$k$-Means Clustering for Persistent Homology

arXiv2210.10003MaRDI QIDQ6414349

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Publication date: 18 October 2022

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 clustering performance directly on persistence diagrams and measures outperform their vectorized representations.

Has companion code repository: https://github.com/pruskileung/ph-kmeans

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