License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SoCG.2023.25
URN: urn:nbn:de:0030-drops-178754
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2023/17875/
Clause, Nate ;
Dey, Tamal K. ;
Mémoli, Facundo ;
Wang, Bei
Meta-Diagrams for 2-Parameter Persistence
Abstract
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.
BibTeX - Entry
@InProceedings{clause_et_al:LIPIcs.SoCG.2023.25,
author = {Clause, Nate and Dey, Tamal K. and M\'{e}moli, Facundo and Wang, Bei},
title = {{Meta-Diagrams for 2-Parameter Persistence}},
booktitle = {39th International Symposium on Computational Geometry (SoCG 2023)},
pages = {25:1--25:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-273-0},
ISSN = {1868-8969},
year = {2023},
volume = {258},
editor = {Chambers, Erin W. and Gudmundsson, Joachim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2023/17875},
URN = {urn:nbn:de:0030-drops-178754},
doi = {10.4230/LIPIcs.SoCG.2023.25},
annote = {Keywords: Multiparameter persistence modules, persistent homology, M\"{o}bius inversion, barcodes, computational topology, topological data analysis}
}
Keywords: |
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Multiparameter persistence modules, persistent homology, Möbius inversion, barcodes, computational topology, topological data analysis |
Collection: |
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39th International Symposium on Computational Geometry (SoCG 2023) |
Issue Date: |
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2023 |
Date of publication: |
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09.06.2023 |