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

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. Our motivation lies in that this metric has the far simpler numerical tests which can lead to computational speedups for high-dimensional data analysis. In this paper, we examine a number of classical complexes under this metric, including the \v{C}ech, Vietoris-Rips, and Alpha complexes. We also introduce two new complexes which we call the Alpha clique and Minibox complexes. We provide results on topological properties of these, as well as computational experiments which show that these can often be used to reduce the number of high-dimensional simplices included in \v{C}ech filtrations and so speed up the computation of persistent homology.