  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.
