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202
Gradient projection for sparse reconstruction: Application to compressed sensing and other inverse problems
 IEEE Journal of Selected Topics in Signal Processing
, 2007
"... Abstract—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 wi ..."
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Cited by 495 (15 self)
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Abstract—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. A. Background I.
Sparse Reconstruction by Separable Approximation
, 2008
"... Finding sparse approximate solutions to large underdetermined linear systems of equations is a common problem in signal/image processing and statistics. Basis pursuit, the least absolute shrinkage and selection operator (LASSO), waveletbased deconvolution and reconstruction, and compressed sensing ( ..."
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Cited by 358 (36 self)
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Finding sparse approximate solutions to large underdetermined linear systems of equations is a common problem in signal/image processing and statistics. Basis pursuit, the least absolute shrinkage and selection operator (LASSO), waveletbased deconvolution and reconstruction, and compressed sensing (CS) are a few wellknown areas in which problems of this type appear. One standard approach is to minimize an objective function that includes a quadratic (ℓ2) error term added to a sparsityinducing (usually ℓ1) regularization term. We present an algorithmic framework for the more general problem of minimizing the sum of a smooth convex function and a nonsmooth, possibly nonconvex regularizer. We propose iterative methods in which each step is obtained by solving an optimization subproblem involving a quadratic term with diagonal Hessian (which is therefore separable in the unknowns) plus the original sparsityinducing regularizer. Our approach is suitable for cases in which this subproblem can be solved much more rapidly than the original problem. In addition to solving the standard ℓ2 − ℓ1 case, our framework yields an efficient solution technique for other regularizers, such as an ℓ∞norm regularizer and groupseparable (GS) regularizers. It also generalizes immediately to the case in which the data is complex rather than real. Experiments with CS problems show that our approach is competitive with the fastest known methods for the standard ℓ2 − ℓ1 problem, as well as being efficient on problems with other separable regularization terms.
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 337 (4 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.
NESTA: A Fast and Accurate FirstOrder Method for Sparse Recovery
, 2009
"... Accurate signal recovery or image reconstruction from indirect and possibly undersampled data is a topic of considerable interest; for example, the literature in the recent field of compressed sensing is already quite immense. Inspired by recent breakthroughs in the development of novel firstorder ..."
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Cited by 168 (2 self)
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Accurate signal recovery or image reconstruction from indirect and possibly undersampled data is a topic of considerable interest; for example, the literature in the recent field of compressed sensing is already quite immense. Inspired by recent breakthroughs in the development of novel firstorder methods in convex optimization, most notably Nesterov’s smoothing technique, this paper introduces a fast and accurate algorithm for solving common recovery problems in signal processing. In the spirit of Nesterov’s work, one of the key ideas of this algorithm is a subtle averaging of sequences of iterates, which has been shown to improve the convergence properties of standard gradientdescent algorithms. This paper demonstrates that this approach is ideally suited for solving largescale compressed sensing reconstruction problems as 1) it is computationally efficient, 2) it is accurate and returns solutions with several correct digits, 3) it is flexible and amenable to many kinds of reconstruction problems, and 4) it is robust in the sense that its excellent performance across a wide range of problems does not depend on the fine tuning of several parameters. Comprehensive numerical experiments on realistic signals exhibiting a large dynamic range show that this algorithm compares favorably with recently proposed stateoftheart methods. We also apply the algorithm to solve other problems for which there are fewer alternatives, such as totalvariation minimization, and
On Augmented Lagrangian methods with general lowerlevel constraints
, 2005
"... Augmented Lagrangian methods with general lowerlevel constraints are considered in the present research. These methods are useful when efficient algorithms exist for solving subproblems where the constraints are only of the lowerlevel type. Two methods of this class are introduced and analyzed. In ..."
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Cited by 74 (7 self)
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Augmented Lagrangian methods with general lowerlevel constraints are considered in the present research. These methods are useful when efficient algorithms exist for solving subproblems where the constraints are only of the lowerlevel type. Two methods of this class are introduced and analyzed. Inexact resolution of the lowerlevel constrained subproblems is considered. Global convergence is proved using the Constant Positive Linear Dependence constraint qualification. Conditions for boundedness of the penalty parameters are discussed. The reliability of the approach is tested by means of an exhaustive comparison against Lancelot. All the problems of the Cute collection are used in this comparison. Moreover, the resolution of location problems in which many constraints of the lowerlevel set are nonlinear is addressed, employing the Spectral Projected Gradient method for solving the subproblems. Problems of this type with more than 3 × 10 6 variables and 14 × 10 6 constraints are solved in this way, using moderate computer time.
LargeScale ActiveSet BoxConstrained Optimization Method with Spectral Projected Gradients
 Computational Optimization and Applications
, 2001
"... A new activeset method for smooth boxconstrained minimization is introduced. The algorithm combines an unconstrained method, including a new linesearch which aims to add many constraints to the working set at a single iteration, with a recently introduced technique (spectral projected gradien ..."
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Cited by 60 (11 self)
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A new activeset method for smooth boxconstrained minimization is introduced. The algorithm combines an unconstrained method, including a new linesearch which aims to add many constraints to the working set at a single iteration, with a recently introduced technique (spectral projected gradient) for dropping constraints from the working set. Global convergence is proved. A computer implementation is fully described and a numerical comparison assesses the reliability of the new algorithm. Keywords: Boxconstrained minimization, numerical methods, activeset strategies, Spectral Projected Gradient. 1
Structure learning in random fields for heart motion abnormality detection
 In CVPR
, 2008
"... Coronary Heart Disease can be diagnosed by assessing the regional motion of the heart walls in ultrasound images of the left ventricle. Even for experts, ultrasound images are difficult to interpret leading to high intraobserver variability. Previous work indicates that in order to approach this pr ..."
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Cited by 54 (8 self)
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Coronary Heart Disease can be diagnosed by assessing the regional motion of the heart walls in ultrasound images of the left ventricle. Even for experts, ultrasound images are difficult to interpret leading to high intraobserver variability. Previous work indicates that in order to approach this problem, the interactions between the different heart regions and their overall influence on the clinical condition of the heart need to be considered. To do this, we propose a method for jointly learning the structure and parameters of conditional random fields, formulating these tasks as a convex optimization problem. We consider blockL1 regularization for each set of features associated with an edge, and formalize an efficient projection method to find the globally optimal penalized maximum likelihood solution. We perform extensive numerical experiments comparing the presented method with related methods that approach the structure learning problem differently. We verify the robustness of our method on echocardiograms collected in routine clinical practice at one hospital. 1.
Optimizing costly functions with simple constraints: A limitedmemory projected quasinewton algorithm
 Proc. of Conf. on Artificial Intelligence and Statistics
, 2009
"... An optimization algorithm for minimizing a smooth function over a convex set is described. Each iteration of the method computes a descent direction by minimizing, over the original constraints, a diagonal plus lowrank quadratic approximation to the function. The quadratic approximation is construct ..."
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Cited by 49 (9 self)
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An optimization algorithm for minimizing a smooth function over a convex set is described. Each iteration of the method computes a descent direction by minimizing, over the original constraints, a diagonal plus lowrank quadratic approximation to the function. The quadratic approximation is constructed using a limitedmemory quasiNewton update. The method is suitable for largescale problems where evaluation of the function is substantially more expensive than projection onto the constraint set. Numerical experiments on onenorm regularized test problems indicate that the proposed method is competitive with stateoftheart methods such as boundconstrained LBFGS and orthantwise descent. We further show that the method generalizes to a wide class of problems, and substantially improves on stateoftheart methods for problems such as learning the structure of Gaussian graphical models and Markov random fields. 1
Inexact Spectral Projected Gradient methods on convex sets
 IMA Journal on Numerical Analysis
, 2003
"... A new method is introduced for large scale convex constrained optimization. The general model algorithm involves, at each iteration, the approximate minimization of a convex quadratic on the feasible set of the original problem and global convergence is obtained by means of nonmonotone line searches ..."
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Cited by 49 (11 self)
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A new method is introduced for large scale convex constrained optimization. The general model algorithm involves, at each iteration, the approximate minimization of a convex quadratic on the feasible set of the original problem and global convergence is obtained by means of nonmonotone line searches. A specific algorithm, the Inexact Spectral Projected Gradient method (ISPG), is implemented using inexact projections computed by Dykstra’s alternating projection method and generates interior iterates. The ISPG method is a generalization of the Spectral Projected Gradient method (SPG), but can be used when projections are difficult to compute. Numerical results for constrained leastsquares rectangular matrix problems are presented. Key words: Convex constrained optimization, projected gradient, nonmonotone line search, spectral gradient, Dykstra’s algorithm. AMS Subject Classification: 49M07, 49M10, 65K, 90C06, 90C20. 1
Structure Learning of Bayesian Networks using Constraints
"... This paper addresses exact learning of Bayesian network structure from data and expert’s knowledge based on score functions that are decomposable. First, it describes useful properties that strongly reduce the time and memory costs of many known methods such as hillclimbing, dynamic programming and ..."
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Cited by 48 (6 self)
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This paper addresses exact learning of Bayesian network structure from data and expert’s knowledge based on score functions that are decomposable. First, it describes useful properties that strongly reduce the time and memory costs of many known methods such as hillclimbing, dynamic programming and sampling variable orderings. Secondly, a branch and bound algorithm is presented that integrates parameter and structural constraints with data in a way to guarantee global optimality with respect to the score function. It is an anytime procedure because, if stopped, it provides the best current solution and an estimation about how far it is from the global solution. We show empirically the advantages of the properties and the constraints, and the applicability of the algorithm to large data sets (up to one hundred variables) that cannot be handled by other current methods (limited to around 30 variables). 1.