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：切片分布选择已被用作一种有效技术，通过最小化切片Wasserstein距离来提升参数估计器在应用中的性能。先前的工作要么利用昂贵的优化来选择切片分布，要么采用需要昂贵采样方法的切片分布。在本研究中，我们提出一种无需优化的切片分布，能够为期望的蒙特卡洛估计提供快速采样。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输入测度下随机向量之间的归一化差异构建而成。基于RPD，我们推导出随机路径切片分布（RPSD）以及两种切片Wasserstein变体，即随机路径投影切片Wasserstein（RPSW）和重要性加权随机路径投影切片Wasserstein（IWRPSW）。随后，我们讨论了RPSW与IWRPSW在拓扑、统计和计算方面的性质。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。