SOT用切片方法加速计算概率分布距离，兼顾精度与效率。
An Introduction to Sliced Optimal Transport
- 通过一维最优传输切片，将复杂高维问题简化为可快速求解的低维问题。
- 支持高效计算分布间距离、重心、核函数等，适用于多种机器学习任务。
- 适合需要快速计算且保留几何结构的研究者，尤其在图像生成等领域有应用。
切片最优传输（Sliced Optimal Transport, SOT）是近年来快速发展的最优传输（OT）分支，利用一维最优传输问题的可处理性，结合最优传输、积分几何和计算统计工具，实现了对概率测度的距离、重心和核函数的快速、可扩展计算，同时保持丰富的几何结构。本文全面综述了SOT的数学基础、方法进展、计算技术及其应用。内容涵盖经典最优传输与一维最优传输的核心概念，积分几何工具如Radon变换在测度投影中的作用，以及基于蒙特卡洛的切片距离估计方法。还探讨了非线性投影、改进的蒙特卡洛近似、一维最优传输的统计估计、加权切片技术及运输计划估计等近期进展。此外，研究了变分问题，包括最小切片Wasserstein估计、重心、梯度流、核构造与嵌入，并拓展至不平衡、部分、多边际及Gromov-Wasserstein设置。应用覆盖机器学习、统计学、计算机图形学与计算机视觉，凸显其作为实用计算工具的广泛适用性。本工作对寻求经典最优传输高效替代方案的研究人员与从业者具有重要参考价值。
原文摘要 · Abstract (English)
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.
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