Slicing distribution selection has been used as an effective technique to improve the performance of parameter estimators based on minimizing sliced Wasserstein distance in applications. Previous works either utilize expensive optimization to select the slicing distribution or use slicing distributions that require expensive sampling methods. In this work, we propose an optimization-free slicing distribution that provides a fast sampling for the Monte Carlo estimation of expectation. In particular, we introduce the random-path projecting direction (RPD) which is constructed by leveraging the normalized difference between two random vectors following the two input measures. From the RPD, we derive the random-path slicing distribution (RPSD) and two variants of sliced Wasserstein, i.e., the Random-Path Projection Sliced Wasserstein (RPSW) and the Importance Weighted Random-Path Projection Sliced Wasserstein (IWRPSW). We then discuss the topological, statistical, and computational properties of RPSW and IWRPSW. Finally, we showcase the favorable performance of RPSW and IWRPSW in gradient flow and the training of denoising diffusion generative models on images.

翻译：切片分布的选择已被用作一种有效技术，通过在应用中最小化切片沃瑟斯坦距离来提升参数估计器的性能。以往研究要么采用昂贵的优化方法选择切片分布，要么使用需要复杂采样方法的切片分布。在本工作中，我们提出了一种无需优化的切片分布，它能实现对期望值的蒙特卡洛估计进行快速采样。具体而言，我们引入了随机路径投影方向（RPD），该方向通过利用两个分别服从输入测度的随机向量之间的归一化差值来构建。基于RPD，我们推导出随机路径切片分布（RPSD）以及两种切片沃瑟斯坦变体，即随机路径投影切片沃瑟斯坦（RPSW）和重要性加权随机路径投影切片沃瑟斯坦（IWRPSW）。随后，我们讨论了RPSW和IWRPSW的拓扑、统计与计算性质。最后，我们展示了RPSW和IWRPSW在梯度流以及图像去噪扩散生成模型训练中的优越性能。