Computing Generalized Rank Invariant for 2-Parameter Persistence Modules via Zigzag Persistence and Its Applications

Abstract

AbstractThe 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.