Unitary Approximate Message Passing for Sparse Bayesian Learning

Sparse Bayesian learning (SBL) can be implemented with low complexity based on the approximate message passing (AMP) algorithm. However, it is vulnerable to 'difficult' measurement matrices, which may cause AMP to diverge. Damped AMP has been used for SBL to alleviate the problem at the cost of reducing convergence speed. In this work, we propose a new SBL algorithm based on structured variational inference, leveraging AMP with a unitary transformation (UAMP). Both single measurement vector and multiple measurement vector problems are investigated. It is shown that, compared to state-of-the-art AMP-based SBL algorithms, the proposed UAMPSBL is more robust and efficient, leading to remarkably better performance.

Approximate Message Passing with Nearest Neighbor Sparsity Pattern Learning

We consider the problem of recovering clustered sparse signals with no prior knowledge of the sparsity pattern. Beyond simple sparsity, signals of interest often exhibits an underlying sparsity pattern which, if leveraged, can improve the reconstruction performance. However, the sparsity pattern is usually unknown a priori. Inspired by the idea of k-nearest neighbor (k-NN) algorithm, we propose an efficient algorithm termed approximate message passing with nearest neighbor sparsity pattern learning (AMP-NNSPL), which learns the sparsity pattern adaptively. AMP-NNSPL specifies a flexible spike and slab prior on the unknown signal and, after each AMP iteration, sets the sparse ratios as the average of the nearest neighbor estimates via expectation maximization (EM). Experimental results on both synthetic and real data demonstrate the superiority of our proposed algorithm both in terms of reconstruction performance and computational complexity.