Parameterized Vietoris-Rips Filtrations via Covers
Abstract
A challenge in computational topology is to deal with large filtered geometric complexes built from point cloud data such as Vietoris-Rips filtrations. This has led to the development of schemes for parallel computation and compression which restrict simplices to lie in open sets in a cover of the data. We extend the method of acyclic carriers to the setting of persistent homology to give detailed bounds on the relationship between Vietoris-Rips filtrations restricted to covers and the full construction. We show how these complexes can be used to study data over a base space and use our results to guide the selection of covers of data. We demonstrate these techniques on a variety of covers, and show the utility of this construction in investigating higher-order homology of a model of high-dimensional image patches.
Cite
@article{arxiv.2205.01539,
title = {Parameterized Vietoris-Rips Filtrations via Covers},
author = {Bradley J. Nelson},
journal= {arXiv preprint arXiv:2205.01539},
year = {2022}
}
Comments
18 pages, 6 figures