  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.
