The Continuous Joint Sparsity Prior for Sparse Representations: Theory and Applications

The classical problem discussed in the literature of compressed sensing is recovering a sparse vector from a relatively small number of linear non-adaptive projections. In this paper, we study the recovery of a continuous set of sparse vectors sharing a common set of locations of their non-zero entries. This model includes the classical sparse representation problem, and also its known extensions. We develop a method for joint recovery of the entire set of sparse vectors by the solution of just one finite dimensional problem. The proposed strategy is exact and does not use heuristics or discretization methods. We then apply our method to two applications: The first is spectrum-blind reconstruction of multi-band analog signals from point-wise samples at a sub-Nyquist rate. The second application is to the well studied multiple-measurement-vectors problem which addresses the recovery of a finite set of sparse vectors.