Algorithms for Optimal Transport and Wasserstein Distances

Abstract

Optimal Transport and Wasserstein Distance are closely related terms that do not only have a long history in the mathematical literature, but also have seen a resurgence in recent years, particularly in the context of the many applications they are used in, which span a variety of scientific fields including - but not limited to - imaging, statistics and machine learning. Due to drastic increases in data volume and a high demand for Wasserstein distance computation, the development of more efficient algorithms in the domain of optimal transport increased in priority and the advancement picked up pace quickly. This thesis is dedicated to algorithms for solving the optimal transport problem and computing Wasserstein distances. After an introduction to the field of optimal transport, there will be an overview of the application areas as well as a summary of the most important methods for computation with a focus on the discrete optimal transport problem. This is followed by a presentation of a benchmark for discrete optimal transport together with a performance test of a selection of algorithms on this data set. Afterwards, two new approaches are introduced: a probabilistic approximation method for Wasserstein distances using subsampling and a clustering method, which aims to generalize multiscale methods to discrete optimal transport problems, including instances with a non-metric cost function.