# Statistical Guarantees of Group-Invariant GANs

arXiv: https://arxiv.org/abs/2305.13517 · HTML: https://mlsanity.com/paper/2305.13517/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 29 of its 57 references that are in the map.
- First posted on arXiv 22 May 2023.

## Builds on (most cited first, 29 of 29)

- [Improved Training of Wasserstein GANs](https://mlsanity.com/paper/1704.00028/) (2017)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Generative Adversarial Text to Image Synthesis](https://mlsanity.com/paper/1605.05396/) (2016)
- [Spectrally-normalized margin bounds for neural networks](https://mlsanity.com/paper/1706.08498/) (2017)
- [Equivariant Diffusion for Molecule Generation in 3D](https://mlsanity.com/paper/2203.17003/) (2022)
- [Generative Image Inpainting with Contextual Attention](https://mlsanity.com/paper/1801.07892/) (2018)
- [Error bounds for approximations with deep ReLU networks](https://mlsanity.com/paper/1610.01145/) (2016)
- [Scaling up GANs for Text-to-Image Synthesis](https://mlsanity.com/paper/2303.05511/) (2023)
- [General E(2)-Equivariant Steerable CNNs](https://mlsanity.com/paper/1911.08251/) (2019)
- [Nearly-tight VC-dimension and pseudodimension bounds for piecewise linear neural networks](https://mlsanity.com/paper/1703.02930/) (2017)
- [Equivariant Flows: Exact Likelihood Generative Learning for Symmetric Densities](https://mlsanity.com/paper/2006.02425/) (2020)
- [A General Theory of Equivariant CNNs on Homogeneous Spaces](https://mlsanity.com/paper/1811.02017/) (2018)
- [StyleGAN-T: Unlocking the Power of GANs for Fast Large-Scale Text-to-Image Synthesis](https://mlsanity.com/paper/2301.09515/) (2023)
- [Universality of Deep Convolutional Neural Networks](https://mlsanity.com/paper/1805.10769/) (2018)
- [Generalization and Equilibrium in Generative Adversarial Nets (GANs)](https://mlsanity.com/paper/1703.00573/) (2017)
- [E(n) Equivariant Normalizing Flows](https://mlsanity.com/paper/2105.09016/) (2021)
- [Sampling using SU(N) gauge equivariant flows](https://mlsanity.com/paper/2008.05456/) (2020)
- [How Well Generative Adversarial Networks Learn Distributions](https://mlsanity.com/paper/1811.03179/) (2018)
- [Neural Network Renormalization Group](https://mlsanity.com/paper/1802.02840/) (2018)
- [Nonlinear Approximation and (Deep) ReLU Networks](https://mlsanity.com/paper/1905.02199/) (2019)
- [Generalization Error of Invariant Classifiers](https://mlsanity.com/paper/1610.04574/) (2016)
- [An error analysis of generative adversarial networks for learning distributions](https://mlsanity.com/paper/2105.13010/) (2021)
- [Distribution Approximation and Statistical Estimation Guarantees of Generative Adversarial Networks](https://mlsanity.com/paper/2002.03938/) (2020)
- [Scalable Normalizing Flows for Permutation Invariant Densities](https://mlsanity.com/paper/2010.03242/) (2020)
- [Group Equivariant Generative Adversarial Networks](https://mlsanity.com/paper/2005.01683/) (2020)
- [On the capacity of deep generative networks for approximating distributions](https://mlsanity.com/paper/2101.12353/) (2021)
- [Structure-preserving GANs](https://mlsanity.com/paper/2202.01129/) (2022)
- [Sample Complexity of Probability Divergences under Group Symmetry](https://mlsanity.com/paper/2302.01915/) (2023)
- [LD-GAN: Low-Dimensional Generative Adversarial Network for Spectral Image Generation with Variance Regularization](https://mlsanity.com/paper/2305.00132/) (2023)

## Built on by (most cited first, 4 of 4)

- [Nonlinear denoising score matching for enhanced learning of structured distributions](https://mlsanity.com/paper/2405.15625/) (2024)
- [Robustness and Structure Preservation in Flow-Based Generative Models via Wasserstein Path-Space Divergences](https://mlsanity.com/paper/2410.01244/) (2024)
- [Statistical Error Bounds for GANs with Nonlinear Objective Functionals](https://mlsanity.com/paper/2406.16834/) (2024)
- [Statistical Learning Guarantees for Group-Invariant Barron Functions](https://mlsanity.com/paper/2509.23474/) (2025)

## Abstract (arXiv)

This work presents the first statistical performance guarantees for group-invariant generative models. Many real data, such as images and molecules, are invariant to certain group symmetries, which can be taken advantage of to learn more efficiently as we rigorously demonstrate in this work. Here we specifically study generative adversarial networks (GANs), and quantify the gains when incorporating symmetries into the model. Group-invariant 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 and in the discriminator approximation error 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 and the discriminator approximation error has a reduced lower bound. Importantly, the overall error reduction cannot be achieved merely through data augmentation on the training data. Numerical results substantiate our theory and highlight the stark contrast between learning with group-invariant GANs and using data augmentation. This work also sheds light on the study of other generative models with group symmetries, such as score-based generative models.