论文标题
WGAN的最佳1-Wasserstein距离
Optimal 1-Wasserstein Distance for WGANs
论文作者
论文摘要
生成对抗网络背后的数学力量提出了挑战性的理论问题。在表征生成分布的几何特性的重要问题上，我们在有限样本和渐近方案中对Wasserstein Gans（WGAN）进行了彻底的分析。我们研究了潜在空间是单变量的特定情况，无论输出空间的维度如何，都会得出有效的结果。我们特别表明，对于固定的样本量，最佳的WGAN与连接的路径紧密相关，以最大程度地减少样品点之间平方欧几里德距离的总和。我们还强调了一个事实，即WGAN能够以给定的收敛速率接近（对于1-wasserstein距离）目标分布，并以给定的收敛速度倾向于无穷大，并提供了生成Lipschitz功能的家族。我们得出在半混凝土环境中传递最佳运输理论的新结果。
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.
