Generalized Sliced Wasserstein Distances
This work provides a theoretical extension of sliced-Wasserstein distances for machine learning applications, but it is incremental as it builds directly on existing concepts.
The authors tackled the problem of defining new distances for probability measures by introducing generalized sliced-Wasserstein (GSW) distances based on the generalized Radon transform, and they demonstrated their numerical performance on generative modeling tasks such as SW flows and SW auto-encoders.
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.