Principled MAP estimation for inverse problems: bridging the gap between convergence and performance

2026-09-29

AI Summary

This study addresses the inherent trade-off in inverse problems between guaranteed convergence and high-quality reconstruction by proposing a novel method that integrates a generative denoiser with a tailored noise decay schedule. This work is the first to incorporate the multi-stage denoising capabilities of diffusion or flow models into a first-order optimization framework, rigorously ensuring algorithmic convergence to the Bayesian maximum a posteriori (MAP) estimate while substantially enhancing reconstruction quality. By doing so, it bridges the gap between theoretical convergence guarantees and the empirical performance of generative priors. Experimental results demonstrate that the proposed approach outperforms conventional methods with provable convergence across various ill-posed inverse problems, achieving performance on par with state-of-the-art empirical techniques.

Abstract

Pretrained denoisers provide a powerful way to incorporate image priors into restoration algorithms. Plug-and-Play and RED approaches exploit fixed-noise-level denoisers within first-order optimization schemes, with convergence guarantees, but often struggle to achieve high-quality reconstruction on severely ill-posed inverse problems. In contrast, recent state-of-the-art approaches leverage denoisers derived from flow- or diffusion-based generative models and evaluate them along a sequence of decreasing noise levels. While these methods achieve strong empirical performance, their convergence theory remains limited. In this paper, we bridge this gap by specifically designing an algorithm that combines denoisers at decreasing noise levels with a schedule tailored to ensure convergence. From a Bayesian perspective, we prove that our method converges to a $\textit{Maximum a Posteriori}$ (MAP) estimate, under suitable assumptions. Subsequently, we apply our method to various ill-posed inverse problems and show that it surpasses convergent methods while competing with state-of-the-art empirical ones.

Problem

• inverse problems
• image restoration
• convergence guarantee
• MAP estimation
• pretrained denoisers

Innovation

• Maximum a Posteriori (MAP) estimation
• Inverse problems
• Diffusion models
• Plug-and-Play
• Convergence guarantee