Projection‐based techniques for high‐dimensional optimal transport problems

Abstract

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

Hilbert Curve Projection Distance for Distribution Comparison

“…The work in [2] considers the Kantorovich duality of Wasserstein distance and converts the problem to a "max-min" game. Besides these two approximation methods, more surrogates of the Wasserstein distance are proposed in recent years [42], e.g., the sliced Wasserstein (SW) distance [6], the generalized sliced Wasserstein (GSW) distance [20], the tree-structured Wasserstein (TSW) distance [23], and so on. Despite the computational efficiency, these surrogates may fail to provide effective approximations for the Wasserstein distance.…”