Geometry-Aware Discretization Error of Diffusion Models

2026-05-08

AI Summary

This work addresses the dominant role of discretization error in reverse-time sampling of diffusion models under a fixed inference budget, a factor overlooked by existing non-asymptotic analyses that are often loose and ignore data structure. By deriving first-order asymptotic expansions for both the weak error of the Euler–Maruyama scheme and the Fréchet discretization error, the paper establishes—for the first time—an explicit connection between this error and intrinsic data geometry, such as the covariance spectrum, as well as the diffusion schedule. Under the exact score assumption, the integration of numerical analysis for stochastic differential equations with asymptotic theory yields a computable, geometry-aware optimization objective in the Gaussian setting. This formulation demonstrates strong predictive performance across diverse image generation and posterior sampling tasks, offering both theoretical grounding and practical tools for geometry-informed schedule design.

Abstract

Practical diffusion sampling is a numerical approximation problem: under a fixed inference budget, one must simulate a reverse-time ODE or SDE using only a limited number of denoising steps, so discretization error is often the dominant source of error. Existing non-asymptotic analyses provide convergence guarantees, but are typically too loose and too insensitive to diffusion parameters to guide practical design: broad families of schedules receive the same rates, which depend on coarse worst-case quantities such as the dimension or the drift Lipschitz constant. We take a less ambitious but more informative route. In the exact-score setting, we derive first-order asymptotic expansions of the Euler-Maruyama weak and Fréchet discretization errors. These formulas hold for general smooth reverse diffusions and become fully explicit under Gaussian data. They show how discretization error adapts to the geometry of the data through the covariance spectrum, and how this geometry interacts with key diffusion parameters, including the diffusion schedules and the diffusion-term coefficient. This yields tractable objectives for geometry-aware parameter optimization. Finally, we show that the qualitative predictions of the Gaussian formulas remain robust across diffusion sampling problems with different geometries, including image generation on different datasets and image posterior sampling.

Problem

• discretization error
• diffusion models
• geometry-aware
• numerical approximation
• covariance spectrum

Innovation

• geometry-aware discretization
• diffusion models
• asymptotic error expansion
• covariance spectrum
• Euler-Maruyama scheme