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# A Fast Algorithm for Recovery of Jointly Sparse Vectors based on the Alternating Direction Methods

*Proceedings of the Fourteenth International Conference on Artificial Intelligence and Statistics*, PMLR 15:461-469, 2011.

#### Abstract

The standard compressive sensing (CS) aims to recover sparse signal from single measurement vector (SMV) which is known as SMV model. In this paper, we consider the recovery of jointly sparse signals in the multiple measurement vector (MMV) scenario where signal is represented as a matrix and the sparsity of signal occurs in a common location set. The sparse MMV model can be formulated as a matrix $(2,1)$-norm minimization problem. However, the $(2,1)$-norm minimization problem is much more difficult to solve than l1-norm minimization. In this paper, we propose a very fast algorithm, called MMV-ADM, for jointly sparse signal recovery in MMV settings based on the alternating direction method (ADM). The MMV-ADM alternately updates the signal matrix, the Lagrangian multiplier and the residue, and all update rules only involve matrix or vector multiplications and summations, so it is simple, easy to implement and much more fast than the state-of-the-art method MMVprox. Numerical simulations show that MMV-ADM is at least dozens of times faster than MMVprox with comparable recovery accuracy.