Fixed-Point and Objective Convergence of Plug-and-Play Algorithms

Abstract

A standard model for image reconstruction involves the minimization of a data-fidelity term along with a regularizer, where the optimization is performed using proximal algorithms such as ISTA and ADMM. In plug-and-play (PnP) regularization, the proximal operator (associated with the regularizer) in ISTA and ADMM is replaced by a powerful image denoiser. Although PnP regularization works surprisingly well in practice, its theoretical convergence -- whether convergence of the PnP iterates is guaranteed and if they minimize some objective function -- is not completely understood even for simple linear denoisers such as nonlocal means. In particular, while there are works where either iterate or objective convergence is established separately, a simultaneous guarantee on iterate and objective convergence is not available for any denoiser to our knowledge. In this paper, we establish both forms of convergence for a special class of linear denoisers. Notably, unlike existing works where the focus is on symmetric denoisers, our analysis covers non-symmetric denoisers such as nonlocal means and almost any convex data-fidelity. The novelty in this regard is that we make use of the convergence theory of averaged operators and we work with a special inner product (and norm) derived from the linear denoiser; the latter requires us to appropriately define the gradient and proximal operators associated with the data-fidelity term. We validate our convergence results using image reconstruction experiments.

Published in IEEE Transactions on Computational Imaging

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• On Maximum-a-Posteriori estimation with Plug & Play priors and stochastic gradient descent
• Plug-and-Play Regularization using Linear Solvers
• On the Contractivity of Plug-and-Play Operators
• TFPnP: Tuning-free Plug-and-Play Proximal Algorithm with Applications to Inverse Imaging Problems
• On the Strong Convexity of PnP Regularization Using Linear Denoisers