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arXiv · 2504.08119

Decomposing Multiparameter Persistence Modules

Tamal K. Dey·Jan Jendrysiak·Michael Kerber
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Abstract

Dey and Xin (J.Appl.Comput.Top., 2022, arXiv:1904.03766) describe an algorithm to decompose finitely presented multiparameter persistence modules using a matrix reduction algorithm. Their algorithm only works for modules whose generators and relations are distinctly graded. We extend their approach to work on \emph{all} finitely presented modules and introduce several improvements that lead to significant speed-ups in practice. Our algorithm is fixed parameter tractable with respect to the maximal number of relations with the same degree and with further optimisation we obtain an $O(n^3)$ algorithm for interval-decomposable modules. In particular, we can decide interval-decomposability in this time. As a by-product to the proofs of correctness we develop a theory of parameter restriction for persistence modules. Our algorithm is implemented as a software library \textsc{aida} which is the first to enable the decomposition of large inputs. We show its capabilities via extensive experimental evaluation.

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Tamal K. Dey, Jan Jendrysiak, Michael Kerber. 2025-11-22. Decomposing Multiparameter Persistence Modules. https://arxiv.org/abs/2504.08119

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