论文标题
扩展持久性的普遍距离
Universal Distances for Extended Persistence
论文作者
论文摘要
扩展的持久性图是分段线性函数的不变性，在函数的扰动下相对于Cohen-Steiner，Edelsbrunner和Harer引入的瓶颈距离，已知该函数稳定。我们解决了普遍性问题，该问题要求扩展持久图上最大的稳定距离，表明瓶颈距离的更具歧视性变体是通用的。我们的结果更普遍地适用于仅在一定程度上认为持久图的设置。我们通过建立功能性结构和相对跨层集合同源性的几种特征性特性来实现我们的结果，这些构造反映了经典的Eilenberg-Steenrod Axioms。最后，我们通过表明后者不是固有的，更不用说通用的瓶颈与束带在实际线路上的交织距离进行了对比。该特定结果具有进一步的暗示，即Reeb图的交织距离也不是内在的。
The extended persistence diagram is an invariant of piecewise linear functions, which is known to be stable under perturbations of functions with respect to the bottleneck distance as introduced by Cohen-Steiner, Edelsbrunner, and Harer. We address the question of universality, which asks for the largest possible stable distance on extended persistence diagrams, showing that a more discriminative variant of the bottleneck distance is universal. Our result applies more generally to settings where persistence diagrams are considered only up to a certain degree. We achieve our results by establishing a functorial construction and several characteristic properties of relative interlevel set homology, which mirror the classical Eilenberg--Steenrod axioms. Finally, we contrast the bottleneck distance with the interleaving distance of sheaves on the real line by showing that the latter is not intrinsic, let alone universal. This particular result has the further implication that the interleaving distance of Reeb graphs is not intrinsic either.
