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.

翻译：生成对抗网络（GAN）旨在通过参数化神经网络（NN）逼近未知分布。尽管GAN已广泛应用于强化学习、半监督学习及计算机视觉任务中，其参数选择通常需要进行穷举搜索，且仅有少数选择方法被证明具有理论最优性。Wasserstein GAN（WGAN）是最具前景的GAN变体之一。现有关于总体WGAN最优参数的研究局限于线性-二次-高斯（LQG）设定，即生成器神经网络为线性且数据服从高斯分布。本文致力于刻画超越LQG设定的总体WGAN最优解特性。作为基础结果，我们推导了当神经网络具有非线性激活函数且数据为非高斯分布时，一维WGAN的闭式最优参数。针对高维数据，我们采用切片Wasserstein框架，证明线性生成器可渐近最优。此外，原始切片WGAN仅约束投影数据的边缘分布而非经典WGAN中的完整分布，因此我们提出另一种新型非投影切片WGAN并证明其渐近最优性。实证研究表明，相较于计算复杂度与数据维度呈三次方关系的经典r-主成分分析（r-PCA）解法，我们的切片WGAN生成器仅需线性复杂度即可实现更优性能。