  In plug-and-play (PnP) regularization, the knowledge of the forward model is
combined with a powerful denoiser to obtain state-of-the-art image
reconstructions. This is typically done by taking a proximal algorithm such as
FISTA or ADMM, and formally replacing the proximal map associated with a
regularizer by nonlocal means, BM3D or a CNN denoiser. Each iterate of the
resulting PnP algorithm involves some kind of inversion of the forward model
followed by denoiser-induced regularization. A natural question in this regard
is that of optimality, namely, do the PnP iterations minimize some f+g, where f
is a loss function associated with the forward model and g is a regularizer?
This has a straightforward solution if the denoiser can be expressed as a
proximal map, as was shown to be the case for a class of linear symmetric
denoisers. However, this result excludes kernel denoisers such as nonlocal
means that are inherently non-symmetric. In this paper, we prove that a broader
class of linear denoisers (including symmetric denoisers and kernel denoisers)
can be expressed as a proximal map of some convex regularizer g. An algorithmic
implication of this result for non-symmetric denoisers is that it necessitates
appropriate modifications in the PnP updates to ensure convergence to a minimum
of f+g. Apart from the convergence guarantee, the modified PnP algorithms are
shown to produce good restorations.
