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

TL;DR

This paper explores the use of the $ ext{l}_{ ext{infinity}}$ metric in topological data analysis, introducing new complexes and algorithms that improve computational efficiency in persistent homology calculations.

Contribution

It introduces the Alpha flag and Minibox complexes, proves their equivalence to  complexes in certain degrees, and provides algorithms to compute Minibox edges, enhancing computational methods.

Findings

Minibox filtrations can speed up persistent homology computations.

Alpha flag and Minibox complexes are equivalent to  complexes in degrees zero and one.

Algorithms for finding Minibox edges in higher dimensions are provided.

Abstract

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.