Accelerating Iterated Persistent Homology Computations with Warm Starts

TL;DR

This paper introduces a method to speed up persistent homology calculations on similar topological spaces by efficiently updating matrix factorizations, significantly reducing computation time in practical applications.

Contribution

It extends existing update schemes to handle cell additions and deletions in filtrations, enabling faster computations through batch processing of changes.

Findings

Achieves practical speedups in feature generation tasks

Reduces computational complexity for small changes in filtrations

Demonstrates efficiency in optimization problems guided by persistent homology

Abstract

Persistent homology is a topological feature used in a variety of applications such as generating features for data analysis and penalizing optimization problems. We develop an approach to accelerate persistent homology computations performed on many similar filtered topological spaces which is based on updating associated matrix factorizations. Our approach improves the update scheme of Cohen-Steiner, Edelsbrunner, and Morozov for permutations by additionally handling addition and deletion of cells in a filtered topological space and by processing changes in a single batch. We show that the complexity of our scheme scales with the number of elementary changes to the filtration which as a result is often less expensive than the full persistent homology computation. Finally, we perform computational experiments demonstrating practical speedups in several situations including feature generation and optimization guided by persistent homology.