An Introduction to Sliced Optimal Transport

2025-08-17

AI Summary

This work addresses the high computational complexity and poor scalability of classical optimal transport (OT). To this end, we propose an efficient probabilistic measure analysis framework based on sliced optimal transport (SOT). Methodologically, we integrate integral geometry with statistical estimation: we introduce nonlinear projections and adaptive weighted slicing strategies to improve Monte Carlo approximation, and extend SOT to unbalanced, multi-marginal, and Gromov–Wasserstein settings. Theoretically and algorithmically, our framework unifies support for sliced Wasserstein distance computation, barycenter estimation, kernel construction, and embedding learning. Experiments demonstrate that the proposed approach preserves the rich geometric structure of OT while achieving significant scalability improvements. Its efficacy and practicality are validated across diverse machine learning, computer vision, and graphics tasks.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Computer Vision: Learning & Optimization for CV
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

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.

Problem

• SOT enables fast computation of distances for probability measures
• SOT combines OT, integral geometry, and computational statistics
• SOT applies to machine learning, statistics, and computer vision

Innovation

• Utilizes one-dimensional OT for fast computation
• Integrates Radon transform for measure projection
• Applies weighted slicing for variational problems