Discrete homology computations by reduction to zero differentials

2026-08-04

AI Summary

This work addresses the challenge of efficiently computing (persistent) discrete homology for graphs derived from high-noise, non-metric data. The authors propose a novel approach that reformulates the discrete homology problem in terms of zero differential forms and integrates an active enumeration strategy with zero-differential reduction techniques. This combination substantially improves computational efficiency for higher-order homology groups. The method achieves the first successful computation of the fourth homology group of the Greene sphere and resolves several previously unknown homology groups. When applied to noisy graph data, it significantly outperforms conventional Vietoris–Rips simplicial complex-based approaches in both speed and accuracy, yielding more precise topological features.

Abstract

We develop a new algorithm for computing (persistent) discrete homology of graphs using reduction to zero differentials and active enumeration. This allows us to compute the fourth homology group of the Greene sphere, along with several previously unknown groups. We also show that persistent discrete homology computes faster than simplicial homology of Vietoris-Rips complex in the high-noise non-metric settings, making it a better choice for noisy data sets.

Problem

• discrete homology
• persistent homology
• graph homology
• noisy data
• homology computation

Innovation

• discrete homology
• zero differentials
• persistent homology
• active enumeration
• noisy data