Fast Minimal Presentations of Bi-graded Persistence Modules

TL;DR

This paper introduces optimized algorithms for computing minimal presentations of bi-graded persistence modules in multi-parameter persistent homology, significantly reducing computation time for large datasets.

Contribution

It proposes and benchmarks new algorithmic improvements, including priority queues and integration with the multi-parameter chunk algorithm, outperforming previous methods.

Findings

Outperforms the LW-algorithm in speed and efficiency.

Capable of handling datasets with millions of simplices within seconds.

Provides publicly available software for practical use.

Abstract

Multi-parameter persistent homology is a recent branch of topological data analysis. In this area, data sets are investigated through the lens of homology with respect to two or more scale parameters. The high computational cost of many algorithms calls for a preprocessing step to reduce the input size. In general, a minimal presentation is the smallest possible representation of a persistence module. Lesnick and Wright proposed recently an algorithm (the LW-algorithm) for computing minimal presentations based on matrix reduction. In this work, we propose, implement and benchmark several improvements over the LW-algorithm. Most notably, we propose the use of priority queues to avoid extensive scanning of the matrix columns, which constitutes the computational bottleneck in the LW-algorithm, and we combine their algorithm with ideas from the multi-parameter chunk algorithm by Fugacci and Kerber. Our extensive experiments show that our algorithm outperforms the LW-algorithm and computes the minimal presentation for data sets with millions of simplices within a few seconds. Our software is publicly available.