Monte Carlo guided Diffusion for Bayesian linear inverse problems

TL;DR

This paper introduces MCGDiff, a Monte Carlo guided diffusion method that leverages score-based generative models and sequential Monte Carlo techniques to improve Bayesian solutions for ill-posed linear inverse problems.

Contribution

It proposes a novel algorithm combining score-based models with SMC to better approximate posteriors in ill-posed inverse problems, with theoretical grounding and superior performance.

Findings

Outperforms baseline methods in numerical simulations

Effectively handles ill-posed inverse problems in Bayesian setting

Provides theoretical analysis of the proposed approach

Abstract

Ill-posed linear inverse problems arise frequently in various applications, from computational photography to medical imaging. A recent line of research exploits Bayesian inference with informative priors to handle the ill-posedness of such problems. Amongst such priors, score-based generative models (SGM) have recently been successfully applied to several different inverse problems. In this study, we exploit the particular structure of the prior defined by the SGM to define a sequence of intermediate linear inverse problems. As the noise level decreases, the posteriors of these inverse problems get closer to the target posterior of the original inverse problem. To sample from this sequence of posteriors, we propose the use of Sequential Monte Carlo (SMC) methods. The proposed algorithm, MCGDiff, is shown to be theoretically grounded and we provide numerical simulations showing that it outperforms competing baselines when dealing with ill-posed inverse problems in a Bayesian setting.