论文标题
有限度量空间的通用家庭，具有相同或微不足道的一维持久性
Generic families of finite metric spaces with identical or trivial 1-dimensional persistence
论文作者
论文摘要
持久性同源性是一种用于分析有限度量空间的流行且有用的工具，可以揭示可用于区分未标记点的集合以及对机器学习管道的输入的功能。持久同源性的著名稳定定理为瓶颈距离的持久性变化提供了上限，但没有给出下限。本文阐明了持续的同源性可能在区分有限的度量空间方面存在的可能局限性，这对于具有相同持久性的非等法点集很明显。我们描述了具有相同甚至微不足道的一维持久性的度量空间中的点集的通用家族。结果激发了更强大的不变性以区分有限点设置为等轴测图。
Persistent homology is a popular and useful tool for analysing finite metric spaces, revealing features that can be used to distinguish sets of unlabeled points and as input into machine learning pipelines. The famous stability theorem of persistent homology provides an upper bound for the change of persistence in the bottleneck distance under perturbations of points, but without giving a lower bound. This paper clarifies the possible limitations persistent homology may have in distinguishing finite metric spaces, which is evident for non-isometric point sets with identical persistence. We describe generic families of point sets in metric spaces that have identical or even trivial one-dimensional persistence. The results motivate stronger invariants to distinguish finite point sets up to isometry.
