Persistent Homology in $\ell_{\infty}$ Metric

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 \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 to \v{C}ech 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.