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 with a power of the group size, and this power depends on the intrinsic dimension of the distribution's support. To our knowledge, this work presents the first statistical estimation for group-invariant generative models, specifically for GANs, and it may shed light on the study of other group-invariant generative models.

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