Interval Multiplicities of Persistence Modules

TL;DR

This work delivers an explicit, rank-based formula for the interval multiplicities $d_M(V_I)$ of persistence modules over finite posets, generalizing the one-parameter case and enabling detection of interval-decomposability and extraction of maximal interval-decomposable summands. The core contribution is a unified formula expressed via ranks of block matrices built from the module’s structure maps, together with a specialized 2D-grid version; these results apply regardless of whether $V_I$ is injective, via almost split sequences when needed. A central practical advance is the essential-cover method, which transfers multiplicity computations from a complex poset to a simpler poset $Z$ (often a zigzag of Dynkin type $\mathbb{A}$), enabling efficient computation with existing zigzag persistence algorithms. The paper complements theoretical results with detailed examples on 2D-grids, Dynkin type $D$, and bipath posets, illustrating how to compute multiplicities directly from filtrations and how to avoid computing the full structure maps in favorable cases. Overall, this framework provides a tractable pathway to assess interval-decomposability and to derive maximal interval approximations in multi-parameter persistence.

Abstract

For any persistence module $M$ over a finite poset $\mathbf{P}$, and any interval $I$ in $\mathbf{P}$, we give a formula of the multiplicity $d_M(V_I)$ of the interval module $V_I$ in the indecomposable decomposition of $M$ in terms of the ranks of matrices consisting of structure linear maps of the module $M$, which gives a generalization of the corresponding formula for 1-dimensional persistence modules. This makes it possible to compute the maximal interval-decomposable direct summand of $M$, which gives us a way to decide whether $M$ is interval-decomposable or not. Moreover, the formula tells us which morphisms of $\mathbf{P}$ are essential to compute the multiplicity $d_M(V_I)$. This suggests us some order-preserving map $ζ\colon Z \to \mathbf{P}$ such that the induced restriction functor $R \colon \operatorname{mod} \mathbf{P} \to \operatorname{mod} Z$ has the property that the multiplicity $d:= d_{R(M)}(R(V_I))$ is equal to $d_M(V_I)$. In this case, we say that $ζ$ essentially covers $I$. If $Z$ can be taken as a poset of Dynkin type $\mathbb{A}$, also known as a zigzag poset, then the calculation of the multiplicity $d$ can be done more efficiently, starting from the filtration level of topological spaces. Thus this even makes it unnecessary to compute the structure linear maps of $M$.

Figures (4)

• Figure 2: An illustration of the formula of persistent Betti numbers and multiplicities in one-parameter persistent homology. The barcodes of the $1$st persistent homology are shown in blue. The red bar $[3,4]$ is the interval that we would like to demonstrate the computation of its multiplicity. The violet bars are intervals along which we take the ranks to recover the multiplicity.
• Figure 3: A $G_{5,2}$-filtration ${\mathcal{F}}$
• Figure 4: A $Z$-filtration ${\mathcal{F}}'$ where $\zeta \colon Z \to \mathbf{P}$ essentially covers $I$
• Figure 5: An illustration of essential cover of $I$

Theorems & Definitions (82)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Remark 2.4
• Definition 2.5: Interval modules
• Theorem 2.6: Krull--Schmidt
• Lemma 2.9
• proof
• Definition 2.10
• Definition 3.1