  For image recovery problems, plug-and-play (PnP) methods have been developed
that replace the proximal step in an optimization algorithm with a call to an
application-specific denoiser, often implemented using a deep neural network.
Although such methods have been successful, they can be improved. For example,
the denoiser is often trained using white Gaussian noise, while PnP's denoiser
input error is often far from white and Gaussian, with statistics that are
difficult to predict from iteration to iteration. PnP methods based on
approximate message passing (AMP) are an exception, but only when the forward
operator behaves like a large random matrix. In this work, we design a PnP
method using the expectation consistent (EC) approximation algorithm, a
generalization of AMP, that offers predictable error statistics at each
iteration, from which a deep-net denoiser can be effectively trained.
