Weighted Sparse Bayesian Learning (WSBL) for Basis Selection in Linear Underdetermined Systems

We propose Weighted SBL (WSBL) for sparse signal recovery, inspired by the Sparse Bayesian Learning (SBL) method. Unlike SBL, where all hyperparameter priors follow Gamma distributions with identical parameters, in WSBL, the hyperparameters are Gamma distributed with distinct parameters. These parameters, guided by some known weights, give more importance to some hyperparameters over others, thus introducing more degrees of freedom to the problem and leading to better recovery performance. The weights can be determined based on a low-resolution estimate of the sparse vector, for example, an estimate obtained via a method that does not encourage sparsity. The choice of the MUSIC estimate as weight is analyzed. Unlike in SBL, the WSBL hyperparameters are upper bounded; this makes it easy to select a threshold to separate zero from non-zero elements in the recovered sparse vector, which makes the iterative recovery process converge faster. Theoretical analysis based on variational approximation theory and also simulation results demonstrate that WSBL results in substantial improvement in terms of probability of detection and probability of false alarm, especially in the low signal to noise ratio regime, as compared to existing approaches, such as SBL, Sparse Bayesian Support knowledge (BSN), and Multiple response Sparse Bayesian Learning (MSBL). The performance of WSBL is evaluated for Direction of Arrival (DOA) estimation in colocated Multiple Input Multiple Output (MIMO) radar.