Efficient SCA Algorithms for Sparse Bayesian Learning

Type-II sparse Bayesian learning (SBL) for the multiple measurement vector model recovers a sparse signal power vector by minimizing the marginal negative log-likelihood via the expectation-maximization (EM) algorithm, referred to as M-SBL. We revisit the covariance learning (CL) problem and develop three parallel first-order algorithms within the successive convex approximation (SCA) framework. CL-SCA minimizes per-coordinate surrogates formed by partially linearizing the log-determinant term, while CL-FP uses a damped projected fixed-point iteration derived from stationarity conditions. Both methods have closed-form updates. We prove that the fixed points of both update maps are exactly the KKT points of the CL problem, that each update direction satisfies a quantitative descent inequality, that the damped iterates achieve sufficient decrease with an $\mathcal{O}(T^{-1/2})$ best-iterate stationarity rate, and we derive a backtracking rule that retains this rate without step-size tuning. If damping factors are bounded away from zero, the residual vanishes along the whole sequence and every limit point is a KKT point. Local analysis motivates CL-Hyb, which combines CL-SCA and CL-FP. Experiments on device activity detection in massive random access show that CL-Hyb-bt matches the best CL solvers' accuracy at a fraction of the runtime and is over an order of magnitude faster than EM.