Interval Multiplicities of Persistence Modules

TL;DR

This paper provides a generalized formula for computing the multiplicities of interval modules in persistence modules over finite posets, enabling efficient decomposition analysis and property study without full module decomposition.

Contribution

It introduces a new formula for interval multiplicities in persistence modules over arbitrary finite posets, extending 1D persistence results and improving computational efficiency.

Findings

Formula for multiplicities using matrix ranks

Method to determine interval-decomposability

Efficient computation via poset maps and resolutions

Abstract

For any persistence module $M$ over a finite poset $\mathbf{P}$, and any interval $I$ of $\mathbf{P}$, 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. As an easy application, the formula enables us to compute the maximal interval-decomposable direct summand of $M$, which gives us a way to decide whether $M$ is interval-decomposable or not. In addition, when a set $V_{I_1}, \dots, V_{I_n}$ of interval direct summands of $M$ gives a specific property of $M$, we can study this property by directly computing the multiplicities $d_M(V_{I_i})$ by our formula without decomposing $M$. 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 $\zeta \colon Z \to \mathbf{P}$ 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 $\zeta$ 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 $M$. Finally, we also give a formula of $d_M(V_I)$ in terms of a projective (or injective) (co)presentation of $M$ rather than its structure linear maps. In the 2D-grid case, this formula is more practical because in that case resolutions of $M$ can be computed from the filtration level of topological spaces.