New divergence measures between persistence diagrams and stability of vectorizations

TL;DR

This paper introduces new divergence measures for persistence diagrams, providing a systematic way to analyze barcodes and ensuring stability of vectorizations, which enhances statistical analysis in topological data analysis.

Contribution

It proposes a novel asymmetric divergence measure that generalizes Wasserstein and bottleneck distances, along with analyzing its induced topology and stability properties.

Findings

Introduces a new divergence measure for persistence diagrams.

Establishes stability of vectorizations under the new divergence.

Analyzes the topology induced by the divergence measure.

Abstract

Given a filtration of simplicial complexes, one usually applies persistent homology and summarizes the results in barcodes. Then, in order to extract statistical information from these barcodes, one needs to compute statistical indicators over the bars of the barcode. An issue with this approach is that usually infinite bars must be deleted or cut to finite ones; however, so far there is no consensus on how to perform this procedure. In this work we propose for the first time a systematic way to analyze barcodes through the use of statistical indicators. Our approach is based on the minimization of a divergence measure that generalizes the standard Wasserstein or bottleneck distance to a new asymmetric distance-like function that we introduce and which is interesting on its own. In particular, we analyze the topology induced by this divergence and the stability of known vectorizations with respect to this topology.