Quantitative Simplification of Filtered Simplicial Complexes

TL;DR

This paper introduces the concept of codensity to quantify the impact of vertex removal on persistent homology in filtered simplicial complexes, providing improved bounds and a practical iterative simplification method.

Contribution

It defines a new invariant called codensity, offers tighter bounds for Vietoris-Rips filtrations, and proposes an iterative simplification approach preserving persistent homology.

Findings

Codensity controls vertex removal effects on persistent homology.

Improved bounds over Gromov-Hausdorff related bounds.

Practical iterative method for complex simplification.

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 persistent homology. We achieve this control through the use of an interleaving type of distance between fitered 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.