On optimal solutions of classical and sliced Wasserstein GANs with non-Gaussian data

The generative adversarial network (GAN) aims to approximate an unknown distribution via a parameterized neural network (NN). While GANs have been widely applied in reinforcement and semi-supervised learning as well as computer vision tasks, selecting their parameters often needs an exhaustive search, and only a few selection methods have been proven to be theoretically optimal. One of the most promising GAN variants is the Wasserstein GAN (WGAN). Prior work on optimal parameters for population WGAN is limited to the linear-quadratic-Gaussian (LQG) setting, where the generator NN is linear, and the data is Gaussian. In this paper, we focus on the characterization of optimal solutions of population WGAN beyond the LQG setting. As a basic result, closed-form optimal parameters for one-dimensional WGAN are derived when the NN has non-linear activation functions, and the data is non-Gaussian. For high-dimensional data, we adopt the sliced Wasserstein framework and show that the linear generator can be asymptotically optimal. Moreover, the original sliced WGAN only constrains the projected data marginal instead of the whole one in classical WGAN, and thus, we propose another new unprojected sliced WGAN and identify its asymptotic optimality. Empirical studies show that compared to the celebrated r-principal component analysis (r-PCA) solution, which has cubic complexity to the data dimension, our generator for sliced WGAN can achieve better performance with only linear complexity.