Projection‐based techniques for high‐dimensional optimal transport problems
Projection‐based techniques for high‐dimensional optimal transport problems
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DOI：
10.1002/wics.1587
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发表时间：
2022-05
期刊：
影响因子：
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通讯作者：
Jingyi Zhang;Ping Ma;Wenxuan Zhong;Cheng Meng-
中科院分区：
文献类型：
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作者：
Jingyi Zhang;Ping Ma;Wenxuan Zhong;Cheng Meng-
最优运输(OT)方法寻求两个概率度量之间的变换映射(或计划)，使得变换具有最小的运输成本。这样的最小运输成本，加上一定的功率变换，称为瓦瑟斯坦距离。近年来，OT方法在统计学、机器学习和计算机科学领域引起了极大的关注，尤其是在深度生成神经网络中。尽管高维Wasserstein距离的估计有着广泛的应用，但由于维度诅咒的存在，高维Wasserstein距离的估计是一个众所周知的具有挑战性的问题。有一些基于投影的尖端技术可以解决高维OT问题。介绍了这类技术的三种主要方法，分别是切片法、迭代投影法和投影稳健OT法。审查结束时讨论了尚未解决的挑战。
Optimal transport (OT) methods seek a transformation map (or plan) between two probability measures, such that the transformation has the minimum transportation cost. Such a minimum transport cost, with a certain power transform, is called the Wasserstein distance. Recently, OT methods have drawn great attention in statistics, machine learning, and computer science, especially in deep generative neural networks. Despite its broad applications, the estimation of high‐dimensional Wasserstein distances is a well‐known challenging problem owing to the curse‐of‐dimensionality. There are some cutting‐edge projection‐based techniques that tackle high‐dimensional OT problems. Three major approaches of such techniques are introduced, respectively, the slicing approach, the iterative projection approach, and the projection robust OT approach. Open challenges are discussed at the end of the review.