Effective Covariance Dynamics in Solvable High-Dimensional GANs

2026-06-25

AI Summary

This work investigates how to effectively learn low-dimensional subspaces spanned by structured latent covariances in high-dimensional GANs, where the covariances exhibit class-dependent, non-diagonal correlations and non-zero means. To this end, the authors construct an analytically tractable linear GAN model, extending high-dimensional solvable analysis for the first time to such complex covariance structures, and characterize the training dynamics via deterministic ordinary differential equations in the high-dimensional limit. Theoretically, they reveal that low-rank correlations enhance the learnability of weak signal directions through a signal amplification mechanism, yet excessive correlation induces instability in recovery; they further derive mode-wise solvability regimes and phase boundaries. Experiments on MNIST, FashionMNIST, and CIFAR-10 demonstrate that incorporating prior covariance significantly improves alignment between the learned generative subspace and the true data subspace.

Abstract

We study a solvable high-dimensional model of generative adversarial network (GAN) training in which a linear generator learns a low-dimensional subspace from data with structured latent covariance. Prior solvable GAN analyses assume unconditional signals with diagonal latent covariance; we extend the multi-feature discriminator setting to class-dependent, correlated, and non-zero-mean latent structure. For the quadratic energy discriminator, all such heterogeneity enters the dynamics through a probability-weighted effective second moment. We prove that the stochastic microscopic training process converges, in the high-dimensional limit, to deterministic ordinary differential equations governed by this effective covariance. In the matched-covariance specialization, the stability analysis yields a mode-wise solvable interval determined by the learning rates and noise level: learning begins when the leading effective eigenvalue crosses the lower threshold, while full recovery requires all relevant effective modes to remain within the interval. This reveals a signal-boosting mechanism: low-rank correlations can lift weak directions above the learnability threshold, whereas overly strong correlations destabilize recovery. Numerical simulations validate the ODE, phase boundary, and boosting mechanism. Experiments on MNIST, FashionMNIST, and CIFAR-10 further show that informed generator covariance improves alignment with the data-driven reference subspace.

Problem

• GAN training
• high-dimensional limit
• latent covariance
• effective covariance
• learnability

Innovation

• effective covariance
• solvable GANs
• high-dimensional limit
• signal boosting
• structured latent covariance