The mathematical forces at work behind Generative Adversarial Networks raise challenging theoretical issues. Motivated by the important question of characterizing the geometrical properties of the generated distributions, we provide a thorough analysis of Wasserstein GANs (WGANs) in both the finite sample and asymptotic regimes. We study the specific case where the latent space is univariate and derive results valid regardless of the dimension of the output space. We show in particular that for a fixed sample size, the optimal WGANs are closely linked with connected paths minimizing the sum of the squared Euclidean distances between the sample points. We also highlight the fact that WGANs are able to approach (for the 1-Wasserstein distance) the target distribution as the sample size tends to infinity, at a given convergence rate and provided the family of generative Lipschitz functions grows appropriately. We derive in passing new results on optimal transport theory in the semi-discrete setting.

翻译：生成对抗网络背后的数学原理引发了具有挑战性的理论问题。受刻画生成分布几何性质这一重要问题的启发，我们对Wasserstein生成对抗网络（WGAN）在有限样本和渐近两种情形下进行了深入分析。我们研究了隐空间为一维变量的特例，并推导出对输出空间任意维度均成立的结论。特别地，我们证明了在固定样本量下，最优WGAN与最小化样本点间欧氏距离平方和的连通路径紧密相关。我们还指出，当样本量趋于无穷时，只要生成Lipschitz函数族适当增长，WGAN能够以给定收敛速率逼近（在1-Wasserstein距离意义下）目标分布。在此过程中，我们推导了半离散环境下最优输运理论的新结果。