Proximity complexes and filtrations are a central construction in topological data analysis. Built using distance functions or more generally metrics, they are often used to infer connectivity information from point clouds. We investigate proximity complexes and filtrations built over the Chebyshev metric, also known as the maximum metric or $\ell_{\infty}$ metric, rather than the classical Euclidean metric. Somewhat surprisingly, the $\ell_{\infty}$ case has not been investigated thoroughly. In this paper, we examine a number of classical complexes under this metric, including the \v{C}ech, Vietoris-Rips, and Alpha complexes. We define two new families of flag complexes, which we call the Alpha flag and Minibox complexes, and prove their equivalence with \v{C}ech complexes in homological dimensions zero and one. Moreover, we provide algorithms for finding Minibox edges for two, three and higher dimensional points. Finally we run computational experiments on random points, which show that Minibox filtrations can often be used to reduce the number of simplices included in \v{C}ech filtrations, and so speed up persistent homology computations.

中文翻译：

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$\ell_{\infty}$ 度量中的持久同源性

邻近复合体和过滤是拓扑数据分析的核心结构。使用距离函数或更一般的度量构建，它们通常用于从点云推断连接信息。我们研究建立在 Chebyshev 度量（也称为最大度量或 $\ell_{\infty}$ 度量）上的邻近复合体和过滤，而不是经典的欧几里德度量。有点令人惊讶的是，$\ell_{\infty}$ 案件尚未得到彻底调查。在本文中，我们研究了该度量下的许多经典配合物，包括 \v{C}ech、Vietoris-Rips 和 Alpha 配合物。我们定义了两个新的标志复合体系列，我们称之为 Alpha 标志和 Minibox 复合体，并证明它们与 0 和 1 同调维度中的 \v{C}ech 复合体的等效性。而且，我们提供了为二维、三维和更高维点寻找 Minibox 边的算法。最后，我们在随机点上运行计算实验，这表明 Minibox 过滤通常可用于减少 \v{C}ech 过滤中包含的单纯形数量，从而加快持久同源性计算。