Log In Sign Up

Persistent Homology in ℓ_∞ Metric

by   Gabriele Beltramo, et al.

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.


DTM-based filtrations

Despite strong stability properties, the persistent homology of filtrati...

A Primer on Persistent Homology of Finite Metric Spaces

TDA (topological data analysis) is a relatively new area of research rel...

ε-net Induced Lazy Witness Complexes on Graphs

Computation of persistent homology of simplicial representations such as...

A higher homotopic extension of persistent (co)homology

Our objective in this article is to show a possibly interesting structur...

An introduction to Topological Data Analysis: fundamental and practical aspects for data scientists

Topological Data Analysis (tda) is a recent and fast growing eld providi...

On the effectiveness of persistent homology

Persistent homology (PH) is one of the most popular methods in Topologic...

Lipschitz (non-)equivalence of the Gromov–Hausdorff distances, including on ultrametric spaces

The Gromov–Hausdorff distance measures the difference in shape between c...