  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.
