Persistence Diagram Bundles: A Multidimensional Generalization of Vineyards
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A persistence diagram (PD) summarizes the persistent homology of a filtration. I introduce the concept of a persistence diagram bundle, which is the space of PDs associated with a fibered filtration function (a set {f_t: 𝒦^t →ℝ}_t ∈𝒯 of filtrations parameterized by a topological space 𝒯). Special cases include vineyards, the persistent homology transform, and fibered barcodes of multiparameter persistence modules. I prove that if 𝒯 is a compact n-dimensional manifold, then for generic fibered filtration functions, 𝒯 is stratified such that within each n-dimensional stratum S, there is a single PD "template" (a list of birth and death simplices) that can be used to obtain PD(f_t) for any t ∈ S. I also show that not every local section can be extended to a global section. Consequently, the points in the PDs do not typically trace out separate manifolds as t ∈𝒯 varies; this is unlike a vineyard, in which the points in the PDs trace out curves ("vines").
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