A Random Matrix Theory Perspective on the Consistency of Diffusion Models

TL;DR

This paper uses random matrix theory to explain why diffusion models trained on different data splits produce similar outputs, revealing how dataset properties influence generative consistency and variability.

Contribution

It develops a novel RMT framework to quantify dataset effects on diffusion models' expectations and fluctuations, providing insights into reproducibility and stability.

Findings

Shared Gaussian statistics predict output similarity across data splits.

Dataset size and spectral properties influence diffusion model variability.

The theory accurately predicts behavior of linear diffusion models on neural architectures.

Abstract

Diffusion models trained on different, non-overlapping subsets of a dataset often produce strikingly similar outputs when given the same noise seed. We trace this consistency to a simple linear effect: the shared Gaussian statistics across splits already predict much of the generated images. To formalize this, we develop a random matrix theory (RMT) framework that quantifies how finite datasets shape the expectation and variance of the learned denoiser and sampling map in the linear setting. For expectations, sampling variability acts as a renormalization of the noise level through a self-consistent relation $\sigma^2 \mapsto \kappa(\sigma^2)$, explaining why limited data overshrink low-variance directions and pull samples toward the dataset mean. For fluctuations, our variance formulas reveal three key factors behind cross-split disagreement: \textit{anisotropy} across eigenmodes, \textit{inhomogeneity} across inputs, and overall scaling with dataset size. Extending deterministic-equivalence tools to fractional matrix powers further allows us to analyze entire sampling trajectories. The theory sharply predicts the behavior of linear diffusion models, and we validate its predictions on UNet and DiT architectures in their non-memorization regime, identifying where and how samples deviates across training data split. This provides a principled baseline for reproducibility in diffusion training, linking spectral properties of data to the stability of generative outputs.