Monte Carlo guided Denoising Diffusion models for Bayesian linear inverse problems

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, scorebased generative models (SGM) have recently been successfully applied to several different inverse problems. In this paper, 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 posterior distributions of these inverse problems get closer to the target posterior of the original inverse problem. To sample from these distributions, 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.