Convergence of Nonconvex PnP-ADMM with MMSE Denoisers

Nonconvex Convergence Analysis of PnP-ADMM for MMSE Priors: Exploring Stability with Expansive CNNs

Chicago Park*, Shirin Shoushtari*, Weijie Gan, Ulugbek S. Kamilov

Computational Imaging Group (CIG)
Washington University in St.Louis, MO, USA

Chicago Park and Shirin Shoushtari contributed equally to this project.


Figure 1: An illustration of convergence of PnP-ADMM with expansive and non-expansive denoisers.


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.



@article{Park.etal2023, author={Park, Chicago and Shoushtari, Shirin and Gan, Weijie and Kamilov, Ulugbek S.}, title={Convergence of Nonconvex PnP-ADMM with MMSE Denoisers}, note={arXiv:2311.18810}, year={2023} }