Persistence Diagrams as Diagrams: A Categorification of the Stability Theorem
Persistence Diagrams as Diagrams: A Categorification of the Stability Theorem
复制标题
持久性图作为图：稳定性定理的分类
DOI：
10.1007/978-3-030-43408-3_3
复制
发表时间：
2016
期刊：
影响因子：
--
通讯作者：
M. Lesnick
中科院分区：
文献类型：
--
作者：
Ulrich Bauer;M. Lesnick
持久化同构是拓扑数据分析的核心工具，它提供了称为条形码（也称为持久化图）的数据的不变量。条形码只是实线上的多组间隔。Edelsbrunner， Jablonski和Mrozek最近的工作提出了一个等价的条形码描述为函子R→Mch，其中R是实数的偏序集范畴，而Mch是对象为集合且其态射为匹配（即偏内射函数）的范畴。这样的函子形成了一个范畴MchR，它的态射是自然变换。因此，这种条形码的解释给了我们一个迄今为止尚未研究的条形码分类结构。我们证明了这种范畴结构导致了著名的持久同调稳定性定理和最近的一个推广称为诱导匹配定理的惊人的简单的重新表述。这些重新表述第一次清楚地表明，这两个结果都可以理解为保留了某些分类结构。我们还表明，这种观点导致了诱导匹配定理证明的一个更系统的变体。
Persistent homology, a central tool of topological data analysis, provides invariants of data called barcodes (also known as persistence diagrams). A barcode is simply a multiset of intervals on the real line. Recent work of Edelsbrunner, Jablonski, and Mrozek suggests an equivalent description of barcodes as functors R →Mch, where R is the poset category of real numbers and Mch is the category whose objects are sets and whose morphisms are matchings (i.e., partial injective functions). Such functors form a category MchR whose morphisms are the natural transformations. Thus, this interpretation of barcodes gives us a hitherto unstudied categorical structure on barcodes. We show that this categorical structure leads to surprisingly simple reformulations of both the well-known stability theorem for persistent homology and a recent generalization called the induced matching theorem. These reformulations make clear for the first time that both of these results can be understood as the preservation of certain categorical structure. We also show that this perspective leads to a more systematic variant of the proof of the induced matching theorem.