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Plug-and-play (PnP) denoising is a popular iterative framework for solvingimaging inverse problems using off-the-shelf image denoisers. Their empiricalsuccess has motivated a line of research that seeks to understand theconvergence of PnP iterates under various assumptions on the denoiser. While asignificant amount of research has gone into establishing the convergence ofthe PnP iteration for different regularity conditions on the denoisers, notmuch is known about the asymptotic properties of the converged solution as thenoise level in the measurement tends to zero, i.e., whether PnP methods areprovably convergent regularization schemes under reasonable assumptions on thedenoiser. This paper serves two purposes: first, we provide an overview of theclassical regularization theory in inverse problems and survey a few notablerecent data-driven methods that are provably convergent regularization schemes.We then continue to discuss PnP algorithms and their established convergenceguarantees. Subsequently, we consider PnP algorithms with linear denoisers andpropose a novel spectral filtering technique to control the strength ofregularization arising from the denoiser. Further, by relating the implicitregularization of the denoiser to an explicit regularization functional, werigorously show that PnP with linear denoisers leads to a convergentregularization scheme. More specifically, we prove that in the limit as thenoise vanishes, the PnP reconstruction converges to the minimizer of aregularization potential subject to the solution satisfying the noiselessoperator equation. The theoretical analysis is corroborated by numericalexperiments for the classical inverse problem of tomographic imagereconstruction.

Convergent regularization in inverse problems and linear plug-and-play denoisers