Filtration Simplification for Persistent Homology via Edge Contraction

TL;DR

This paper introduces two edge contraction methods for simplifying filtrations in persistent homology, providing bounds on diagram perturbations and enabling efficient multiple contractions, with experiments on manifolds.

Contribution

It proposes novel contraction operators that bound persistence diagram perturbations and can be efficiently composed for complex simplicial complexes.

Findings

Bound the perturbation in persistence diagrams using contraction operators

Operators work for 2-manifolds and arbitrary d-complexes

Efficient composition of multiple contractions demonstrated on manifolds

Abstract

Persistent homology is a popular data analysis technique that is used to capture the changing topology of a filtration associated with some simplicial complex $K$. These topological changes are summarized in persistence diagrams. We propose two contraction operators which when applied to $K$ and its associated filtration, bound the perturbation in the persistence diagrams. The first assumes that the underlying space of $K$ is a $2$-manifold and ensures that simplices are paired with the same simplices in the contracted complex as they are in the original. The second is for arbitrary $d$-complexes, and bounds the bottleneck distance between the initial and contracted $p$-dimensional persistence diagrams. This is accomplished by defining interleaving maps between persistence modules which arise from chain maps defined over the filtrations. In addition, we show how the second operator can efficiently compose across multiple contractions. We conclude with experiments demonstrating the second operator's utility on manifolds.