  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.
