Towards Inversion-Free Sparse Bayesian Learning: A Universal Approach

Sparse Bayesian Learning (SBL) has emerged as a powerful tool for sparse signal recovery, due to its superior performance. However, the practical implementation of SBL faces a significant computational complexity associated with matrix inversion. Despite numerous efforts to alleviate this issue, existing methods are often limited to specifically structured sparse signals. This paper aims to provide a universal inversion-free approach to accelerate existing SBL algorithms. We unify the optimization of SBL variants with different priors within the expectation-maximization (EM) framework, where a lower bound of the likelihood function is maximized. Due to the linear Gaussian model foundation of SBL, updating this lower bound requires maximizing a quadratic function, which involves matrix inversion. Thus, we employ the minorization-maximization (MM) framework to derive two novel lower bounds that diagonalize the quadratic coefficient matrix, thereby eliminating the need for any matrix inversions. We further investigate their properties, including convergence guarantees under the MM framework and the slow convergence rate due to reduced curvature. The proposed approach is applicable to various types of structured sparse signals, such as row-sparse, block-sparse, and burst-sparse signals. Our simulations on synthetic and real data demonstrate remarkably shorter running time compared to state-of-the-art methods while achieving comparable recovery performance.