A compact formulation for the l21 mixed-norm minimization problem

We present an equivalent, compact reformulation of the ℓ2,1 mixed-norm minimization problem for joint sparse signal reconstruction from multiple measurement vectors (MMVs). The reformulation builds upon a compact parameterization, which models the row-norms of the sparse signal representation as parameters of interest, resulting in a significant reduction of the MMV problem size. Given the sparse vector of row-norms, the joint sparse signal can be computed from the MMVs in closed form. For the special case of uniform linear sampling, we present an extension of the compact formulation for gridless parameter estimation by means of semidefinite programming. Furthermore, we derive in this case from our compact problem formulation the exact equivalence between the ℓ2,1 mixed-norm minimization and the atomic-norm minimization.