eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-06-09
25:1
25:16
10.4230/LIPIcs.SoCG.2023.25
article
Meta-Diagrams for 2-Parameter Persistence
Clause, Nate
1
Dey, Tamal K.
2
Mémoli, Facundo
1
Wang, Bei
3
Ohio State University, Columbus, OH, USA
Purdue University, West Lafayette, IN, USA
University of Utah, Salt Lake City, UT, USA
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol258-socg2023/LIPIcs.SoCG.2023.25/LIPIcs.SoCG.2023.25.pdf
Multiparameter persistence modules
persistent homology
Möbius inversion
barcodes
computational topology
topological data analysis