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Decomposing Multiparameter Persistence Modules

Authors Tamal K. Dey , Jan Jendrysiak , Michael Kerber

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  • Mathematics of computing → Algebraic topology
Keywords
  • Topological Data Analysis
  • Multiparameter Persistence Modules
  • Persistence
  • Decomposition

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Abstract

Dey and Xin (J.Appl.Comput.Top., 2022) 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 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 of the same degree and with further optimisation we obtain an O(n³) time 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 aida, the first to enable the decomposition of large inputs. We show its capabilities via extensive experimental evaluation.

Cite As Get BibTex

Tamal K. Dey, Jan Jendrysiak, and Michael Kerber. Decomposing Multiparameter Persistence Modules. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 41:1-41:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.SoCG.2025.41

Author Details

Tamal K. Dey
  • Department of Computer Science, Purdue University, West Lafayette, IN, USA
Jan Jendrysiak
  • Graz University of Technology, Austria
Michael Kerber
  • Graz University of Technology, Austria

Funding

  • Dey, Tamal K.: Supported by NSF grants NSF grants CCF-2437030 and DMS-2301360.
  • Jendrysiak, Jan: Supported by Austrian Science Fund (FWF) grant P 33765-N and W1230.
  • Kerber, Michael: Supported by the Austrian Science Fund (FWF) grant P 33765-N.

Acknowledgements

The second author wants to thank Klaus Lux for his insights and discussions about the Meat-Axe algorithm and Algorithmic Representation Theory, as well as Martin Kalck for pointers to relevant established theory.

Supplementary Materials

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