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151
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 73 (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
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 61 (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.
Learning with Structured Sparsity
"... This paper investigates a new learning formulation called structured sparsity, which is a natural extension of the standard sparsity concept in statistical learning and compressive sensing. By allowing arbitrary structures on the feature set, this concept generalizes the group sparsity idea. A gener ..."
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Cited by 58 (5 self)
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This paper investigates a new learning formulation called structured sparsity, which is a natural extension of the standard sparsity concept in statistical learning and compressive sensing. By allowing arbitrary structures on the feature set, this concept generalizes the group sparsity idea. A general theory is developed for learning with structured sparsity, based on the notion of coding complexity associated with the structure. Moreover, a structured greedy algorithm is proposed to efficiently solve the structured sparsity problem. Experiments demonstrate the advantage of structured sparsity over standard sparsity. 1.
Bayesian Compressed Sensing via Belief Propagation
, 2010
"... Compressive sensing (CS) is an emerging field based on the revelation that a small collection of linear projections of a sparse signal contains enough information for stable, subNyquist signal acquisition. When a statistical characterization of the signal is available, Bayesian inference can comple ..."
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Cited by 52 (12 self)
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Compressive sensing (CS) is an emerging field based on the revelation that a small collection of linear projections of a sparse signal contains enough information for stable, subNyquist signal acquisition. When a statistical characterization of the signal is available, Bayesian inference can complement conventional CS methods based on linear programming or greedy algorithms. We perform asymptotically optimal Bayesian inference using belief propagation (BP) decoding, which represents the CS encoding matrix as a graphical model. Fast computation is obtained by reducing the size of the graphical model with sparse encoding matrices. To decode a length signal containing large coefficients, our CSBP decoding algorithm uses ( log ()) measurements and ( log 2 ()) computation. Finally, although we focus on a twostate mixture Gaussian model, CSBP is easily adapted to other signal models.
An augmented Lagrangian approach to the constrained optimization formulation of imaging inverse problems
 IEEE Trans. Image Process
, 2011
"... Abstract—We propose a new fast algorithm for solving one of the standard approaches to illposed linear inverse problems (IPLIP), where a (possibly nonsmooth) regularizer is minimized under the constraint that the solution explains the observations sufficiently well. Although the regularizer and con ..."
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Cited by 40 (4 self)
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Abstract—We propose a new fast algorithm for solving one of the standard approaches to illposed linear inverse problems (IPLIP), where a (possibly nonsmooth) regularizer is minimized under the constraint that the solution explains the observations sufficiently well. Although the regularizer and constraint are usually convex, several particular features of these problems (huge dimensionality, nonsmoothness) preclude the use of offtheshelf optimization tools and have stimulated a considerable amount of research. In this paper, we propose a new efficient algorithm to handle one class of constrained problems (often known as basis pursuit denoising) tailored to image recovery applications. The proposed algorithm, which belongs to the family of augmented Lagrangian methods, can be used to deal with a variety of imaging IPLIP, including deconvolution and reconstruction from compressive observations (such as MRI), using either totalvariation or waveletbased (or, more generally, framebased) regularization. The proposed algorithm is an instance of the socalled alternating direction method of multipliers, for which convergence sufficient conditions are known; we show that these conditions are satisfied by the proposed algorithm. Experiments on a set of image restoration and reconstruction benchmark problems show that the proposed algorithm is a strong contender for the stateoftheart. Index Terms—Convex optimization, frames, image reconstruction, image restoration, inpainting, totalvariation. A. Problem Formulation
1Bit Compressive Sensing
"... Abstract—Compressive sensing is a new signal acquisition technology with the potential to reduce the number of measurements required to acquire signals that are sparse or compressible in some basis. Rather than uniformly sampling the signal, compressive sensing computes inner products with a randomi ..."
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Cited by 32 (9 self)
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Abstract—Compressive sensing is a new signal acquisition technology with the potential to reduce the number of measurements required to acquire signals that are sparse or compressible in some basis. Rather than uniformly sampling the signal, compressive sensing computes inner products with a randomized dictionary of test functions. The signal is then recovered by a convex optimization that ensures the recovered signal is both consistent with the measurements and sparse. Compressive sensing reconstruction has been shown to be robust to multilevel quantization of the measurements, in which the reconstruction algorithm is modified to recover a sparse signal consistent to the quantization measurements. In this paper we consider the limiting case of 1bit measurements, which preserve only the sign information of the random measurements. Although it is possible to reconstruct using the classical compressive sensing approach by treating the 1bit measurements as ±1 measurement values, in this paper we reformulate the problem by treating the 1bit measurements as sign constraints and further constraining the optimization to recover a signal on the unit sphere. Thus the sparse signal is recovered within a scaling factor. We demonstrate that this approach performs significantly better compared to the classical compressive sensing reconstruction methods, even as the signal becomes less sparse and as the number of measurements increases. I.
Templates for Convex Cone Problems with Applications to Sparse Signal Recovery
, 2010
"... This paper develops a general framework for solving a variety of convex cone problems that frequently arise in signal processing, machine learning, statistics, and other fields. The approach works as follows: first, determine a conic formulation of the problem; second, determine its dual; third, app ..."
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Cited by 31 (3 self)
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This paper develops a general framework for solving a variety of convex cone problems that frequently arise in signal processing, machine learning, statistics, and other fields. The approach works as follows: first, determine a conic formulation of the problem; second, determine its dual; third, apply smoothing; and fourth, solve using an optimal firstorder method. A merit of this approach is its flexibility: for example, all compressed sensing problems can be solved via this approach. These include models with objective functionals such as the totalvariation norm, ‖W x‖1 where W is arbitrary, or a combination thereof. In addition, the paper also introduces a number of technical contributions such as a novel continuation scheme, a novel approach for controlling the step size, and some new results showing that the smooth and unsmoothed problems are sometimes formally equivalent. Combined with our framework, these lead to novel, stable and computationally efficient algorithms. For instance, our general implementation is competitive with stateoftheart methods for solving intensively studied problems such as the LASSO. Further, numerical experiments show that one can solve the Dantzig selector problem, for which no efficient largescale solvers exist, in a few hundred iterations. Finally, the paper is accompanied with a software release. This software is not a single, monolithic solver; rather, it is a suite of programs and routines designed to serve as building blocks for constructing complete algorithms. Keywords. Optimal firstorder methods, Nesterov’s accelerated descent algorithms, proximal algorithms, conic duality, smoothing by conjugation, the Dantzig selector, the LASSO, nuclearnorm minimization.
Alternating direction algorithms for ℓ1problems in compressive sensing
, 2009
"... Abstract. In this paper, we propose and study the use of alternating direction algorithms for several ℓ1norm minimization problems arising from sparse solution recovery in compressive sensing, including the basis pursuit problem, the basispursuit denoising problems of both unconstrained and constr ..."
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Cited by 23 (2 self)
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Abstract. In this paper, we propose and study the use of alternating direction algorithms for several ℓ1norm minimization problems arising from sparse solution recovery in compressive sensing, including the basis pursuit problem, the basispursuit denoising problems of both unconstrained and constrained forms, as well as others. We present and investigate two classes of algorithms derived from either the primal or the dual forms of the ℓ1problems. The construction of the algorithms consists of two main steps: (1) to reformulate an ℓ1problem into one having partially separable objective functions by adding new variables and constraints; and (2) to apply an exact or inexact alternating direction method to the resulting problem. The derived alternating direction algorithms can be regarded as firstorder primaldual algorithms because both primal and dual variables are updated at each and every iteration. Convergence properties of these algorithms are established or restated when they already exist. Extensive numerical results in comparison with several stateoftheart algorithms are given to demonstrate that the proposed algorithms are efficient, stable and robust. Moreover, we present numerical results to emphasize two practically important but perhaps overlooked points. One point is that algorithm speed should always be evaluated relative to appropriate solution accuracy; another is that whenever erroneous measurements possibly exist, the ℓ1norm fidelity should be the fidelity of choice in compressive sensing. Key words. Sparse solution recovery, compressive sensing, ℓ1minimization, primal, dual, alternating direction method
A fast algorithm for sparse reconstruction based on shrinkage, subspace optimization and continuation
 SIAM Journal on Scientific Computing
, 2010
"... Abstract. We propose a fast algorithm for solving the ℓ1regularized minimization problem minx∈R n µ‖x‖1 + ‖Ax − b ‖ 2 2 for recovering sparse solutions to an undetermined system of linear equations Ax = b. The algorithm is divided into two stages that are performed repeatedly. In the first stage a ..."
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Cited by 22 (7 self)
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Abstract. We propose a fast algorithm for solving the ℓ1regularized minimization problem minx∈R n µ‖x‖1 + ‖Ax − b ‖ 2 2 for recovering sparse solutions to an undetermined system of linear equations Ax = b. The algorithm is divided into two stages that are performed repeatedly. In the first stage a firstorder iterative method called “shrinkage ” yields an estimate of the subset of components of x likely to be nonzero in an optimal solution. Restricting the decision variables x to this subset and fixing their signs at their current values reduces the ℓ1norm ‖x‖1 to a linear function of x. The resulting subspace problem, which involves the minimization of a smaller and smooth quadratic function, is solved in the second phase. Our code FPC AS embeds this basic twostage algorithm in a continuation (homotopy) approach by assigning a decreasing sequence of values to µ. This code exhibits stateoftheart performance both in terms of its speed and its ability to recover sparse signals. It can even recover signals that are not as sparse as required by current compressive sensing theory.
Convex structure learning in loglinear models: Beyond pairwise potentials
 In Proceedings of International Workshop on Artificial Intelligence and Statistics
, 2010
"... Previous work has examined structure learning in loglinear models with `1regularization, largely focusing on the case of pairwise potentials. In this work we consider the case of models with potentials of arbitrary order, but that satisfy a hierarchical constraint. We enforce the hierarchical const ..."
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Cited by 18 (1 self)
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Previous work has examined structure learning in loglinear models with `1regularization, largely focusing on the case of pairwise potentials. In this work we consider the case of models with potentials of arbitrary order, but that satisfy a hierarchical constraint. We enforce the hierarchical constraint using group `1regularization with overlapping groups. An active set method that enforces hierarchical inclusion allows us to tractably consider the exponential number of higherorder potentials. We use a spectral projected gradient method as a subroutine for solving the overlapping group `1regularization problem, and make use of a sparse version of Dykstra's algorithm to compute the projection. Our experiments indicate that this model gives equal or better test set likelihood compared to previous models. 1