The Wasserstein distance and its variations, e.g., the sliced-Wasserstein (SW) distance, have recently drawn attention from the machine learning community. The SW distance, specifically, was shown to have similar properties to the Wasserstein distance, while being much simpler to compute, and is therefore used in various applications including generative modeling and general supervised/unsupervised learning. In this paper, we first clarify the mathematical connection between the SW distance and the Radon transform. We then utilize the generalized Radon transform to define a new family of distances for probability measures, which we call generalized sliced-Wasserstein (GSW) distances. We also show that, similar to the SW distance, the GSW distance can be extended to a maximum GSW (max-GSW) distance. We then provide the conditions under which GSW and max-GSW distances are indeed distances. Finally, we compare the numerical performance of the proposed distances on several generative modeling tasks, including SW flows and SW auto-encoders.

使用广义Radon变换定义了一类新的概率测度距离，称为广义sliced-Wasserstein(GSW)距离，并给出GSW和max-GSW距离是否为距离的条件；并在几个生成建模任务中比较了所提出距离的数值性能。

广义切片Wasserstein距离