Convergence of Nonconvex PnP-ADMM with MMSE Denoisers
It addresses the stability issue in inverse problem solving for researchers in computational imaging and optimization, but is incremental as it extends prior theoretical arguments from PnP-ISTA to PnP-ADMM.
This paper provides a theoretical explanation for the empirical convergence of PnP-ADMM with expansive CNNs by interpreting the CNN prior as an MMSE denoiser, and numerically shows that expansive DRUNet denoisers outperform nonexpansive DnCNN denoisers, motivating their use.
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.