MAP-convergent Denoising in Nonconvex PnP for Inverse Problems

Plug-and-Play (PnP) has become an increasingly popular framework for inverse problems, enabling optimization formulations with learning-based priors in proximal splitting methods. Score-based generative models have recently emerged as strong learned priors, but it remains challenging to efficiently utilize them in splitting-based PnP frameworks. Specifically, PnP frameworks often use denoisers as implicit priors in algorithms motivated by Maximum a Posteriori (MAP) estimation, but the annealed nature of score-based denoisers makes the theoretical connection between MAP and PnP challenging. Another difficulty comes from the geometry of the optimization iterates, which can deviate significantly from the noisy data that the score-based model is trained on. We consider MAP estimation with score-based PnP through primal-dual splitting and show that under suitable conditions, the primal-dual iterates converge to the stationary set of the corresponding nonconvex and nonsmooth MAP objective function. Additionally, we quantify the geometry mismatch introduced by the splitting method, while also revealing a tradeoff between convergence and geometrical alignment of the iterates. Our primal-dual approach yields updates which admit efficient closed-form solutions for structured nonlinear problems, including phase retrieval and sparse-view transmission computed tomography. Experiments on linear and nonlinear inverse problems show competitive reconstruction quality with fewer score evaluations and shorter runtimes compared to the state-of-the-art.