تقرير
Meta-Diagrams for 2-Parameter Persistence
| العنوان: | Meta-Diagrams for 2-Parameter Persistence |
|---|---|
| المؤلفون: | Clause, Nate, Dey, Tamal K., Mémoli, Facundo, Wang, Bei |
| سنة النشر: | 2023 |
| المجموعة: | Computer Science Mathematics |
| مصطلحات موضوعية: | Mathematics - Algebraic Topology, Computer Science - Computational Geometry |
| الوصف: | 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 M\"{o}bius 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 $O(n^3)$ time. This implies an improvement upon the existing algorithm for computing the signed barcode, which has $O(n^4)$ runtime. This also allows us to improve the existing upper bound on the number of rectangles in the rank decomposition of $M$ from $O(n^4)$ to $O(n^3)$. 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. Comment: 22 pages, 8 figures. Full version of the paper that is to appear in the Proceedings of the 39th International Symposium on Computational Geometry (SoCG 2023) |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2303.08270 |
| رقم الأكسشن: | edsarx.2303.08270 |
| قاعدة البيانات: | arXiv |
كن أول من يترك تعليقا!