Universal Local Error and Realized Amplification for the First-Order EDM Predictor

2026-10-07

AI Summary

This study presents a theoretical analysis of local discretization errors and their cross-step amplification in first-order EDM diffusion samplers. Methodologically, it decouples local errors from amplification effects via the 2-Wasserstein distance, establishing a universal quadratic local error bound independent of the data distribution. By incorporating network parameterization properties, a high-noise contraction criterion is derived, while a low-noise amplification metric based on the empirical transport distribution is introduced to overcome worst-case Lipschitz constraints. These components collectively yield a global error bound of $O(e^{\Lambda_K}/K)$. The theoretical stability mechanisms and error decay rates are subsequently validated through experiments on one-dimensional Gaussian mixtures and CIFAR-10.

Abstract

We analyze the first-order deterministic diffusion sampler of Karras et al. (2022), termed EDM, in 2-Wasserstein distance by separating two sources of error: local discretization error and its amplification by subsequent learned steps. We prove that local error admits a universal bound: for any data distribution with finite second moment, the one-step discretization error is quadratic in the step size, with an explicit constant that does not depend on the data distribution. Error propagation, in contrast, depends on the learned network. At high noise levels, we exploit the network parametrization of EDM to derive an explicit contraction criterion. At low noise levels, we measure propagation through the amplification realized on the distributions transported by the sampler; this realized amplification can be arbitrarily smaller than the worst-case Lipschitz constant. This analysis yields an $O(e^{Λ_K}/K)$ global discretization error for $K$ sampling steps, where $Λ_K$ is the low-noise log-amplification. Experiments on a one-dimensional Gaussian mixture show how measured amplification accounts for slower error decay on finite sampling grids. Diagnostics on a pretrained CIFAR-10 model illustrate related stability mechanisms without certifying the global assumptions.

Problem

• Diffusion sampler
• Discretization error
• Error amplification
• 2-Wasserstein distance
• EDM

Innovation

• Deterministic Diffusion Sampler
• Local Discretization Error
• Realized Amplification
• 2-Wasserstein Distance
• Error Propagation