Swap, Shift and Trim to Edge Collapse a Filtration

TL;DR

This paper revisits and extends an edge collapse algorithm for simplifying filtrations in persistent homology computations, introducing new methods and analyzing their effectiveness through experiments.

Contribution

It provides a clearer explanation of the edge collapse algorithm, proposes extensions like zigzag filtration simplification, and evaluates their performance experimentally.

Findings

Edge collapse effectively reduces filtration complexity.

Extensions improve simplification of zigzag filtrations.

Experimental results demonstrate the method's practical benefits.

Abstract

Boissonnat and Pritam introduced an algorithm to reduce a filtration of flag (or clique) complexes, which can in particular speed up the computation of its persistent homology. They used so-called edge collapse to reduce the input flag filtration and their reduction method required only the 1-skeleton of the filtration. In this paper we revisit the usage of edge collapse for efficient computation of persistence homology. We first give a simple and intuitive explanation of the principles underlying that algorithm. This in turn allows us to propose various extensions including a zigzag filtration simplification algorithm. We finally show some experiments to better understand how it behaves.