An Introduction to Sliced Optimal Transport

TL;DR

Sliced Optimal Transport (SOT) offers a scalable and computationally efficient framework for measuring distances and barycenters of probability measures by leveraging one-dimensional OT problems and integral geometry techniques.

Contribution

This paper provides a comprehensive review of SOT, including its mathematical foundations, recent methodological advances, computational methods, and diverse applications.

Findings

SOT enables fast computation of optimal transport distances.

Recent advances include non-linear projections and improved Monte Carlo methods.

SOT has broad applications in machine learning and data analysis.

Abstract

Sliced Optimal Transport (SOT) is a rapidly developing branch of optimal transport (OT) that exploits the tractability of one-dimensional OT problems. By combining tools from OT, integral geometry, and computational statistics, SOT enables fast and scalable computation of distances, barycenters, and kernels for probability measures, while retaining rich geometric structure. This paper provides a comprehensive review of SOT, covering its mathematical foundations, methodological advances, computational methods, and applications. We discuss key concepts of OT and one-dimensional OT, the role of tools from integral geometry such as Radon transform in projecting measures, and statistical techniques for estimating sliced distances. The paper further explores recent methodological advances, including non-linear projections, improved Monte Carlo approximations, statistical estimation techniques for one-dimensional optimal transport, weighted slicing techniques, and transportation plan estimation methods. Variational problems, such as minimum sliced Wasserstein estimation, barycenters, gradient flows, kernel constructions, and embeddings are examined alongside extensions to unbalanced, partial, multi-marginal, and Gromov-Wasserstein settings. Applications span machine learning, statistics, computer graphics and computer visions, highlighting SOT's versatility as a practical computational tool. This work will be of interest to researchers and practitioners in machine learning, data sciences, and computational disciplines seeking efficient alternatives to classical OT.