On Association between Absolute and Relative Zigzag Persistence

TL;DR

This paper establishes a weak duality for non-repetitive zigzag filtrations, enabling near-linear algorithms for computing certain persistence modules and improving computational efficiency in zigzag persistence analysis.

Contribution

It introduces a weak duality for non-repetitive zigzag filtrations and develops an efficient algorithm for computing persistence modules, reducing zigzag persistence to standard persistence computations.

Findings

Weak duality for non-repetitive zigzag filtrations.

Near-linear algorithm for computing persistence modules.

Performance gains in zigzag persistence computation.

Abstract

Duality results connecting persistence modules for absolute and relative homology provides a fundamental understanding into persistence theory. In this paper, we study similar associations in the context of zigzag persistence. Our main finding is a weak duality for the so-called non-repetitive zigzag filtrations in which a simplex is never added again after being deleted. The technique used to prove the duality for non-zigzag persistence does not extend straightforwardly to our case. Accordingly, taking a different route, we prove the weak duality by converting a non-repetitive filtration to an up-down filtration by a sequence of diamond switches. We then show an application of the weak duality result which gives a near-linear algorithm for computing the $p$-th and a subset of the $(p-1)$-th persistence for a non-repetitive zigzag filtration of a simplicial $p$-manifold. Utilizing the fact that a non-repetitive filtration admits an up-down filtration as its canonical form, we further reduce the problem of computing zigzag persistence for non-repetitive filtrations to the problem of computing standard persistence for which several efficient implementations exist. Our experiment shows that this achieves substantial performance gain. Our study also identifies repetitive filtrations as instances that fundamentally distinguish zigzag persistence from the standard persistence.