Pattern-Coupled Sparse Bayesian Learning for Recovery of Block-Sparse Signals

We consider the problem of recovering block-sparse signals whose cluster patterns are unknown a priori. Block-sparse signals with nonzero coefficients occurring in clusters arise naturally in many practical scenarios. However, the knowledge of the block partition is usually unavailable in practice. In this paper, we develop a new sparse Bayesian learning method for recovery of block-sparse signals with unknown cluster patterns. A pattern-coupled hierarchical Gaussian prior is introduced to characterize the pattern dependencies among neighboring coefficients, where a set of hyperparameters are employed to control the sparsity of signal coefficients. The proposed hierarchical model is similar to that for the conventional sparse Bayesian learning. However, unlike the conventional sparse Bayesian learning framework in which each individual hyperparameter is associated independently with each coefficient, in this paper, the prior for each coefficient not only involves its own hyperparameter, but also its immediate neighbor hyperparameters. In doing this way, the sparsity patterns of neighboring coefficients are related to each other and the hierarchical model has the potential to encourage structured-sparse solutions. The hyperparameters are learned by maximizing their posterior probability. We exploit an expectation-maximization (EM) formulation to develop an iterative algorithm that treats the signal as hidden variables and iteratively maximizes a lower bound on the posterior probability. In the M-step, a simple suboptimal solution is employed to replace a gradient-based search to maximize the lower bound. Numerical results are provided to illustrate the effectiveness of the proposed algorithm.