Plug-and-Play Methods Provably Converge with Properly Trained Denoisers (1905.05406v1)

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.

• The paper establishes convergence guarantees for PnP-FBS and PnP-ADMM methods by enforcing a Lipschitz condition with a novel real spectral normalization technique.
• The authors validate their theory with numerical experiments, demonstrating superior performance in Poisson denoising, single photon imaging, and compressed sensing MRI.
• This work advances the integration of deep learning and model-based optimization, offering robust solutions for real-world imaging challenges in medical and scientific fields.

Insightful Overview of "Plug-and-Play Methods Provably Converge with Properly Trained Denoisers"

The paper "Plug-and-Play Methods Provably Converge with Properly Trained Denoisers" addresses a core challenge in the domain of plug-and-play (PnP) image restoration techniques: the theoretical convergence analysis of these non-convex frameworks. PnP methods are notable for integrating modern, pre-trained denoisers into optimization problems without the need for an explicit model of the image prior. This flexibility is particularly advantageous when the availability of data for end-to-end training is limited.

Key Contributions and Theoretical Foundations

The principal contribution of the paper is the establishment of theoretical guarantees for the convergence of two specific PnP methods: PnP-FBS (Forward-Backward Splitting) and PnP-ADMM (Alternating Direction Method of Multipliers). This convergence is shown under a novel Lipschitz condition applied to the denoisers involved. Specifically, the authors introduce a real spectral normalization (realSN) technique to train deep learning-based denoisers so they adhere to the stipulated Lipschitz continuity.

The convergence analysis is executed through contractive iterations, a method ensuring that the iterates generated by the PnP algorithms contract towards a fixed point. The authors demonstrate that the analyses of these PnP methods under the generalized Lipschitz condition do not require diminishing step sizes, a property that simplifies the deployment in practical applications.

Numerical Validation and Experimental Setup

To bolster their theoretical findings, the authors present extensive experimental validations, demonstrating the efficacy of the proposed realSN and showcasing the performance of various PnP algorithms across several image restoration tasks, including Poisson denoising, single photon imaging, and compressed sensing MRI. Notably, the results reveal that PnP-ADMM with realSN-DnCNN consistently achieves superior performance, highlighting its robustness and effectiveness in diverse experimental conditions.

Specific Findings

The paper's experimental section includes several significant results:

• Poisson Denoising: The PnP-ADMM framework, when paired with realSN-DnCNN, exhibits a notable PSNR improvement over traditional denoisers like BM3D.
• Single Photon Imaging: The ADMM-based methods outperform their FBS counterparts when utilizing the same denoiser configuration, confirming the theoretical predictions about convergence efficacy.
• Compressed Sensing MRI: With the application of PnP techniques, particularly those incorporating realSN-DnCNN, notable improvements over baseline methods such as total variation (TV) and zero-filling are observed.

Implications and Future Prospects

This paper’s framework substantiates the possibility of leveraging sophisticated pretrained denoisers in plug-and-play methodologies, with rigorous convergence assurances. The proposed real spectral normalization marks a significant step toward integrating deep learning models with optimization frameworks, ensuring stability and reliability.

In the practical field, these findings have implications for fields like medical imaging, where comprehensive data for end-to-end training is scarce, and for areas such as astronomy and microscopy, where precise denoising plays a critical role. Theoretically, this work may spur further research into the design of learning algorithms for denoisers that meet specific mathematical properties, bolstering the efficiency and applicability of PnP methodologies.

Looking ahead, the principles established could be extended to other non-linear inverse problems or adapted to incorporate safety and fairness constraints seamlessly into machine learning models. Thus, this work lays a robust foundation for future explorations into the intersection of model-based optimization and data-driven learning methodologies.