Sparse Bayesian Learning with Stein's Unbiased Risk Estimator based Hyperparameter Optimization

Sparse Bayesian Learning (SBL) is a popular compressed sensing technique in which the sparsifying prior for the unknowns in the underdetermined linear system is modeled as a Gaussian scale mixture. This leads to a number of hyperparameters which involve at least the variance profile and the noise variance, but also possible parameters in the variance profile priors. These hyperparameters are typically determined by Type I or Type II Maximum Likelihood (ML) estimation. In this paper we introduce SURE SBL in which the hyperparameter optimization (and not estimation) is based on Stein's Unbiased Risk Estimator (SURE). Indeed the ultimate performance criterion is usually the Mean Squared Error (MSE) of the sparse parameters or the resulting signal model. We review the SURE approach and its use in the world of automatic control. Then we apply the SURE approach to the MSE of the sparse parameters (linear model input) and find that it yields the same hyperparameter optimization as by Type II ML. We finally propose the SURE approach at the level of the output of the linear model, where it leads to new hyperparameter adjustments.