The Jordan type of a multiparameter persistence module

TL;DR

This work defines a Jordan-type invariant for multiparameter persistence modules by restricting to a sequence of finite slices $\mathcal S$ and analyzing the induced nilpotent action $\mathbf T_{M,\mathcal S}$. It proves the completeness of multirank invariants for finite zigzag posets, resolving Thomas's zigzag conjecture, and introduces the Jordan filtered rank invariant $\mathbf{rk^{fil}_{\mathcal S}}(M)$, which refines the classical rank invariant and encodes the entire Jordan structure via the degree-$i$ modules $M^i_{\mathcal S}$. A general stability framework is developed, showing that erosion and persistence landscape distances between the Jordan-filtered invariants are bounded above by the interleaving distance, with a concrete template to derive erosion-type stability from interleavings. These results yield robust, finer-grained invariants for MPH data and provide theoretical guarantees for their stability under perturbations, with applications to computing invariants like bigraded Betti numbers in zigzag and grid settings.

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 $N$.

Theorems & Definitions (38)

• Theorem 1
• Theorem 2
• Definition 3
• Example 4
• Proposition 5
• proof
• Remark 6
• Corollary 7
• proof
• Example 8