论文标题
虚拟持久图，签名的措施，瓦斯泰因距离和巴拉克空间
Virtual persistence diagrams, signed measures, Wasserstein distances, and Banach spaces
论文作者
论文摘要
持久图是拓扑数据分析中的一个重要摘要，由一组有序对组成，每个对都具有积极的多重性。持久图是通过Mobius倒置获得的，可以使用称为Wasserstein距离的一个参数家族来比较。在某些情况下，Mobius倒置会产生一组有序对，可能具有负多重性。我们称这些虚拟持久图。 Divol和Lacombe最近表明，在有序成对的半平面上有一个wasserstein距离，可以概括持续图的Wasserstein距离，而经典的Wasserstein距离与最佳运输理论相比。在这项工作之后，我们定义了在任意度量空间上的持久图和ra尺度的兼容瓦斯坦距离。我们表明，1-wasserstein距离扩展到虚拟持久图和签署的措施。此外，我们表征了库奇的持续图在瓦斯堡距离方面的完成。我们还为具有1-Wasserstein Norm的Banach空间提供了普遍的结构。具有1-wasserstein距离的持久图将嵌入到这个Banach空间中。
Persistence diagrams, an important summary in topological data analysis, consist of a set of ordered pairs, each with positive multiplicity. Persistence diagrams are obtained via Mobius inversion and may be compared using a one-parameter family of metrics called Wasserstein distances. In certain cases, Mobius inversion produces sets of ordered pairs which may have negative multiplicity. We call these virtual persistence diagrams. Divol and Lacombe recently showed that there is a Wasserstein distance for Radon measures on the half plane of ordered pairs that generalizes both the Wasserstein distance for persistence diagrams and the classical Wasserstein distance from optimal transport theory. Following this work, we define compatible Wasserstein distances for persistence diagrams and Radon measures on arbitrary metric spaces. We show that the 1-Wasserstein distance extends to virtual persistence diagrams and to signed measures. In addition, we characterize the Cauchy completion of persistence diagrams with respect to the Wasserstein distances. We also give a universal construction of a Banach space with a 1-Wasserstein norm. Persistence diagrams with the 1-Wasserstein distance isometrically embed into this Banach space.
