Computing Generalized Rank Invariant for 2-Parameter Persistence Modules via Zigzag Persistence and Its Applications
Computing Generalized Rank Invariant for 2-Parameter Persistence Modules via Zigzag Persistence and Its Applications
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DOI：
10.4230/lipics.socg.2022.34
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发表时间：
2021-11
影响因子：
0.8
通讯作者：
T. Dey;Woojin Kim;F. Mémoli
中科院分区：
文献类型：
--
作者：
T. Dey;Woojin Kim;F. Mémoli
多参数持久性的一般性等级已成为定义有趣的同源结构（例如广义持久性图）的重要组成部分，但是，其有效计算尚未在文献中研究，我们在文献中表明，在A $ \ textbf的有限级别上，Z 2 $ Z 2 -iS Z 2 -iS Z 2 - 被诱导因此，在i追踪其边界中，我们可以通过限制到该路径获得的曲折模块来计算i的通用等级。 ）$$ \ omega \ in [2,2.373）$$ω∈[2， 2.373）是矩阵乘法的指数。
The notion of generalized rank in the context of multiparameter persistence has become an important ingredient for defining interesting homological structures such as generalized persistence diagrams. However, its efficient computation has not yet been studied in the literature. We show that the generalized rank over a finite interval I of a $$\textbf{Z}^2$$ Z 2 -indexed persistence module M is equal to the generalized rank of the zigzag module that is induced on a certain path in I tracing mostly its boundary. Hence, we can compute the generalized rank of M over I by computing the barcode of the zigzag module obtained by restricting to that path. If M is the homology of a bifiltration F of $$t$$ t simplices (while accounting for multi-criticality) and I consists of $$t$$ t points, this computation takes $$O(t^\omega )$$ O ( t ω ) time where $$\omega \in [2,2.373)$$ ω ∈ [ 2 , 2.373 ) is the exponent of matrix multiplication. We apply this result to obtain an improved algorithm for the following problem. Given a bifiltration inducing a module M , determine whether M is interval decomposable and, if so, compute all intervals supporting its indecomposable summands.