The sliced Wasserstein (SW) distance has been widely recognized as a statistically effective and computationally efficient metric between two probability measures. A key component of the SW distance is the slicing distribution. There are two existing approaches for choosing this distribution. The first approach is using a fixed prior distribution. The second approach is optimizing for the best distribution which belongs to a parametric family of distributions and can maximize the expected distance. However, both approaches have their limitations. A fixed prior distribution is non-informative in terms of highlighting projecting directions that can discriminate two general probability measures. Doing optimization for the best distribution is often expensive and unstable. Moreover, designing the parametric family of the candidate distribution could be easily misspecified. To address the issues, we propose to design the slicing distribution as an energy-based distribution that is parameter-free and has the density proportional to an energy function of the projected one-dimensional Wasserstein distance. We then derive a novel sliced Wasserstein metric, energy-based sliced Waserstein (EBSW) distance, and investigate its topological, statistical, and computational properties via importance sampling, sampling importance resampling, and Markov Chain methods. Finally, we conduct experiments on point-cloud gradient flow, color transfer, and point-cloud reconstruction to show the favorable performance of the EBSW.

翻译：切片Wasserstein（SW）距离已被广泛认为是两个概率测度之间统计有效且计算高效的度量。SW距离的关键组成部分是切片分布。目前存在两种选择该分布的方法：第一种是使用固定的先验分布；第二种是优化出属于某参数分布族且能最大化期望距离的最佳分布。然而，这两种方法均有其局限性。固定先验分布在突出能够区分两个通用概率测度的投影方向方面缺乏信息性，而优化最佳分布通常计算昂贵且不稳定，且候选分布参数族的设计容易产生误设。为解决这些问题，我们提出将切片分布设计为基于能量的分布——该分布无需参数且其密度与投影后一维Wasserstein距离的能量函数成正比。由此推导出新型切片Wasserstein度量——基于能量的切片Wasserstein（EBSW）距离，并通过重要性采样、采样重要性重采样及马尔可夫链方法研究其拓扑、统计及计算特性。最后，我们在点云梯度流、颜色迁移与点云重建实验中展示了EBSW的优越性能。