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

The standard compressive sensing (CS) aims to recover sparse signal from single measurement vector which is known as SMV model. By contrast, recovery of sparse signals from multiple measurement vectors is called MMV model. In this paper, we consider the recovery of jointly sparse signals in the MMV model where multiple signal measurements are represented as a matrix and the sparsity of signal occurs in common locations. The sparse MMV model can be formulated as a matrix (2, 1)-norm minimization problem, which is much more difficult to solve than the l1-norm minimization in standard CS. In this paper, we propose a very fast algorithm, called MMV-ADM, to solve the jointly sparse signal recovery problem in MMV settings based on the alternating direction method (ADM). The MMVADM alternately updates the recovered 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 faster 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. Appearing in Proceedings of the 14 International Conference on Artificial Intelligence and Statistics (AISTATS) 2011, Fort Lauderdale, FL, USA. Volume 15 of JMLR: W&CP 15. Copyright 2011 by the authors.