论文标题
持续图束：葡萄园的多维概括
Persistence Diagram Bundles: A Multidimensional Generalization of Vineyards
论文作者
论文摘要
我介绍了持久图（PD）捆绑的概念，这是用于光纤过滤功能的PDS的空间（一组$ \ {f_p：\ Mathcal {k}^p \ to \ Mathbb {r} \} \} _ {r} _ {p \ {特殊情况包括葡萄园，持续的综合转换和用于多参数持续模块的纤维条形码。我证明，如果$ b $是一种平稳的紧凑型歧管，那么对于通用的纤维过滤功能，$ b $进行分层，以使每个层中$ y \ y \ subseteq b $，则可以使用单个PD“模板”（“出生”和“死亡”简单列表），可用于为FELTRATION $ f_p $ $ f_p $ $ pd $ pd $ pd $ pd。如果$ b $紧凑，那么有很多层次，因此$ b $上的通用纤维过滤的PD捆绑由持续的同源性在$ b $中有限的许多点确定。我还表明，并非每个本地部分都可以扩展到全球部分（从$ b $到PDS的总空间$ e $，因此$ s（p）\ in \ textrm {pd}（f_p）$ in B $中的所有$ p \）。因此，PD捆绑包不一定是“藤蔓” $γ：b \ to e $的结合；这与葡萄园不同。如上所述，当存在分层时，我构建了一个蜂窝支架，该细胞支架存储足够的数据以构建部分并确定是否可以将给定的本地部分扩展到全局部分。
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" $γ: B \to E$; 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.
