On Approximation of $2$D Persistence Modules by Interval-decomposables

In this work, we propose a new invariant for $2$D persistence modules called the compressed multiplicity and show that it generalizes the notions of the dimension vector and the rank invariant. In addition, for a $2$D persistence module $M$, we propose an "interval-decomposable replacement" $\delta^{\ast}(M)$ (in the split Grothendieck group of the category of persistence modules), which is expressed by a pair of interval-decomposable modules, that is, its positive and negative parts. We show that $M$ is interval-decomposable if and only if $\delta^{\ast}(M)$ is equal to $M$ in the split Grothendieck group. Furthermore, even for modules $M$ not necessarily interval-decomposable, $\delta^{\ast}(M)$ preserves the dimension vector and the rank invariant of $M$. In addition, we provide an algorithm to compute $\delta^{\ast}(M)$ (a high-level algorithm in the general case, and a detailed algorithm for the size $2\times n$ case).