Interval Replacements of Persistence Modules
Hideto Asashiba, Etienne Gauthier, Enhao Liu
TL;DR
This work develops interval replacements for persistence modules over finite posets by introducing compression systems and the interval rank invariant, thereby extending interval-decomposable approximations beyond 2D grids. It provides explicit formulas for the $I$-multiplicity and the interval rank under any compression system using structure maps and Auslander–Reiten theory, and proves that interval replacement preserves these invariants (Main results A and B). The essential-cover technique (Main result C) reduces invariant computations to restricted subposets, enabling practical calculation and yielding conditions under which different compression systems share identical invariants (Main result D). The framework connects to the generalized rank invariant of Kim–Mémoli, recovers zigzag-based computations in the 2D-grid setting, and includes a computational implementation to compute interval rank invariants and replacements for both total and source-sink compressions. Through a sequence of detailed constructions and examples, the paper demonstrates both the power and limitations of the interval-rank approach for summarizing and comparing multi-parameter persistence modules.
Abstract
We define two notions. The first one is a $rank\ compression\ system$ $ξ$ for a finite poset $\mathbf{P}$ that assigns each interval subposet $I$ to an order-preserving map $ξ_I \colon I^ξ \to \mathbf{P}$ satisfying some conditions, where $I^ξ$ is a connected finite poset. An example is given by the $total$ compression system that assigns each $I$ to the inclusion of $I$ into $\mathbf{P}$. The second one is an $I$-$rank$ of a persistence module $M$ under $ξ$, the family of which is called the $interval\ rank\ invariant$ of $M$ under $ξ$. A compression system $ξ$ makes it possible to define the $interval\ replacement$ (also called the interval-decomposable approximation) not only for 2D persistence modules but also for any persistence modules over any finite poset. We will show that the forming of the interval replacement preserves the interval rank invariant, which is a stronger property than the preservation of the usual rank invariant. Moreover, to know what is preserved by the replacement explicitly, we will give a formula of the $I$-rank of $M$ under $ξ$ in terms of the structure linear maps of $M$ for any compression system $ξ$. The formula leads us to a concept of essential cover, which gives us a sufficient condition for the $I$-rank of $M$ under $ξ$ to coincide with that under another compression system $ζ$. This is applied to the case where $ξ= \mathrm{tot}$, the value of $I$-rank under which is equal to the generalized rank invariant introduced by Kim--M{é}moli, to give an alternative proof of their theorem computing the generalized rank invariant by using a zigzag path.