Analysis of convergence for the alternating direction method applied to joint sparse recovery

The sparse representation of a multiple measurement vector (MMV) is an important problem in compressed sensing theory, the old alternating direction method (ADM) is an optimization algorithm that has recently become very popular due to its capabilities to solve large-scale or distributed problems. The MMV-ADM algorithm to solve the MMV problem by ADM has been proposed by H. Lu, et al. (2011)[24], but the theoretical result about the convergence of matrix iteration sequence generated by the algorithm is left as a future research topic. In this paper, based on the subdifferential property of the two-norm for vector, a shrink operator associated with matrix is established. By using the operator, a convergence theorem is proved, which shows the MMV-ADM algorithm can recover the jointly sparse vectors.