Achieving Bayes MMSE performance in the sparse signal + Gaussian white noise model when the noise level is unknown

Recent work on Approximate Message Passing algorithms in compressed sensing focuses on `ideal´ algorithms which at each iteration face a subproblem of recovering an unknown sparse signal in Gaussian white noise. The noise level in each subproblem changes from iteration to iteration in a way that depends on the underlying signal (which we don´t know!). For such algorithms to be used in practice, it seems we need an estimator that achieves the MMSE when the noise level is unknown. In this paper we solve this problem using convex optimization, Stein Unbiased Risk Estimates and Huber Splines.