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.

我们提出了一种无需优化的切片分布方法，用于蒙特卡洛估计期望，该方法借助两个输入测量值之间的标准化差异构建随机路径投影方向（RPD），进而导出了随机路径切片分布（RPSD）和两种切片Wasserstein的变体，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）。最后，我们展示了RPSW和IWRPSW在图像的梯度流和去噪扩散生成模型的训练中的良好性能。

随机路径投影方向的切片Wasserstein