2024/03/12 by Tamal K. Dey, Xin Cheng, Dey, Tamal K. +1
Computer Science · #Algebraic Topology (math.AT) #Computational Geometry (cs.CG) #FOS: Computer and information sciences #FOS: Mathematics #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2403.08110
openalex publication_date 2024/03/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For a P-indexed persistence module \sf M, the (generalized) rank of \sf M is defined as the rank of the limit-to-colimit map for the diagram of vector spaces of \sf M over the poset P. For 2-parameter persistence modules, recently a zigzag persistence based algorithm has been proposed that takes advantage of the fact that generalized rank for 2-parameter modules is equal to the number of full intervals in a zigzag module defined on the boundary of the poset. Analogous definition of boundary for d-parameter persistence modules or general P-indexed persistence modules does not seem plausible. To overcome this difficulty, we first unfold a given P-indexed module \sf M into a zigzag module \sf MZZ and then check how many full interval modules in a decomposition of \sf MZZ can be folded back to remain full in a decomposition of \sf M. This number determines the generalized rank of \sf M. For special cases of degree-d homology for d-complexes, we obtain a more efficient algorithm including a linear time algorithm for degree-1 homology in graphs.