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Iterative Algorithm
Linear Inverse Problem
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An iterative thresholding algorithm for linear inverse problems with a sparsity constraint
An iterative thresholding algorithm for linear inverse problems with a sparsity constraint,10.1002/cpa.20042,Communications on Pure and Applied Mathem
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An iterative thresholding algorithm for linear inverse problems with a sparsity constraint
(
Citations: 593
)
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Ingrid Daubechies
,
Michel Defrise
,
Christine De Mol
We consider linear inverse problems where the solution is assumed to have a sparse expansion on an arbitrary preassigned orthonormal basis. We prove that replacing the usual quadratic regularizing penalties by weighted l^ppenalties on the coefficients of such expansions, with 1 < or = p < or =2, still regularizes the problem. If p < 2, regularized solutions of such l^ppenalized problems will have sparser expansions, with respect to the basis under consideration. To compute the corresponding regularized solutions we propose an
iterative algorithm
that amounts to a Landweber iteration with thresholding (or nonlinear shrinkage) applied at each iteration step. We prove that this algorithm converges in norm. We also review some potential applications of this method.
Journal:
Communications on Pure and Applied Mathematics  COMMUN PURE APPL MATH
, vol. 57, no. 11, pp. 14131457, 2004
DOI:
10.1002/cpa.20042
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Citation Context
(374)
...(
2004
), Genkin et al...
Yi Yang
,
et al.
An Efficient Algorithm for Computing The HHSVM and Its Generalizations
...gif"/> , the shrinkage operator is defined as The convergence of the iterative shrinkage operation <0003" reftype="dispformula">(3) has been studied in
7
9
13
under suitable conditions on λ and the Hessian ∇
^{2}
f
...
Zaiwen Wen
,
et al.
On the convergence of an activeset method for ℓ1 minimization
...For a proof, observe that the SBITE algorithm corresponds to the shooting algorithm (Fu,
1998
) for lasso, the BCR algorithm (Sardy et al,
2000
) for basis pursuit (Chen et al,
1999
) and to the iterative algorithm of Yuan and Lin (
2006
) to minimize penalized least squares problems of the form Note that the last three estimators above (4, 5, 6) converge after one iteration, and (<0010" reftype="dispformula">10) can also be solved by another class of iterative algorithms developed for inverse problems (Daubechies et al,
2004
)...
Sylvain Sardy
.
Smooth blockwise iterative thresholding: a smooth fixed point estimato...
...
^{[}
^{14}
^{]}
was adopted by Yu and Wang in 2010...
Qian Xue
,
et al.
ITERATIVE FILTERING RECONSTRUCTION ALGORITHM FOR GAMMA COMPUTERIZED TO...
...On the other hand, inspired by iterative thresholding ideas [
24
, 30, 20], we have now available a great number of rstorder methods, see [31, 9, 34, 35] and the many earlier references therein, which may be faster but not necessarily accurate...
Stephen Becker
,
et al.
NESTA: A Fast and Accurate FirstOrder Method for Sparse Recovery
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F. ABRAMOVICH
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Journal:
Biometrika
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Nonlinear wavelet image processing: variational problems, compression, and noise removal through wavelet shrinkage
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Antonin Chambolle
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Journal:
IEEE Transactions on Image Processing
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Wavelet methods in Numerical Analysis
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A. Cohen
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Adaptive Wavelet Galerkin Methods for Linear Inverse Problems
(
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Albert Cohen
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Marc Hoffmann
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Markus Reiss
Journal:
Siam Journal on Numerical Analysis  SIAM J NUMER ANAL
, vol. 42, no. 4, pp. 14791501, 2004
A modified expectation maximization algorithm for penalized likelihood estimation in emission tomography
(
Citations: 158
)
Alvaro R. De Pierro
Journal:
IEEE Transactions on Medical Imaging  TMI
, vol. 14, no. 1, pp. 132137, 1995
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Citations
(593)
An Efficient Algorithm for Computing The HHSVM and Its Generalizations
Yi Yang
,
Hui Zou
Journal:
Journal of Computational and Graphical Statistics  J COMPUT GRAPH STAT
, vol. justaccep, no. justaccep, 2012
A coordinate majorization descent algorithm for ℓ1 penalized learning
Yi Yang
,
Hui Zou
Journal:
Journal of Statistical Computation and Simulation  J STAT COMPUT SIM
, vol. aheadofp, no. aheadofp, pp. 112, 2012
On the convergence of an activeset method for ℓ1 minimization
Zaiwen Wen
,
Wotao Yin
,
Hongchao Zhang
,
Donald Goldfarb
Journal:
Optimization Methods & Software  OPTIM METHOD SOFTW
, vol. 27, no. 6, pp. 11271146, 2012
Smooth blockwise iterative thresholding: a smooth fixed point estimator based on the likelihood’s block gradient
Sylvain Sardy
Journal:
Journal of The American Statistical Association  J AMER STATIST ASSN
, vol. justaccep, no. justaccep, 2012
ITERATIVE FILTERING RECONSTRUCTION ALGORITHM FOR GAMMA COMPUTERIZED TOMOGRAPHY: APPLICATION TO TWOPHASE FLOW VISUALIZATION
Qian Xue
,
Huaxiang Wang
,
Kuihong Hao
,
Shan Xin
Journal:
Instrumentation Science & Technology  INSTRUM SCI TECHNOL
, vol. 40, no. 1, pp. 116, 2012