Wasserstein distance (WD) and the associated optimal transport plan have been proven useful in many applications where probability measures are at stake. In this paper, we propose a new proxy of the squared WD, coined min-SWGG, that is based on the transport map induced by an optimal one-dimensional projection of the two input distributions. We draw connections between min-SWGG and Wasserstein generalized geodesics in which the pivot measure is supported on a line. We notably provide a new closed form for the exact Wasserstein distance in the particular case of one of the distributions supported on a line allowing us to derive a fast computational scheme that is amenable to gradient descent optimization. We show that min-SWGG is an upper bound of WD and that it has a complexity similar to as Sliced-Wasserstein, with the additional feature of providing an associated transport plan. We also investigate some theoretical properties such as metricity, weak convergence, computational and topological properties. Empirical evidences support the benefits of min-SWGG in various contexts, from gradient flows, shape matching and image colorization, among others.

翻译：Wasserstein距离（WD）及其相关的最优传输映射已在涉及概率测度的众多应用中展现出重要价值。本文提出了一种新型平方WD代理度量——min-SWGG，该度量基于两个输入分布的最优一维投影所诱导的传输映射。我们建立了min-SWGG与以直线支撑测度为枢轴的Wasserstein广义测地线之间的关联，并特别针对一个分布支撑在直线上的特殊情况，给出了精确Wasserstein距离的新闭式解，从而推导出适用于梯度下降优化的快速计算方案。理论分析表明：min-SWGG是WD的上界，其计算复杂度与切片Wasserstein相当，同时额外提供关联的传输映射。我们还研究了度量性、弱收敛性、计算及拓扑性质等理论特性。实验证据支持min-SWGG在梯度流、形状匹配和图像着色等多元场景中的优势。