k-means clustering for persistent homology
File(s)
Published version
Author(s)
Cao, Yueqi
Leung, Prudence
Monod, Anthea
Type
Journal Article
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.
Date Issued
2025-03-01
Date Acceptance
2023-12-29
Citation
Advances in Data Analysis and Classification, 2025, 19 (1)
ISSN
1862-5347
Publisher
Springer
Journal / Book Title
Advances in Data Analysis and Classification
Volume
19
Issue
1
Copyright Statement
© The Author(s) 2024 Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
License URL
Subjects
Alexandrov geometry
Karush-Kuhn-Tucker optimization
k-means clustering
Mathematics
Persistence diagrams
Persistent homology
Physical Sciences
Science & Technology
SHAPE
SIZE FUNCTIONS
STATISTICS
Statistics & Probability
Publication Status
Published
Date Publish Online
2024-01-31