Despite the remarkable empirical successes of Generative Adversarial Networks (GANs), the theoretical guarantees for their statistical accuracy remain rather pessimistic. In particular, the data distributions on which GANs are applied, such as natural images, are often hypothesized to have an intrinsic low-dimensional structure in a typically high-dimensional feature space, but this is often not reflected in the derived rates in the state-of-the-art analyses. In this paper, we attempt to bridge the gap between the theory and practice of GANs and their bidirectional variant, Bi-directional GANs (BiGANs), by deriving statistical guarantees on the estimated densities in terms of the intrinsic dimension of the data and the latent space. We analytically show that if one has access to $n$ samples from the unknown target distribution and the network architectures are properly chosen, the expected Wasserstein-1 distance of the estimates from the target scales as $O\left( n^{-1/d_\mu } \right)$ for GANs and $O\left( n^{-1/(d_\mu+\ell)} \right)$ for BiGANs, where $d_\mu$ and $\ell$ are the upper Wasserstein-1 dimension of the data-distribution and latent-space dimension, respectively. The theoretical analyses not only suggest that these methods successfully avoid the curse of dimensionality, in the sense that the exponent of $n$ in the error rates does not depend on the data dimension but also serve to bridge the gap between the theoretical analyses of GANs and the known sharp rates from optimal transport literature. Additionally, we demonstrate that GANs can effectively achieve the minimax optimal rate even for non-smooth underlying distributions, with the use of larger generator networks.

翻译：尽管生成对抗网络（GANs）在经验上取得了显著成功，但其统计精度的理论保证仍较悲观。特别地，GANs所应用的数据分布（如自然图像）通常被假设在高维特征空间中具有低维本征结构，但在现有最先进分析方法推导的收敛速率中，这一特性往往未被体现。本文尝试弥合GANs及其双向变体（BiGANs）理论与实践的鸿沟，通过基于数据本征维度和隐空间维度推导估计密度的统计保证。我们分析证明：若从未知目标分布获得$n$个样本，且网络架构选择得当，则GANs估计结果与目标分布的期望Wasserstein-1距离按$O\left( n^{-1/d_\mu } \right)$标度，而BiGANs按$O\left( n^{-1/(d_\mu+\ell)} \right)$标度，其中$d_\mu$和$\ell$分别为数据分布的上Wasserstein-1维度和隐空间维度。理论分析不仅表明这些方法成功规避了维度灾难（即误差率中$n$的指数不依赖于数据维度），还弥合了GANs理论分析与最优传输文献中已知锐化速率之间的差距。此外，我们证明当使用更大的生成器网络时，GANs即使对非光滑底层分布也能有效实现极小化最优速率。