In topological data analysis, we want to discern topological and geometric structure of data, and to understand whether or not certain features of data are significant as opposed to simply random noise. While progress has been made on statistical techniques for single-parameter persistence, the case of two-parameter persistence, which is highly desirable for real-world applications, has been less studied. This paper provides an accessible introduction to two-parameter persistent homology and presents results about matching distance between 2-D persistence modules obtained from families of point clouds. Results include observations of how differences in geometric structure of point clouds affect the matching distance between persistence modules. We offer these results as a starting point for the investigation of more complex data.

中文翻译：

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拓扑数据分析中匹配距离的分布

在拓扑数据分析中，我们想要辨别数据的拓扑结构和几何结构，并了解数据的某些特征是否与简单的随机噪声相反。虽然在单参数持久性的统计技术方面取得了进展，但对实际应用非常理想的双参数持久性的情况研究较少。本文提供了对双参数持久同源性的可访问性介绍，并展示了从点云族中获得的二维持久性模块之间的匹配距离的结果。结果包括观察点云几何结构的差异如何影响持久性模块之间的匹配距离。我们提供这些结果作为调查更复杂数据的起点。