A likelihood-based hyperparameter-free algorithm for robust block-sparse recovery

In this paper, a novel hyperparameter-free algorithm, which requires no a priori knowledge of block-sparsity level, is proposed for robust recovery of block-sparse signals with single/multiple measurement vector(s). The algorithm is based on the minimization of the negative log-likelihood function via majorization-minimization, and the block-sparsity is further induced by performing a Hölder inequality based relaxation. Two alternative noise variance updating rules are established upon the assumptions of equal and unequal noise variances, respectively. Furthermore, the proposed algorithm is proved to be theoretically equivalent to iteratively optimizing the combination of linear minimum mean square error criterion and weighted block-sparse penalty, and also equivalent to the iterative reweighted versions of covariance-based LASSO-type group sparse regression algorithms. Through the simulation results, we show that proper parameter setting is potential to improve the robustness against inaccurate knowledge of block partition, while the provided two noise variance updating rules are well applicable to white Gaussian noise case and impulsive noise case, respectively. Moreover, compared to some existing algorithms, the proposed algorithm offers superior recovery performance with incoherent dictionary, as well as greater robustness against highly coherent dictionary.