Globally sparse and locally dense signal recovery for compressed sensing

Sparsity regularized least squares are very popular for the solution of the underdetermined linear inverse problem. One of the recent progress is that structural information is incorporated to the sparse signal recovery for compressed sensing. Sparse group signal model, which is also called block-sparse signal, is one example in this way. In this paper, the internal structure of each group is further defined to get the globally sparse and locally dense group signal model. It assumes that most of the entries in the active groups are nonzero. To estimate this newly defined signal, minimization of the ℓ1 norm of the total variation is incorporated to the group Lasso which is the combination of a sparsity constraint and a data fitting constraint. The newly proposed optimization model is called globally sparse and locally dense group Lasso. The added total variation based constraint can encourage local dense distribution in each group. Theoretical analysis is performed to give a class of theoretical sufficient conditions to guarantee successful recovery. Simulations demonstrate the proposed method׳s performance gains against Lasso and group Lasso.