Projection-based techniques for high-dimensional optimal transport problems

Optimal transport (OT) methods seek a transformation map (or plan) between two probability measures, such that the transformation has the minimum transportation cost. Such a minimum transport cost, with a certain power transform, is called the Wasserstein distance. Recently, OT methods have drawn great attention in statistics, machine learning, and computer science, especially in deep generative neural networks. Despite its broad applications, the estimation of high-dimensional Wasserstein distances is a well-known challenging problem owing to the curse-of-dimensionality. There are some cutting-edge projection-based techniques that tackle high-dimensional OT problems. Three major approaches of such techniques are introduced, respectively, the slicing approach, the iterative projection approach, and the projection robust OT approach. Open challenges are discussed at the end of the review. This article is categorized under: Statistical and Graphical Methods of Data Analysis > Dimension Reduction Statistical Learning and Exploratory Methods of the Data Sciences > Manifold Learning

Automated summary

Optimal transport methods aim to find a transformation map that minimizes the transportation cost between two probability measures, known as the Wasserstein distance. Recently, these methods have gained attention in statistics, machine learning, and computer science, particularly in deep generative neural networks. However, estimating high-dimensional Wasserstein distances is a challenging problem due to the curse-of-dimensionality. Advanced projection-based techniques, such as the slicing approach, iterative projection approach, and projection robust OT approach, have been developed to tackle these high-dimensional OT problems. The article concludes by discussing open challenges in the field.