Figure 1: An illustration of convergence of PnP-ADMM with expansive and non-expansive denoisers.
Abstract
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.
Convergence Behaviour of PnP-ADMM and PnP-FISTA
Figure 2: Comparison of PnP-ADMM and PnP-FISTA, each using a non-expansive DnCNN denoiser and an expansive DRUNet denoiser. The figure plots the evolution of \( \|x^{k} - x^{k+1}\|_2 / \|x^{k+1}\|_2 \), while the right one that of PSNR (dB).
PnP-ADMM and PnP-FISTA for Image Deblurring
Figure 3: Comparison of four different methods for deblurring a color image with a noise level of 0.03. The reconstruction performance is quantified using PSNR and SSIM in the top-left corner of each image. Note the improved performance of PnP-ADMM using an expansive DRUNet denoiser compared to nonexpansive DnCNN denoiser.
Table 1: Performance of PnP-ADMM and PnP-FISTA using two priors on image deblurring at different levels of noise.
Paper
