In this report, a novel efficient algorithm for recovery of jointly sparse signals (sparse matrix) from multiple incomplete measurements has been presented, in particular, the NESTA-based MMV optimization method. In a nutshell, the jointly sparse recovery is obviously superior to applying standard sparse reconstruction methods to each channel individually. Moreover several efforts have been made to improve the NESTA-based MMV algorithm, in particular, (1) the NESTA-based MMV algorithm for partially known support to greatly improve the convergence rate, (2) the detection of partial (or all) locations of unknown jointly sparse signals by using so-called MUSIC algorithm; (3) the iterative NESTA-based algorithm by combing hard thresholding technique to decrease the numbers of measurements. It has been shown that by using proposed approach one can recover the unknown sparse matrix X with () Spark A -sparsity from () Spark A measurements, predicted in Ref. [1], where the measurement matrix denoted by A satisfies the so-called restricted isometry property (RIP). Under a very mild condition on the sparsity of X and characteristics of the A, the iterative hard threshold (IHT)-based MMV method has been shown to be also a very good candidate.
improve the NESTA-based MMV algorithm, in particular, (1) the NESTA-based MMV algorithm for partially known support to greatly improve the convergence rate, (2) the detection of partial (or all) locations of unknown jointly sparse signals by using so-called MUSIC algorithm; (3) the iterative NESTA-based algorithm by combing hard thresholding technique to decrease the numbers of measurements. It has been shown that by using proposed approach one can recover the unknown sparse matrix X with ( ) Spark A -sparsity from ( ) Spark A measurements, predicted in Ref. [1],
where the measurement matrix denoted by A satisfies the so-called restricted isometry property (RIP). Under a very mild condition on the sparsity of X and characteristics of the A , the iterative hard threshold (IHT)-based MMV method has been shown to be also a very good candidate.
INDEX TERMS: compressive sensing, SMV (single measurement vector), MMV (multiple measurement vector), Nesterov’s method, iterative hard threshold algorithm, MUSIC, restricted isometry property (RIP)
Recovery of sparse signals from a small number of measurements is a fundamental problem in many practical applications such as medical imaging, seismic exploration, communication, image denoising, analog-to-digital conversion, and so on.
The well-known compressed sensing, developed by Candes, Tao and Donoho et al, studies information acquisition methods as well as efficient computational algorithms.
By exploiting colorful results developed within the framework of compressive sensing, we can reconstruct a sparse vector x by solving the highly underdetermined linear equations y Ax = under minimal 1
A satisfies some properties such as restricted isometry property (RIP), null-space property (NSP), and so on. Though determining the sparest vector x consistent with the data y Ax = is generally an NP-hard problem, many suboptimal algorithms have been formulated to attack this problem, for example, greedy algorithm, basis pursuit (BP), Bayesian algorithm, and so on.
The single measurement sparse solution problem has been extensively studied in the past. In many practical applications such as dynamic medical imaging, neromagnetic inverse problem, beam forming, electromagnetic inverse source, communication, and so on, the recovery of jointly sparse signal or MMV problem, the variation of the compressive sensing or sparse linear inverse problem, is an important topic; in particular, to deal with the computation of sparse solution when there are multiple measurement vectors (MMV) and the solutions are assumed to have a common sparsity profile or jointly sparse. The most widely studied approaches to the MMV problem are based on solving the convex optimization problem , min
where the mixed ,
1,: , 2,: , , ,:
,: X j is the jth row of X .
Up to now, many efforts have made to attack this problem. Cotter et al. considered the
Chen and Huo considered the uniqueness under 0 p = via the spark of measurement matrix A and equivalence between the minimization problem with 1 p = and 0 p = [1].
Further, the orthogonal matching pursuit (OMP) algorithm for MMV has also been developed [1]. Tropp dealt with (1) for 2 p = and q = ∞ . Mishali and Eldar proposed the ReMBo algorithm which reduces MMV to a series of SMV problems. Eldar and Rauhut proposed the OMP algorithm with hard threshold technique and analyzed the average case for jointly sparse signal recovery [4]. Berg and Friedlander studied performance of 1,1 l and 1,2 l for different structure of sparse X [5].
In this presentation, we consider in depth the extension of a class of algorithm-NESTA algorithm-to the multiple measurement vectors available, and solutions with a common sparsity structure must be computed, especially, NESTA-based MMV algorithm. Inspired by recent breakthroughs in the development of novel first-order methods in convex optimization, the cost functions appropriate to NESTA-based MMV are developed, and algorithms are derived based on their minimization. Further several approaches to improve the NESTA-based MMV algorithm to decrease the number of measurements and increase the convergence rate have been proposed. This report demonstrates that this approach is ideally suited for solving large-scale MMV reconstruction problems.
In this section, the basic idea of NESTA-based MMV algorithm has been provided; moreover, several approaches to improve it have been discussed. We will refer the reader to [7] for detailed discussions about proposed approaches.
Similar done by Nesterov’s method for single-measurement problem [9] [10], the NESAT-based MMV algorithm minimize the smooth convex function f on the convex
where the primal feasible set p Q is defined by { } : :
To exploit fully the structure of unknown matrix X , we introduce X α = Ψ with sparse transformation matrix Ψ . The smoothed version of convex function ( )
where
is the dual feasible set.
To control flexibly the inherent structure of X , the smoothed conve
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