Statistical Guarantees of Group-Invariant GANs

TL;DR

This work provides the first statistical guarantees for group-invariant GANs by introducing $\Sigma$-invariant generators and discriminators via symmetry-based symmetrization. It proves that learning a $\Sigma$-invariant target distribution $\mu$ on a compact domain in $\mathbb{R}^d$ achieves an expected Wasserstein-1 error of order $\left(|\Sigma|n\right)^{-1/d}$, and, when $\mu$ is supported on a $d^*$-dimensional manifold, of order $\left(|\Sigma|n\right)^{-1/d^*}$, effectively simulating $|\Sigma|n$ i.i.d. samples. The analysis decomposes the error into invariant-discriminator approximation, invariant-generator approximation, and statistical terms, showing that symmetry reduces sample complexity and discriminator lower-bounds, beyond what data augmentation can accomplish. Numerical experiments with $C_4$-symmetric Gaussian mixtures corroborate the theory, demonstrating superior performance of invariant GANs over non-invariant models and augmentation alone. The results open avenues for extending these guarantees to continuous groups and other symmetry-preserving generative models, highlighting the practical impact on data-efficient learning in symmetry-rich domains.

Abstract

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.

Figures (4)

• Figure 1: Examples of Assumption \ref{['assumption:new']}. Left: Mirror reflection in $\mathbb{R}^2$. $\mathcal{X}_0 = [0,1]\times[0,1]$. The group actions are generated by $\sigma(x,y) = (-x,y)$. Right: Circular rotations within the unit disk in $\mathbb{R}^2$. The group actions are generated by the $\pi/2$-rotation with respect to the origin. $\mathcal{X}_0 = (\rho\cos\theta,\rho\sin\theta),~~\rho\in[0,1],~~\theta\in[0,\pi/2)$. $A_0(\epsilon)$ are filled with yellow color in both subfigures.
• Figure 2: Heat map of 5000 GAN generated samples learned from a 2D Gaussian mixture. Top row: GAN with no symmetry; second row: GAN with $C_2$ (partial) symmetry; third row: GAN with $C_4$ (full) symmetry; bottom row: GAN with no symmetry but with $C_4$ augmentation on the training data. Brighter parts refer to higher density.
• Figure 3: Heat map of 5000 GAN generated samples learned from a 2D Gaussian mixtur embedded in a higher-dimensional ambient space $\mathbb{R}^{12}$. The figure shows the 2D projection of the generated samples onto the intrinsic support plane ($d^*=2$) of the distribution. Top row: GAN with no symmetry; second row: GAN with $C_2$ (partial) symmetry; third row: GAN with $C_4$ (full) symmetry; bottom row: GAN with no symmetry but with $C_4$ augmentation on the training data. Compared to \ref{['fig:student-t_2d']}, this result qualitatively suggests that the convergence depends only on the intrinsic dimension, as discussed in \ref{['theorem:lowdimensional']}.
• Figure 4: Wasserstein-1 distance between 10000 samples drawn from the generated and the target distribution with different GAN implementations, over 20 runs. Left: 2D example. Right: 12D example. $C_4$ GAN achieves the best performance in both cases.

Theorems & Definitions (50)

• Lemma 1: paraphrased from birrell2022structure
• Remark 1
• Definition 1: Fundamental domain
• Definition 2: Covering number
• Remark 2
• Example 1: Mirror reflection in $\mathbb{R}^2$
• Example 2: Rotations in $\mathbb{R}^2$
• Remark 3
• Definition 3: $\Sigma$-invariant discriminators
• Remark 4