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## A fast iterative shrinkage-thresholding algorithm with application to . . . (2009)

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2682 | Atomic decomposition by basis pursuit
- Chen, Donoho, et al.
- 1998
(Show Context)
Citation Context ...rm regularization criterion is that most images have a sparse representation in the wavelet domain. The presence of the l1 term is used to induce sparsity in the optimal solution of (1.3); see, e.g., =-=[11, 8]-=-. Another important advantage of the l1-based regularization (1.3) over the l2-based regularization (1.2) is that as opposed to the latter, l1 regularization is less sensitive to outliers, which in im... |

983 | Adapting to unknown smoothness via wavelet shrinkage
- DONOHO, JOHNSTONE
- 1995
(Show Context)
Citation Context ... where ‖x‖1 stands for the sum of the absolute values of the components of x, see e.g., [1, 2, 3, 4]. The presence of the l1 term is used to induce sparsity in the optimal solution, of (2), see e.g., =-=[5, 6]-=-. In image deblurring for example, A is often chosen as A = RW where R is the blurring matrix and W contains a wavelet basis. The underlying philosophy here in dealing with the l1 norm regularization ... |

860 | Numerical Methods for Least Squares Problems - Björck - 1996 |

752 |
Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables
- Ortega, C
- 1970
(Show Context)
Citation Context ...od introduced in section 4. For that purpose we first need to recall the first pillar, which is the following well-known and fundamental property for a smooth function in the class C 1,1 ; see, e.g., =-=[29, 2]-=-. Lemma 2.1. Let f : R n → R be a continuously differentiable function with Lipschitz continuous gradient and Lipschitz constant L(f). Then, for any L ≥ L(f), (2.7) f(x) ≤ f(y)+〈x − y, ∇f(y)〉 + L 2 ‖x... |

732 | An iterative thresholding algorithm for linear inverse problems with a sparsity constraint
- Daubechies, Defrise, et al.
(Show Context)
Citation Context ...x − b‖ 2 + λ‖x‖1}, (2) This research is partially supported by the Israel Science Foundation, ISF grant #489-06. where ‖x‖1 stands for the sum of the absolute values of the components of x, see e.g., =-=[1, 2, 3, 4]-=-. The presence of the l1 term is used to induce sparsity in the optimal solution, of (2), see e.g., [5, 6]. In image deblurring for example, A is often chosen as A = RW where R is the blurring matrix ... |

731 |
Regularization of Inverse Problems
- Engl, Hanke, et al.
- 1996
(Show Context)
Citation Context ...a few. The interdisciplinary nature of inverse problems is evident through a vast literature which includes a large body of mathematical and algorithmic developments; see, for instance, the monograph =-=[13]-=- and the references therein. A basic linear inverse problem leads us to study a discrete linear system of the form (1.1) Ax = b + w, where A ∈ Rm×n and b ∈ Rm are known, w is an unknown noise (or pert... |

717 |
Rank-Deficient and Discrete Ill-Posed Problems
- Hansen
- 1998
(Show Context)
Citation Context ...arameter λ>0providesatradeoff between fidelity to the measurements and noise sensitivity. Common choices for L are the identity or a matrix approximating the first or second order derivative operator =-=[19, 21, 17]-=-. Another regularization method that has attracted a revived interest and considerable amount of attention in the signal processing literature is l1 regularization in which one seeks to find the solut... |

520 | Gradient projection for sparse reconstruction: application to compressed sensing and other inverse problems
- Figueiredo, Novak, et al.
- 2007
(Show Context)
Citation Context ...x − b‖ 2 + λ‖x‖1}, (2) This research is partially supported by the Israel Science Foundation, ISF grant #489-06. where ‖x‖1 stands for the sum of the absolute values of the components of x, see e.g., =-=[1, 2, 3, 4]-=-. The presence of the l1 term is used to induce sparsity in the optimal solution, of (2), see e.g., [5, 6]. In image deblurring for example, A is often chosen as A = RW where R is the blurring matrix ... |

498 | Signal recovery by proximal forward-backward splitting,”Multiscale Modeling
- Combettes, Wajs
- 2005
(Show Context)
Citation Context ...ooth part followed by a shrinkage operation. The convergence analysis of ISTA has been well studied in the literature under various contexts and frameworks, including various modifications, see e.g., =-=[1, 3, 9]-=- and references therein. The advantage of ISTA is in its simplicity. However, ISTA has also been recognized as a slow method. Traditionally, the convergence analysis of iterative algorithms focuses on... |

388 | Gradient methods for minimizing composite objective function
- Nesterov
(Show Context)
Citation Context ...us linear inverse problems. However, for both of these two recent methods [11, 12], global rate of convergence have not been established. More recently, a different speed-up of ISTA was introduced in =-=[13]-=- for problem (5)with a O(1/k 2 ) global rate of convergence. Although the method in [13] and FISTA shares the same O(1/k 2 ) complexity result, the two schemes are very much different. In particular, ... |

362 | Sparse reconstruction by separable approximation
- Wright, Nowak, et al.
(Show Context)
Citation Context ...intheclassofiterative shrinkage-thresholding algorithms (ISTA), where each iteration involves matrix-vector multiplication involving A and AT followed by a shrinkage/soft-threshold step; 1 see, e.g., =-=[7, 15, 10, 34, 18, 35]-=-. Specifically, the general step of ISTA is ( (1.4) xk+1 = T λt xk − 2tA T (Axk − b) ) , where t is an appropriate stepsize and T α : R n → R n is the shrinkage operator defined by (1.5) T α(x)i =(|xi... |

347 | An EM algorithm for wavelet-based image restoration
- Figueiredo, Nowak
- 2003
(Show Context)
Citation Context ...x − b‖ 2 + λ‖x‖1}, (2) This research is partially supported by the Israel Science Foundation, ISF grant #489-06. where ‖x‖1 stands for the sum of the absolute values of the components of x, see e.g., =-=[1, 2, 3, 4]-=-. The presence of the l1 term is used to induce sparsity in the optimal solution, of (2), see e.g., [5, 6]. In image deblurring for example, A is often chosen as A = RW where R is the blurring matrix ... |

282 |
A method for solving a convex programming problem with convergence rate 1/k2
- Nesterov
- 1983
(Show Context)
Citation Context ...d rate of O(1/k2 ). We recall that when g(x) ≡ 0 the general model (6) consists of minimizing a smooth convex function and ISTA reduces to the gradient method. In this smooth setting it was proven in =-=[14]-=- that there exists a gradient method with an O(1/k2 ) complexity result which is an ”optimal” first-order method for smooth problems. The remarkable fact is that the method developed in [14] does not ... |

275 | Regularization tools: A matlab package for analysis and solution of discrete ill-posed problems. Numerical algorithms
- Hansen
- 1994
(Show Context)
Citation Context ...e. 5.2. Example 2: A simple test image. In this example we will further show the benefit of FISTA. The 256 × 256 simple test image was extracted from the function blur from the regularization toolbox =-=[20]-=-. The image then undergoes the same blurring and noise-adding procedure described in the previous example. The original and observed images are given in Figure 3. The algorithms were tested with regul... |

256 |
A.: Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications
- Ben-Tal, Nemirovski
- 2001
(Show Context)
Citation Context ...age processing applications correspond to sharp edges. The convex optimization problem (1.3) can be cast as a second order cone programming problem and thus could be solved via interior point methods =-=[1]-=-. However, in most applications, e.g., in image deblurring, the problem is not only large scale (can reach millions of decision variables) but also involves dense matrix data, which often precludes th... |

255 | Nonlinear wavelet image processing: Variational problems, compression, and noise removal through wavelet shrinkage
- Chambolle, DeVore, et al.
- 1998
(Show Context)
Citation Context ...intheclassofiterative shrinkage-thresholding algorithms (ISTA), where each iteration involves matrix-vector multiplication involving A and AT followed by a shrinkage/soft-threshold step; 1 see, e.g., =-=[7, 15, 10, 34, 18, 35]-=-. Specifically, the general step of ISTA is ( (1.4) xk+1 = T λt xk − 2tA T (Axk − b) ) , where t is an appropriate stepsize and T α : R n → R n is the shrinkage operator defined by (1.5) T α(x)i =(|xi... |

255 |
Introduction to Optimization
- Polyak
- 1987
(Show Context)
Citation Context ...t methods for solving (U) is the gradient algorithm which generates a sequence {xk} via (2.1) x0 ∈ R n , xk = xk−1 − tk∇f(xk−1), where tk > 0 is a suitable stepsize. It is very well known (see, e.g., =-=[31, 2]-=-) that the gradient iteration (2.1) can be viewed as a proximal regularization [24] of the linearized function f at xk−1, and written equivalently as xk =argmin x { f(xk−1)+〈x − xk−1, ∇f(xk−1)〉 + 1 ‖x... |

216 |
The use of the l-curve in the regularization of discrete ill-posed problems
- Hansen, O’Leary
- 1993
(Show Context)
Citation Context ...arameter λ>0providesatradeoff between fidelity to the measurements and noise sensitivity. Common choices for L are the identity or a matrix approximating the first or second order derivative operator =-=[19, 21, 17]-=-. Another regularization method that has attracted a revived interest and considerable amount of attention in the signal processing literature is l1 regularization in which one seeks to find the solut... |

214 | Analysis of multiresolution image denoising schemes using generalized-gaussian and complexity priors, in
- Moulin, Liu
- 1999
(Show Context)
Citation Context ...nsional minimization problem, e.g., with g(·) beingthepth power of the lp norm of x, withp≥1. For such computation and other separable regularizers, see, for instance, the general formulas derived in =-=[25, 7, 9]-=-. A possible drawback of this basic scheme is that the Lipschitz constant L(f) is not always known or computable. For instance, the Lipschitz constant in the l1 regularization problem (1.3) depends on... |

179 | A new TwIST: Two-step iterative shrinkage/thresholding algorithms for image restoration
- Bioucas-Dias, Figueiredo
- 2007
(Show Context)
Citation Context ...1/k 2 ), k being the iteration number. Moreover, FISTA shares the same simplicity and computational demand of ISTA. Recently, several accelerations of ISTA have been proposed in the literature, e.g., =-=[11, 12]-=-. The recent scheme of [11], called TwIST, uses at each step the last two iterations and is also based on a ”gradient” type step followed by a shrinkage operation. Within another line of analysis, the... |

106 |
Régularisation d’inéquations variationnelles par approximations successives
- Martinet
- 1970
(Show Context)
Citation Context ...a (2.1) x0 ∈ R n , xk = xk−1 − tk∇f(xk−1), where tk > 0 is a suitable stepsize. It is very well known (see, e.g., [31, 2]) that the gradient iteration (2.1) can be viewed as a proximal regularization =-=[24]-=- of the linearized function f at xk−1, and written equivalently as xk =argmin x { f(xk−1)+〈x − xk−1, ∇f(xk−1)〉 + 1 ‖x − xk−1‖ 2 } . 2tk Adopting this same basic gradient idea to the nonsmooth l1 regul... |

84 | Tikhonov regularization and total least squares
- Golub, Hansen, et al.
- 1999
(Show Context)
Citation Context ...arameter λ>0providesatradeoff between fidelity to the measurements and noise sensitivity. Common choices for L are the identity or a matrix approximating the first or second order derivative operator =-=[19, 21, 17]-=-. Another regularization method that has attracted a revived interest and considerable amount of attention in the signal processing literature is l1 regularization in which one seeks to find the solut... |

68 | Zibulevsky M. Coordinate and subspace optimization methods for linear least squares with non-quadratic regularization
- Elad, Matalon
(Show Context)
Citation Context ...1/k 2 ), k being the iteration number. Moreover, FISTA shares the same simplicity and computational demand of ISTA. Recently, several accelerations of ISTA have been proposed in the literature, e.g., =-=[11, 12]-=-. The recent scheme of [11], called TwIST, uses at each step the last two iterations and is also based on a ”gradient” type step followed by a shrinkage operation. Within another line of analysis, the... |

53 |
Ergodic convergence to a zero of the sum of monotone operators in Hilbert space
- Passty
- 1979
(Show Context)
Citation Context ...shrinkage operator defined by (1.5) T α(x)i =(|xi|−α)+sgn (xi). In the optimization literature, this algorithm can be traced back to the proximal forwardbackward iterative scheme introduced in [6] and=-=[30]-=- within the general framework of splitting methods; see [14, Chapter 12] and the references therein for a very good introduction to this approach, including convergence results. Another interesting re... |

49 |
A fixed-point continuation method for l1 regularized minimization with applications to compressed sensing,” Manuscript 2007 [Online]. Available: http://www.dsp.ece.rice.edu/cs
- Hale, Yin, et al.
(Show Context)
Citation Context ...intheclassofiterative shrinkage-thresholding algorithms (ISTA), where each iteration involves matrix-vector multiplication involving A and AT followed by a shrinkage/soft-threshold step; 1 see, e.g., =-=[7, 15, 10, 34, 18, 35]-=-. Specifically, the general step of ISTA is ( (1.4) xk+1 = T λt xk − 2tA T (Axk − b) ) , where t is an appropriate stepsize and T α : R n → R n is the shrinkage operator defined by (1.5) T α(x)i =(|xi... |

18 |
Problem complexity and method efficiency
- Nemirovski, Yudin
- 1983
(Show Context)
Citation Context ...nd which was introduced and developed by Nesterov in 1983 [27] for minimizing a smooth convex function, and proven to be an “optimal” first order (gradient) method in the sense of complexity analysis =-=[26]-=-. Here, the problem under consideration is convex but nonsmooth, due to the l1 term. Despite the presence of a nonsmooth regularizer in the objective function, we prove that we can construct a faster ... |

14 |
A fast iterative thresholding algorithm for wavelet-regularized deconvolution
- Vonesch, Unser
(Show Context)
Citation Context ...elatively cheap matrix-vector multiplications involving A and AT . One of the most popular methods to solve problem (2) is in the class of iterative shrinkage/thresholding algorithms (ISTA), see e.g. =-=[7, 1, 3, 8]-=-. Specifically, the general step of ISTA is ( xk − 2tkA T (Axk − b) ) (3) xk+1 = T λtk where tk is an appropriate stepsize and T α : R n → R n is the shrinkage operator defined by T α(x)i =(|xi|−α)+sg... |

14 |
On the weak convergence of an ergodic iteration for the solution of variational inequalities for monotone operators in hilbert space
- Bruck
- 1977
(Show Context)
Citation Context ...is the shrinkage operator defined by (1.5) T α(x)i =(|xi|−α)+sgn (xi). In the optimization literature, this algorithm can be traced back to the proximal forwardbackward iterative scheme introduced in =-=[6]-=- and[30] within the general framework of splitting methods; see [14, Chapter 12] and the references therein for a very good introduction to this approach, including convergence results. Another intere... |

4 |
Iterative soft-thresholding converges linearly. submitted
- Bredies, Lorenz
- 2007
(Show Context)
Citation Context ...ditions under which the sequence {xk} converges to a solution of (1.3). The advantage of ISTA is in its simplicity. However, ISTA has also been recognized as a slow method. The very recent manuscript =-=[5]-=- provides further rigorous grounds to that claim by proving that under some assumptions on the operator A the sequence {xk} produced by ISTA shares an asymptotic rate of convergence that can be very s... |

1 |
Astronomical image representation by the curevelet transform, Astron
- Starck, Donoho, et al.
(Show Context)
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