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:基因反转网络背后的数学力量提出了具有挑战性的理论问题。我们以对所产生分布的几何特性进行定性这一重要问题为动力,在有限的样本和无药可治的系统中对瓦塞尔斯坦GANs(WGANs)进行了透彻的分析。我们研究了潜伏空间是单体的,无论输出空间的尺寸如何,其结果都是有效的具体案例。我们特别表明,对于固定的样本大小,最佳WGAN与连接路径密切相关,最大限度地减少各采样点之间的正方形欧西里德距离之和。我们还着重指出,WGANs能够(在1-瓦瑟斯坦距离方面)接近标的分布,因为样本大小趋向于不完全性,以一定的趋同速度,并使基因型Lipschitz功能的组合能够适当地增长。我们从半分立方形环境中传递关于最佳运输理论的新结果。