2022/10/11 by Hickok, Abigail · 1 citation
#55N31 #55R99 #57N80 #Algebraic Topology (math.AT) #Computational Geometry (cs.CG) #FOS: Computer and information sciences #FOS: Mathematics
paper · doi:10.48550/arxiv.2210.05124
I introduce the concept of a persistence diagram (PD) bundle, which is the space of PDs for a fibered filtration function (a set \fp: Kp → ℝ\p ∈ B of filtrations that is parameterized by a topological space B). Special cases include vineyards, the persistent homology transform, and fibered barcodes for multiparameter persistence modules. I prove that if B is a smooth compact manifold, then for a generic fibered filtration function, B is stratified such that within each stratum Y ⊆ B, there is a single PD "template" (a list of "birth" and "death" simplices) that can be used to obtain the PD for the filtration fp for any p ∈ Y. If B is compact, then there are finitely many strata, so the PD bundle for a generic fibered filtration on B is determined by the persistent homology at finitely many points in B. I also show that not every local section can be extended to a global section (a continuous map s from B to the total space E of PDs such that s(p) ∈ \textrmPD(fp) for all p ∈ B). Consequently, a PD bundle is not necessarily the union of "vines" γ: B → E; this is unlike a vineyard. When there is a stratification as described above, I construct a cellular sheaf that stores sufficient data to construct sections and determine whether a given local section can be extended to a global section.