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 \infty metric, rather than the classical Euclidean metric. Somewhat surprisingly, the \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 Cech, 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 Cech filtrations  and so speed up the computation of persistent homology .
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 \infty metric, rather than the classical Euclidean metric. Somewhat surprisingly, the \infty case has not been investigated thoroughly. In this paper, we examine a number of classical complexes under this metric, including the Cech, 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 \v{C which show that Minibox filtrations can often be used to reduce the number of simplices included in Cech filtrations , and so speed up persistent homology computations .
D
Our motivation lies in that this metric has the far simpler numerical tests which can lead to computational speedups for high-dimensional data analysis.
R
also introduce two new complexes
define two new families of flag complexes,
R
clique
flag
R
. We provide results on topological properties of these, as well as computational experiments
, and prove their equivalence with \v{C
R
these
Minibox filtrations
D
high-dimensional
A
,
R
the computation of persistent homology
persistent homology computations