Computing Optimal Persistent Cycles for Levelset Zigzag on Manifold-like Complexes

TL;DR

This paper introduces polynomial-time algorithms for computing optimal persistent cycles in levelset zigzag persistence, capturing the evolution of topological features in manifold-like complexes, with practical applications demonstrated through empirical results.

Contribution

It extends persistent homology to levelset zigzag persistence, providing the first polynomial-time algorithms for optimal cycle sequences in this setting.

Findings

Algorithms successfully compute optimal cycle sequences.

Empirical results show high-quality cycles in practice.

Applicable to weak pseudomanifolds in real-world data.

Abstract

In standard persistent homology, a persistent cycle born and dying with a persistence interval (bar) associates the bar with a concrete topological representative, which provides means to effectively navigate back from the barcode to the topological space. Among the possibly many, optimal persistent cycles bring forth further information due to having guaranteed quality. However, topological features usually go through variations in the lifecycle of a bar which a single persistent cycle may not capture. Hence, for persistent homology induced from PL functions, we propose levelset persistent cycles consisting of a sequence of cycles that depict the evolution of homological features from birth to death. Our definition is based on levelset zigzag persistence which involves four types of persistence intervals as opposed to the two types in standard persistence. For each of the four types, we present a polynomial-time algorithm computing an optimal sequence of levelset persistent $p$-cycles for the so-called weak $(p+1)$-pseudomanifolds. Given that optimal cycle problems for homology are NP-hard in general, our results are useful in practice because weak pseudomanifolds do appear in applications. Our algorithms draw upon an idea of relating optimal cycles to min-cuts in a graph that was exploited earlier for standard persistent cycles. Notice that levelset zigzag poses non-trivial challenges for the approach because a sequence of optimal cycles instead of a single one needs to be computed in this case. We show some empirical evidence that optimal cycles produced by our implemented software have nice quality.