Sparse Bayesian Learning For Block-sparse Signal Recovery

We consider the problem of recovering block-sparse signals whose structures 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 structure 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. Specifically, a pattern-coupled hierarchical Gaussian prior model is introduced to characterize the statistical dependencies among coefficients, in which a set of hyperparameters are employed to control the sparsity of signal coefficients. 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 the hyperparameters of its immediate neighbors. 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, along with the sparse signal, are learned by maximizing their posterior probability via an expectation-maximization(EM) algorithm.We also discuss how to extend the Bayesian inference method to the scenario where the noise variance is unknown. In this case, we treats the signal x as the hidden variables and employ the expectation-maximization(EM) formulation to extimate the sparse signal and the noise covariance iteratively.Besides a modified iterative reweighted algorithm is also proposed to solve the block sparse signal recovery problem. Likewise, by using this algorithm, the sparsity pattern of the sparse signal is still not required for reliably recovering the sparse signal.Other than the single measurement vector case, this thesis also study the scenario where multiple measurement vectors are available. However, unlike the conventional multiple measurement algorithms, the sparse patterns of the signals are not stable, instead, they change slowly over time. To solve this time-varying sparse signal recovery problem, we reformulate the multiple measurement vector problem to single measurement case.Numerical results show that the proposed algorithm presents uniform superiority over other existing methods in a series of experiments, including noiseless case, noisy case, image or audio signals, as well as experiments concerning slowly time varying signals.