Exact Evaluation of the Accuracy of Diffusion Models for Inverse Problems with Gaussian Data Distributions

Abstract

Used as priors for Bayesian inverse problems, diffusion models have recently attracted considerable attention in the literature. Their flexibility and high variance enable them to generate multiple solutions for a given task, such as inpainting, super-resolution, and deblurring. However, there is still a lack of understanding about how accurately these conditional diffusion algorithms perform conditional sampling. In this article, we investigate the errors induced by these models when applied to a Gaussian data distribution for which the score function is exactly known. Within this constrained context, we are able to precisely analyze the discrepancy between the theoretical resolution of inverse problems via conditional sampling and the practical distributions generated by conditional diffusion models. This is done by characterizing all the involved iterative Gaussian processes and by computing the exact Wasserstein distance between the distributions of the diffusion model samplers and the ideal conditional distribution associated with the inverse problem. Our findings allow for the comparison of two major algorithms from the literature, Deep Posterior Sampling (DPS) and Pseudo-inverse Guided Diffusion Models ($\Pi$GDM), and the introduction of the new paradigm Conditional Gaussian Diffusion Models (CGDM) that is shown to be more accurate for Gaussian data distributions.