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Meta-Diagrams for 2-Parameter Persistence

Authors Nate Clause, Tamal K. Dey, Facundo Mémoli, Bei Wang

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ACM Subject Classification
  • Theory of computation → Computational geometry
  • Mathematics of computing → Topology
Keywords
  • Multiparameter persistence modules
  • persistent homology
  • Möbius inversion
  • barcodes
  • computational topology
  • topological data analysis

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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.

Cite As Get BibTex

Nate Clause, Tamal K. Dey, Facundo Mémoli, and Bei Wang. Meta-Diagrams for 2-Parameter Persistence. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 25:1-25:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.SoCG.2023.25

Author Details

Nate Clause
  • Ohio State University, Columbus, OH, USA
Tamal K. Dey
  • Purdue University, West Lafayette, IN, USA
Facundo Mémoli
  • Ohio State University, Columbus, OH, USA
Bei Wang
  • University of Utah, Salt Lake City, UT, USA

Funding

  • Clause, Nate: NC is partially supported by NSF CCF 1839356 and NSF DMS 1547357.
  • Dey, Tamal K.: TD is partially supported by NSF CCF 2049010.
  • Mémoli, Facundo: FM is partially supported by BSF 2020124, NSF CCF 1740761, NSF CCF 1839358, and NSF IIS 1901360.
  • Wang, Bei: BW is partially supported by NSF IIS 2145499, NSF IIS 1910733, and DOE DE SC0021015.

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