Persistent 1-Cycles: Definition, Computation, and Its Application

TL;DR

This paper introduces a polynomial-time algorithm for computing meaningful persistent 1-cycles in homology, addressing the NP-hardness of optimal cycles, and demonstrates its effectiveness on diverse datasets.

Contribution

It defines persistent 1-cycles based on interval module decomposition, proposes an efficient algorithm, and analyzes stability issues in persistent homology.

Findings

Efficient polynomial-time algorithm for persistent 1-cycles.

Effective application to 3D point clouds, mineral structures, and images.

Insights into stability limitations of persistent 1-cycles.

Abstract

Persistence diagrams, which summarize the birth and death of homological features extracted from data, are employed as stable signatures for applications in image analysis and other areas. Besides simply considering the multiset of intervals included in a persistence diagram, some applications need to find representative cycles for the intervals. In this paper, we address the problem of computing these representative cycles, termed as persistent 1-cycles, for $\text{H}_1$-persistent homology with $\mathbb{Z}_2$ coefficients. The definition of persistent cycles is based on the interval module decomposition of persistence modules, which reveals the structure of persistent homology. After showing that the computation of the optimal persistent 1-cycles is NP-hard, we propose an alternative set of meaningful persistent 1-cycles that can be computed with an efficient polynomial time algorithm. We also inspect the stability issues of the optimal persistent 1-cycles and the persistent 1-cycles computed by our algorithm with the observation that the perturbations of both cannot be properly bounded. We design a software which applies our algorithm to various datasets. Experiments on 3D point clouds, mineral structures, and images show the effectiveness of our algorithm in practice.