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距离（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。其核心组成部分是切片分布。现有两种选取该分布的方法：第一种使用固定先验分布；第二种通过优化从参数化分布族中寻找能最大化期望距离的最佳分布。然而这两种方法均有局限性：固定先验分布无法突出能区分一般概率测度的投影方向，而优化最佳分布通常成本高昂且不稳定，且候选分布的参数化族设计易出现错误设定。为解决上述问题，我们提出将切片分布设计为无参数化且密度与投影一维Wasserstein距离的能量函数成正比的能量基分布。进而推导出新型切片Wasserstein度量——基于能量的切片Wasserstein距离（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。最后在点云梯度流、色彩传递和点云重建实验中的结果表明EBSW具有优越性能。