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125
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 163 (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
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.
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 104 (16 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.
Sparse Recovery Using Sparse Matrices
"... 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 ..."
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Cited by 74 (12 self)
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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.
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
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 46 (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.
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.
Construction of a Large Class of Deterministic Sensing Matrices that Satisfy a Statistical Isometry Property
"... Compressed Sensing aims to capture attributes of ksparse signals using very few measurements. In the standard Compressed Sensing paradigm, the N × C measurement matrix Φ is required to act as a near isometry on the set of all ksparse signals (Restricted Isometry Property or RIP). If Φ satisfies th ..."
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Cited by 39 (3 self)
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Compressed Sensing aims to capture attributes of ksparse signals using very few measurements. In the standard Compressed Sensing paradigm, the N × C measurement matrix Φ is required to act as a near isometry on the set of all ksparse signals (Restricted Isometry Property or RIP). If Φ satisfies the RIP, then Basis Pursuit or Matching Pursuit recovery algorithms can be used to recover any ksparse vector α from the N measurements Φα. Although it is known that certain probabilistic processes generate N × C matrices that satisfy RIP with high probability, there is no practical algorithm for verifying whether a given sensing matrix Φ has this property, crucial for the feasibility of the standard recovery algorithms. In contrast this paper provides simple criteria that guarantee that a deterministic sensing matrix satisfying these criteria acts as a near isometry on an overwhelming majority of ksparse signals; in particular, most such signals have a unique representation in the measurement domain. Probability still plays a critical role, but it enters the signal model rather than the construction of the sensing matrix. An essential element in our construction is that we require the columns of the sensing matrix to form a group under pointwise multiplication. The construction allows recovery methods for which the expected performance is sublinear in C, and only quadratic in N, as compared to the superlinear complexity in C of the Basis Pursuit or Matching Pursuit algorithms; the focus on expected performance is more typical of mainstream signal processing than the worstcase analysis that prevails in standard Compressed Sensing. Our framework encompasses many families of deterministic sensing matrices, including those formed from discrete chirps, DelsarteGoethals codes, and extended BCH codes.