An Introduction to Sliced Optimal Transport
Khai Nguyen
TL;DR
This work surveys Sliced Optimal Transport (SOT), a framework that reduces high-dimensional OT to a family of one-dimensional problems via the Radon transform and its generalizations. It systematically builds from foundational measure theory and 1D OT to the sliced distance SW, and then surveys broad advances including generalized Radon transforms, non-Euclidean projections, product-space extensions, and efficient numerical schemes. The contributions cover methodological foundations, computational techniques, and variants such as GSW, PMSW, H2SW, and CHSW, with connections to transport maps like Knothe and IDT, and practical considerations for images, shapes, and manifolds. By integrating theory, algorithms, and applications, the paper highlights SOT as a scalable, flexible alternative to classical OT across ML, graphics, and data science, while outlining open questions in injectivity, transport maps, and higher-order extensions.
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.