FISTA Iterates Converge Linearly for Denoiser-Driven Regularization

The effectiveness of denoising-driven regularization for image reconstruction has been widely recognized. Two prominent algorithms in this area are Plug-and-Play (textttPnP) and Regularization-by-Denoising (textttRED). We consider two specific algorithms textttPnP-FISTA and textttRED-APG, where regularization is performed by replacing the proximal operator in the textttFISTA algorithm with a powerful denoiser. The iterate convergence of textttFISTA is known to be challenging with no universal guarantees. Yet, we show that for linear inverse problems and a class of linear denoisers, global linear convergence of the iterates of textttPnP-FISTA and textttRED-APG can be established through simple spectral analysis.