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71
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 71 (1 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
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 (2 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.
A SELECTIVE OVERVIEW OF VARIABLE SELECTION IN HIGH DIMENSIONAL FEATURE SPACE
, 2010
"... High dimensional statistical problems arise from diverse fields of scientific research and technological development. Variable selection plays a pivotal role in contemporary statistical learning and scientific discoveries. The traditional idea of best subset selection methods, which can be regarded ..."
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Cited by 23 (4 self)
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High dimensional statistical problems arise from diverse fields of scientific research and technological development. Variable selection plays a pivotal role in contemporary statistical learning and scientific discoveries. The traditional idea of best subset selection methods, which can be regarded as a specific form of penalized likelihood, is computationally too expensive for many modern statistical applications. Other forms of penalized likelihood methods have been successfully developed over the last decade to cope with high dimensionality. They have been widely applied for simultaneously selecting important variables and estimating their effects in high dimensional statistical inference. In this article, we present a brief account of the recent developments of theory, methods, and implementations for high dimensional variable selection. What limits of the dimensionality such methods can handle, what the role of penalty functions is, and what the statistical properties are rapidly drive the advances of the field. The properties of nonconcave penalized likelihood and its roles in high dimensional statistical modeling are emphasized. We also review some recent advances in ultrahigh dimensional variable selection, with emphasis on independence screening and twoscale methods.
ℓ1 Trend Filtering
, 2007
"... The problem of estimating underlying trends in time series data arises in a variety of disciplines. In this paper we propose a variation on HodrickPrescott (HP) filtering, a widely used method for trend estimation. The proposed ℓ1 trend filtering method substitutes a sum of absolute values (i.e., ..."
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Cited by 18 (6 self)
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The problem of estimating underlying trends in time series data arises in a variety of disciplines. In this paper we propose a variation on HodrickPrescott (HP) filtering, a widely used method for trend estimation. The proposed ℓ1 trend filtering method substitutes a sum of absolute values (i.e., an ℓ1norm) for the sum of squares used in HP filtering to penalize variations in the estimated trend. The ℓ1 trend filtering method produces trend estimates that are piecewise linear, and therefore is well suited to analyzing time series with an underlying piecewise linear trend. The kinks, knots, or changes in slope, of the estimated trend can be interpreted as abrupt changes or events in the underlying dynamics of the time series. Using specialized interiorpoint methods, ℓ1 trend filtering can be carried out with not much more effort than HP filtering; in particular, the number of arithmetic operations required grows linearly with the number of data points. We describe the method and some of its basic properties, and give some illustrative examples. We show how the method is related to ℓ1 regularization based methods in sparse signal recovery and feature selection, and list some extensions of the basic method.
An efficient algorithm for compressed MR imaging using total variation and wavelets
 in IEEE Conference on Computer Vision and Pattern Recognition
, 2008
"... Compressed sensing, an emerging multidisciplinary field involving mathematics, probability, optimization, and signal processing, focuses on reconstructing an unknown signal from a very limited number of samples. Because information such as boundaries of organs is very sparse in most MR images, compr ..."
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Cited by 18 (2 self)
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Compressed sensing, an emerging multidisciplinary field involving mathematics, probability, optimization, and signal processing, focuses on reconstructing an unknown signal from a very limited number of samples. Because information such as boundaries of organs is very sparse in most MR images, compressed sensing makes it possible to reconstruct the same MR image from a very limited set of measurements significantly reducing the MRI scan duration. In order to do that however, one has to solve the difficult problem of minimizing nonsmooth functions on large data sets. To handle this, we propose an efficient algorithm that jointly minimizes the ℓ1 norm, total variation, and a least squares measure, one of the most powerful models for compressive MR imaging. Our algorithm is based upon an iterative operatorsplitting framework. The calculations are accelerated by continuation and takes advantage of fast wavelet and Fourier transforms enabling our code to process MR images from actual real life applications. We show that faithful MR images can be reconstructed from a subset that represents a mere 20 percent of the complete set of measurements. 1.
Jointsparse recovery from multiple measurements
, 2009
"... The jointsparse recovery problem aims to recover, from sets of compressed measurements, unknown sparse matrices with nonzero entries restricted to a subset of rows. This is an extension of the singlemeasurementvector (SMV) problem widely studied in compressed sensing. We analyze the recovery prop ..."
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Cited by 15 (0 self)
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The jointsparse recovery problem aims to recover, from sets of compressed measurements, unknown sparse matrices with nonzero entries restricted to a subset of rows. This is an extension of the singlemeasurementvector (SMV) problem widely studied in compressed sensing. We analyze the recovery properties for two types of recovery algorithms. First, we show that recovery using sumofnorm minimization cannot exceed the uniform recovery rate of sequential SMV using ℓ1 minimization, and that there are problems that can be solved with one approach but not with the other. Second, we analyze the performance of the ReMBo algorithm [M. Mishali and Y. Eldar, IEEE Trans. Sig. Proc., 56 (2008)] in combination with ℓ1 minimization, and show how recovery improves as more measurements are taken. From this analysis it follows that having more measurements than number of nonzero rows does not improve the potential theoretical recovery rate. 1
Sparse signal reconstruction via iterative support detection
 Siam Journal on Imaging Sciences, issue
, 2010
"... Abstract. We present a novel sparse signal reconstruction method, iterative support detection (ISD), aiming to achieve fast reconstruction and a reduced requirement on the number of measurements compared to the classical ℓ1 minimization approach. ISD addresses failed reconstructions of ℓ1 minimizati ..."
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Cited by 12 (4 self)
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Abstract. We present a novel sparse signal reconstruction method, iterative support detection (ISD), aiming to achieve fast reconstruction and a reduced requirement on the number of measurements compared to the classical ℓ1 minimization approach. ISD addresses failed reconstructions of ℓ1 minimization due to insufficient measurements. It estimates a support set I from a current reconstruction and obtains a new reconstruction by solving the minimization problem min { ∑ i/∈I xi  : Ax = b}, and it iterates these two steps for a small number of times. ISD differs from the orthogonal matching pursuit method, as well as its variants, because (i) the index set I in ISD is not necessarily nested or increasing, and (ii) the minimization problem above updates all the components of x at the same time. We generalize the null space property to the truncated null space property and present our analysis of ISD based on the latter. We introduce an efficient implementation of ISD, called thresholdISD, for recovering signals with fast decaying distributions of nonzeros from compressive sensing measurements. Numerical experiments show that thresholdISD has significant advantages over the classical ℓ1 minimization approach, as well as two stateoftheart algorithms: the iterative reweighted ℓ1 minimization algorithm (IRL1) and the iterative reweighted leastsquares algorithm (IRLS). MATLAB code is available for download from
Sparse signal recovery with temporally correlated source vectors using sparse Bayesian learning
 IEEE J. Sel. Topics Signal Process
, 2011
"... Abstract — We address the sparse signal recovery problem in the context of multiple measurement vectors (MMV) when elements in each nonzero row of the solution matrix are temporally correlated. Existing algorithms do not consider such temporal correlation and thus their performance degrades signific ..."
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Cited by 12 (3 self)
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Abstract — We address the sparse signal recovery problem in the context of multiple measurement vectors (MMV) when elements in each nonzero row of the solution matrix are temporally correlated. Existing algorithms do not consider such temporal correlation and thus their performance degrades significantly with the correlation. In this work, we propose a block sparse Bayesian learning framework which models the temporal correlation. We derive two sparse Bayesian learning (SBL) algorithms, which have superior recovery performance compared to existing algorithms, especially in the presence of high temporal correlation. Furthermore, our algorithms are better at handling highly underdetermined problems and require less rowsparsity on the solution matrix. We also provide analysis of the global and local minima of their cost function, and show that the SBL cost function has the very desirable property that the global minimum is at the sparsest solution to the MMV problem. Extensive experiments also provide some interesting results that motivate future theoretical research on the MMV model.
LassoKalman smoother for tracking sparse signals
 in Asilomar Conf. on Signals, Systems and Computers 2009
, 2009
"... Abstract—Fixedinterval smoothing of timevarying vector processes is an estimation approach with welldocumented merits for tracking applications. The optimal performance in the linear GaussMarkov model is achieved by the Kalman smoother (KS), which also admits an efficient recursive implementatio ..."
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Cited by 6 (0 self)
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Abstract—Fixedinterval smoothing of timevarying vector processes is an estimation approach with welldocumented merits for tracking applications. The optimal performance in the linear GaussMarkov model is achieved by the Kalman smoother (KS), which also admits an efficient recursive implementation. The present paper deals with vector processes for which it is known a priori that many of their entries equal to zero. In this context, the process to be tracked is sparse, and the performance of sparsityagnostic KS schemes degrades considerably. On the other hand, it is shown here that a sparsityaware KS exhibits complexity which grows exponentially in the vector dimension. To obtain a tractable alternative, the KS cost is regularized with the sparsitypromoting ℓ1 norm of the vector process – a relaxation also used in linear regression problems to obtain the leastabsolute shrinkage and selection operator (Lasso). The Lasso (L)KS derived in this work is not only capable of tracking sparse timevarying vector processes, but can also afford an efficient recursive implementation based on the alternating direction method of multipliers (ADMoM). Finally, a weighted (W)LKS is also introduced to cope with the bias of the LKS, and simulations are provided to validate the performance of the novel algorithms. I.