A signal recovery scheme is developed for linear observation systems based on expectation consistent (EC) mean field approximation. Approximate message passing (AMP) is known to be consistent with the results obtained using the replica theory, which is supposed to be exact in the large system limit, when each entry of the observation matrix is independently generated from an identical distribution. However, this is not necessarily the case for general matrices. We show that EC recovery exhibits consistency with the replica theory for a wider class of random observation matrices. This is numerically confirmed by experiments for the Bayesian optimal signal recovery of compressed sensing using random row-orthogonal matrices.
To the best of our knowledge, computationally feasible algorithms approximately performing the Bayesian recovery were initially developed for a simple perceptron (linear classifier) @cite_13 and later for CDMA @cite_20 @cite_32 . Recently, a similar idea was applied for CS @cite_26 @cite_29 @cite_3 as approximate message passing (AMP), and was summarized as a general formulation termed generalized approximate message passing (GAMP) @cite_1 . However, these studies rely on the assumption that each entry of @math , @math , is i.i.d., and the appropriateness for the employment to other ensembles is not guaranteed. In fact, the necessity for considering a certain characteristic feature of @math in constructing the approximation was pointed out in @cite_30 , and its significance was tested for the simple perceptron @cite_28 , CDMA @cite_4 , and MIMO @cite_27 . Here, we show how this approach is employed for the signal recovery of linear observations and examine its significance for an example of CS.
@cite_30 @cite_26 @cite_4 @cite_28 @cite_29 @cite_1 @cite_32 @cite_3 @cite_27 @cite_13 @cite_20
2020282020 2543631487 2026933032 1510028821
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. The approximate message passing (AMP) algorithm originally proposed by Donoho, Maleki, and Montanari yields a computationally attractive solution to the usual l 1 -regularized least-squares problem faced in compressed sensing, whose solution is known to be robust to the signal distribution. When the signal is drawn i.i.d from a marginal distribution that is not least-favorable, better performance can be attained using a Bayesian variation of AMP. The latter, however, assumes that the distribution is perfectly known. In this paper, we navigate the space between these two extremes by modeling the signal as i.i.d Bernoulli-Gaussian (BG) with unknown prior sparsity, mean, and variance, and the noise as zero-mean Gaussian with unknown variance, and we simultaneously reconstruct the signal while learning the prior signal and noise parameters. To accomplish this task, we embed the BG-AMP algorithm within an expectation-maximization (EM) framework. Numerical experiments confirm the excellent performance of our proposed EM-BG-AMP on a range of signal types.12 When recovering a sparse signal from noisy compressive linear measurements, the distribution of the signal's non-zero coefficients can have a profound effect on recovery mean-squared error (MSE). If this distribution was a priori known, then one could use computationally efficient approximate message passing (AMP) techniques for nearly minimum MSE (MMSE) recovery. In practice, however, the distribution is unknown, motivating the use of robust algorithms like LASSO-which is nearly minimax optimal-at the cost of significantly larger MSE for non-least-favorable distributions. As an alternative, we propose an empirical-Bayesian technique that simultaneously learns the signal distribution while MMSE-recovering the signal-according to the learned distribution-using AMP. In particular, we model the non-zero distribution as a Gaussian mixture and learn its parameters through expectation maximization, using AMP to implement the expectation step. Numerical experiments on a wide range of signal classes confirm the state-of-the-art performance of our approach, in both reconstruction error and runtime, in the high-dimensional regime, for most (but not all) sensing operators. In this paper, we extend the generalized approximate message passing (G-AMP) approach, originally proposed for high-dimensional generalized-linear regression in the context of compressive sensing, to the generalized-bilinear case, which enables its application to matrix completion, robust PCA, dictionary learning, and related matrix-factorization problems. Here, in Part I of a two-part paper, we derive our Bilinear G-AMP (BiG-AMP) algorithm as an approximation of the sum-product belief propagation algorithm in the high-dimensional limit, where central-limit theorem arguments and Taylor-series approximations apply, and under the assumption of statistically independent matrix entries with known priors. In addition, we propose an adaptive damping mechanism that aids convergence under finite problem sizes, an expectation-maximization (EM)-based method to automatically tune the parameters of the assumed priors, and two rank-selection strategies. In Part II of the paper, we will discuss the specializations of EM-BiG-AMP to the problems of matrix completion, robust PCA, and dictionary learning, and we will present the results of an extensive empirical study comparing EM-BiG-AMP to state-of-the-art algorithms on each problem.