On the Strong Convexity of PnP Regularization Using Linear Denoisers

In the Plug-and-Play (PnP) method, a denoiser is used as a regularizer within classical proximal algorithms for image reconstruction. It is known that a broad class of linear denoisers can be expressed as the proximal operator of a convex regularizer. Consequently, the associated PnP algorithm can be linked to a convex optimization problem $\mathcal {P}$. For such a linear denoiser, we prove that $\mathcal {P}$ exhibits strong convexity for linear inverse problems. Specifically, we show that the strong convexity of $\mathcal {P}$ can be used to certify objective and iterative convergence of any PnP algorithm derived from classical proximal methods.