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129
Message passing algorithms for compressed sensing: I. motivation and construction
 Proc. ITW
, 2010
"... Abstract—In a recent paper, the authors proposed a new class of lowcomplexity 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 tw ..."
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Cited by 170 (19 self)
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Abstract—In a recent paper, the authors proposed a new class of lowcomplexity 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. I. GENERAL AMP AND STATE EVOLUTION We consider the model
Structured compressed sensing: From theory to applications
 IEEE TRANS. SIGNAL PROCESS
, 2011
"... Compressed sensing (CS) is an emerging field that has attracted considerable research interest over the past few years. Previous review articles in CS limit their scope to standard discretetodiscrete measurement architectures using matrices of randomized nature and signal models based on standard ..."
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Cited by 98 (15 self)
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Compressed sensing (CS) is an emerging field that has attracted considerable research interest over the past few years. Previous review articles in CS limit their scope to standard discretetodiscrete measurement architectures using matrices of randomized nature and signal models based on standard sparsity. In recent years, CS has worked its way into several new application areas. This, in turn, necessitates a fresh look on many of the basics of CS. The random matrix measurement operator must be replaced by more structured sensing architectures that correspond to the characteristics of feasible acquisition hardware. The standard sparsity prior has to be extended to include a much richer class of signals and to encode broader data models, including continuoustime signals. In our overview, the theme is exploiting signal and measurement structure in compressive sensing. The prime focus is bridging theory and practice; that is, to pinpoint the potential of structured CS strategies to emerge from the math to the hardware. Our summary highlights new directions as well as relations to more traditional CS, with the hope of serving both as a review to practitioners wanting to join this emerging field, and as a reference for researchers that attempts to put some of the existing ideas in perspective of practical applications.
1 Sparse Recovery Using Sparse Matrices
"... Abstract—We survey algorithms for sparse recovery problems that are based on sparse random matrices. Such matrices has several attractive properties: they support algorithms with low computational complexity, and make it easy to perform incremental updates to signals. We discuss applications to seve ..."
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Cited by 72 (13 self)
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Abstract—We survey algorithms for sparse recovery problems that are based on sparse random matrices. Such matrices has several attractive properties: they support algorithms with low computational complexity, and make it easy to perform incremental updates to signals. We discuss applications to several areas, including compressive sensing, data stream computing and group testing. I.
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 64 (29 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 57 (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
, 1112
"... 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 52 (4 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
The effect of spatial coupling on compressive sensing
 in Communication, Control, and Computing (Allerton
"... Abstract — Recently, it was observed that spatiallycoupled LDPC code ensembles approach the Shannon capacity for a class of binaryinput memoryless symmetric (BMS) channels. The fundamental reason for this was attributed to a threshold saturation phenomena derived in [1]. In particular, it was show ..."
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Cited by 47 (9 self)
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Abstract — Recently, it was observed that spatiallycoupled LDPC code ensembles approach the Shannon capacity for a class of binaryinput memoryless symmetric (BMS) channels. The fundamental reason for this was attributed to a threshold saturation phenomena derived in [1]. In particular, it was shown that the belief propagation (BP) threshold of the spatially coupled codes is equal to the maximum a posteriori (MAP) decoding threshold of the underlying constituent codes. In this sense, the BP threshold is saturated to its maximum value. Moreover, it has been empirically observed that the same phenomena also occurs when transmitting over more general classes of BMS channels. In this paper, we show that the effect of spatial coupling is not restricted to the realm of channel coding. The effect of coupling also manifests itself in compressed sensing. Specifically, we show that spatiallycoupled measurement matrices have an improved sparsity to sampling threshold for reconstruction algorithms based on verification decoding. For BPbased reconstruction algorithms, this phenomenon is also tested empirically via simulation. At the block lengths accessible via simulation, the effect is quite small and it seems that spatial coupling is not providing the gains one might expect. Based on the threshold analysis, however, we believe this warrants further study. I.
Compressive Imaging using Approximate Message Passing and a
 MarkovTree Prior, Proc. Asilomar Conf. on Signals, Systems, and Computers
, 2010
"... Abstract—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, ..."
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Cited by 43 (8 self)
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Abstract—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. 1 I.
Graphical Models Concepts in Compressed Sensing
"... 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 ℓ1 penalized leastsquares (known as LASSO or BPDN). We discuss how to deri ..."
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Cited by 38 (2 self)
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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 ℓ1 penalized leastsquares (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 highdimensional 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. 1
1 Compressive Video Sampling with Approximate Message Passing Decoding
"... In this paper, we apply compressed sensing to video compression. Compressed sensing (CS) techniques exploit the observation that one needs much fewer random measurements than given by the ShannonNyquist sampling theory to recover an object if this object is compressible (i.e., sparse in the spatial ..."
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Cited by 38 (2 self)
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In this paper, we apply compressed sensing to video compression. Compressed sensing (CS) techniques exploit the observation that one needs much fewer random measurements than given by the ShannonNyquist sampling theory to recover an object if this object is compressible (i.e., sparse in the spatial domain or in a transform domain). In the CS framework, we can achieve sensing, compression and denoising simultaneously. We propose a fast and simple online encoding by application of pseudorandom downsampling of the twodimensional fast Fourier transform to video frames. For offline decoding, we apply a modification of the recently proposed approximate message passing (AMP) algorithm. The AMP method has been derived using the statistical concept of ’state evolution’, and it has been shown to considerably accelerate the convergence rate in special CSdecoding applications. We shall prove that the AMP method can be rewritten as a forwardbackward splitting algorithm. This new representation enables us to give conditions that ensure convergence of the AMP method and to modify the algorithm in order to achieve higher robustness. The success of reconstruction methods for video decoding also essentially depends on the chosen transform, where sparsity of the video signals is assumed. We propose to incorporate the 3D dualtree complex wavelet transform that possesses sufficiently good properties of directional selectivity and shift invariance while being computationally less expensive and less redundant than other directional 3D wavelet transforms.