The Jordan type of a multiparameter persistence module

TL;DR

This paper introduces the Jordan type of multiparameter persistence modules, proving its completeness in certain cases and demonstrating its finer discriminative power over classical invariants, with implications for stability and functoriality.

Contribution

It defines the Jordan type for multiparameter persistence modules, proves its completeness for finite zigzag posets, and establishes its functoriality and stability properties.

Findings

Multirank invariants are complete for finite zigzag posets.

Jordan filtered rank invariants are strictly finer than classical rank invariants.

Distances between invariants are bounded by interleaving distance.

Abstract

Let $\mathscr{P}$ be a poset and $\mathcal{S}$ a sequence of $n$ finite substes of $\mathscr{P}$. The Jordan type of a $\mathscr{P}$-persistence module $M$ at $\mathcal{S}$, denoted by $\mathsf{J}_{\mathcal{S}}(M) \in \mathbb{N}^n$, is defined as the Jordan type of a nilpotent operator $\mathbf{T}_{M, \mathcal{S}}$, which is constructed from $M$ and $\mathcal{S}$. When $n=2$, we recover the notion of multirank previously introduced and studied in [Tho19]. We first prove that the multirank invariants are complete for persistence modules over finite zigzag posets. This proves a conjecture of Thomas in the zigzag case. The nilpotent operator $\mathbf{T}_{M, \mathcal{S}}$ is functorial in $M$. When $\mathscr{P}=\mathbb{Z}^d$ or $\mathbb{R}^d$, this functoriality allows us to define the Jordan filtered rank invariant of $M$ at $\mathscr{S}$. We demonstrate that these invariants are strictly finer than the classical rank invariants. We next prove that for any two $\mathscr{P}$-persistence modules $M$ and $N$, the landscape and erosion distances between their Jordan filtered rank invariants are bounded from above by the interleaving distance between $M$ and $N$.