Plug-and-Play Methods Provably Converge with Properly Trained Denoisers
This provides theoretical foundations for PnP methods, addressing a key gap for researchers and practitioners using these techniques in inverse problems and imaging.
The paper tackles the lack of theoretical convergence guarantees for plug-and-play (PnP) methods, which integrate denoising priors into optimization algorithms, by proving convergence for PnP-FBS and PnP-ADMM under a Lipschitz condition and proposing real spectral normalization to train denoisers that satisfy this condition.
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.