Bottleneck Profiles and Discrete Prokhorov Metrics for Persistence Diagrams

TL;DR

This paper introduces bottleneck profiles and discrete Prokhorov metrics to improve comparison of persistence diagrams in topological data analysis, offering stability guarantees and computational algorithms.

Contribution

It proposes a novel framework for analyzing persistence diagrams using bottleneck profiles and discrete Prokhorov metrics, addressing limitations of existing distances.

Findings

Discrete Prokhorov metrics generalize Bottleneck distance

Metrics satisfy stability properties

Algorithms for computing new metrics are provided

Abstract

In topological data analysis (TDA), persistence diagrams have been a succesful tool. To compare them, Wasserstein and Bottleneck distances are commonly used. We address the shortcomings of these metrics and show a way to investigate them in a systematic way by introducing bottleneck profiles. This leads to a notion of discrete Prokhorov metrics for persistence diagrams as a generalization of the Bottleneck distance. They satisfy a stability result and bounds with respect to Wasserstein metrics. We provide algorithms to compute the newly introduced quantities and end with an discussion about experiments.