Diffusion models for Gaussian distributions: Exact solutions and Wasserstein errors

Emile Pierret, Bruno Galerne

Keywords: Diffusion models, image generation, differential equations, discretization schemes

TL;DR: A study of the convergence of diffusion models for Gaussian distributions.

Abstract:

Diffusion or score-based models recently showed high performance in image generation.
They rely on a forward and a backward stochastic differential equations (SDE). The sampling of a data distribution is achieved by solving numerically the backward SDE or its associated flow ODE.
Studying the convergence of these models necessitates to control four different types of error: the initialization error, the truncation error, the discretization and the score approximation.
In this paper, we study theoretically the behavior of diffusion models and their numerical implementation when the data distribution is Gaussian.
In this restricted framework where the score function is a linear operator, we derive the analytical solutions of the backward SDE and the probability flow ODE.
We prove that these solutions and their discretizations are all Gaussian processes, which allows us to compute exact Wasserstein errors induced by each error type for any sampling scheme.
Monitoring convergence directly in the data space instead of relying on Inception features, our experiments show that the recommended numerical schemes from the diffusion models literature are also the best sampling schemes for Gaussian distributions.

Primary Area: generative models

Code Of Ethics: I acknowledge that I and all co-authors of this work have read and commit to adhering to the ICLR Code of Ethics.

Submission Guidelines: I certify that this submission complies with the submission instructions as described on https://iclr.cc/Conferences/2025/AuthorGuide.

Anonymous Url: I certify that there is no URL (e.g., github page) that could be used to find authors’ identity.

No Acknowledgement Section: I certify that there is no acknowledgement section in this submission for double blind review.

Submission Number: 6981