A New Non-archimedean Metric on Persistent Homology

TL;DR

This paper introduces a novel non-archimedean cophenetic metric for persistent homology that enhances hierarchical clustering and visualization across all homology degrees, with verified statistical and topological benefits.

Contribution

It presents the first cophenetic metric applicable to all persistent homology degrees, improving clustering analysis and topological visualization in topological data analysis.

Findings

Cophenetic metric improves clustering quality as shown by silhouette scores.

Hierarchical clustering with cophenetic metric yields statistically verifiable topological insights.

Rooted trees effectively display inter-relations of persistent homology classes across degrees.

Abstract

In this article, we define a new non-archimedean metric structure, called cophenetic metric, on persistent homology classes of all degrees. We then show that zeroth persistent homology together with the cophenetic metric and hierarchical clustering algorithms with a number of different metrics do deliver statistically verifiable commensurate topological information based on experimental results we obtained on different datasets. We also observe that the resulting clusters coming from cophenetic distance do shine in terms of different evaluation measures such as silhouette score and the Rand index. Moreover, since the cophenetic metric is defined for all homology degrees, one can now display the inter-relations of persistent homology classes in all degrees via rooted trees.