Harder-Narasimhan filtrations of persistence modules: metric stability

TL;DR

This paper extends the concept of Harder-Narasimhan filtrations to infinite multiparameter persistence modules, introducing a stable skyscraper invariant that refines existing invariants and is robust under perturbations.

Contribution

It generalizes the skyscraper invariant to infinite settings and proves its stability, enhancing the analysis of multiparameter persistence modules.

Findings

Extended skyscraper invariant to infinite modules over al^n and b^n

Proved erosion-type stability for the extended invariant

Demonstrated the invariant's discriminating power over previous invariants

Abstract

The Harder-Narasimhan types are a family of discrete isomorphism invariants for representations of finite quivers. Previously (arXiv:2303.16075), we evaluated their discriminating power in the context of persistence modules over a finite poset, including multiparameter persistence modules (over a finite grid). In particular, we introduced the skyscraper invariant and proved it was strictly finer than the rank invariant. In order to study the stability of the skyscraper invariant, we extend its definition from the finite to the infinite setting and consider multiparameter persistence modules over $\mathbb Z ^n$ and $\mathbb R^n$. We then establish an erosion-type stability result for this version of the skyscraper invariant.