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Bayesian methods to solve imaging inverse problems usually combine anexplicit data likelihood function with a prior distribution that explicitlymodels expected properties of the solution. Many kinds of priors have beenexplored in the literature, from simple ones expressing local properties tomore involved ones exploiting image redundancy at a non-local scale. In adeparture from explicit modelling, several recent works have proposed andstudied the use of implicit priors defined by an image denoising algorithm.This approach, commonly known as Plug & Play (PnP) regularisation, can deliverremarkably accurate results, particularly when combined with state-of-the-artdenoisers based on convolutional neural networks. However, the theoreticalanalysis of PnP Bayesian models and algorithms is difficult and works on thetopic often rely on unrealistic assumptions on the properties of the imagedenoiser. This papers studies maximum-a-posteriori (MAP) estimation forBayesian models with PnP priors. We first consider questions related toexistence, stability and well-posedness, and then present a convergence prooffor MAP computation by PnP stochastic gradient descent (PnP-SGD) underrealistic assumptions on the denoiser used. We report a range of imagingexperiments demonstrating PnP-SGD as well as comparisons with other PnPschemes.

On Maximum-a-Posteriori estimation with Plug & Play priors and stochastic gradient descent