Quantitative Simplification of Filtered Simplicial Complexes

Abstract

We introduce a new invariant defined on the vertices of a given filtered simplicial complex, called codensity, which controls the impact of removing vertices on the persistent homology of this filtered complex. We achieve this control through the use of an interleaving type of distance between filtered simplicial complexes. We study the special case of Vietoris–Rips filtrations and show that our bounds offer a significant improvement over the immediate bounds coming from considerations related to the Gromov–Hausdorff distance. Based on these ideas we give an iterative method for the practical simplification of filtered simplicial complexes. As a byproduct of our analysis we identify a notion of core of a filtered simplicial complex which admits the interpretation as a minimalistic simplicial filtration which retains all the persistent homology information.