论文标题
通信模块和持久性滑轮：对单参数持续同源性的统一观点
Correspondence Modules and Persistence Sheaves: A Unifying Perspective on One-Parameter Persistent Homology
论文作者
论文摘要
我们开发了一个统一的框架，用于使用对应模块的概念来处理各种持续的同源架构。在这种表述中，向量空间之间的形态是由部分线性关系给出的，而不是线性映射。在一维情况下，这使我们能够：（i）将持久性模块和曲折模块视为相同类型的代数对象； （ii）在连续参数上给出锯齿形结构的分类公式； （iii）构造与空间和映射相关的条形码，这些条形图和映射富含几何信息。一参数持久性的结构分析在产生类似捆的结构的对应模块的级别上进行，称为持久性束。在某些可驯服的假设下，我们证明了持久性滑轮和对应模块的间隔分解定理，以及从间隔分解获得的持久图的等轴测定理。应用程序包括：（a）将Sublevelset过滤的持续同源性和超级速度过滤与真实价值函数的LevelSet同源模块相关联的Mayer-Vietoris序列，以及（b）沿负面滑动线构建2-参数仪持久模块的切片。
We develop a unifying framework for the treatment of various persistent homology architectures using the notion of correspondence modules. In this formulation, morphisms between vector spaces are given by partial linear relations, as opposed to linear mappings. In the one-dimensional case, among other things, this allows us to: (i) treat persistence modules and zigzag modules as algebraic objects of the same type; (ii) give a categorical formulation of zigzag structures over a continuous parameter; and (iii) construct barcodes associated with spaces and mappings that are richer in geometric information. A structural analysis of one-parameter persistence is carried out at the level of sections of correspondence modules that yield sheaf-like structures, termed persistence sheaves. Under some tameness hypotheses, we prove interval decomposition theorems for persistence sheaves and correspondence modules, as well as an isometry theorem for persistence diagrams obtained from interval decompositions. Applications include: (a) a Mayer-Vietoris sequence that relates the persistent homology of sublevelset filtrations and superlevelset filtrations to the levelset homology module of a real-valued function and (b) the construction of slices of 2-parameter persistence modules along negatively sloped lines.
