论文标题
$ \ ell _ {\ infty} $ metric中的持续同源性
Persistent Homology in $\ell_{\infty}$ Metric
论文作者
论文摘要
接近络合物和过滤是拓扑数据分析中的中心结构。它们使用距离功能或更一般的指标构建，通常用于从点云中推断连接信息。在这里，我们研究了建立在Chebyshev公制上的邻近络合物和过滤，也称为最大度量或$ \ ell _ {\ infty} $公制，而不是经典的Euclidean Metric。令人惊讶的是，尚未对$ \ ell _ {\ infty} $ case进行彻底调查。在本文中，我们检查了该指标下的许多经典络合物，包括čech，越野河流和α络合物。我们定义了两个新的国旗复合体系列，我们称之为Alpha Flag和Minibox Complexs，并证明它们等于同源度零，一个。此外，我们提供了寻找两个，三个和高维点的迷你贝克斯边缘的算法。最后，我们在随机点上介绍了计算实验，这表明Minibox过滤通常可以用来以零同源度的持续同源性计算加快持续的同源性计算，并通过减少过滤中的简单数量来加快持续的同源度计算。
Proximity complexes and filtrations are central constructions in topological data analysis. Built using distance functions, or more generally metrics, they are often used to infer connectivity information from point clouds. Here 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 Čech, 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 to Čech complexes in homological degrees zero and one. Moreover, we provide algorithms for finding Minibox edges of two, three, and higher-dimensional points. Finally, we present computational experiments on random points, which shows that Minibox filtrations can often be used to speed up persistent homology computations in homological degrees zero and one by reducing the number of simplices in the filtration.
