Results 1  10
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131
Just Relax: Convex Programming Methods for Identifying Sparse Signals in Noise
, 2006
"... This paper studies a difficult and fundamental problem that arises throughout electrical engineering, applied mathematics, and statistics. Suppose that one forms a short linear combination of elementary signals drawn from a large, fixed collection. Given an observation of the linear combination that ..."
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Cited by 386 (1 self)
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This paper studies a difficult and fundamental problem that arises throughout electrical engineering, applied mathematics, and statistics. Suppose that one forms a short linear combination of elementary signals drawn from a large, fixed collection. Given an observation of the linear combination that has been contaminated with additive noise, the goal is to identify which elementary signals participated and to approximate their coefficients. Although many algorithms have been proposed, there is little theory which guarantees that these algorithms can accurately and efficiently solve the problem. This paper studies a method called convex relaxation, which attempts to recover the ideal sparse signal by solving a convex program. This approach is powerful because the optimization can be completed in polynomial time with standard scientific software. The paper provides general conditions which ensure that convex relaxation succeeds. As evidence of the broad impact of these results, the paper describes how convex relaxation can be used for several concrete signal recovery problems. It also describes applications to channel coding, linear regression, and numerical analysis.
Stable recovery of sparse overcomplete representations in the presence of noise
 IEEE TRANS. INFORM. THEORY
, 2006
"... Overcomplete representations are attracting interest in signal processing theory, particularly due to their potential to generate sparse representations of signals. However, in general, the problem of finding sparse representations must be unstable in the presence of noise. This paper establishes t ..."
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Cited by 361 (21 self)
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Overcomplete representations are attracting interest in signal processing theory, particularly due to their potential to generate sparse representations of signals. However, in general, the problem of finding sparse representations must be unstable in the presence of noise. This paper establishes the possibility of stable recovery under a combination of sufficient sparsity and favorable structure of the overcomplete system. Considering an ideal underlying signal that has a sufficiently sparse representation, it is assumed that only a noisy version of it can be observed. Assuming further that the overcomplete system is incoherent, it is shown that the optimally sparse approximation to the noisy data differs from the optimally sparse decomposition of the ideal noiseless signal by at most a constant multiple of the noise level. As this optimalsparsity method requires heavy (combinatorial) computational effort, approximation algorithms are considered. It is shown that similar stability is also available using the basis and the matching pursuit algorithms. Furthermore, it is shown that these methods result in sparse approximation of the noisy data that contains only terms also appearing in the unique sparsest representation of the ideal noiseless sparse signal.
Sharp thresholds for highdimensional and noisy sparsity recovery using l1constrained quadratic programmming (Lasso)
, 2006
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On sparse reconstruction from Fourier and Gaussian measurements
 Communications on Pure and Applied Mathematics
, 2006
"... Abstract. This paper improves upon best known guarantees for exact reconstruction of a sparse signal f from a small universal sample of Fourier measurements. The method for reconstruction that has recently gained momentum in the Sparse Approximation Theory is to relax this highly nonconvex problem ..."
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Cited by 160 (8 self)
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Abstract. This paper improves upon best known guarantees for exact reconstruction of a sparse signal f from a small universal sample of Fourier measurements. The method for reconstruction that has recently gained momentum in the Sparse Approximation Theory is to relax this highly nonconvex problem to a convex problem, and then solve it as a linear program. We show that there exists a set of frequencies Ω such that one can exactly reconstruct every rsparse signal f of length n from its frequencies in Ω, using the convex relaxation, and Ω has size k(r, n) = O(r log(n)·log 2 (r) log(r log n)) = O(r log 4 n). A random set Ω satisfies this with high probability. This estimate is optimal within the log log n and log 3 r factors. We also give a relatively short argument for a similar problem with k(r, n) � r[12 + 8 log(n/r)] Gaussian measurements. We use methods of geometric functional analysis and probability theory in Banach spaces, which makes our arguments quite short. 1.
Sparsest solutions of underdetermined linear systems via ℓ
"... We present a condition on the matrix of an underdetermined linear system which guarantees that the solution of the system with minimal ℓqquasinorm is also the sparsest one. This generalizes, and sightly improves, a similar result for the ℓ1norm. We then introduce a simple numerical scheme to compu ..."
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Cited by 117 (10 self)
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We present a condition on the matrix of an underdetermined linear system which guarantees that the solution of the system with minimal ℓqquasinorm is also the sparsest one. This generalizes, and sightly improves, a similar result for the ℓ1norm. We then introduce a simple numerical scheme to compute solutions with minimal ℓqquasinorm, and we study its convergence. Finally, we display the results of some experiments which indicate that the ℓqmethod performs better than other available methods. 1
Computational methods for sparse solution of linear inverse problems
, 2009
"... The goal of sparse approximation problems is to represent a target signal approximately as a linear combination of a few elementary signals drawn from a fixed collection. This paper surveys the major practical algorithms for sparse approximation. Specific attention is paid to computational issues, ..."
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Cited by 108 (0 self)
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The goal of sparse approximation problems is to represent a target signal approximately as a linear combination of a few elementary signals drawn from a fixed collection. This paper surveys the major practical algorithms for sparse approximation. Specific attention is paid to computational issues, to the circumstances in which individual methods tend to perform well, and to the theoretical guarantees available. Many fundamental questions in electrical engineering, statistics, and applied mathematics can be posed as sparse approximation problems, making these algorithms versatile and relevant to a wealth of applications.
Geometric approach to error correcting codes and reconstruction of signals
 INT. MATH. RES. NOT
, 2005
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Blocksparse signals: Uncertainty relations and efficient recovery
 IEEE TRANS. SIGNAL PROCESS
, 2010
"... We consider efficient methods for the recovery of blocksparse signals — i.e., sparse signals that have nonzero entries occurring in clusters—from an underdetermined system of linear equations. An uncertainty relation for blocksparse signals is derived, based on a blockcoherence measure, which we ..."
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Cited by 93 (15 self)
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We consider efficient methods for the recovery of blocksparse signals — i.e., sparse signals that have nonzero entries occurring in clusters—from an underdetermined system of linear equations. An uncertainty relation for blocksparse signals is derived, based on a blockcoherence measure, which we introduce. We then show that a blockversion of the orthogonal matching pursuit algorithm recovers block ksparse signals in no more than k steps if the blockcoherence is sufficiently small. The same condition on blockcoherence is shown to guarantee successful recovery through a mixed `2=`1optimization approach. This complements previous recovery results for the blocksparse case which relied on small blockrestricted isometry constants. The significance of the results presented in this paper lies in the fact that making explicit use of blocksparsity can provably yield better reconstruction properties than treating the signal as being sparse in the conventional sense, thereby ignoring the additional structure in the problem.
Sparse reconstruction by convex relaxation: Fourier and Gaussian measurements
 CISS 2006 (40th Annual Conference on Information Sciences and Systems
, 2006
"... Abstract — This paper proves best known guarantees for exact reconstruction of a sparse signal f from few nonadaptive universal linear measurements. We consider Fourier measurements (random sample of frequencies of f) and random Gaussian measurements. The method for reconstruction that has recently ..."
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Cited by 93 (7 self)
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Abstract — This paper proves best known guarantees for exact reconstruction of a sparse signal f from few nonadaptive universal linear measurements. We consider Fourier measurements (random sample of frequencies of f) and random Gaussian measurements. The method for reconstruction that has recently gained momentum in the Sparse Approximation Theory is to relax this highly nonconvex problem to a convex problem, and then solve it as a linear program. What are best guarantees for the reconstruction problem to be equivalent to its convex relaxation is an open question. Recent work shows that the number of measurements k(r, n) needed to exactly reconstruct any rsparse signal f of length n from its linear measurements with convex relaxation is usually O(r polylog(n)). However, known guarantees involve huge constants, in spite of very good performance of the algorithms in practice. In attempt to reconcile theory with practice, we prove the first guarantees for universal measurements (i.e. which work for all sparse functions) with reasonable constants. For Gaussian measurements, k(r, n) � 11.7 r ˆ 1.5 + log(n/r) ˜ , which is optimal up to constants. For Fourier measurements, we prove the best known bound k(r, n) = O(r log(n) · log 2 (r) log(r log n)), which is optimal within the log log n and log 3 r factors. Our arguments are based on the
Necessary and sufficient conditions on sparsity pattern recovery
, 2009
"... The paper considers the problem of detecting the sparsity pattern of a ksparse vector in R n from m random noisy measurements. A new necessary condition on the number of measurements for asymptotically reliable detection with maximum likelihood (ML) estimation and Gaussian measurement matrices is ..."
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Cited by 64 (11 self)
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The paper considers the problem of detecting the sparsity pattern of a ksparse vector in R n from m random noisy measurements. A new necessary condition on the number of measurements for asymptotically reliable detection with maximum likelihood (ML) estimation and Gaussian measurement matrices is derived. This necessary condition for ML detection is compared against a sufficient condition for simple maximum correlation (MC) or thresholding algorithms. The analysis shows that the gap between thresholding and ML can be described by a simple expression in terms of the total signaltonoise ratio (SNR), with the gap growing with increasing SNR. Thresholding is also compared against the more sophisticated lasso and orthogonal matching pursuit (OMP) methods. At high SNRs, it is shown that the gap between lasso and OMP over thresholding is described by the range of powers of the nonzero component values of the unknown signals. Specifically, the key benefit of lasso and OMP over thresholding is the ability of lasso and OMP to detect signals with relatively small components.