On Approximation of $2$D Persistence Modules by Intervaldecomposables
Abstract
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 "intervaldecomposable replacement" $\delta^{\ast}(M)$ (in the split Grothendieck group of the category of persistence modules), which is expressed by a pair of intervaldecomposable modules, that is, its positive and negative parts. We show that $M$ is intervaldecomposable if and only if $\delta^{\ast}(M)$ is equal to $M$ in the split Grothendieck group. Furthermore, even for modules $M$ not necessarily intervaldecomposable, $\delta^{\ast}(M)$ preserves the dimension vector and the rank invariant of $M$. In addition, we provide an algorithm to compute $\delta^{\ast}(M)$ (a highlevel algorithm in the general case, and a detailed algorithm for the size $2\times n$ case).
 Publication:

arXiv eprints
 Pub Date:
 November 2019
 DOI:
 10.48550/arXiv.1911.01637
 arXiv:
 arXiv:1911.01637
 Bibcode:
 2019arXiv191101637A
 Keywords:

 Mathematics  Representation Theory;
 Mathematics  Algebraic Topology
 EPrint:
 43 pages, changed term "intervaldecomposable approximation" to "intervaldecomposable replacement"