Many practical applications in topological data analysis arise from data in the form of point clouds, which then yield simplicial complexes. The combinatorial structure of simplicial complexes captures the topological relationships between the elements of the complex. In addition to the combinatorial structure, simplicial complexes possess a geometric realization that provides a concrete way to visualize the complex and understand its geometric properties. This work presents an amended Hausdorff distance as an extended metric that integrates geometric proximity with the topological features of simplicial complexes. We also present a version of the simplicial Hausdorff metric for filtered complexes and show results on its computational complexity. In addition, we discuss concerns about the monotonicity of the measurement functions involved in the setup of the simplicial complexes.

翻译：拓扑数据分析中的许多实际应用源自点云形式的数据，这些数据随后产生单纯复形。单纯复形的组合结构捕捉了复形元素之间的拓扑关系。除了组合结构外，单纯复形还具有几何实现，这为可视化复形和理解其几何性质提供了具体方法。本文提出了一种修正的豪斯多夫距离作为扩展度量，该度量将几何邻近性与单纯复形的拓扑特征相结合。我们还提出了适用于过滤复形的单纯豪斯多夫度量版本，并展示了其计算复杂度的结果。此外，我们讨论了在单纯复形构建过程中涉及的测量函数单调性问题。