Bayesian approximate message passing (BAMP) is an efficient method in compressed sensing that is nearly optimal in the minimum mean squared error (MMSE) sense. Multiple measurement vector (MMV)-BAMP performs joint recovery of multiple vectors with identical support and accounts for correlations in the signal of interest and in the noise. In this paper, we show how to reduce the complexity of vector BAMP via a simple joint decorrelation (diagonalization) transform of the signal and noise vectors, which also facilitates the subsequent performance analysis. We prove that the corresponding state evolution is equivariant with respect to the joint decorrelation transform and preserves diagonality of the residual noise covariance for the Bernoulli–Gauss prior. We use these results to analyze the dynamics and the mean squared error (MSE) performance of BAMP via the replica method, and thereby understand the impact of signal correlation and number of jointly sparse signals. Finally, we evaluate an application of MMV-BAMP for single-pixel imaging with correlated color channels and thereby explore the performance gain of joint recovery compared to conventional BAMP reconstruction as well as group lasso.
@cite_30: 
@cite_29: This paper revisits the sparse multiple measurement vector (MMV) problem, where the aim is to recover a set of jointly sparse multichannel vectors from incomplete measurements. This problem is an extension of single channel sparse recovery, which lies at the heart of compressed sensing. Inspired by the links to array signal processing, a new family of MMV algorithms is considered that highlight the role of rank in determining the difficulty of the MMV recovery problem. The simplest such method is a discrete version of MUSIC which is guaranteed to recover the sparse vectors in the full rank MMV setting, under mild conditions. This idea is extended to a rank aware pursuit algorithm that naturally reduces to Order Recursive Matching Pursuit (ORMP) in the single measurement case while also providing guaranteed recovery in the full rank setting. In contrast, popular MMV methods such as Simultaneous Orthogonal Matching Pursuit (SOMP) and mixed norm minimization techniques are shown to be rank blind in terms of worst case analysis. Numerical simulations demonstrate that the rank aware techniques are significantly better than existing methods in dealing with multiple measurements.
@cite_42: This work is motivated by a distributed compressed sensing (DCS) scenario where multiple sensors independently perform compressed sensing and the sparse signals share a common support. The heterogeneous case is considered where the numbers of measurements and the noise levels at different sensors may be different. To analyse the performance, we focus on a probability model for sparse signals and use the state evolution tool developed for the approximate message passing (AMP) technique. In the noise free case, we are able to quantify the asymptotic rate region for exact recovery. The rate region has a shape that is significantly different from that by information theoretic analysis and provides a guidance for resource allocation in practice. It shows that an equal allocation of the number of measurements across sensors is strictly suboptimal. Finally, we also study the effect of the correlation among nonzero components from different sparse signals, which appears in many practical scenarios.
@cite_44: 
@cite_43: This paper surveys recent work in applying ideas from graphical models and message passing algorithms to solve large scale regularized regression problems. In particular, the focus is on compressed sensing reconstruction via 1 penalized least-squares (known as LASSO or BPDN). We discuss how to derive fast approximate message passing algorithms to solve this problem. Surprisingly, the analysis of such algorithms allows to prove exact high-dimensional limit results for the LASSO risk. This paper will appear as a chapter in a book on ‘Compressed Sensing’ edited by Yonina Eldar and Gitta Kutynok.
@cite_5: Compressed sensing (CS) demonstrates that sparse signals can be estimated from underdetermined linear systems. Distributed CS (DCS) further reduces the number of measurements by considering joint sparsity within signal ensembles. DCS with jointly sparse signals has applications in multisensor acoustic sensing, magnetic resonance imaging with multiple coils, remote sensing, and array signal processing. Multimeasurement vector (MMV) problems consider the estimation of jointly sparse signals under the DCS framework. Two related MMV settings are studied. In the first setting, each signal vector is measured by a different independent and identically distributed (i.i.d.) measurement matrix, while in the second setting, all signal vectors are measured by the same i.i.d. matrix. Replica analysis is performed for these two MMV settings, and the minimum mean squared error (MMSE), which turns out to be identical for both settings, is obtained as a function of the noise variance and number of measurements. To showcase the application of MMV models, the MMSE's of complex CS problems with both real and complex measurement matrices are also analyzed. Multiple performance regions for MMV are identified where the MMSE behaves differently as a function of the noise variance and the number of measurements. Belief propagation (BP) is a CS signal estimation framework that often achieves the MMSE asymptotically. A phase transition for BP is identified. This phase transition, verified by numerical results, separates the regions where BP achieves the MMSE and where it is suboptimal. Numerical results also illustrate that more signal vectors in the jointly sparse signal ensemble lead to a better phase transition.
@cite_20: We address the problem of finding sparse solutions to an underdetermined system of equations when there are multiple measurement vectors having the same, but unknown, sparsity structure. The single measurement sparse solution problem has been extensively studied in the past. Although known to be NP-hard, many single-measurement suboptimal algorithms have been formulated that have found utility in many different applications. Here, we consider in depth the extension of two classes of algorithms-Matching Pursuit (MP) and FOCal Underdetermined System Solver (FOCUSS)-to the multiple measurement case so that they may be used in applications such as neuromagnetic imaging, where multiple measurement vectors are available, and solutions with a common sparsity structure must be computed. Cost functions appropriate to the multiple measurement problem are developed, and algorithms are derived based on their minimization. A simulation study is conducted on a test-case dictionary to show how the utilization of more than one measurement vector improves the performance of the MP and FOCUSS classes of algorithm, and their performances are compared.
@cite_38: This paper considers the reconstruction of structured-sparse signals from noisy linear observations. In particular, the support of the signal coefficients is parameterized by hidden binary pattern, and a structured probabilistic prior (e.g., Markov random chain field tree) is assumed on the pattern. Exact inference is discussed and an approximate inference scheme, based on loopy belief propagation (BP), is proposed. The proposed scheme iterates between exploitation of the observation-structure and exploitation of the pattern-structure, and is closely related to noncoherent turbo equalization, as used in digital communication receivers. An algorithm that exploits the observation structure is then detailed based on approximate message passing ideas. The application of EXIT charts is discussed, and empirical phase transition plots are calculated for Markov-chain structured sparsity.1
@cite_4: A simultaneous sparse approximation problem requests a good approximation of several input signals at once using different linear combinations of the same elementary signals. At the same time, the problem balances the error in approximation against the total number of elementary signals that participate. These elementary signals typically model coherent structures in the input signals, and they are chosen from a large, linearly dependent collection.The first part of this paper proposes a greedy pursuit algorithm, called simultaneous orthogonal matching pursuit (S-OMP), for simultaneous sparse approximation. Then it presents some numerical experiments that demonstrate how a sparse model for the input signals can be identified more reliably given several input signals. Afterward, the paper proves that the S-OMP algorithm can compute provably good solutions to several simultaneous sparse approximation problems.The second part of the paper develops another algorithmic approach called convex relaxation, and it provides theoretical results on the performance of convex relaxation for simultaneous sparse approximation.
@cite_52: We address the problem of joint sparsity pattern recovery based on low dimensional multiple measurement vectors (MMVs) in resource constrained distributed networks. We assume that distributed nodes observe sparse signals which share the same sparsity pattern and each node obtains measurements via a low dimensional linear operator. When the measurements are collected at distributed nodes in a communication network, it is often required that joint sparse recovery be performed under inherent resource constraints such as communication bandwidth and transmit processing power. We present two approaches to take the communication constraints into account while performing common sparsity pattern recovery. First, we explore the use of a shared multiple access channel (MAC) in forwarding observations residing at each node to a fusion center. With MAC, while the bandwidth requirement does not depend on the number of nodes, the fusion center has access to only a linear combination of the observations. We discuss the conditions under which the common sparsity pattern can be estimated reliably. Second, we develop two collaborative algorithms based on Orthogonal Matching Pursuit (OMP), to jointly estimate the common sparsity pattern in a decentralized manner with a low communication overhead. In the proposed algorithms, each node exploits collaboration among neighboring nodes by sharing a small amount of information for fusion at different stages in estimating the indices of the true support in a greedy manner. Efficiency and effectiveness of the proposed algorithms are demonstrated via simulations along with a comparison with the most related existing algorithms considering the trade-off between the performance gain and the communication overhead.
@cite_46: 
@cite_17: The paper quantifies and compares the exact asymptotic performance of multiple measurement vector (MMV) and distributed sensing (DS) models. Both models assume multiple measurement instances yk = Akxk + wk, k = 1, 2, …, K. The difference is that MMV involves identical measurement matrices whereas DS allows different matrices for different measurement instances. It has been recognized that DS works better than MMV empirically. However, the quantification of the performance difference is not available in the literature. Our contribution is to quantify the asymptotic performance of MMV and DS in the asymptotic regime that the dimensions of the measurement matrices approach infinity proportionally but the number of measurement instances K remains a constant. The case study and numerical results justify the accuracy of the performance quantification. The analysis technique is based on the state evolution for approximate message passing.
@cite_16: In a recent paper, the authors proposed a new class of low-complexity iterative thresholding algorithms for reconstructing sparse signals from a small set of linear measurements [1]. The new algorithms are broadly referred to as AMP, for approximate message passing. This is the first of two conference papers describing the derivation of these algorithms, connection with the related literature, extensions of the original framework, and new empirical evidence. In particular, the present paper outlines the derivation of AMP from standard sum-product belief propagation, and its extension in several directions. We also discuss relations with formal calculations based on statistical mechanics methods.
@cite_34: A simultaneous sparse approximation problem requests a good approximation of several input signals at once using different linear combinations of the same elementary signals. At the same time, the problem balances the error in approximation against the total number of elementary signals that participate. These elementary signals typically model coherent structures in the input signals, and they are chosen from a large, linearly dependent collection.The first part of this paper presents theoretical and numerical results for a greedy pursuit algorithm, called simultaneous orthogonal matching pursuit.The second part of the paper develops another algorithmic approach called convex relaxation. This method replaces the combinatorial simultaneous sparse approximation problem with a closely related convex program that can be solved efficiently with standard mathematical programming software. The paper develops conditions under which convex relaxation computes good solutions to simultaneous sparse approximation problems.
@cite_33: Gaussian and quadratic approximations of message passing algorithms on graphs have attracted considerable recent attention due to their computational simplicity, analytic tractability, and wide applicability in optimization and statistical inference problems. This paper presents a systematic framework for incorporating such approximate message passing (AMP) methods in general graphical models. The key concept is a partition of dependencies of a general graphical model into strong and weak edges, with the weak edges representing small, linearizable couplings of variables. AMP approximations based on the central limit theorem can be readily applied to aggregates of many weak edges and integrated with standard message passing updates on the strong edges. The resulting algorithm, which we call hybrid generalized approximate message passing (HyGAMP), can yield significantly simpler implementations of sum-product and max-sum loopy belief propagation. By varying the partition of strong and weak edges, a performance-complexity tradeoff can be achieved. Group sparsity and multinomial logistic regression problems are studied as examples of the proposed methodology.
@cite_9: In this work, a Bayesian approximate message passing algorithm is proposed for solving the multiple measurement vector (MMV) problem in compressive sensing, in which a collection of sparse signal vectors that share a common support are recovered from undersampled noisy measurements. The algorithm, AMP-MMV, is capable of exploiting temporal correlations in the amplitudes of non-zero coefficients, and provides soft estimates of the signal vectors as well as the underlying support. Central to the proposed approach is an extension of recently developed approximate message passing techniques to the amplitude-correlated MMV setting. Aided by these techniques, AMP-MMV offers a computational complexity that is linear in all problem dimensions. In order to allow for automatic parameter tuning, an expectation-maximization algorithm that complements AMP-MMV is described. Finally, a detailed numerical study demonstrates the power of the proposed approach and its particular suitability for application to high-dimensional problems.
@cite_53: In a recent paper, the authors proposed a new class of low-complexity iterative thresholding algorithms for reconstructing sparse signals from a small set of linear measurements [1]. The new algorithms are broadly referred to as AMP, for approximate message passing. This is the second of two conference papers describing the derivation of these algorithms, connection with related literature, extensions of original framework, and new empirical evidence. This paper describes the state evolution formalism for analyzing these algorithms, and some of the conclusions that can be drawn from this formalism. We carried out extensive numerical simulations to confirm these predictions. We present here a few representative results.
@cite_1: Compressed sensing (CS) demonstrates that sparse signals can be recovered from underdetermined linear measurements. We focus on the joint sparse recovery problem where multiple signals share the same common sparse support sets, and they are measured through the same sensing matrix. Leveraging a recent information theoretic characterization of single signal CS, we formulate the optimal minimum mean square error (MMSE) estimation problem, and derive a belief propagation algorithm, its relaxed version, for the joint sparse recovery problem and an approximate message passing algorithm. In addition, using density evolution, we provide a sufficient condition for exact recovery.
@cite_45: Given a large overcomplete dictionary of basis vectors, the goal is to simultaneously represent L>1 signal vectors using coefficient expansions marked by a common sparsity profile. This generalizes the standard sparse representation problem to the case where multiple responses exist that were putatively generated by the same small subset of features. Ideally, the associated sparse generating weights should be recovered, which can have physical significance in many applications (e.g., source localization). The generic solution to this problem is intractable and, therefore, approximate procedures are sought. Based on the concept of automatic relevance determination, this paper uses an empirical Bayesian prior to estimate a convenient posterior distribution over candidate basis vectors. This particular approximation enforces a common sparsity profile and consistently places its prominent posterior mass on the appropriate region of weight-space necessary for simultaneous sparse recovery. The resultant algorithm is then compared with multiple response extensions of matching pursuit, basis pursuit, FOCUSS, and Jeffreys prior-based Bayesian methods, finding that it often outperforms the others. Additional motivation for this particular choice of cost function is also provided, including the analysis of global and local minima and a variational derivation that highlights the similarities and differences between the proposed algorithm and previous approaches.
Several methods for jointly sparse recovery have been proposed in the literature @cite_20 @cite_4 @cite_34 @cite_45 @cite_29 @cite_38 @cite_52 @cite_9 @cite_44 @cite_42 @cite_17 @cite_1 . amp was introduced in @cite_46 @cite_16 @cite_53 as a large system relaxation of loopy belief propagation to solve a random linear system with sparsity constraint. Scalar bamp , its Bayesian version @cite_30 @cite_43 , uses the signal prior explicitly and is an efficient approximate mmse estimator. The turbo bamp methods in @cite_38 @cite_9 @cite_44 , and their generalization in @cite_33 for clustered sparse signals, improve the recovery performance by exchanging extrinsic information about the current support estimate in each message passing iteration. In @cite_42 @cite_17 @cite_5 , joint sparsity is directly enforced by an appropriate vector estimator (denoiser) function for the bg prior.