Geometry of Wasserstein and Transportation Cost Metrics

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.