Convergence and Objective Function for Plug-and-Play Iterations

Determine whether Alternating Direction Method of Multipliers (ADMM)-based Plug-and-Play Prior iterations—formed by decoupling data fidelity from regularization and replacing the proximal step of the regularizer with a black-box denoiser D—converge, and identify the explicit objective function (if any) that these iterations optimize under clearly stated assumptions on the denoiser and the data fidelity term.

Background

Plug-and-Play (PnP) methods replace the proximal operator of a regularizer in operator-splitting schemes (such as ADMM) with a denoiser, enabling the use of powerful black-box denoising algorithms within iterative reconstruction. This flexibility yields strong empirical results but complicates theoretical analysis.

The authors highlight that, due to the black-box nature of the denoiser D, it is uncertain both whether the PnP iterations converge and, if they do, to what objective function they correspond. Establishing convergence guarantees together with a precise objective remains a central theoretical challenge.