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1,516
A Singular Value Thresholding Algorithm for Matrix Completion
, 2008
"... This paper introduces a novel algorithm to approximate the matrix with minimum nuclear norm among all matrices obeying a set of convex constraints. This problem may be understood as the convex relaxation of a rank minimization problem, and arises in many important applications as in the task of reco ..."
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Cited by 555 (22 self)
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This paper introduces a novel algorithm to approximate the matrix with minimum nuclear norm among all matrices obeying a set of convex constraints. This problem may be understood as the convex relaxation of a rank minimization problem, and arises in many important applications as in the task of recovering a large matrix from a small subset of its entries (the famous Netflix problem). Offtheshelf algorithms such as interior point methods are not directly amenable to large problems of this kind with over a million unknown entries. This paper develops a simple firstorder and easytoimplement algorithm that is extremely efficient at addressing problems in which the optimal solution has low rank. The algorithm is iterative and produces a sequence of matrices {X k, Y k} and at each step, mainly performs a softthresholding operation on the singular values of the matrix Y k. There are two remarkable features making this attractive for lowrank matrix completion problems. The first is that the softthresholding operation is applied to a sparse matrix; the second is that the rank of the iterates {X k} is empirically nondecreasing. Both these facts allow the algorithm to make use of very minimal storage space and keep the computational cost of each iteration low. On
Gradient projection for sparse reconstruction: Application to compressed sensing and other inverse problems
 IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING
, 2007
"... Many problems in signal processing and statistical inference involve finding sparse solutions to underdetermined, or illconditioned, linear systems of equations. A standard approach consists in minimizing an objective function which includes a quadratic (squared ℓ2) error term combined with a spa ..."
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Cited by 539 (17 self)
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Many problems in signal processing and statistical inference involve finding sparse solutions to underdetermined, or illconditioned, linear systems of equations. A standard approach consists in minimizing an objective function which includes a quadratic (squared ℓ2) error term combined with a sparsenessinducing (ℓ1) regularization term.Basis pursuit, the least absolute shrinkage and selection operator (LASSO), waveletbased deconvolution, and compressed sensing are a few wellknown examples of this approach. This paper proposes gradient projection (GP) algorithms for the boundconstrained quadratic programming (BCQP) formulation of these problems. We test variants of this approach that select the line search parameters in different ways, including techniques based on the BarzilaiBorwein method. Computational experiments show that these GP approaches perform well in a wide range of applications, often being significantly faster (in terms of computation time) than competing methods. Although the performance of GP methods tends to degrade as the regularization term is deemphasized, we show how they can be embedded in a continuation scheme to recover their efficient practical performance.
Sparse MRI: The Application of Compressed Sensing for Rapid MR Imaging
 MAGNETIC RESONANCE IN MEDICINE 58:1182–1195
, 2007
"... The sparsity which is implicit in MR images is exploited to significantly undersample kspace. Some MR images such as angiograms are already sparse in the pixel representation; other, more complicated images have a sparse representation in some transform domain–for example, in terms of spatial finit ..."
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Cited by 538 (11 self)
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The sparsity which is implicit in MR images is exploited to significantly undersample kspace. Some MR images such as angiograms are already sparse in the pixel representation; other, more complicated images have a sparse representation in some transform domain–for example, in terms of spatial finitedifferences or their wavelet coefficients. According to the recently developed mathematical theory of compressedsensing, images with a sparse representation can be recovered from randomly undersampled kspace data, provided an appropriate nonlinear recovery scheme is used. Intuitively, artifacts due to random undersampling add as noiselike interference. In the sparse transform domain the significant coefficients stand out above the interference. A nonlinear thresholding scheme can recover the sparse coefficients, effectively recovering the image itself. In this article, practical incoherent undersampling schemes are developed and analyzed by means of their aliasing interference. Incoherence is introduced by pseudorandom variabledensity undersampling of phaseencodes. The reconstruction is performed by minimizing the ℓ1 norm of a transformed image, subject to data fidelity constraints. Examples demonstrate improved spatial resolution and accelerated acquisition for multislice fast spinecho brain imaging and 3D contrast enhanced angiography.
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 483 (2 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.
From Sparse Solutions of Systems of Equations to Sparse Modeling of Signals and Images
, 2007
"... A fullrank matrix A ∈ IR n×m with n < m generates an underdetermined system of linear equations Ax = b having infinitely many solutions. Suppose we seek the sparsest solution, i.e., the one with the fewest nonzero entries: can it ever be unique? If so, when? As optimization of sparsity is combin ..."
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Cited by 427 (36 self)
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A fullrank matrix A ∈ IR n×m with n < m generates an underdetermined system of linear equations Ax = b having infinitely many solutions. Suppose we seek the sparsest solution, i.e., the one with the fewest nonzero entries: can it ever be unique? If so, when? As optimization of sparsity is combinatorial in nature, are there efficient methods for finding the sparsest solution? These questions have been answered positively and constructively in recent years, exposing a wide variety of surprising phenomena; in particular, the existence of easilyverifiable conditions under which optimallysparse solutions can be found by concrete, effective computational methods. Such theoretical results inspire a bold perspective on some important practical problems in signal and image processing. Several wellknown signal and image processing problems can be cast as demanding solutions of undetermined systems of equations. Such problems have previously seemed, to many, intractable. There is considerable evidence that these problems often have sparse solutions. Hence, advances in finding sparse solutions to underdetermined systems energizes research on such signal and image processing problems – to striking effect. In this paper we review the theoretical results on sparse solutions of linear systems, empirical
Probing the Pareto frontier for basis pursuit solutions
, 2008
"... The basis pursuit problem seeks a minimum onenorm solution of an underdetermined leastsquares problem. Basis pursuit denoise (BPDN) fits the leastsquares problem only approximately, and a single parameter determines a curve that traces the optimal tradeoff between the leastsquares fit and the ..."
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Cited by 365 (5 self)
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The basis pursuit problem seeks a minimum onenorm solution of an underdetermined leastsquares problem. Basis pursuit denoise (BPDN) fits the leastsquares problem only approximately, and a single parameter determines a curve that traces the optimal tradeoff between the leastsquares fit and the onenorm of the solution. We prove that this curve is convex and continuously differentiable over all points of interest, and show that it gives an explicit relationship to two other optimization problems closely related to BPDN. We describe a rootfinding algorithm for finding arbitrary points on this curve; the algorithm is suitable for problems that are large scale and for those that are in the complex domain. At each iteration, a spectral gradientprojection method approximately minimizes a leastsquares problem with an explicit onenorm constraint. Only matrixvector operations are required. The primaldual solution of this problem gives function and derivative information needed for the rootfinding method. Numerical experiments on a comprehensive set of test problems demonstrate that the method scales well to large problems.
Singlepixel imaging via compressive sampling
 IEEE Signal Processing Magazine
"... Humans are visual animals, and imaging sensors that extend our reach – cameras – have improved dramatically in recent times thanks to the introduction of CCD and CMOS digital technology. Consumer digital cameras in the megapixel range are now ubiquitous thanks to the happy coincidence that the semi ..."
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Cited by 296 (19 self)
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Humans are visual animals, and imaging sensors that extend our reach – cameras – have improved dramatically in recent times thanks to the introduction of CCD and CMOS digital technology. Consumer digital cameras in the megapixel range are now ubiquitous thanks to the happy coincidence that the semiconductor material of choice for largescale electronics integration (silicon) also happens to readily convert photons at visual wavelengths into electrons. On the contrary, imaging at wavelengths where silicon is blind is considerably more complicated, bulky, and expensive. Thus, for comparable resolution, a $500 digital camera for the visible becomes a $50,000 camera for the infrared. In this paper, we present a new approach to building simpler, smaller, and cheaper digital cameras that can operate efficiently across a much broader spectral range than conventional siliconbased cameras. Our approach fuses a new camera architecture based on a digital micromirror device (DMD – see Sidebar: Spatial Light Modulators) with the new mathematical theory and algorithms of compressive sampling (CS – see Sidebar: Compressive Sampling in a Nutshell). CS combines sampling and compression into a single nonadaptive linear measurement process [1–4]. Rather than measuring pixel samples of the scene under view, we measure inner products
Compressed sensing and best kterm approximation
 J. Amer. Math. Soc
, 2009
"... Compressed sensing is a new concept in signal processing where one seeks to minimize the number of measurements to be taken from signals while still retaining the information necessary to approximate them well. The ideas have their origins in certain abstract results from functional analysis and app ..."
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Cited by 282 (10 self)
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Compressed sensing is a new concept in signal processing where one seeks to minimize the number of measurements to be taken from signals while still retaining the information necessary to approximate them well. The ideas have their origins in certain abstract results from functional analysis and approximation theory by Kashin [23] but were recently brought into the forefront by the work of Candès, Romberg and Tao [7, 5, 6] and Donoho [9] who constructed concrete algorithms and showed their promise in application. There remain several fundamental questions on both the theoretical and practical side of compressed sensing. This paper is primarily concerned about one of these theoretical issues revolving around just how well compressed sensing can approximate a given signal from a given budget of fixed linear measurements, as compared to adaptive linear measurements. More precisely, we consider discrete signals x ∈ IR N, allocate n < N linear measurements of x, and we describe the range of k for which these measurements encode enough information to recover x in the sense of ℓp to the accuracy of best kterm approximation. We also consider the problem of having such accuracy only with high probability.
Sparse solution of underdetermined linear equations by stagewise orthogonal matching pursuit
, 2006
"... Finding the sparsest solution to underdetermined systems of linear equations y = Φx is NPhard in general. We show here that for systems with ‘typical’/‘random ’ Φ, a good approximation to the sparsest solution is obtained by applying a fixed number of standard operations from linear algebra. Our pr ..."
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Cited by 274 (22 self)
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Finding the sparsest solution to underdetermined systems of linear equations y = Φx is NPhard in general. We show here that for systems with ‘typical’/‘random ’ Φ, a good approximation to the sparsest solution is obtained by applying a fixed number of standard operations from linear algebra. Our proposal, Stagewise Orthogonal Matching Pursuit (StOMP), successively transforms the signal into a negligible residual. Starting with initial residual r0 = y, at the sth stage it forms the ‘matched filter ’ Φ T rs−1, identifies all coordinates with amplitudes exceeding a speciallychosen threshold, solves a leastsquares problem using the selected coordinates, and subtracts the leastsquares fit, producing a new residual. After a fixed number of stages (e.g. 10), it stops. In contrast to Orthogonal Matching Pursuit (OMP), many coefficients can enter the model at each stage in StOMP while only one enters per stage in OMP; and StOMP takes a fixed number of stages (e.g. 10), while OMP can take many (e.g. n). StOMP runs much faster than competing proposals for sparse solutions, such as ℓ1 minimization and OMP, and so is attractive for solving largescale problems. We use phase diagrams to compare algorithm performance. The problem of recovering a ksparse vector x0 from (y, Φ) where Φ is random n × N and y = Φx0 is represented by a point (n/N, k/n)