Plug-and-Play Alternating Direction Method of Multipliers (PnP-ADMM) is a widely-used algorithm for solving inverse problems by integrating physical measurement models and convolutional neural network (CNN) priors. PnP-ADMM has been theoretically proven to converge for convex data-fidelity terms and nonexpansive CNNs. It has however been observed that PnP-ADMM often empirically converges even for expansive CNNs. This paper presents a theoretical explanation for the observed stability of PnP-ADMM based on the interpretation of the CNN prior as a minimum mean-squared error (MMSE) denoiser. Our explanation parallels a similar argument recently made for the iterative shrinkage/thresholding algorithm variant of PnP (PnP-ISTA) and relies on the connection between MMSE denoisers and proximal operators. We also numerically evaluate the performance gap between PnP-ADMM using a nonexpansive DnCNN denoiser and expansive DRUNet denoiser, thus motivating the use of expansive CNNs.

翻译：插电式交替方向乘子法（PnP-ADMM）是一种广泛用于求解逆问题的算法，它通过融合物理测量模型与卷积神经网络（CNN）先验来实现。当数据保真项为凸函数且CNN为非扩张型时，PnP-ADMM理论上已被证明收敛。然而，实际观测表明，即使对于扩张型CNN，PnP-ADMM也常能经验性收敛。本文基于将CNN先验解释为最小均方误差（MMSE）去噪器的观点，为PnP-ADMM的观测稳定性提供了理论解释。该解释与近期针对PnP迭代收缩/阈值算法变体（PnP-ISTA）提出的论证思路类似，并依赖于MMSE去噪器与近端算子之间的关联。此外，我们通过数值实验评估了使用非扩张型DnCNN去噪器与扩张型DRUNet去噪器时PnP-ADMM的性能差距，从而论证了采用扩张型CNN的合理性。