[Paper Review] Plug-and-Play Methods Provably Converge with Properly Trained Denoisers

Ernest K. Ryu, Jialin Liu+4 more|arXiv (Cornell University)|May 14, 2019

Sparse and Compressive Sensing Techniques100 citations

TL;DR

The paper proves convergence of PnP-FBS and PnP-ADMM under a Lipschitz condition on the denoiser and introduces real spectral normalization (realSN) to train denoisers to satisfy this condition, with experiments validating theory.

ABSTRACT

Plug-and-play (PnP) is a non-convex framework that integrates modern denoising priors, such as BM3D or deep learning-based denoisers, into ADMM or other proximal algorithms. An advantage of PnP is that one can use pre-trained denoisers when there is not sufficient data for end-to-end training. Although PnP has been recently studied extensively with great empirical success, theoretical analysis addressing even the most basic question of convergence has been insufficient. In this paper, we theoretically establish convergence of PnP-FBS and PnP-ADMM, without using diminishing stepsizes, under a certain Lipschitz condition on the denoisers. We then propose real spectral normalization, a technique for training deep learning-based denoisers to satisfy the proposed Lipschitz condition. Finally, we present experimental results validating the theory.

Motivation & Objective

• Motivate the use of denoisers within proximal optimization frameworks (PnP) for image restoration.
• Establish convergence of PnP-FBS and PnP-ADMM without diminishing stepsizes under a Lipschitz assumption on the denoiser.
• Introduce real spectral normalization (realSN) to train deep denoisers to satisfy the Lipschitz condition.
• Present theoretical results and experimental validation across Poisson denoising, single-photon imaging, and CS-MRI.

Proposed method

• Model PnP-FBS and PnP-ADMM as fixed-point iterations via denoisers and proximal operators.
• Impose Assumption (A): (H_sigma - I) is Lipschitz with constant epsilon, enabling contraction proofs.
• Derive contraction factors for PNP-FBS and PNP-DRS, leading to convergence conditions on the step parameter alpha.
• Show equivalence between PNP-ADMM and PNP-DRS for analytical tractability.
• Introduce real spectral normalization (realSN) to train denoisers so that the Lipschitz assumption holds in practice.
• Implement and evaluate on Gaussian-denoising framed denoisers (DnCNN and SimpleCNN) with RealSN, and BM3D as a baseline.

Experimental results

Research questions

• RQ1 Under what conditions do Plug-and-Play Forward-Backward Splitting (PnP-FBS) and PnP-ADMM converge when denoisers are not nonexpansive or differentiable?
• RQ2 Can a Lipschitz-type constraint on denoisers (Assumption A) guarantee contraction and hence convergence of PnP iterations without diminishing stepsizes?
• RQ3 How can denoisers be trained to satisfy the Lipschitz condition effectively in practice?
• RQ4 Do realSN-trained denoisers improve convergence and empirical denoising performance in Poisson denoising, single-photon imaging, and CS-MRI?

Key findings

• PnP-FBS and PnP-DRS are contractive under Assumption (A) and strong convexity of f, yielding geometric convergence to fixed points.
• Convergence conditions for PnP-FBS require a beta-interval for alpha, with existence when epsilon < 2 mu /(L - mu).
• Convergence for PnP-DRS and PnP-ADMM hold with explicit bounds involving epsilon and mu, including epsilon < 1 for ADMM fixed-point results.
• Real spectral normalization (realSN) improves the Lipschitz bound of deep denoisers, making Assumption (A) more realistic in practice.
• Experiments on Poisson denoising show RealSN improves convergence behavior and, with ADMM, achieves competitive PSNR versus BM3D; ADMM generally outperforms FBS in several tasks.
• RealSN-enhanced denoisers yield better reconstruction performance in CS-MRI and single-photon imaging when plugged into PnP frameworks.