Efficient and Robust Persistent Homology for Measures

TL;DR

This paper introduces a robust and efficient method for topological data analysis of probability measures in metric spaces, extending persistent homology techniques to handle noise and outliers effectively.

Contribution

It extends the distance to a measure concept to general metric spaces and provides an efficient approximation method for sub-level sets, improving robustness and computational efficiency.

Findings

Method is robust to noise and outliers.

Efficient approximation of sub-level sets using unions of metric balls.

Demonstrated effectiveness through practical examples.

Abstract

We extend the notion of the distance to a measure from Euclidean space to probability measures on general metric spaces as a way to do topological data analysis in a way that is robust to noise and outliers. We then give an efficient way to approximate the sub-level sets of this function by a union of metric balls and extend previous results on sparse Rips filtrations to this setting. This robust and efficient approach to topological data analysis is illustrated with several examples from an implementation.