2020/06/15 by Haibin Hang, Washington Mio, Hang, Haibin +1 · 2 citations
Computer Science · Mathematics · #18F20 (Secondary) #55N31 (Primary) #62R40 (Primary) #Algebraic Topology (math.AT) #Commutative Algebra and Its Applications #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2006.08557
openalex publication_date 2020/06/15 · openalex created_date 2023/01/06 · openalex updated_date 2026/07/28
We develop a unifying framework for the treatment of various persistent homology architectures using the notion of correspondence modules. In this formulation, morphisms between vector spaces are given by partial linear relations, as opposed to linear mappings. In the one-dimensional case, among other things, this allows us to: (i) treat persistence modules and zigzag modules as algebraic objects of the same type; (ii) give a categorical formulation of zigzag structures over a continuous parameter; and (iii) construct barcodes associated with spaces and mappings that are richer in geometric information. A structural analysis of one-parameter persistence is carried out at the level of sections of correspondence modules that yield sheaf-like structures, termed persistence sheaves. Under some tameness hypotheses, we prove interval decomposition theorems for persistence sheaves and correspondence modules, as well as an isometry theorem for persistence diagrams obtained from interval decompositions. Applications include: (a) a Mayer-Vietoris sequence that relates the persistent homology of sublevelset filtrations and superlevelset filtrations to the levelset homology module of a real-valued function and (b) the construction of slices of 2-parameter persistence modules along negatively sloped lines.