We investigate the impact of the input dimension on the generalization error in generative adversarial networks (GANs). In particular, we first provide both theoretical and practical evidence to validate the existence of an optimal input dimension (OID) that minimizes the generalization error. Then, to identify the OID, we introduce a novel framework called generalized GANs (G-GANs), which includes existing GANs as a special case. By incorporating the group penalty and the architecture penalty developed in the paper, G-GANs have several intriguing features. First, our framework offers adaptive dimensionality reduction from the initial dimension to a dimension necessary for generating the target distribution. Second, this reduction in dimensionality also shrinks the required size of the generator network architecture, which is automatically identified by the proposed architecture penalty. Both reductions in dimensionality and the generator network significantly improve the stability and the accuracy of the estimation and prediction. Theoretical support for the consistent selection of the input dimension and the generator network is provided. Third, the proposed algorithm involves an end-to-end training process, and the algorithm allows for dynamic adjustments between the input dimension and the generator network during training, further enhancing the overall performance of G-GANs. Extensive experiments conducted with simulated and benchmark data demonstrate the superior performance of G-GANs. In particular, compared to that of off-the-shelf methods, G-GANs achieves an average improvement of 45.68% in the CT slice dataset, 43.22% in the MNIST dataset and 46.94% in the FashionMNIST dataset in terms of the maximum mean discrepancy or Frechet inception distance. Moreover, the features generated based on the input dimensions identified by G-GANs align with visually significant features.

翻译：本文研究了生成对抗网络（GANs）中输入维度对泛化误差的影响。具体而言，我们首先从理论和实践两方面验证了存在一个最优输入维度（OID）能够最小化泛化误差。随后，为识别该OID，我们提出了一种名为广义生成对抗网络（G-GANs）的新框架，该框架将现有GANs作为特例纳入其中。通过引入本文提出的群组惩罚与架构惩罚，G-GANs具备若干显著特性：第一，该框架可实现从初始维度到生成目标分布所需维度的自适应降维；第二，这种维度缩减同时会压缩生成器网络架构所需的规模，而该规模可由所提出的架构惩罚自动识别。维度和生成器网络的双重缩减显著提升了估计与预测的稳定性及精度。本文为输入维度与生成器网络的一致性选择提供了理论支撑。第三，所提算法采用端到端训练过程，并允许在训练过程中对输入维度与生成器网络进行动态调整，从而进一步增强了G-GANs的整体性能。基于模拟数据与基准数据的广泛实验表明，G-GANs具有优越性能。特别地，与现成方法相比，G-GANs在CT切片数据集、MNIST数据集和FashionMNIST数据集上的最大均值差异或Frechet初始距离分别平均提升了45.68%、43.22%和46.94%。此外，基于G-GANs识别出的输入维度所生成的特征与视觉显著特征高度吻合。