Results 1  10
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55
Compressed sensing
 IEEE Trans. Inform. Theory
"... Abstract—Suppose is an unknown vector in (a digital image or signal); we plan to measure general linear functionals of and then reconstruct. If is known to be compressible by transform coding with a known transform, and we reconstruct via the nonlinear procedure defined here, the number of measureme ..."
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Cited by 1730 (18 self)
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Abstract—Suppose is an unknown vector in (a digital image or signal); we plan to measure general linear functionals of and then reconstruct. If is known to be compressible by transform coding with a known transform, and we reconstruct via the nonlinear procedure defined here, the number of measurements can be dramatically smaller than the size. Thus, certain natural classes of images with pixels need only = ( 1 4 log 5 2 ()) nonadaptive nonpixel samples for faithful recovery, as opposed to the usual pixel samples. More specifically, suppose has a sparse representation in some orthonormal basis (e.g., wavelet, Fourier) or tight frame (e.g., curvelet, Gabor)—so the coefficients belong to an ball for 0 1. The most important coefficients in that expansion allow reconstruction with 2 error ( 1 2 1
Sparse solutions to linear inverse problems with multiple measurement vectors
 IEEE Trans. Signal Processing
, 2005
"... Abstract—We address the problem of finding sparse solutions to an underdetermined system of equations when there are multiple measurement vectors having the same, but unknown, sparsity structure. The single measurement sparse solution problem has been extensively studied in the past. Although known ..."
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Cited by 131 (10 self)
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Abstract—We address the problem of finding sparse solutions to an underdetermined system of equations when there are multiple measurement vectors having the same, but unknown, sparsity structure. The single measurement sparse solution problem has been extensively studied in the past. Although known to be NPhard, many single–measurement suboptimal algorithms have been formulated that have found utility in many different applications. Here, we consider in depth the extension of two classes of algorithms–Matching Pursuit (MP) and FOCal Underdetermined System Solver (FOCUSS)–to the multiple measurement case so that they may be used in applications such as neuromagnetic imaging, where multiple measurement vectors are available, and solutions with a common sparsity structure must be computed. Cost functions appropriate to the multiple measurement problem are developed, and algorithms are derived based on their minimization. A simulation study is conducted on a testcase dictionary to show how the utilization of more than one measurement vector improves the performance of the MP and FOCUSS classes of algorithm, and their performances are compared. I.
A Sparse Signal Reconstruction Perspective for Source Localization With Sensor Arrays
 M.S. thesis, Mass. Inst. Technol
, 2003
"... Abstract—We present a source localization method based on a sparse representation of sensor measurements with an overcomplete basis composed of samples from the array manifold. We enforce sparsity by imposing penalties based on the 1norm. A number of recent theoretical results on sparsifying proper ..."
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Cited by 112 (4 self)
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Abstract—We present a source localization method based on a sparse representation of sensor measurements with an overcomplete basis composed of samples from the array manifold. We enforce sparsity by imposing penalties based on the 1norm. A number of recent theoretical results on sparsifying properties of 1 penalties justify this choice. Explicitly enforcing the sparsity of the representation is motivated by a desire to obtain a sharp estimate of the spatial spectrum that exhibits superresolution. We propose to use the singular value decomposition (SVD) of the data matrix to summarize multiple time or frequency samples. Our formulation leads to an optimization problem, which we solve efficiently in a secondorder cone (SOC) programming framework by an interior point implementation. We propose a grid refinement method to mitigate the effects of limiting estimates to a grid of spatial locations and introduce an automatic selection criterion for the regularization parameter involved in our approach. We demonstrate the effectiveness of the method on simulated data by plots of spatial spectra and by comparing the estimator variance to the Cramér–Rao bound (CRB). We observe that our approach has a number of advantages over other source localization techniques, including increased resolution, improved robustness to noise, limitations in data quantity, and correlation of the sources, as well as not requiring an accurate initialization. Index Terms—Directionofarrival estimation, overcomplete representation, sensor array processing, source localization, sparse representation, superresolution. I.
Just relax: Convex programming methods for subset selection and sparse approximation
, 2004
"... Abstract. Subset selection and sparse approximation problems request a good approximation of an input signal using a linear combination of elementary signals, yet they stipulate that the approximation may only involve a few of the elementary signals. This class of problems arises throughout electric ..."
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Cited by 91 (4 self)
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Abstract. Subset selection and sparse approximation problems request a good approximation of an input signal using a linear combination of elementary signals, yet they stipulate that the approximation may only involve a few of the elementary signals. This class of problems arises throughout electrical engineering, applied mathematics and statistics, but small theoretical progress has been made over the last fifty years. Subset selection and sparse approximation both admit natural convex relaxations, but the literature contains few results on the behavior of these relaxations for general input signals. This report demonstrates that the solution of the convex program frequently coincides with the solution of the original approximation problem. The proofs depend essentially on geometric properties of the ensemble of elementary signals. The results are powerful because sparse approximation problems are combinatorial, while convex programs can be solved in polynomial time with standard software. Comparable new results for a greedy algorithm, Orthogonal Matching Pursuit, are also stated. This report should have a major practical impact because the theory applies immediately to many realworld signal processing problems. 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 77 (8 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 60 (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.
Fast solution of ℓ1norm minimization problems when the solution may be sparse
, 2006
"... The minimum ℓ1norm solution to an underdetermined system of linear equations y = Ax, is often, remarkably, also the sparsest solution to that system. This sparsityseeking property is of interest in signal processing and information transmission. However, generalpurpose optimizers are much too slo ..."
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Cited by 47 (1 self)
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The minimum ℓ1norm solution to an underdetermined system of linear equations y = Ax, is often, remarkably, also the sparsest solution to that system. This sparsityseeking property is of interest in signal processing and information transmission. However, generalpurpose optimizers are much too slow for ℓ1 minimization in many largescale applications. The Homotopy method was originally proposed by Osborne et al. for solving noisy overdetermined ℓ1penalized least squares problems. We here apply it to solve the noiseless underdetermined ℓ1minimization problem min ‖x‖1 subject to y = Ax. We show that Homotopy runs much more rapidly than generalpurpose LP solvers when sufficient sparsity is present. Indeed, the method often has the following kstep solution property: if the underlying solution has only k nonzeros, the Homotopy method reaches that solution in only k iterative steps. When this property holds and k is small compared to the problem size, this means that ℓ1 minimization problems with ksparse solutions can be solved in a fraction of the cost of solving one fullsized linear system. We demonstrate this kstep solution property for two kinds of problem suites. First,
Toeplitz compressed sensing matrices with applications to sparse channel estimation,” submitted
 Online]. Available: http://www.ece.wisc.edu/ ∼nowak/sub08 toep.pdf
"... Compressed sensing (CS) has recently emerged as a powerful signal acquisition paradigm. In essence, CS enables the recovery of highdimensional sparse signals from relatively few linear observations in the form of projections onto a collection of test vectors. Existing results show that if the entri ..."
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Cited by 43 (8 self)
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Compressed sensing (CS) has recently emerged as a powerful signal acquisition paradigm. In essence, CS enables the recovery of highdimensional sparse signals from relatively few linear observations in the form of projections onto a collection of test vectors. Existing results show that if the entries of the test vectors are independent realizations of certain zeromean random variables, then with high probability the unknown signals can be recovered by solving a tractable convex optimization. This work extends CS theory to settings where the entries of the test vectors exhibit structured statistical dependencies. It follows that CS can be effectively utilized in linear, timeinvariant system identification problems provided the impulse response of the system is (approximately or exactly) sparse. An immediate application is in wireless multipath channel estimation. It is shown here that timedomain probing of a multipath channel with a random binary sequence, along with utilization of CS reconstruction techniques, can provide significant improvements in estimation accuracy compared to traditional leastsquares based linear channel estimation strategies. Abstract extensions of the main results are also discussed, where the theory of equitable graph coloring is employed to establish the utility of CS in settings where the test vectors exhibit more general statistical dependencies. Index Terms circulant matrices, compressed sensing, Hankel matrices, restricted isometry property, sparse channel estimation, Toeplitz matrices, wireless communications. I.
Linear convergence of iterative softthresholding
 J. Fourier Anal. Appl
"... ABSTRACT. In this article a unified approach to iterative softthresholding algorithms for the solution of linear operator equations in infinite dimensional Hilbert spaces is presented. We formulate the algorithm in the framework of generalized gradient methods and present a new convergence analysis ..."
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Cited by 34 (9 self)
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ABSTRACT. In this article a unified approach to iterative softthresholding algorithms for the solution of linear operator equations in infinite dimensional Hilbert spaces is presented. We formulate the algorithm in the framework of generalized gradient methods and present a new convergence analysis. As main result we show that the algorithm converges with linear rate as soon as the underlying operator satisfies the socalled finite basis injectivity property or the minimizer possesses a socalled strict sparsity pattern. Moreover it is shown that the constants can be calculated explicitly in special cases (i.e. for compact operators). Furthermore, the techniques also can be used to establish linear convergence for related methods such as the iterative thresholding algorithm for joint sparsity and the accelerated gradient projection method. 1.
A simple test to check the optimality of sparse signal approximations
 Tech. Rep., IRISA, 2004, in preparation. V  720
"... Approximating a signal or an image with a sparse linear expansion from an overcomplete dictionary of atoms is an extremely useful tool to solve many signal processing problems. Finding the sparsest approximation of a signal from an arbitrary dictionary is an NPhard problem. Despite of this, several ..."
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Cited by 30 (2 self)
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Approximating a signal or an image with a sparse linear expansion from an overcomplete dictionary of atoms is an extremely useful tool to solve many signal processing problems. Finding the sparsest approximation of a signal from an arbitrary dictionary is an NPhard problem. Despite of this, several algorithms have been proposed that provide suboptimal solutions. However, it is generally difficult to know how close the computed solution is to being “optimal”, and whether another algorithm could provide a better result. In this paper we provide a simple test to check whether the output of a sparse approximation algorithm is nearly optimal, in the sense that no significantly different linear expansion from the dictionary can provide both a smaller approximation error and a better sparsity. As a byproduct of our theorems, we obtain results on the identifiability of sparse overcomplete models in the presence of noise, for a fairly large class of sparse priors. 1.