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.

翻译：持久同源性是拓扑数据分析中的核心方法，它通过持久图提取并总结数据集中的拓扑特征；因其在众多领域的广泛应用而近来备受欢迎。然而，其代数构造导致持久图构成的度量空间具有高度复杂的几何结构。本文证明了$k$-均值聚类算法在持久图空间中的收敛性，并在Karush--Kuhn--Tucker框架下建立了优化问题解的理论性质。此外，我们对持久同源性的多种表示进行了数值实验，包括持久图的嵌入、持久图本身及其推广形式（作为持久度量），发现直接在持久图和持久度量上进行$k$-均值聚类的性能优于其向量化表示。