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## NESTA: A Fast and Accurate First-Order Method for Sparse Recovery (2009)

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Citations: | 168 - 2 self |

### Citations

5274 | Variational Analysis,
- Rockafellar, Wets
- 1998
(Show Context)
Citation Context ...m a posteriori estimate in a Bayesian setting. In statistics, the same problem is more well-known as the lasso [49] (LSτ ) minimize ‖b − Ax‖ℓ2 subject to ‖x‖ℓ1 ≤ τ. (1.4) Standard optimization theory =-=[47]-=- asserts that these three problems are of course equivalent provided that ɛ, λ, τ obey some special relationships. With the exception of the case where the matrix A is orthogonal, this functional depe... |

4061 | Regression shrinkage and selection via the LASSO.
- Tibshirani
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(Show Context)
Citation Context ...is popular in signal and image processing because of its loose interpretation as a maximum a posteriori estimate in a Bayesian setting. In statistics, the same problem is more well-known as the lasso =-=[49]-=- (LSτ ) minimize ‖b − Ax‖ℓ2 subject to ‖x‖ℓ1 ≤ τ. (1.4) Standard optimization theory [47] asserts that these three problems are of course equivalent provided that ɛ, λ, τ obey some special relationshi... |

3571 | Compressed sensing,”
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(Show Context)
Citation Context ...pproximations of nonsmooth functions, ℓ1 minimization, duality in convex optimization, continuation methods, compressed sensing, total-variation minimization. 1. Introduction. Compressed sensing (CS) =-=[13, 14, 25]-=- is a novel sampling theory, which is based on the revelation that one can exploit sparsity or compressibility when acquiring signals of general interest. In a nutshell, compressed sensing designs non... |

3163 |
A Wavelet Tour of Signal Processing,
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Citation Context ...ed by an extra term, namely 2 CW where CW is the cost of applying W or W ∗ to a vector. In practical situations, there is often a fast algorithm for applying W and W ∗ , e.g. a fast wavelet transform =-=[39]-=-, a fast curvelet transform [11], a fast short-time Fourier transform [39] and so on, which makes this a low-cost extra step 3 . 6.2. Numerical results for nonstandard ℓ1 minimization. Because NESTA i... |

2694 | Atomic decomposition by basis pursuit.
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(Show Context)
Citation Context ... frequently discussed approach considers solving this problem in Lagrangian form, i.e. (QP λ) minimize λ‖x‖ℓ1 1 + 2 ‖b − Ax‖2ℓ2 , (1.3) and is also known as the basis pursuit denoising problem (BPDN) =-=[18]-=-. This problem is popular in signal and image processing because of its loose interpretation as a maximum a posteriori estimate in a Bayesian setting. In statistics, the same problem is more well-know... |

2585 | Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,”
- Candes, Romberg, et al.
- 2006
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Citation Context ...pproximations of nonsmooth functions, ℓ1 minimization, duality in convex optimization, continuation methods, compressed sensing, total-variation minimization. 1. Introduction. Compressed sensing (CS) =-=[13, 14, 25]-=- is a novel sampling theory, which is based on the revelation that one can exploit sparsity or compressibility when acquiring signals of general interest. In a nutshell, compressed sensing designs non... |

1487 | Near-optimal signal recovery from random projections: Universal encoding strategies?” Information Theory,
- Candes, Tao
- 2006
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Citation Context ...pproximations of nonsmooth functions, ℓ1 minimization, duality in convex optimization, continuation methods, compressed sensing, total-variation minimization. 1. Introduction. Compressed sensing (CS) =-=[13, 14, 25]-=- is a novel sampling theory, which is based on the revelation that one can exploit sparsity or compressibility when acquiring signals of general interest. In a nutshell, compressed sensing designs non... |

1035 | A fast iterative shrinkage-thresholding algorithm for linear inverse problems,”
- Beck, Teboulle
- 2009
(Show Context)
Citation Context ...optimal [41] two decades earlier. As a consequence of this breakthrough, a few recent works have followed up with improved techniques for some very special problems in signal or image processing, see =-=[3, 21, 52, 1]-=- for example, or for minimizing composite functions such as ℓ1-regularized least-squares problems [44]. In truth, these novel algorithms demonstrate great promise; they are fast, accurate and robust i... |

860 | The dantzig selector: statistical estimation when p is much larger than n.
- Candes, Tao
- 2007
(Show Context)
Citation Context ...ated problems, which do not have the special ℓ1 + ℓ2 2 structure. One example might be the Dantzig selector, which is a convenient and flexible estimator for recovering sparse signals from noisy data =-=[15]-=-: minimize ‖x‖ℓ1 subject to ‖A∗ (b − AX)‖ℓ∞ ≤ δ. (7.2) This is of course equivalent to the unconstrained problem minimize λ‖x‖ℓ1 + ‖A∗ (b − AX)‖ℓ∞ for some value of λ. Clearly, one could apply Nestero... |

749 | CoSaMP: Iterative signal recovery from incomplete and inaccurate samples. Applied and Computational Harmonic Analysis,
- Needell, Tropp
- 2008
(Show Context)
Citation Context ...ficantly outperforms standard FPC. All parameters were set to default values. 5.1.7. FPC Active Set (FPC-AS) [53]. In 2009, inspired by both first-order algorithms, such as FPC, and greedy algorithms =-=[28, 40]-=-, Wen et al. [53] extend FPC into the two-part algorithm FPC Active Set to solve (QPλ). In the first stage, FPC-AS calls an improved version of FPC that allows the step-size to be updated dynamically,... |

736 | An iterative thresholding algorithm for linear inverse problems with a sparsity constraint,”
- Daubechies, Defrise, et al.
- 2004
(Show Context)
Citation Context ...int methods [10, 36, 48] are accurate but problematic for they need to solve large systems of linear equations to compute the Newton steps. On the other hand, inspired by iterative thresholding ideas =-=[24, 30, 20]-=-, we have now available a great number of first-order methods, see [31, 9, 34, 35] and the many earlier references therein, which may be faster but not necessarily accurate. Indeed, these methods are ... |

698 | Regularization paths for generalized linear models via coordinate descent. - Friedman, Hastie, et al. - 2010 |

525 | Gradient projection for sparse reconstruction: application to compressed sensing and other inverse problems,”
- Figueiredo, Nowak, et al.
- 2007
(Show Context)
Citation Context ...rge systems of linear equations to compute the Newton steps. On the other hand, inspired by iterative thresholding ideas [24, 30, 20], we have now available a great number of first-order methods, see =-=[31, 9, 34, 35]-=- and the many earlier references therein, which may be faster but not necessarily accurate. Indeed, these methods are shown to converge slowly, and typically need a very large number of iterations whe... |

525 | Sparse MRI: the application of compressed sensing for rapid MR imaging Magn.
- Lustig, Donoho, et al.
- 2007
(Show Context)
Citation Context ... to recover a signal accurately, engineers are changing the way they think about signal acquisition in areas ranging from analog-to-digital conversion [23], digital optics, magnetic resonance imaging =-=[38]-=-, seismics [37] and astronomy [8]. In this field, a signal x0 ∈ Rn is acquired by collecting data of the form b = Ax 0 + z, where x 0 is the signal of interest (or its coefficient sequence in a repres... |

516 | Introductory Lectures on Convex Optimization. A Basic Course, - Nesterov - 2004 |

505 | Smooth minimization of non-smooth functions. - Nesterov - 2005 |

498 | Signal recovery by proximal forward-backward splitting.
- Combettes, Wajs
- 2005
(Show Context)
Citation Context ...int methods [10, 36, 48] are accurate but problematic for they need to solve large systems of linear equations to compute the Newton steps. On the other hand, inspired by iterative thresholding ideas =-=[24, 30, 20]-=-, we have now available a great number of first-order methods, see [31, 9, 34, 35] and the many earlier references therein, which may be faster but not necessarily accurate. Indeed, these methods are ... |

392 | Gradient methods for minimizing composite objective function - Nesterov - 2007 |

364 | Sparse reconstruction by separable approximation,
- Wright, Nowak, et al.
- 2009
(Show Context)
Citation Context ...n the final step) was changed. Future releases of GPSR will probably contain a similarly updated continuation stopping criteria. 1920 5.1.3. Sparse reconstruction by separable approximation (SpaRSA) =-=[54]-=-. SpaRSA is an algorithm to minimize composite functions φ(x) = f(x) + λc(x) composed of a smooth term f and a separable non-smooth term c, e.g. (QP λ). At every step, a subproblem of the form minimiz... |

362 |
The split Bregman method for L1-regularized problems,”
- Goldstein, Osher
- 2009
(Show Context)
Citation Context ...g 31 minimize ‖x‖T V subject to ‖b − Ax‖ℓ2 ≤ ɛ. (6.4) To be sure, a number of efficient TV-minimization algorithms have been proposed to solve (6.4) in the special case A = I (denoising problem), see =-=[17, 22, 33]-=-. In comparison, only a few methods have been proposed to solve the more general problem (6.4) even when A is a projector. Known methods include interior point methods (ℓ1magic) [10], proximal-subgrad... |

352 | Probing the Pareto frontier for basis pursuit solutions,”
- Berg, Friedlander
- 2008
(Show Context)
Citation Context ...ction 5 presents a comprehensive series of numerical experiments which illustrate the behavior of several state-of-the-art methods including interior point methods [36], projected gradient techniques =-=[34, 51, 31]-=-, fixed point continuation and iterative thresholding algorithms [34, 56, 3]. It is important to consider that most of these methods have been perfected after several years of research [36, 31], and d... |

349 | An EM algorithm for wavelet-based image restoration,”
- Figueiredo, Nowak
- 2003
(Show Context)
Citation Context ...int methods [10, 36, 48] are accurate but problematic for they need to solve large systems of linear equations to compute the Newton steps. On the other hand, inspired by iterative thresholding ideas =-=[24, 30, 20]-=-, we have now available a great number of first-order methods, see [31, 9, 34, 35] and the many earlier references therein, which may be faster but not necessarily accurate. Indeed, these methods are ... |

305 |
Two point step size gradient methods
- Barzilai, Borwein
- 1988
(Show Context)
Citation Context ... vector of ones, and v belongs to the nonnegative orthant, v[i] ≥ 0 for all i. The projection onto Q is then trivial. Different techniques for choosing the stepsize αk (backtracking, Barzilai-Borwein =-=[2]-=-, and so on) are discussed in [31]. The code is available at http://www.lx.it.pt/~mtf/GPSR/. In the forthcoming experiments, the parameters are set to their default values. GPSR also implements contin... |

288 | A method of solving a convex programming problem with convergence rate o (1/k2). - Nesterov - 1983 |

280 | An InteriorPoint Method for Large-Scale l1-Regularized Least Squares,” - Kim, Koh, et al. - 2007 |

270 | Sparse solution of underdetermined linear equations by stagewise orthogonal matching pursuit
- Donoho, Tsaig, et al.
(Show Context)
Citation Context ...ficantly outperforms standard FPC. All parameters were set to default values. 5.1.7. FPC Active Set (FPC-AS) [53]. In 2009, inspired by both first-order algorithms, such as FPC, and greedy algorithms =-=[28, 40]-=-, Wen et al. [53] extend FPC into the two-part algorithm FPC Active Set to solve (QPλ). In the first stage, FPC-AS calls an improved version of FPC that allows the step-size to be updated dynamically,... |

239 | A new approach to variable selection in least squares problems.
- Osborne, Presnell, et al.
- 2000
(Show Context)
Citation Context ...olutions to the problem (QPλ) and, hence, the solutions to (1.1) and (1.4) may be found by solving a sequence a ℓ1 penalized least-squares problems. The point of this is that it has been noticed (see =-=[34, 45, 27]-=-) that solving (1.3) (resp. the lasso (1.4)) is faster when λ is large (resp. τ is low). This observation greatly motivates the use of continuation for solving (1.3) for a fixed λf . The idea is simpl... |

210 | Nonmonotone spectral projected gradient methods on convex sets,”
- Birgin, Martınez, et al.
- 2000
(Show Context)
Citation Context ...de is available at http://www.stanford.edu/~boyd/l1_ls/. 5.1.5. Spectral projected gradient (SPGL1) [51]. In 2008, van den Berg et al. adapted the spectral projection gradient algorithm introduced in =-=[6]-=- to solve the LASSO (LSτ ). Interestingly, they introduced a clever root finding procedure such that solving a few instances of (LSτ ) for different values of τ enables them to equivalently solve (BPɛ... |

203 | Fast discrete curvelet transforms. Multiscale Modeling and Simulation, - Candes, Demanet, et al. - 2006 |

180 | A new TwIST: two-step iterative shrinkage/thresholding algorithms for image restoration,”
- Bioucas-Dias, Figueiredo
- 2007
(Show Context)
Citation Context ...on, only a few methods have been proposed to solve the more general problem (6.4) even when A is a projector. Known methods include interior point methods (ℓ1magic) [10], proximal-subgradient methods =-=[5, 19]-=-, Split-Bregman [33], and the very recently introduced RecPF4 [55], which operates in the special case of partial Fourier measurements. Roughly, proximal gradient methods approach the solution to (6.4... |

173 |
On accelerated proximal gradient methods for convexconcave optimization,” submitted to
- Tseng
- 2008
(Show Context)
Citation Context ...k from moving too far away from the center x c p. The point xk, at which the gradient of f is evaluated, is a weighted average between zk and yk. In truth, this is motivated by a theoretical analysis =-=[43, 50]-=-, which shows that if αk = 1/2(k + 1) and τk = 2/(k + 3), then the algorithm converges to with the convergence rate x ⋆ = argmin f(x) x∈Qp f(yk) − f(x ⋆ ) ≤ 4Lpp(x⋆ ) (k + 1) 2 . (2.3) σp6 This decay... |

169 | Fast Discrete Curvelet Transforms.
- Candes, Demanet, et al.
- 2005
(Show Context)
Citation Context ... where CW is the cost of applying W or W ∗ to a vector. In practical situations, there is often a fast algorithm for applying W and W ∗ , e.g. a fast wavelet transform [39], a fast curvelet transform =-=[11]-=-, a fast short-time Fourier transform [39] and so on, which makes this a low-cost extra step 3 . 6.2. Numerical results for nonstandard ℓ1 minimization. Because NESTA is one of very few algorithms tha... |

147 | Analysis versus synthesis in signal priors’, - Elad, Milanfar, et al. - 2007 |

144 | Enhancing sparsity by reweighted ℓ1 minimization - Candes, Wakin, et al. - 2008 |

127 | Fast image recovery using variable splitting and constrained optimization,” Image Processing, - Afonso, Bioucas-Dias, et al. - 2010 |

106 |
A method for unconstrained convex minimization problem with the rate of convergence O(1/k2).
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Citation Context ...per which couples smoothing techniques (see [4] and the references therein) with an improved gradient method to derive first-order methods which achieve a convergence rate he had proved to be optimal =-=[41]-=- two decades earlier. As a consequence of this breakthrough, a few recent works have followed up with improved techniques for some very special problems in signal or image processing, see [3, 21, 52, ... |

94 | Fast Linearized Bregman Iteration for Compressed Sensing
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- 2008
(Show Context)
Citation Context ...rge systems of linear equations to compute the Newton steps. On the other hand, inspired by iterative thresholding ideas [24, 30, 20], we have now available a great number of first-order methods, see =-=[31, 9, 34, 35]-=- and the many earlier references therein, which may be faster but not necessarily accurate. Indeed, these methods are shown to converge slowly, and typically need a very large number of iterations whe... |

84 | Near-ideal model selection by ℓ1 minimization - Candès, Plan - 2009 |

83 | Bregman iterative algorithms for ℓ1 minimization with applications to compressed sensing
- Yin, Osher, et al.
(Show Context)
Citation Context ...rate the behavior of several state-of-the-art methods including interior point methods [36], projected gradient techniques [34, 51, 31], fixed point continuation and iterative thresholding algorithms =-=[34, 56, 3]-=-. It is important to consider that most of these methods have been perfected after several years of research [36, 31], and did not exist two years ago. For example, the Fixed Point Continuation method... |

80 | Accelerated projected gradient method for linear inverse problems with sparsity constraints - Daubechies, Fornasier, et al. - 2008 |

74 | J.-C.: A proximal decomposition method for solving convex variational inverse problems.
- Combettes, Pesquet
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Citation Context ...on, only a few methods have been proposed to solve the more general problem (6.4) even when A is a projector. Known methods include interior point methods (ℓ1magic) [10], proximal-subgradient methods =-=[5, 19]-=-, Split-Bregman [33], and the very recently introduced RecPF4 [55], which operates in the special case of partial Fourier measurements. Roughly, proximal gradient methods approach the solution to (6.4... |

73 | Efficient schemes for total variation minimization under constraints in image processing. - Weiss, Aubert, et al. - 2009 |

70 | An efficient primal-dual hybrid gradient algorithm for total variation image restoration, Cam Reports 08-34 UCLA, Center for Applied Mathematics - Zhu, Chan |

68 | Zibulevsky M. Coordinate and subspace optimization methods for linear least squares with non-quadratic regularization - Elad, Matalon |

63 | Fixed-point continuation for ℓ1-minimization: Methodology and convergence - Hale, Yin, et al. |

51 | A fast algorithm for sparse reconstruction based on shrinkage, subspace optimization and continuation.
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Citation Context ...ormance. In the numerical tests, the Barzilai-Borwein version (referred to as FPC-BB) significantly outperforms standard FPC. All parameters were set to default values. 5.1.7. FPC Active Set (FPC-AS) =-=[53]-=-. In 2009, inspired by both first-order algorithms, such as FPC, and greedy algorithms [28, 40], Wen et al. [53] extend FPC into the two-part algorithm FPC Active Set to solve (QPλ). In the first stag... |

50 |
Fixed-point continuation for L1-minimization: Methodology and convergence
- Hale, Yin, et al.
(Show Context)
Citation Context ... sparse reconstruction algorithms. To repeat ourselves, many of these methodshave been improved after several years of research [36, 31], and many did not exist two years ago [34, 51]. For instance, =-=[35]-=- was submitted for publication less than three months before we put the final touches on this paper. Finally, our focus is on rapid algorithms so that we are interested in methods which can take advan... |

49 |
A fixed-point continuation method for l1regularized minimization with applications to compressed sensing
- Hale, Yin, et al.
- 2007
(Show Context)
Citation Context ...rge systems of linear equations to compute the Newton steps. On the other hand, inspired by iterative thresholding ideas [24, 30, 20], we have now available a great number of first-order methods, see =-=[31, 9, 34, 35]-=- and the many earlier references therein, which may be faster but not necessarily accurate. Indeed, these methods are shown to converge slowly, and typically need a very large number of iterations whe... |

41 | Compressed sensing in astronomy,”
- Bobin, Starck, et al.
- 2008
(Show Context)
Citation Context ...ngineers are changing the way they think about signal acquisition in areas ranging from analog-to-digital conversion [23], digital optics, magnetic resonance imaging [38], seismics [37] and astronomy =-=[8]-=-. In this field, a signal x0 ∈ Rn is acquired by collecting data of the form b = Ax 0 + z, where x 0 is the signal of interest (or its coefficient sequence in a representation where it is assumed to b... |

39 | Some first-order algorithms for total variation based image restoration
- Aujol
- 2009
(Show Context)
Citation Context ...optimal [41] two decades earlier. As a consequence of this breakthrough, a few recent works have followed up with improved techniques for some very special problems in signal or image processing, see =-=[3, 21, 52, 1]-=- for example, or for minimizing composite functions such as ℓ1-regularized least-squares problems [44]. In truth, these novel algorithms demonstrate great promise; they are fast, accurate and robust i... |

32 |
PDCO: Primal-Dual Interior Method for Convex Objectives
- Saunders
- 2002
(Show Context)
Citation Context ...f the algorithms that have been proposed are unable to solve these problems accurately with low computational complexity. On the one hand, standard second-order methods such as interior-point methods =-=[10, 36, 48]-=- are accurate but problematic for they need to solve large systems of linear equations to compute the Newton steps. On the other hand, inspired by iterative thresholding ideas [24, 30, 20], we have no... |

28 |
Compressed wavefield extrapolation
- Lin, Herrmann
(Show Context)
Citation Context ...ignal accurately, engineers are changing the way they think about signal acquisition in areas ranging from analog-to-digital conversion [23], digital optics, magnetic resonance imaging [38], seismics =-=[37]-=- and astronomy [8]. In this field, a signal x0 ∈ Rn is acquired by collecting data of the form b = Ax 0 + z, where x 0 is the signal of interest (or its coefficient sequence in a representation where ... |

25 | Convergence of the linearized Bregman iteration for ℓ1-norm minimization - Cai, Osher, et al. |

24 |
minimization for non-smooth functions
- Smooth
- 2005
(Show Context)
Citation Context ...arely found in applications—while compressible signals are ubiquitous—it is important to have an accurate first-order method to handle realistic signals. 1.1. Contributions. A few years ago, Nesterov =-=[43]-=- published a seminal paper which couples smoothing techniques (see [4] and the references therein) with an improved gradient method to derive first-order methods which achieve a convergence rate he ha... |

21 | Berwin A Turlach. A new approach to variable selection in least squares problems. IMA journal of numerical analysis - Osborne, Presnell |

15 | Algorithms and software for total variation image reconstruction via first-order methods,” submitted to Numerical Algorithms
- Dahl, Hansen, et al.
- 2008
(Show Context)
Citation Context ...optimal [41] two decades earlier. As a consequence of this breakthrough, a few recent works have followed up with improved techniques for some very special problems in signal or image processing, see =-=[3, 21, 52, 1]-=- for example, or for minimizing composite functions such as ℓ1-regularized least-squares problems [44]. In truth, these novel algorithms demonstrate great promise; they are fast, accurate and robust i... |

15 |
Analysis versus synthesis
- Elad, Milanfar, et al.
(Show Context)
Citation Context ...nation of these, or a general arbitrary dictionary of waveforms (note that this class of recovery problems also include weighted ℓ1 methods [16]). This is particularly interesting because recent work =-=[29]-=- suggests the potential advantage of this analysis-based approach over the classical basis pursuit in solving important inverse problems [29]. A consequence of these properties is that NESTA, and more... |

15 |
methods for minimizing composite objective functions
- Gradient
- 2012
(Show Context)
Citation Context ...ximal subgradient algorithm, which only uses two sequences of iterates. In some sense, FISTA is a simplified version of the algorithm previously introduced by Nesterov to minimize composite functions =-=[44]-=-. The theoretical rate of convergence of FISTA is similar to NESTA’s, and has been shown to decay as O(1/k 2 ). For each test, FISTA is run twice: it is first run until the relative variation in the f... |

12 | On the performance of algorithms for the minimization of ℓ1-penalized functionals - Loris |

9 | A new TwIST: two-step iterative shrinkage /thresholding algorithms for image restoration - Figueiredo - 2007 |

8 | Fifteen years of reproducible research in computational harmonic analysis
- DONOHO, MALEKI, et al.
- 2008
(Show Context)
Citation Context ... This is an example among many others. Another might be the minimization of a sum of two norms, e.g. an ℓ1 and a TV norm, under data constraints.7.2. Software. In the spirit of reproducible research =-=[26]-=-, a Matlab version of NESTA will be made available at: http://www.acm.caltech.edu/~nesta/ Acknowledgements. S. Becker wishes to thank Peter Stobbe for the use of his Hadamard Transform and Gabor frame... |

7 |
Near-ideal model selection by ℓ1 minimization, Annals of Statistics 37 (5A
- Candès, Plan
- 2009
(Show Context)
Citation Context ...ciently sparse and if the nonzero entries of x 0 are sufficiently large, the solution x ⋆ to (QP λ) is given by x ⋆ [I] = (A[I] ∗ A[I]) −1 (A[I] ∗ b − λ sgn(x 0 [I])), (4.3) x ⋆ [I c ] = 0, (4.4) see =-=[12]-=- for example. In this expression, x[I] is the vector with indices in I and A[I] is the submatrix with columns indices in I. To evaluate NESTA’s accuracy, we set n = 262,144, m = n/8, and s = m/100 (th... |

7 |
Fast solution of ℓ1 minimization problems when solution may be sparse
- Donoho, Tsaig
- 2006
(Show Context)
Citation Context ...olutions to the problem (QPλ) and, hence, the solutions to (1.1) and (1.4) may be found by solving a sequence a ℓ1 penalized least-squares problems. The point of this is that it has been noticed (see =-=[34, 45, 27]-=-) that solving (1.3) (resp. the lasso (1.4)) is faster when λ is large (resp. τ is low). This observation greatly motivates the use of continuation for solving (1.3) for a fixed λf . The idea is simpl... |

7 | Accelerating gradient projection methods for ℓ1-constrained signal recovery by steplength selection rules, Applied and Computational Harmonic Analysis 27 (2 - Loris, Bertero, et al. - 2009 |

4 |
Solver for l1-regularized least squares problems
- Koh, Kim, et al.
(Show Context)
Citation Context ...f the algorithms that have been proposed are unable to solve these problems accurately with low computational complexity. On the one hand, standard second-order methods such as interior-point methods =-=[10, 36, 48]-=- are accurate but problematic for they need to solve large systems of linear equations to compute the Newton steps. On the other hand, inspired by iterative thresholding ideas [24, 30, 20], we have no... |

3 | A fast alternating direction method for TVℓ1-ℓ2 signal reconstruction from partial Fourier data - Yang, Zhang, et al. |

2 |
A fast and accurate first-order algorithm for compressed sensing
- Bobin, Candès
- 2009
(Show Context)
Citation Context ...→0 fµ(x) = ‖x‖ℓ1 ) and the speed of convergence (the convergence rate is proportional to µ). With noiseless data, µ is directly linked to the desired accuracy. To illustrate this, we have observed in =-=[7]-=- that when the true signal x 0 is exactly sparse and is actually the minimum solution under the equality constraints Ax 0 = b, the ℓ∞ error on the nonzero entries is on the order of µ. The link betwee... |

2 |
Enhancing sparsity by reweighted ℓ1 minimization, tech
- Candès, Wakin, et al.
- 2008
(Show Context)
Citation Context ...form, an undecimated wavelet transform and so on, or a combination of these, or a general arbitrary dictionary of waveforms (note that this class of recovery problems also include weighted ℓ1 methods =-=[16]-=-). This is particularly interesting because recent work [29] suggests the potential advantage of this analysis-based approach over the classical basis pursuit in solving important inverse problems [29... |

2 |
A fast and exact algorithm for total-variation minimization, IbPRIA 3522
- Darbon, Sigelle
- 2005
(Show Context)
Citation Context ...g 31 minimize ‖x‖T V subject to ‖b − Ax‖ℓ2 ≤ ɛ. (6.4) To be sure, a number of efficient TV-minimization algorithms have been proposed to solve (6.4) in the special case A = I (denoising problem), see =-=[17, 22, 33]-=-. In comparison, only a few methods have been proposed to solve the more general problem (6.4) even when A is a projector. Known methods include interior point methods (ℓ1magic) [10], proximal-subgrad... |

1 |
An algorithm for total-variation minimization and applications
- Chambolle
(Show Context)
Citation Context ...g 31 minimize ‖x‖T V subject to ‖b − Ax‖ℓ2 ≤ ɛ. (6.4) To be sure, a number of efficient TV-minimization algorithms have been proposed to solve (6.4) in the special case A = I (denoising problem), see =-=[17, 22, 33]-=-. In comparison, only a few methods have been proposed to solve the more general problem (6.4) even when A is a projector. Known methods include interior point methods (ℓ1magic) [10], proximal-subgrad... |

1 | Efficient schemes for total variation mini- under constraints in image processing - Weiss, Blanc-Féraud, et al. - 2009 |

1 |
A fast TV ℓ1-ℓ2 minimization algorithm for signal reconstruction from partial Fourier data
- Yang, Zhang, et al.
- 2008
(Show Context)
Citation Context ...roblem (6.4) even when A is a projector. Known methods include interior point methods (ℓ1magic) [10], proximal-subgradient methods [5, 19], Split-Bregman [33], and the very recently introduced RecPF4 =-=[55]-=-, which operates in the special case of partial Fourier measurements. Roughly, proximal gradient methods approach the solution to (6.4) by iteratively updating the current estimate xk as follows: xk+1... |

1 | Analog-to-Information Receiver Development Program (A-to-I), DARPA Broad Agency Announcement BAA08-03, 2007; available online from http://www.darpa.mil/mto/Solicitations - DARPA |