  A trend in compressed sensing (CS) is to exploit structure for improved
reconstruction performance. In the basic CS model, exploiting the clustering
structure among nonzero elements in the solution vector has drawn much
attention, and many algorithms have been proposed. However, few algorithms
explicitly consider correlation within a cluster. Meanwhile, in the multiple
measurement vector (MMV) model correlation among multiple solution vectors is
largely ignored. Although several recently developed algorithms consider the
exploitation of the correlation, these algorithms need to know a priori the
correlation structure, thus limiting their effectiveness in practical problems.
  Recently, we developed a sparse Bayesian learning (SBL) algorithm, namely
T-SBL, and its variants, which adaptively learn the correlation structure and
exploit such correlation information to significantly improve reconstruction
performance. Here we establish their connections to other popular algorithms,
such as the group Lasso, iterative reweighted $\ell_1$ and $\ell_2$ algorithms,
and algorithms for time-varying sparsity. We also provide strategies to improve
these existing algorithms.
