Hyperparameter optimization of sparse bayesian learning based on Stein’s unbiased risk estimator

Sparse Bayesian Learning (SBL) stands as a widely utilized compressed sensing technique wherein the sparsityinducing prior for the unknowns within the underdetermined linear system is characterized by a Gaussian scale mixture. This formulation results in several hyperparameters, which encompass the variance profile, noise variance, and potentially other parameters within the variance profile priors. Traditionally, these hyperparameters are determined via Type I or Type II Maximum Likelihood (ML) estimation methods. In this paper, we introduce SURE SBL, wherein the optimization of hyperparameters (as opposed to mere estimation) relies on Stein’s Unbiased Risk Estimator (SURE). Notably, the primary performance criterion typically centers on the Mean Squared Error (MSE) of the sparse parameters or the resultant signal model. We conduct a review of the SURE approach. Subsequently, we apply the SURE approach to assess the MSE of the sparse parameters (the input to the linear model) and observe that it produces identical hyperparameter optimization outcomes as those obtained via Type II ML. Furthermore, we propose extending the SURE approach to the output level of the linear model. Remarkably, in the context of the large system limit, this extension yields equivalent hyperparameter optimization outcomes concerning the input to the linear model; however, when measurement noise is present, the results obtained by the two kinds of SURE optimizers diverge from those obtained through MSE optimization.