Results 1  10
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278
Atomic decomposition by basis pursuit
 SIAM Journal on Scientific Computing
, 1998
"... Abstract. The timefrequency and timescale communities have recently developed a large number of overcomplete waveform dictionaries — stationary wavelets, wavelet packets, cosine packets, chirplets, and warplets, to name a few. Decomposition into overcomplete systems is not unique, and several meth ..."
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Cited by 1660 (43 self)
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Abstract. The timefrequency and timescale communities have recently developed a large number of overcomplete waveform dictionaries — stationary wavelets, wavelet packets, cosine packets, chirplets, and warplets, to name a few. Decomposition into overcomplete systems is not unique, and several methods for decomposition have been proposed, including the method of frames (MOF), Matching pursuit (MP), and, for special dictionaries, the best orthogonal basis (BOB). Basis Pursuit (BP) is a principle for decomposing a signal into an “optimal ” superposition of dictionary elements, where optimal means having the smallest l 1 norm of coefficients among all such decompositions. We give examples exhibiting several advantages over MOF, MP, and BOB, including better sparsity and superresolution. BP has interesting relations to ideas in areas as diverse as illposed problems, in abstract harmonic analysis, total variation denoising, and multiscale edge denoising. BP in highly overcomplete dictionaries leads to largescale optimization problems. With signals of length 8192 and a wavelet packet dictionary, one gets an equivalent linear program of size 8192 by 212,992. Such problems can be attacked successfully only because of recent advances in linear programming by interiorpoint methods. We obtain reasonable success with a primaldual logarithmic barrier method and conjugategradient solver.
CoSaMP: Iterative signal recovery from incomplete and inaccurate samples
 California Institute of Technology, Pasadena
, 2008
"... Abstract. Compressive sampling offers a new paradigm for acquiring signals that are compressible with respect to an orthonormal basis. The major algorithmic challenge in compressive sampling is to approximate a compressible signal from noisy samples. This paper describes a new iterative recovery alg ..."
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Cited by 345 (6 self)
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Abstract. Compressive sampling offers a new paradigm for acquiring signals that are compressible with respect to an orthonormal basis. The major algorithmic challenge in compressive sampling is to approximate a compressible signal from noisy samples. This paper describes a new iterative recovery algorithm called CoSaMP that delivers the same guarantees as the best optimizationbased approaches. Moreover, this algorithm offers rigorous bounds on computational cost and storage. It is likely to be extremely efficient for practical problems because it requires only matrix–vector multiplies with the sampling matrix. For compressible signals, the running time is just O(N log 2 N), where N is the length of the signal. 1.
Snopt: An SQP Algorithm For LargeScale Constrained Optimization
, 1997
"... Sequential quadratic programming (SQP) methods have proved highly effective for solving constrained optimization problems with smooth nonlinear functions in the objective and constraints. Here we consider problems with general inequality constraints (linear and nonlinear). We assume that first deriv ..."
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Cited by 328 (18 self)
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Sequential quadratic programming (SQP) methods have proved highly effective for solving constrained optimization problems with smooth nonlinear functions in the objective and constraints. Here we consider problems with general inequality constraints (linear and nonlinear). We assume that first derivatives are available, and that the constraint gradients are sparse.
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 291 (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.
An interiorpoint method for largescale l1regularized logistic regression
 Journal of Machine Learning Research
, 2007
"... Logistic regression with ℓ1 regularization has been proposed as a promising method for feature selection in classification problems. In this paper we describe an efficient interiorpoint method for solving largescale ℓ1regularized logistic regression problems. Small problems with up to a thousand ..."
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Cited by 153 (6 self)
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Logistic regression with ℓ1 regularization has been proposed as a promising method for feature selection in classification problems. In this paper we describe an efficient interiorpoint method for solving largescale ℓ1regularized logistic regression problems. Small problems with up to a thousand or so features and examples can be solved in seconds on a PC; medium sized problems, with tens of thousands of features and examples, can be solved in tens of seconds (assuming some sparsity in the data). A variation on the basic method, that uses a preconditioned conjugate gradient method to compute the search step, can solve very large problems, with a million features and examples (e.g., the 20 Newsgroups data set), in a few minutes, on a PC. Using warmstart techniques, a good approximation of the entire regularization path can be computed much more efficiently than by solving a family of problems independently.
Basis Pursuit
, 1994
"... The TimeFrequency and TimeScale communities have recently developed an enormous number of overcomplete signal dictionaries  wavelets, wavelet packets, cosine packets, wilson bases, chirplets, warped bases, and hyperbolic cross bases being a few examples. Basis Pursuit is a technique for decompos ..."
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Cited by 119 (15 self)
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The TimeFrequency and TimeScale communities have recently developed an enormous number of overcomplete signal dictionaries  wavelets, wavelet packets, cosine packets, wilson bases, chirplets, warped bases, and hyperbolic cross bases being a few examples. Basis Pursuit is a technique for decomposing a signal into an "optimal" superposition of dictionary elements. The optimization criterion is the l 1 norm of coefficients. The method has several advantages over Matching Pursuit and Best Ortho Basis, including superresolution and stability. 1 Introduction Over the last five years or so, there has been an explosion of awareness of alternatives to traditional signal representations. Instead of just representing objects as superpositions of sinusoids (the traditional Fourier representation) we now have available alternate dictionaries  signal representation schemes  of which the Wavelets dictionary is only the most wellknown. Wavelet dictionaries, Gabor dictionaries, Multiscale...
Calibration as Parameter Estimation in Sensor Networks
, 2002
"... We describe an adhoc localization system for sensor networks and explain why traditional calibration methods are inadequate for this system. Building upon previous work, we frame calibration as a parameter estimation problem; we parameterize each device and choose the values of those parameters tha ..."
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Cited by 116 (7 self)
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We describe an adhoc localization system for sensor networks and explain why traditional calibration methods are inadequate for this system. Building upon previous work, we frame calibration as a parameter estimation problem; we parameterize each device and choose the values of those parameters that optimize the overall system performance. This method reduces our average error from 74.6% without calibration to 10.1%. We propose ways to expand this technique to a method of autocalibration for localization as well as to other sensor network applications.
Iterative Solution of Linear Systems
 Acta Numerica
, 1992
"... this paper is as follows. In Section 2, we present some background material on general Krylov subspace methods, of which CGtype algorithms are a special case. We recall the outstanding properties of CG and discuss the issue of optimal extensions of CG to nonHermitian matrices. We also review GMRES ..."
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Cited by 100 (8 self)
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this paper is as follows. In Section 2, we present some background material on general Krylov subspace methods, of which CGtype algorithms are a special case. We recall the outstanding properties of CG and discuss the issue of optimal extensions of CG to nonHermitian matrices. We also review GMRES and related methods, as well as CGlike algorithms for the special case of Hermitian indefinite linear systems. Finally, we briefly discuss the basic idea of preconditioning. In Section 3, we turn to Lanczosbased iterative methods for general nonHermitian linear systems. First, we consider the nonsymmetric Lanczos process, with particular emphasis on the possible breakdowns and potential instabilities in the classical algorithm. Then we describe recent advances in understanding these problems and overcoming them by using lookahead techniques. Moreover, we describe the quasiminimal residual algorithm (QMR) proposed by Freund and Nachtigal (1990), which uses the lookahead Lanczos process to obtain quasioptimal approximate solutions. Next, a survey of transposefree Lanczosbased methods is given. We conclude this section with comments on other related work and some historical remarks. In Section 4, we elaborate on CGNR and CGNE and we point out situations where these approaches are optimal. The general class of Krylov subspace methods also contains parameterdependent algorithms that, unlike CGtype schemes, require explicit information on the spectrum of the coefficient matrix. In Section 5, we discuss recent insights in obtaining appropriate spectral information for parameterdependent Krylov subspace methods. After that, 4 R.W. Freund, G.H. Golub and N.M. Nachtigal
Efficiently combining positions and normals for precise 3d geometry
 ACM Transactions on Graphics (Proc. SIGGRAPH
, 2005
"... not use color information in order to focus on geometric aspects. Note how our method eliminates noise from the range image while introducing real detail. The surface normals are of the same quality or better than those from photometric stereo, while most of the lowfrequency bias has been eliminate ..."
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Cited by 89 (8 self)
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not use color information in order to focus on geometric aspects. Note how our method eliminates noise from the range image while introducing real detail. The surface normals are of the same quality or better than those from photometric stereo, while most of the lowfrequency bias has been eliminated. Range scanning, manual 3D editing, and other modeling approaches can provide information about the geometry of surfaces in the form of either 3D positions (e.g., triangle meshes or range images) or orientations (normal maps or bump maps). We present an algorithm that combines these two kinds of estimates to produce a new surface that approximates both. Our formulation is linear, allowing it to operate efficiently on complex meshes commonly used in graphics. It also treats high and lowfrequency components separately, allowing it to optimally combine outputs from data sources such as stereo triangulation and photometric stereo, which have different errorvs.frequency characteristics. We demonstrate the ability of our technique to both recover highfrequency details and avoid lowfrequency bias, producing surfaces that are more widely applicable than position or orientation data alone. 1