Meta-Diagrams for 2-Parameter Persistence

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

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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

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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)


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.

Subject Classification

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


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