Meta-Diagrams for 2-Parameter Persistence.
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| Title: | Meta-Diagrams for 2-Parameter Persistence. |
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| Authors: | Clause, Nate1 (AUTHOR) nate.clause@gmail.com, Dey, Tamal K.2 (AUTHOR) tamaldey@purdue.edu, Mémoli, Facundo1 (AUTHOR) facundo.memoli@gmail.com, Wang, Bei3 (AUTHOR) beiwang@sci.utah.edu |
| Source: | Discrete & Computational Geometry. Dec2025, Vol. 74 Issue 4, p872-898. 27p. |
| Subjects: | Algorithms, Möbius function, Mathematical invariants |
| 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. The equivalence leads to 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 (n 4) time. 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. [ABSTRACT FROM AUTHOR] |
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| Database: | Engineering Source |
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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. The equivalence leads to 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 (n 4) time. 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. [ABSTRACT FROM AUTHOR] |
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| ISSN: | 01795376 |
| DOI: | 10.1007/s00454-025-00741-6 |
