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k-means clustering for persistent homology

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Listed:
  • Yueqi Cao

    (Imperial College London, South Kensington Campus)

  • Prudence Leung

    (Imperial College London, South Kensington Campus)

  • Anthea Monod

    (Imperial College London, South Kensington Campus)

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 both simulated and real data of 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.

Suggested Citation

  • Yueqi Cao & Prudence Leung & Anthea Monod, 2025. "k-means clustering for persistent homology," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 19(1), pages 95-119, March.
    • Handle: RePEc:spr:advdac:v:19:y:2025:i:1:d:10.1007_s11634-023-00578-y
    DOI: 10.1007/s11634-023-00578-y
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    References listed on IDEAS

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    1. Lorin Crawford & Anthea Monod & Andrew X. Chen & Sayan Mukherjee & Raúl Rabadán, 2020. "Predicting Clinical Outcomes in Glioblastoma: An Application of Topological and Functional Data Analysis," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 115(531), pages 1139-1150, July.
    2. Umar Islambekov & Yulia R. Gel, 2019. "Unsupervised space–time clustering using persistent homology," Environmetrics, John Wiley & Sons, Ltd., vol. 30(4), June.
    3. Robert Thorndike, 1953. "Who belongs in the family?," Psychometrika, Springer;The Psychometric Society, vol. 18(4), pages 267-276, December.
    4. Lawrence Hubert & Phipps Arabie, 1985. "Comparing partitions," Journal of Classification, Springer;The Classification Society, vol. 2(1), pages 193-218, December.
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