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163
Generalized Approximate Message Passing for Estimation with Random Linear Mixing
, 2012
"... We consider the estimation of an i.i.d. random vector observed through a linear transform followed by a componentwise, probabilistic (possibly nonlinear) measurement channel. A novel algorithm, called generalized approximate message passing (GAMP), is presented that provides computationally effici ..."
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Cited by 123 (18 self)
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We consider the estimation of an i.i.d. random vector observed through a linear transform followed by a componentwise, probabilistic (possibly nonlinear) measurement channel. A novel algorithm, called generalized approximate message passing (GAMP), is presented that provides computationally efficient approximate implementations of maxsum and sumproblem loopy belief propagation for such problems. The algorithm extends earlier approximate message passing methods to incorporate arbitrary distributions on both the input and output of the transform and can be applied to a wide range of problems in nonlinear compressed sensing and learning. Extending an analysis by Bayati and Montanari, we argue that the asymptotic componentwise behavior of the GAMP method under large, i.i.d. Gaussian transforms is described by a simple set of state evolution (SE) equations. From the SE equations, one can exactly predict the asymptotic value of virtually any componentwise performance metric including meansquared error or detection accuracy. Moreover, the analysis is valid for arbitrary input and output distributions, even when the corresponding optimization problems are nonconvex. The results match predictions by Guo and Wang for relaxed belief propagation on large sparse matrices and, in certain instances, also agree with the optimal performance predicted by the replica method. The GAMP methodology thus provides a computationally efficient methodology, applicable to a large class of nonGaussian estimation problems with precise asymptotic performance guarantees.
Compressed Sensing: Theory and Applications
, 2012
"... Compressed sensing is a novel research area, which was introduced in 2006, and since then has already become a key concept in various areas of applied mathematics, computer science, and electrical engineering. It surprisingly predicts that highdimensional signals, which allow a sparse representati ..."
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Cited by 120 (30 self)
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Compressed sensing is a novel research area, which was introduced in 2006, and since then has already become a key concept in various areas of applied mathematics, computer science, and electrical engineering. It surprisingly predicts that highdimensional signals, which allow a sparse representation by a suitable basis or, more generally, a frame, can be recovered from what was previously considered highly incomplete linear measurements by using efficient algorithms. This article shall serve as an introduction to and a survey about compressed sensing. Key Words. Dimension reduction. Frames. Greedy algorithms. Illposed inverse problems. `1 minimization. Random matrices. Sparse approximation. Sparse recovery.
Turbo reconstruction of structured sparse signals
 in Proc. 44th Annual Conf. Information Sciences and Systems
, 2010
"... Abstract—This paper considers the reconstruction of structuredsparse 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 ..."
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Cited by 59 (26 self)
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Abstract—This paper considers the reconstruction of structuredsparse 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 observationstructure and exploitation of the patternstructure, 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 Markovchain structured sparsity. 1 I.
A Singleletter Characterization of Optimal Noisy Compressed Sensing
"... Abstract—Compressed sensing deals with the reconstruction of a highdimensional signal from far fewer linear measurements, where the signal is known to admit a sparse representation in a certain linear space. The asymptotic scaling of the number of measurements needed for reconstruction as the dimen ..."
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Cited by 56 (16 self)
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Abstract—Compressed sensing deals with the reconstruction of a highdimensional signal from far fewer linear measurements, where the signal is known to admit a sparse representation in a certain linear space. The asymptotic scaling of the number of measurements needed for reconstruction as the dimension of the signal increases has been studied extensively. This work takes a fundamental perspective on the problem of inferring about individual elements of the sparse signal given the measurements, where the dimensions of the system become increasingly large. Using the replica method, the outcome of inferring about any fixed collection of signal elements is shown to be asymptotically decoupled, i.e., those elements become independent conditioned on the measurements. Furthermore, the problem of inferring about each signal element admits a singleletter characterization in the sense that the posterior distribution of the element, which is a sufficient statistic, becomes asymptotically identical to the posterior of inferring about the same element in scalar Gaussian noise. The result leads to simple characterization of all other elemental metrics of the compressed sensing problem, such as the mean squared error and the error probability for reconstructing the support set of the sparse signal. Finally, the singleletter characterization is rigorously justified in the special case of sparse measurement matrices where belief propagation becomes asymptotically optimal. I.
InformationTheoretically Optimal Compressed Sensing via Spatial Coupling and Approximate Message Passing
, 2011
"... We study the compressed sensing reconstruction problem for a broad class of random, banddiagonal sensing matrices. This construction is inspired by the idea of spatial coupling in coding theory. As demonstrated heuristically and numerically by Krzakala et al. [KMS+ 11], message passing algorithms ca ..."
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Cited by 51 (5 self)
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We study the compressed sensing reconstruction problem for a broad class of random, banddiagonal sensing matrices. This construction is inspired by the idea of spatial coupling in coding theory. As demonstrated heuristically and numerically by Krzakala et al. [KMS+ 11], message passing algorithms can effectively solve the reconstruction problem for spatially coupled measurements with undersampling rates close to the fraction of nonzero coordinates. We use an approximate message passing (AMP) algorithm and analyze it through the state evolution method. We give a rigorous proof that this approach is successful as soon as the undersampling rate δ exceeds the (upper) Rényi information dimension of the signal, d(pX). More precisely, for a sequence of signals of diverging dimension n whose empirical distribution converges to pX, reconstruction is with high probability successful from d(pX) n + o(n) measurements taken according to a band diagonal matrix. For sparse signals, i.e. sequences of dimension n and k(n) nonzero entries, this implies reconstruction from k(n)+o(n) measurements. For ‘discrete ’ signals, i.e. signals whose coordinates take a fixed finite set of values, this implies reconstruction from o(n) measurements. The result
Estimation with Random Linear Mixing, Belief Propagation and Compressed Sensing
, 2010
"... We apply Guo and Wang’s relaxed belief propagation (BP) method to the estimation of a random vector from linear measurements followed by a componentwise probabilistic measurement channel. Relaxed BP uses a Gaussian approximation in standard BP to obtain significant computational savings for dense ..."
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Cited by 43 (10 self)
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We apply Guo and Wang’s relaxed belief propagation (BP) method to the estimation of a random vector from linear measurements followed by a componentwise probabilistic measurement channel. Relaxed BP uses a Gaussian approximation in standard BP to obtain significant computational savings for dense measurement matrices. The main contribution of this paper is to extend the relaxed BP method and analysis to general (nonAWGN) output channels. Specifically, we present detailed equations for implementing relaxed BP for general channels and show that relaxed BP has an identical asymptotic large sparse limit behavior as standard BP, as predicted by the Guo and Wang’s state evolution (SE) equations. Applications are presented to compressed sensing and estimation with bounded noise.
Compressive Imaging using Approximate Message Passing and a Markovtree Prior
 PROC. ASILOMAR CONF. ON SIGNALS, SYSTEMS, AND COMPUTERS
, 2010
"... We propose a novel algorithm for compressive imaging that exploits both the sparsity and persistence across scales found in the 2D wavelet transform coefficients of natural images. Like other recent works, we model wavelet structure using a hidden Markov tree (HMT) but, unlike other works, ours is b ..."
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Cited by 41 (6 self)
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We propose a novel algorithm for compressive imaging that exploits both the sparsity and persistence across scales found in the 2D wavelet transform coefficients of natural images. Like other recent works, we model wavelet structure using a hidden Markov tree (HMT) but, unlike other works, ours is based on loopy belief propagation (LBP). For LBP, we adopt a recently proposed “turbo ” message passing schedule that alternates between exploitation of HMT structure and exploitation of compressivemeasurement structure. For the latter, we leverage Donoho, Maleki, and Montanari’s recently proposed approximate message passing (AMP) algorithm. Experiments on a large image database show that our turbo LBP approach maintains stateoftheart reconstruction performance at half the complexity.
Accurate Prediction of Phase Transitions in Compressed Sensing via a Connection to Minimax Denoising
, 2012
"... Compressed sensing posits that, within limits, one can undersample a sparse signal and yet reconstruct it accurately. Knowing the precise limits to such undersampling is important both for theory and practice. We present a formula that characterizes the allowed undersampling of generalized sparse ob ..."
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Cited by 41 (5 self)
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Compressed sensing posits that, within limits, one can undersample a sparse signal and yet reconstruct it accurately. Knowing the precise limits to such undersampling is important both for theory and practice. We present a formula that characterizes the allowed undersampling of generalized sparse objects. The formula applies to Approximate Message Passing (AMP) algorithms for compressed sensing, which are here generalized to employ denoising operators besides the traditional scalar soft thresholding denoiser. This paper gives several examples including scalar denoisers not derived from convex penalization – the firm shrinkage nonlinearity and the minimax nonlinearity – and also nonscalar denoisers – block thresholding, monotone regression, and total variation minimization. Let the variables ε = k/N and δ = n/N denote the generalized sparsity and undersampling fractions for sampling the kgeneralizedsparse Nvector x0 according to y = Ax0. Here A is an n × N measurement matrix whose entries are iid standard Gaussian. The formula states that the phase transition curve δ = δ(ε) separating successful from unsuccessful reconstruction of x0
Rényi Information Dimension: Fundamental Limits . . .
, 2010
"... In Shannon theory, lossless source coding deals with the optimal compression of discrete sources. Compressed sensing is a lossless coding strategy for analog sources by means of multiplication by realvalued matrices. In this paper we study almost lossless analog compression for analog memoryless so ..."
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Cited by 41 (9 self)
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In Shannon theory, lossless source coding deals with the optimal compression of discrete sources. Compressed sensing is a lossless coding strategy for analog sources by means of multiplication by realvalued matrices. In this paper we study almost lossless analog compression for analog memoryless sources in an informationtheoretic framework, in which the compressor or decompressor is constrained by various regularity conditions, in particular linearity of the compressor and Lipschitz continuity of the decompressor. The fundamental limit is shown to be the information dimension proposed by Rényi in 1959.
ExpectationMaximization GaussianMixture Approximate Message Passing
"... Abstract—When recovering a sparse signal from noisy compressive linear measurements, the distribution of the signal’s nonzero coefficients can have a profound affect on recovery meansquared error (MSE). If this distribution was apriori known, one could use efficient approximate message passing (AM ..."
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Cited by 40 (12 self)
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Abstract—When recovering a sparse signal from noisy compressive linear measurements, the distribution of the signal’s nonzero coefficients can have a profound affect on recovery meansquared error (MSE). If this distribution was apriori known, one could use efficient approximate message passing (AMP) techniques for nearly minimum MSE (MMSE) recovery. In practice, though, 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 nonleastfavorable distributions. As an alternative, we propose an empiricalBayesian technique that simultaneously learns the signal distribution while MMSErecovering the signal—according to the learned distribution—using AMP. In particular, we model the nonzero distribution as a Gaussian mixture, and learn its parameters through expectation maximization, using AMP to implement the expectation step. Numerical experiments confirm the stateoftheart performance of our approach on a range of 1 2 signal classes. I.