On approximation of 2D persistence modules by interval-decomposables

In this work, we propose a new invariant for 2D 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 2D persistence module M, we propose an “interval-decomposable replacement” ${\delta }^{⁎}\left(M\right)$ (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 ${\delta }^{⁎}\left(M\right)$ is equal to M in the split Grothendieck group. Furthermore, even for modules M not necessarily interval-decomposable, ${\delta }^{⁎}\left(M\right)$ preserves the dimension vector and the rank invariant of M. In addition, we provide an algorithm to compute ${\delta }^{⁎}\left(M\right)$ (a high-level algorithm in the general case, and a detailed algorithm for the size $2\times n$ case).