate Betti numbers from persistence diagrams in topological data analysis, accounting for sampling randomness and complex diagram structures.

Contribution

It presents a novel Bayesian method leveraging features' lifetimes and random partitions to estimate topological Betti numbers from persistence diagrams.

Findings

Effective Bayesian clustering for Betti number estimation.

Simulation results demonstrate the method's accuracy.

Addresses challenges of randomness and complex structures in persistence diagrams.

Abstract

Persistent homology is a common technique in topological data analysis providing geometrical and topological information about the sample space. All this information, known as topological features, is summarized in persistence diagrams, and the main interest is in identifying the most persisting ones since they correspond to the Betti number values. Given the randomness inherent in the sampling process, and the complex structure of the space where persistence diagrams take values, estimation of Betti numbers is not straightforward. The approach followed in this work makes use of features' lifetimes and provides a full Bayesian clustering model, based on random partitions, in order to estimate Betti numbers. A simulation study is also presented.