Bayesian optimal compressed sensing without priors: Parametric sure approximate message passing

It has been shown that the Bayesian optimal approximate message passing (AMP) technique achieves the minimum mean-squared error (MMSE) optimal compressed sensing (CS) recovery. However, the prerequisite of the signal prior makes it often impractical. To address this dilemma, we propose the parametric SURE-AMP algorithm. The key feature is it uses the Stein's unbiased risk estimate (SURE) based parametric family of MMSE estimator for the CS denoising. Given that the optimization of the estimator and the calculation of its mean squared error purely depend on the noisy data, there is no need of the signal prior. The weighted sum of piecewise kernel functions is used to form the parametric estimator. Numerical experiments on both Bernoulli-Gaussian and k-dense signal justify our proposal.