$k$-Means Clustering for Persistent Homology

TL;DR

This paper proves the convergence of the $k$-means clustering algorithm in the complex space of persistence diagrams used in topological data analysis, and demonstrates its effectiveness through numerical experiments.

Contribution

It establishes theoretical convergence and properties of $k$-means clustering directly on persistence diagram space, a novel approach in topological data analysis.

Findings

$k$-means converges on persistence diagram space.

Clustering on diagrams outperforms vectorized representations.

Numerical experiments validate theoretical results.

Abstract

Persistent homology is a methodology central to topological data analysis that extracts and summarizes the topological features within a dataset as a persistence diagram; it has recently gained much popularity from its myriad successful applications to many domains. However, its algebraic construction induces a metric space of persistence diagrams with a highly complex geometry. In this paper, we prove convergence of the $k$-means clustering algorithm on persistence diagram space and establish theoretical properties of the solution to the optimization problem in the Karush--Kuhn--Tucker framework. Additionally, we perform numerical experiments on various representations of persistent homology, including embeddings of persistence diagrams as well as diagrams themselves and their generalizations as persistence measures; we find that $k$-means clustering performance directly on persistence diagrams and measures outperform their vectorized representations.