Geometry of Wasserstein and Transportation Cost Metrics

瓦瑟斯坦-1度量，也称为推土机度量，是对概率分布之间的距离的测量，量化了将一种分布的质量传输到另一种分布所需的最低成本。这一指标不仅出现在纯数学中，而且经常出现在应用数学和计算机科学中。例如，数字图像可以抽象地建模为2维区域上的概率分布，并且沃瑟斯坦距离提供了在该模型下图像的相似性的自然测量。尽管它们无处不在，但这些度量通常很难计算，而且它们的几何属性的很大方面仍然知之甚少。这个项目旨在推进这一理论，特别强调通过更简单的度量来逼近Wasserstein度量，例如欧几里得空间上的经典p-度量。该项目还通过组织有不同发言者和参与者的会议和研讨会来增加未被充分代表的群体对数学的参与。在计算两个分布之间的沃瑟斯坦距离时，传输质量的成本取决于定义分布的基础度量空间的几何形状。该项目的一个主要目标是对其Wasserstein-1度量允许双Lipschitz嵌入到Banach空间L1的度量空间进行分类。这个问题的一个方面将涉及证明某些度量空间的L1-嵌入不存在，而在这一方面所使用的方法将主要基于对分形的分析的概念。挑战的另一边将是证明对于其他度量空间确实存在L1-嵌入，为此，计划结合几何测度论和度量几何的工具。最后，现有结果的证明方法在很大程度上依赖于Banach空间的线性理论。该项目的另一个主要目标是开发非线性方法，对现有结果产生新的见解，并提供解决通过线性技术无法实现的问题的方法。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

The “Wasserstein-1 metric”, also known as earth mover's metric, is a measurement of distance between probability distributions quantifying the least cost needed to transport the mass of one distribution onto the other. This metric appears not only in pure mathematics, but frequently in applied mathematics and computer science as well. For example, a digital image may be modeled abstractly as a probability distribution over a 2-dimensional region, and the Wasserstein distance provides a natural measurement of similarity of images under this model. Despite their pervasiveness, these metrics are often difficult to compute, and large aspects of their geometric properties remain poorly understood. This project aims to advance this theory, with a particular emphasis on approximations of Wasserstein metrics through simpler metrics, such as the classical p-metrics on Euclidean spaces. The project also serves to increase the participation of underrepresented groups in mathematics through organization of conferences and seminars with diverse speakers and participants.In calculating the Wasserstein distance between two distributions, the cost of transporting mass depends on the geometry of the underlying metric space on which the distributions are defined. A main goal of the project is to classify those metric spaces whose Wasserstein-1 metric admits a biLipschitz embedding into the Banach space L1. One side of this question will involve showing the nonexistence of L1-embeddings for certain metric spaces, and on this side the methodology to be used will largely be based on concepts from analysis on fractals. The other side of the challenge will be to show that L1-embeddings do exist for other metric spaces, and towards this end an incorporation of tools from geometric measure theory and metric geometry is planned. Finally, the current methods of proof for existing results rely heavily on the linear theory of Banach spaces. Another main goal of this project is to develop nonlinear methods that yield new insights into existing results as well as provide approaches toward solutions of questions unattainable via linear techniques.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.