Persistent Homology in ℓ_∞ Metric
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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 ℓ_∞ metric, rather than the classical Euclidean metric. Somewhat surprisingly, the ℓ_∞ 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 with Čech 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 Čech filtrations, and so speed up persistent homology computations.
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