It is known that the minimum-mean-squared-error (MMSE) denoiser under Gaussian noise can be written as a proximal operator, which suffices for asymptotic convergence of plug-and-play (PnP) methods but does not reveal the structure of the induced regularizer or give convergence rates. We show that the MMSE denoiser corresponds to a regularizer that can be written explicitly as an upper Moreau envelope of the negative log-marginal density, which in turn implies that the regularizer is 1-weakly convex. Using this property, we derive (to the best of our knowledge) the first sublinear convergence guarantee for PnP proximal gradient descent with an MMSE denoiser. We validate the theory with a one-dimensional synthetic study that recovers the implicit regularizer. We also validate the theory with imaging experiments (deblurring and computed tomography), which exhibit the predicted sublinear behavior.

翻译：已知高斯噪声下的最小均方误差（MMSE）去噪器可表示为近端算子，这一性质虽足以保证即插即用（PnP）方法的渐进收敛性，但未能揭示其诱导正则化器的结构，也无法给出收敛速率。本文证明MMSE去噪器对应的正则化器可显式表示为负对数边际密度的上Moreau包络，进而表明该正则化器具有1-弱凸性。利用这一性质，我们推导出（据我们所知）首个针对MMSE去噪器的PnP近端梯度下降法的次线性收敛保证。通过一维合成实验对隐式正则化器进行恢复，验证了理论结果。同时通过成像实验（去模糊与计算机断层扫描）验证了理论，实验结果呈现出预期的次线性收敛行为。