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.
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.
@cite_30 @cite_29 @cite_42 @cite_44 @cite_43 @cite_5 @cite_20 @cite_38 @cite_4 @cite_52 @cite_46 @cite_17 @cite_16 @cite_34 @cite_33 @cite_9 @cite_53 @cite_1 @cite_45
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This paper considers the problem of sparse signal recovery when the decoder has prior information on the sparsity pattern of the data. The data vector x=[x1,...,xN]T has a randomly generated sparsity pattern, where the i-th entry is non-zero with probability pi. Given knowledge of these probabilities, the decoder attempts to recover x based on M random noisy projections. Information-theoretic limits on the number of measurements needed to recover the support set of x perfectly are given, and it is shown that significantly fewer measurements can be used if the prior distribution is sufficiently non-uniform. Furthermore, extensions of Basis Pursuit, LASSO, and Orthogonal Matching Pursuit which exploit the prior information are presented. The improved performance of these methods over their standard counterparts is demonstrated using simulations. 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. Recovery of the sparsity pattern (or support) of an unknown sparse vector from a limited number of noisy linear measurements is an important problem in compressed sensing. In the high-dimensional setting, it is known that recovery with a vanishing fraction of errors is impossible if the measurement rate and the per-sample signal-to-noise ratio (SNR) are finite constants, independent of the vector length. In this paper, it is shown that recovery with an arbitrarily small but constant fraction of errors is, however, possible, and that in some cases computationally simple estimators are near-optimal. Bounds on the measurement rate needed to attain a desired fraction of errors are given in terms of the SNR and various key parameters of the unknown vector for several different recovery algorithms. The tightness of the bounds, in a scaling sense, as a function of the SNR and the fraction of errors, is established by comparison with existing information-theoretic necessary bounds. Near optimality is shown for a wide variety of practically motivated signal models. Most of the existing sparse-recovery methods assume a static system: the signal is a finite-length vector for which a fixed set of measurements and sparse representation are available and an l1 problem is solved for the reconstruction. However, the same representation and reconstruction framework is not readily applicable in a streaming system: the signal changes over time, and it is measured and reconstructed sequentially over small intervals. This is particularly desired when dividing signals into disjoint blocks and processing each block separately is infeasible or inefficient. In this paper, we discuss two streaming systems and a new homotopy algorithm for quickly solving the associated l1 problems: 1) recovery of smooth, time-varying signals for which, instead of using block transforms, we use lapped orthogonal transforms for sparse representation and 2) recovery of sparse, time-varying signals that follows a linear dynamic model. For both systems, we iteratively process measurements over a sliding interval and solve a weighted l1-norm minimization problem for estimating sparse coefficients. Since we estimate overlapping portions of the signal while adding and removing measurements, instead of solving a new l1 program as the system changes, we use available signal estimates as starting point in a homotopy formulation and update the solution in a few simple steps. We demonstrate with numerical experiments that our proposed streaming recovery framework provides better reconstruction compared to the methods that represent and reconstruct signals as independent, disjoint blocks, and that our proposed homotopy algorithm updates the solution faster than the current state-of-the-art solvers.