Complex Matching Distance and Stability for Minimal Projective Resolutions, with Applications to Persistence

TL;DR

This paper introduces a stability framework for minimal projective resolutions of modules over finite metric posets, extending classical persistence stability to multiparameter cases using complex matching distances.

Contribution

It develops a new complex matching distance and proves a stability theorem linking it to the G"ulen-McCleary distance, with applications to persistence diagrams and multiparameter persistence.

Findings

Extended the stability bounds to multiparameter persistence diagrams.

Introduced a new complex matching distance on projective complexes.

Reproduced classical bottleneck stability in the one-parameter case.

Abstract

We develop a stability theory for minimal projective resolutions of $\mathbf{P}$-modules, where $\mathbf{P}$ is a finite metric poset. We use the G\"ulen-McCleary distance on $\mathbf{P}$-modules together with a new complex matching distance on bounded complexes of finitely generated projective $\mathbf{P}$-modules. The latter yields an extended metric on homotopy classes of such complexes and restricts to minimal projective resolutions. Our main theorem shows that this induced distance on minimal projective resolutions is bounded above by the G\"ulen-McCleary distance. As an application, we pass to the interval poset and kernel construction, interpreting persistence diagrams as minimal projective resolutions of kernel modules. This gives a corresponding stability inequality, which in the one-parameter case recovers classical bottleneck stability and in the multiparameter case extends to signed diagrams arising from minimal projective resolutions.