Persistence Diagram Bundles: A Multidimensional Generalization of Vineyards

TL;DR

This paper introduces persistence diagram bundles as a multidimensional generalization of vineyards, providing a stratification framework and sheaf-theoretic tools to analyze the structure and sections of these bundles in topological data analysis.

Contribution

It formalizes the concept of PD bundles, establishes stratification results for generic fibered filtrations, and develops sheaf-theoretic methods to study sections and their extensions.

Findings

PD bundles are stratified into finitely many strata for generic functions.

Not all local sections extend to global sections, indicating complex bundle structure.

A cellular sheaf can be constructed to analyze sections and their extendability.

Abstract

I introduce the concept of a persistence diagram (PD) bundle, which is the space of PDs for a fibered filtration function (a set $\{f_p: \mathcal{K}^p \to \mathbb{R}\}_{p \in B}$ 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 $Y \subseteq B$, there is a single PD "template" (a list of "birth" and "death" simplices) that can be used to obtain the PD for the filtration $f_p$ for any $p \in Y$. 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 $s(p) \in \textrm{PD}(f_p)$ for all $p \in B$). Consequently, a PD bundle is not necessarily the union of "vines" $\gamma: B \to E$; 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.