Plug-and-Play Methods Provably Converge Even with Improperly Trained Denoisers: Convergence by Architectural Design

2026-09-27

AI Summary

This study addresses the limitation of plug-and-play methods, where directly substituting proximal operators compromises variational interpretability and precludes convergence guarantees. To overcome this, we propose the Learned Proximal Network (LPN), which leverages architectural design to ensure that the denoiser strictly corresponds to the proximal operator of a regularizer. Furthermore, we extend the theoretical framework to a broader class of activation functions, analytically characterize the mean-induced regularization mechanism, and develop an operator scaling technique with provable convergence guarantees. Consequently, this work restores both the variational interpretation and convergence properties of the algorithm while maintaining state-of-the-art reconstruction quality, thereby providing rigorous theoretical foundations for plug-and-play approaches.

Abstract

Plug-and-Play (PnP) methods replace proximal operators with learned denoisers, which produce state-of-the-art reconstruction quality, but sacrifice the variational interpretation and convergence guarantees of proximal methods. Learned Proximal Networks (LPNs) address this problem by designing the denoiser architecture so that it is exactly the proximal operator of a regularizer. In this work, we clarify the theoretical foundations of LPNs and extend the framework to a broader class of activation functions. We then study two practical mechanisms for controlling the learned prior: (i) averaging the LPN with the identity, a common heuristic in PnP methods, and (ii) directly scaling the implicit regularizer induced by the proximal operator. In particular, we characterize the regularizer induced by averaging and develop a convergent method for evaluating the scaled proximal operator.

Problem

• Plug-and-Play
• convergence guarantees
• proximal operators
• variational interpretation

Innovation

• Plug-and-Play
• Learned Proximal Networks
• Convergence Guarantee
• Architectural Design
• Implicit Regularizer