  Plug-and-play priors (PnP) is a methodology for regularized image
reconstruction that specifies the prior through an image denoiser. While PnP
algorithms are well understood for denoisers performing maximum a posteriori
probability (MAP) estimation, they have not been analyzed for the minimum mean
squared error (MMSE) denoisers. This letter addresses this gap by establishing
the first theoretical convergence result for the iterative
shrinkage/thresholding algorithm (ISTA) variant of PnP for MMSE denoisers. We
show that the iterates produced by PnP-ISTA with an MMSE denoiser converge to a
stationary point of some global cost function. We validate our analysis on
sparse signal recovery in compressive sensing by comparing two types of
denoisers, namely the exact MMSE denoiser and the approximate MMSE denoiser
obtained by training a deep neural net.
