Monte Carlo guided Diffusion for Bayesian linear inverse problems
This addresses Bayesian inference for inverse problems in fields like computational photography and medical imaging, offering an incremental improvement over existing methods.
The paper tackles ill-posed linear inverse problems, such as in medical imaging, by proposing MCGDiff, a method that combines score-based generative models with Sequential Monte Carlo to sample from a sequence of posteriors, showing it outperforms baselines in numerical simulations.
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.