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$k$-Means Clustering for Persistent Homology
- Publication Year :
- 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.<br />Comment: 20 pages, 6 figures
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.2210.10003
- Document Type :
- Working Paper