In this paper, we study the stability of commonly used filtration functions in topological data analysis under small pertubations of the underlying nonrandom point cloud. Relying on these stability results, we then develop a test procedure to detect and determine structural breaks in a sequence of topological data objects obtained from weakly dependent data. The proposed method applies for instance to statistics of persistence diagrams of $\mathbb{R}^d$-valued Bernoulli shift systems under the \v{C}ech or Vietoris-Rips filtration.

翻译：本文研究了在底层非随机点云发生微小扰动时，拓扑数据分析中常用过滤函数的稳定性。基于这些稳定性结果，我们进一步开发了一种检验程序，用于检测并确定由弱相依数据生成的拓扑数据对象序列中的结构断点。所提出的方法可应用于例如$\mathbb{R}^d$值伯努利移位系统在Čech或Vietoris-Rips过滤下的持续图统计量。