2026-07-25T01:58:05Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:17875 2024-03-06T11:00:35Z ddc:004 open_access

Meta-Diagrams for 2-Parameter Persistence Clause, Nate Dey, Tamal K. Mémoli, Facundo Wang, Bei Multiparameter persistence modules persistent homology Möbius inversion barcodes computational topology topological data analysis We first introduce the notion of meta-rank for a 2-parameter persistence module, an invariant that captures the information behind images of morphisms between 1D slices of the module. We then define the meta-diagram of a 2-parameter persistence module to be the Möbius inversion of the meta-rank, resulting in a function that takes values from signed 1-parameter persistence modules. We show that the meta-rank and meta-diagram contain information equivalent to the rank invariant and the signed barcode. This equivalence leads to computational benefits, as we introduce an algorithm for computing the meta-rank and meta-diagram of a 2-parameter module M indexed by a bifiltration of n simplices in O(n³) time. This implies an improvement upon the existing algorithm for computing the signed barcode, which has O(n⁴) time complexity. This also allows us to improve the existing upper bound on the number of rectangles in the rank decomposition of M from O(n⁴) to O(n³). In addition, we define notions of erosion distance between meta-ranks and between meta-diagrams, and show that under these distances, meta-ranks and meta-diagrams are stable with respect to the interleaving distance. Lastly, the meta-diagram can be visualized in an intuitive fashion as a persistence diagram of diagrams, which generalizes the well-understood persistence diagram in the 1-parameter setting. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Nate Clause and Tamal K. Dey and Facundo Mémoli and Bei Wang 2023 Is Part Of LIPIcs, Volume 258, 39th International Symposium on Computational Geometry (SoCG 2023) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.SoCG.2023.25 urn:nbn:de:0030-drops-178754 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.25 eng https://creativecommons.org/licenses/by/4.0/legalcode