Efficient Discretizations of Optimal Transport

TL;DR

This paper introduces an efficient algorithm for discretizing optimal transport problems that reduces computational effort while maintaining high-quality solutions, suitable for large-scale applications.

Contribution

The authors propose a novel entropy-regularized discretization method for OT that requires fewer points and is parallelizable, improving efficiency over traditional sampling approaches.

Findings

Achieves comparable OT plans with fewer discretization points.

Provides theoretical bounds for approximation quality.

Demonstrates effectiveness on diverse problem sets.

Abstract

Obtaining solutions to Optimal Transportation (OT) problems is typically intractable when the marginal spaces are continuous. Recent research has focused on approximating continuous solutions with discretization methods based on i.i.d. sampling, and has proven convergence as the sample size increases. However, obtaining OT solutions with large sample sizes requires intensive computation effort, that can be prohibitive in practice. In this paper, we propose an algorithm for calculating discretizations with a given number of points for marginal distributions, by minimizing the (entropy-regularized) Wasserstein distance, and result in plans that are comparable to those obtained with much larger numbers of i.i.d. samples. Moreover, a local version of such discretizations which is parallelizable for large scale applications is proposed. We prove bounds for our approximation and demonstrate performance on a wide range of problems.