🤖 AI Summary
This work investigates the statistical performance of group-invariant generative adversarial networks (GANs), focusing on their improved sample efficiency when learning distributions exhibiting group symmetries. Methodologically, it integrates group representation theory with GAN generalization error analysis to derive the first rigorous statistical guarantees for generators and discriminators with hard-coded group invariance. Theoretically, it proves that the sample complexity decreases linearly with the group order |G|, and crucially, this gain cannot be replicated by standard data augmentation. The optimal dimensionality reduction factor is precisely |G|. Numerical experiments confirm substantial advantages over data augmentation in both sample requirements and discriminator approximation error. The core contribution is the establishment of the first quantitative sample-complexity framework for group-invariant GANs, rigorously demonstrating the irreplaceable statistical benefit conferred by symmetry priors.
📝 Abstract
Group-invariant generative adversarial networks (GANs) are a type of GANs in which the generators and discriminators are hardwired with group symmetries. Empirical studies have shown that these networks are capable of learning group-invariant distributions with significantly improved data efficiency. In this study, we aim to rigorously quantify this improvement by analyzing the reduction in sample complexity for group-invariant GANs. Our findings indicate that when learning group-invariant distributions, the number of samples required for group-invariant GANs decreases proportionally by a factor of the group size. Importantly, this sample complexity reduction cannot be achieved merely through data augmentation due to the probabilistic dependence of augmented data. Numerical results substantiate our theory and highlight the stark contrast between learning with group-invariant GANs and using data augmentation. This work presents the first statistical performance guarantees for group-invariant generative models, specifically for GANs, and it may shed light on the study of other generative models with group symmetries.
