Parameterized Vietoris-Rips Filtrations via Covers

TL;DR

This paper introduces a method to analyze large Vietoris-Rips filtrations by using covers and acyclic carriers, enabling more efficient computation and better understanding of data topology.

Contribution

It extends acyclic carriers to persistent homology, providing bounds on restricted filtrations and guiding cover selection for data analysis.

Findings

Bounds on the relationship between restricted and full filtrations

Effective cover selection for data topology analysis

Application to high-dimensional image patch data

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.