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

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

5 Nov 2019arXiv:1911.01637links table onlyarchive 2025-07-28

Hideto Asashiba, Emerson G. Escolar, Ken Nakashima, Michio Yoshiwaki

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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" δ^∗(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 δ^∗(M) is equal to M in the split Grothendieck group. Furthermore, even for modules M not necessarily interval-decomposable, δ^∗(M) preserves the dimension vector and the rank invariant of M. In addition, we provide an algorithm to compute δ^∗(M) (a high-level algorithm in the general case, and a detailed algorithm for the size 2×n case).

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