Effective Covariance Dynamics in Solvable High-Dimensional GANs

Mode collapse remains a fundamental challenge in training generative adversarial networks (GANs). While existing works have primarily focused on inter-mode collapse, such as mode dropping, intra-mode collapse-where many latent variables map to the same or highly similar outputs-has received significantly less attention. In this work, we propose a pairing regularizer jointly optimized with the generator to mitigate the many-to-one collapse by enforcing local consistency between latent variables and generated samples. We show that the effect of pairing regularization depends on the dominant failure mode of training. In collapse-prone regimes with limited exploration, pairing encourages structured local exploration, leading to improved coverage and higher recall. In contrast, under stabilized training with sufficient exploration, pairing refines the generator's induced data density by discouraging redundant mappings, thereby improving precision without sacrificing recall. Extensive experiments on both toy distributions and real-image benchmarks demonstrate that the proposed regularizer effectively complements existing stabilization techniques by directly addressing intra-mode collapse.

Despite their empirical success across a wide range of generative tasks, the fundamental principles underlying the ability of diffusion models to learn data distributions are poorly understood. In this work, we develop a new mathematical framework that explains how diffusion models can effectively learn low-dimensional distributions from a finite number of training samples without suffering from the curse of dimensionality. Specifically, motivated by the intrinsic low-dimensional structure of image data, we theoretically analyze a setting in which the data distribution is modeled as a mixture of low-rank Gaussians. Under suitable network parameterization, we show that optimizing the training objective of diffusion models is equivalent to solving the canonical subspace clustering problem over the training samples, where each subspace basis corresponds to the low-rank covariance of a Gaussian component. This equivalence allows us to show that the sample complexity for learning the underlying distribution scales linearly with the intrinsic dimension of the data, rather than exponentially with the ambient dimension. Our theoretical findings are further supported by empirical evidence that demonstrates phase transition phenomena in generalization on both synthetic and real-world image datasets. Moreover, we establish a correspondence between the learned subspace bases and semantic attributes of image data, providing a principled foundation for controllable image generation.

A central error measure in Gaussian DDPMs is the path-space KL divergence between the exact reverse chain and the learned Gaussian reverse process. This quantity is especially relevant for procedures such as classifier guidance, which perturb the entire reverse trajectory rather than only the terminal sample. Prior analyses show that standard isotropic reverse covariances suffer an unavoidable $Ω(1/T)$ path-KL error as the number of denoising steps $T$ grows. We show that matching the full posterior covariance breaks this barrier, yielding an order-wise improvement that reduces the path KL to $O(1/T^2)$. To make full covariance matching practical, we introduce the Lanczos Gaussian sampler (LGS), a training-free, matrix-free method for sampling from the optimal reverse covariance using only covariance-vector products, which are available through Jacobian-vector products of the posterior mean. LGS avoids dense covariance storage and auxiliary covariance models. We prove that LGS approximation error decays exponentially in the number of Lanczos steps, where each Lanczos step requires a single Jacobian-vector product. Empirically, using only just three such steps improves sample quality over strong diagonal-covariance baselines, including OCM-DDPM, across standard image benchmarks. This identifies full covariance matching as both theoretically valuable and practically accessible for fast DDPM sampling.