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.

翻译：群不变生成对抗网络（Group-invariant GANs）是一种生成器和判别器被硬编码为具有群对称性的生成对抗网络。实证研究表明，这类网络能够以显著提升的数据效率学习群不变分布。在本研究中，我们旨在通过分析群不变GANs样本复杂度的降低，严格量化这一提升。我们的发现表明：在学习群不变分布时，群不变GANs所需样本数量随群规模幂次成比例减少，且该幂次取决于分布支撑集的内在维度。据我们所知，本研究首次为群不变生成模型（特别是GANs）提供了统计估计，这可能为其他群不变生成模型的研究提供启发。