  We consider the problem of robust compressed sensing whose objective is to
recover a high-dimensional sparse signal from compressed measurements corrupted
by outliers. A new sparse Bayesian learning method is developed for robust
compressed sensing. The basic idea of the proposed method is to identify and
remove the outliers from sparse signal recovery. To automatically identify the
outliers, we employ a set of binary indicator hyperparameters to indicate which
observations are outliers. These indicator hyperparameters are treated as
random variables and assigned a beta process prior such that their values are
confined to be binary. In addition, a Gaussian-inverse Gamma prior is imposed
on the sparse signal to promote sparsity. Based on this hierarchical prior
model, we develop a variational Bayesian method to estimate the indicator
hyperparameters as well as the sparse signal. Simulation results show that the
proposed method achieves a substantial performance improvement over existing
robust compressed sensing techniques.
