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45
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 discrete-to-discrete 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 discrete-to-discrete 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 continuous-time 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.
Asymptotic analysis of MAP estimation via the replica method and applications to compressed sensing
, 2009
"... The replica method is a non-rigorous but widely-accepted technique from statistical physics used in the asymptotic analysis of large, random, nonlinear problems. This paper applies the replica method to non-Gaussian maximum a posteriori (MAP) estimation. It is shown that with random linear measureme ..."
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Cited by 77 (9 self)
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The replica method is a non-rigorous but widely-accepted technique from statistical physics used in the asymptotic analysis of large, random, nonlinear problems. This paper applies the replica method to non-Gaussian maximum a posteriori (MAP) estimation. It is shown that with random linear measurements and Gaussian noise, the asymptotic behavior of the MAP estimate of ann-dimensional vector “decouples ” asnscalar MAP estimators. The result is a counterpart to Guo and Verdú’s replica analysis of minimum mean-squared error estimation. The replica MAP analysis can be readily applied to many estimators used in compressed sensing, including basis pursuit, lasso, linear estimation with thresholding, and zero norm-regularized estimation. In the case of lasso estimation the scalar estimator reduces to a soft-thresholding operator, and for zero normregularized estimation it reduces to a hard-threshold. Among other benefits, the replica method provides a computationally-tractable method for exactly computing various performance metrics including mean-squared error and sparsity pattern recovery probability.
Information-Theoretically 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 non-zero 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) non-zero 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
Support recovery with sparsely sampled free random matrices
- in Proc. IEEE Int. Symp. Inf. Theory, Saint
, 2011
"... Abstract—Consider a Bernoulli-Gaussian complex n-vector whose components are Vi = XiBi, with Xi ∼ CN (0, Px) and binary Bi mutually independent and iid across i. This random q-sparse vector is multiplied by a square random matrix U, and a randomly chosen subset, of average size np, p ∈ [0, 1], of th ..."
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Cited by 27 (1 self)
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Abstract—Consider a Bernoulli-Gaussian complex n-vector whose components are Vi = XiBi, with Xi ∼ CN (0, Px) and binary Bi mutually independent and iid across i. This random q-sparse vector is multiplied by a square random matrix U, and a randomly chosen subset, of average size np, p ∈ [0, 1], of the resulting vector components is then observed in additive Gaussian noise. We extend the scope of conventional noisy compressive sampling models where U is typically a matrix with iid components, to allow U satisfying a certain freeness condition. This class of matrices encompasses Haar matrices and other unitarily invariant matrices. We use the replica method and the decoupling principle of Guo and Verdú, as well as a number of information theoretic bounds, to study the input-output mutual information and the support recovery error rate in the limit of n → ∞. We also extend the scope of the large deviation approach of Rangan, Fletcher and Goyal and characterize the performance of a class of estimators encompassing thresholded linear MMSE and ℓ1 relaxation.
The pros and cons of compressive sensing for wideband signal acquisition: Noise folding vs. dynamic range
, 2011
"... Compressive sensing (CS) exploits the sparsity present in many common signals to reduce the number of measurements needed for digital acquisition. With this reduction would come, in theory, commensurate reductions in the size, weight, power consumption, and/or monetary cost of both signal sensors an ..."
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Cited by 26 (5 self)
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Compressive sensing (CS) exploits the sparsity present in many common signals to reduce the number of measurements needed for digital acquisition. With this reduction would come, in theory, commensurate reductions in the size, weight, power consumption, and/or monetary cost of both signal sensors and any associated communication links. This paper examines the use of CS in the design of a wideband radio receiver in a noisy environment. We formulate the problem statement for such a receiver and establish a reasonable set of requirements that a receiver should meet to be practically useful. We then evaluate the performance of a CS-based receiver in two ways: via a theoretical analysis of the expected performance, with a particular emphasis on noise and dynamic range, and via simulations that compare the CS receiver against the performance expected from a conventional implementation. On the one hand, we show that CS-based systems that aim to reduce the number of acquired measurements are somewhat sensitive to signal noise, exhibiting a 3dB SNR loss per octave of subsampling which parallels the classic noise-folding phenomenon. On the other hand, we demonstrate that since they sample at a lower rate, CS-based systems can potentially attain a significantly larger dynamic range. Hence, we conclude that while a CS-based system has inherent limitations that do impose some restrictions on its potential applications, it also has attributes that make it highly desirable in a number of important practical settings. 1
On the fundamental limits of adaptive sensing
, 2011
"... Suppose we can sequentially acquire arbitrary linear measurements of an n-dimensional vector x resulting in the linear model y = Ax + z, where z represents measurement noise. If the signal is known to be sparse, one would expect the following folk theorem to be true: choosing an adaptive strategy wh ..."
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Cited by 25 (3 self)
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Suppose we can sequentially acquire arbitrary linear measurements of an n-dimensional vector x resulting in the linear model y = Ax + z, where z represents measurement noise. If the signal is known to be sparse, one would expect the following folk theorem to be true: choosing an adaptive strategy which cleverly selects the next row of A based on what has been previously observed should do far better than a nonadaptive strategy which sets the rows of A ahead of time, thus not trying to learn anything about the signal in between observations. This paper shows that the folk theorem is false. We prove that the advantages offered by clever adaptive strategies and sophisticated estimation procedures—no matter how intractable—over classical compressed acquisition/recovery schemes are, in general, minimal.
Y Eldar, Noise folding in compressed sensing
- IEEE Signal Process. Lett
, 2011
"... Abstract—The literature on compressed sensing has focused almost entirely on settings where the signal is noiseless and the measurements are contaminated by noise. In practice, however, the signal itself is often subject to random noise prior to measurement. We briefly study this setting and show th ..."
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Cited by 24 (4 self)
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Abstract—The literature on compressed sensing has focused almost entirely on settings where the signal is noiseless and the measurements are contaminated by noise. In practice, however, the signal itself is often subject to random noise prior to measurement. We briefly study this setting and show that, for the vast majority of measurement schemes employed in compressed sensing, the two models are equivalent with the important difference that the signal-to-noise ratio (SNR) is divided by a factor proportional to,whereis the dimension of the signal and is the number of observations. Since is often large, this leads to noise folding which can have a severe impact on the SNR. Index Terms—Analog noise versus digital noise, compressed sensing, matching pursuit, noise folding, sparse signals. I.
Near-Optimal Adaptive Compressed Sensing
, 1306
"... Abstract—This paper proposes a simple adaptive sensing and group testing algorithm for sparse signal recovery. The algorithm, termed Compressive Adaptive Sense and Search (CASS), is shown to be near-optimal in that it succeeds at the lowest possible signal-to-noise-ratio (SNR) levels. Like tradition ..."
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Cited by 19 (0 self)
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Abstract—This paper proposes a simple adaptive sensing and group testing algorithm for sparse signal recovery. The algorithm, termed Compressive Adaptive Sense and Search (CASS), is shown to be near-optimal in that it succeeds at the lowest possible signal-to-noise-ratio (SNR) levels. Like traditional compressed sensing based on random non-adaptive design matrices, the CASS algorithm requires only k log n measurements to recover a k-sparse signal of dimension n. However,CASSsucceedsatSNR levels that are a factor log n less than required by standard compressed sensing. From the point of view of constructing and implementing the sensing operation as well as computing the reconstruction, the proposed algorithm is substantially less computationally intensive than standard compressed sensing. CASS is also demonstrated to perform considerably better in practice through simulation. To the best of our knowledge, this is the first demonstration of an adaptive compressed sensing algorithm with near-optimal theoretical guarantees and excellent practical performance. This paper also shows that methods like compressed sensing, group testing, and pooling have an advantage beyond simply reducing the number of measurements or tests – adaptive versions of such methods can also improve detection and estimation performance when compared to non-adaptive direct (uncompressed) sensing. I.
Sequentially Designed Compressed Sensing
"... A sequential adaptive compressed sensing procedure for signal support recovery is proposed and analyzed. The procedure is based on the principle of distilled sensing, and makes used of sparse sensing matrices to perform sketching observations able to quickly identify irrelevant signal components. I ..."
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Cited by 18 (5 self)
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A sequential adaptive compressed sensing procedure for signal support recovery is proposed and analyzed. The procedure is based on the principle of distilled sensing, and makes used of sparse sensing matrices to perform sketching observations able to quickly identify irrelevant signal components. It is shown that adaptive compressed sensing enables recovery of weaker sparse signals than those that can be recovered using traditional non-adaptive compressed sensing approaches.
Sharp Support Recovery from Noisy Random Measurements by ℓ1 minimization
, 2011
"... In this paper, we investigate the theoretical guarantees of penalized ℓ1-minimization (also called Basis Pursuit Denoising or Lasso) in terms of sparsity pattern recovery (support and sign consistency) from noisy measurements with non-necessarily random noise, when the sensing operator belongs to th ..."
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Cited by 11 (5 self)
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In this paper, we investigate the theoretical guarantees of penalized ℓ1-minimization (also called Basis Pursuit Denoising or Lasso) in terms of sparsity pattern recovery (support and sign consistency) from noisy measurements with non-necessarily random noise, when the sensing operator belongs to the Gaussian ensemble (i.e. random design matrix with i.i.d. Gaussian entries). More precisely, we derive sharp non-asymptotic bounds on the sparsity level and (minimal) signal-to-noise ratio that ensure support identification for most signals and most Gaussian sensing matrices by solving the Lasso with an appropriately chosen regularization parameter. Our first purpose is to establish conditions allowing exact sparsity pattern recovery when the signal is strictly sparse. Then, these conditions are extended to cover the compressible or nearly sparse case. In these two results, the role of the minimal signal-to-noise ratio is crucial. Our third main result gets rid of this assumption in the strictly sparse case, but this time, the Lasso allows only partial recovery of the support. We also provide in this case a sharp ℓ2-consistency result on the coefficient vector. The results of the present work have several distinctive features compared to previous ones. One of them is that the leading constants involved in all the bounds are sharp and explicit. This is illustrated by some numerical experiments where it is indeed shown that the sharp sparsity level threshold identified by our theoretical results below which sparsistency of the Lasso solution is guaranteed meets the one empirically observed.
