A Proximal Markov Chain Monte Carlo Method for Bayesian Inference in Imaging Inverse Problems: When Langevin Meets Moreau

Modern imaging methods rely strongly on Bayesian inference techniques to solve challenging imaging problems. Currently, the predominant Bayesian computational approach is convex optimization, which scales very efficiently to high-dimensional image models and delivers accurate point estimation results. However, in order to perform more complex analyses, for example, image uncertainty quantification or model selection, it is often necessary to use more computationally intensive Bayesian computation techniques such as Markov chain Monte Carlo methods. This paper presents a new and highly efficient Markov chain Monte Carlo methodology to perform Bayesian computation for high-dimensional models that are log-concave and nonsmooth, a class of models that is central in imaging sciences. The methodology is based on a regularized unadjusted Langevin algorithm that exploits tools from convex analysis, namely, Moreau--Yosida envelopes and proximal operators, to construct Markov chains with favorable convergence properties. In addition to scaling efficiently to high dimensions, the method can be applied in a straightforward manner to models that are currently solved using proximal optimization algorithms. We provide a detailed theoretical analysis of the proposed methodology, including asymptotic and nonasymptotic convergence results with easily verifiable conditions, and explicit bounds on the convergence rates. The proposed methodology is demonstrated with five experiments related to image deconvolution and tomographic reconstruction with total-variation and $\ell_1$ priors, where we conduct a range of challenging Bayesian analyses related to uncertainty quantification, hypothesis testing, and model selection in the absence of ground truth.