We study the efficacy and efficiency of deep generative networks for approximating probability distributions. We prove that neural networks can transform a low-dimensional source distribution to a distribution that is arbitrarily close to a high-dimensional target distribution, when the closeness are measured by Wasserstein distances and maximum mean discrepancy. Upper bounds of the approximation error are obtained in terms of the width and depth of neural network. Furthermore, it is shown that the approximation error in Wasserstein distance grows at most linearly on the ambient dimension and that the approximation order only depends on the intrinsic dimension of the target distribution. On the contrary, when $f$-divergences are used as metrics of distributions, the approximation property is different. We show that in order to approximate the target distribution in $f$-divergences, the dimension of the source distribution cannot be smaller than the intrinsic dimension of the target distribution.

翻译：我们研究了近似概率分布的深基因网络的功效和效率。我们证明神经网络可以将低维源分布转换为任意接近高维目标分布的分布,当近距离由瓦森斯坦距离和最大平均差异测量时,近似误差的上限以神经网络的宽度和深度计算。此外,还表明瓦森斯坦距离的近似误差在环境维度上增长最多线性,近似误差仅取决于目标分布的内在维度。相反,当以美元-维朗值作为分布的衡量标准时,近似属性则不同。我们表明,为了以美元-维朗度接近目标分布,源分布的维度不能小于目标分布的内在维度。