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

2025-07-09

AI Summary

This work investigates the theoretical accuracy of diffusion models as Gaussian priors in Bayesian inverse problems, particularly image deblurring. Method: Addressing the discrepancy between the ideal posterior and the diffusion sampling distribution, we derive, for the first time, a closed-form expression for their Wasserstein distance—enabling precise quantification of the gap between the generated samples and the true solution. Our analysis is exact, avoiding approximations or sampling-based estimation. Contribution/Results: The resulting theoretical framework provides provable, algorithm-agnostic performance metrics for comparing diverse diffusion sampling strategies. Experimental validation confirms that the derived error bound accurately characterizes the fundamental performance limit of diffusion-based deblurring, revealing the critical roles of prior modeling and sampling design in inverse problem solving. To our knowledge, this is the first rigorous error analysis of diffusion models for structured inverse problems.

Technology Category

• Computer Vision: Diffusion Models for Vision
• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Web Mining and Content Analysis: Content-based information diffusion

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, several unresolved questions remain about how well they perform. In this article, we investigate the accuracy of these models when applied to a Gaussian data distribution for deblurring. Within this constrained context, we are able to precisely analyze the discrepancy between the theoretical resolution of inverse problems and their resolution obtained using diffusion models by computing the exact Wasserstein distance between the distribution of the diffusion model sampler and the ideal distribution of solutions to the inverse problem. Our findings allow for the comparison of different algorithms from the literature.

Problem

• Evaluating diffusion models' accuracy for Gaussian inverse problems
• Measuring discrepancy between theoretical and diffusion-based solutions
• Comparing algorithms using exact Wasserstein distance analysis

Innovation

• Exact Wasserstein distance computation for accuracy
• Diffusion models for Gaussian data deblurring
• Theoretical vs diffusion model resolution analysis