A Unified Primal-Dual Algorithm Framework Based on Bregman Iteration
In this paper, we propose a unified primal-dual algorithm framework for two classes of problems that arise from various signal and image processing applications. We also show the connections to existing methods, in particular Bregman iteration (Osher et al., Multiscale Model. Simul. 4(2):460–489, 20...
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| Published in | Journal of scientific computing Vol. 46; no. 1; pp. 20 - 46 |
|---|---|
| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
Boston
Springer US
01.01.2011
Springer Nature B.V |
| Subjects | |
| Online Access | Get full text |
| ISSN | 0885-7474 1573-7691 1573-7691 |
| DOI | 10.1007/s10915-010-9408-8 |
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| Abstract | In this paper, we propose a unified primal-dual algorithm framework for two classes of problems that arise from various signal and image processing applications. We also show the connections to existing methods, in particular Bregman iteration (Osher et al., Multiscale Model. Simul. 4(2):460–489,
2005
) based methods, such as linearized Bregman (Osher et al., Commun. Math. Sci. 8(1):93–111,
2010
; Cai et al., SIAM J. Imag. Sci. 2(1):226–252,
2009
, CAM Report 09-28, UCLA, March
2009
; Yin, CAAM Report, Rice University,
2009
) and split Bregman (Goldstein and Osher, SIAM J. Imag. Sci., 2,
2009
). The convergence of the general algorithm framework is proved under mild assumptions. The applications to
ℓ
1
basis pursuit, TV−
L
2
minimization and matrix completion are demonstrated. Finally, the numerical examples show the algorithms proposed are easy to implement, efficient, stable and flexible enough to cover a wide variety of applications. |
|---|---|
| AbstractList | In this paper, we propose a unified primal-dual algorithm framework for two classes of problems that arise from various signal and image processing applications. We also show the connections to existing methods, in particular Bregman iteration (Osher et al., Multiscale Model. Simul. 4(2):460--489, 2005) based methods, such as linearized Bregman (Osher et al., Commun. Math. Sci. 8(1):93--111, 2010; Cai et al., SIAM J. Imag. Sci. 2(1):226--252, 2009, CAM Report 09-28, UCLA, March 2009; Yin, CAAM Report, Rice University, 2009) and split Bregman (Goldstein and Osher, SIAM J. Imag. Sci., 2, 2009). The convergence of the general algorithm framework is proved under mild assumptions. The applications to l 1 basis pursuit, TV-L 2 minimization and matrix completion are demonstrated. Finally, the numerical examples show the algorithms proposed are easy to implement, efficient, stable and flexible enough to cover a wide variety of applications. In this paper, we propose a unified primal-dual algorithm framework for two classes of problems that arise from various signal and image processing applications. We also show the connections to existing methods, in particular Bregman iteration (Osher et al., Multiscale Model. Simul. 4(2):460–489, 2005 ) based methods, such as linearized Bregman (Osher et al., Commun. Math. Sci. 8(1):93–111, 2010 ; Cai et al., SIAM J. Imag. Sci. 2(1):226–252, 2009 , CAM Report 09-28, UCLA, March 2009 ; Yin, CAAM Report, Rice University, 2009 ) and split Bregman (Goldstein and Osher, SIAM J. Imag. Sci., 2, 2009 ). The convergence of the general algorithm framework is proved under mild assumptions. The applications to ℓ 1 basis pursuit, TV− L 2 minimization and matrix completion are demonstrated. Finally, the numerical examples show the algorithms proposed are easy to implement, efficient, stable and flexible enough to cover a wide variety of applications. In this paper, we propose a unified primal-dual algorithm framework for two classes of problems that arise from various signal and image processing applications. We also show the connections to existing methods, in particular Bregman iteration (Osher et al., Multiscale Model. Simul. 4(2):460–489, 2005) based methods, such as linearized Bregman (Osher et al., Commun. Math. Sci. 8(1):93–111, 2010; Cai et al., SIAM J. Imag. Sci. 2(1):226–252, 2009, CAM Report 09-28, UCLA, March 2009; Yin, CAAM Report, Rice University, 2009) and split Bregman (Goldstein and Osher, SIAM J. Imag. Sci., 2, 2009). The convergence of the general algorithm framework is proved under mild assumptions. The applications to ℓ1 basis pursuit, TV−L2 minimization and matrix completion are demonstrated. Finally, the numerical examples show the algorithms proposed are easy to implement, efficient, stable and flexible enough to cover a wide variety of applications. |
| Author | Osher, Stanley Zhang, Xiaoqun Burger, Martin |
| Author_xml | – sequence: 1 givenname: Xiaoqun surname: Zhang fullname: Zhang, Xiaoqun email: xiaoqun.zhang@gmail.com organization: Department of Mathematics, Shanghai Jiao Tong University – sequence: 2 givenname: Martin surname: Burger fullname: Burger, Martin organization: Institute for Computational and Applied Mathematics, Westfälische Wilhelms-Universität – sequence: 3 givenname: Stanley surname: Osher fullname: Osher, Stanley organization: Department of Mathematics, UCLA |
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| Keywords | Inexact Uzawa methods Bregman iteration minimization Proximal point iteration Saddle point |
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CAAM Report, Rice University (2009) Becker, S., Bobin, J., Candès, E.: Nesta: a fast and accurate first-order method for sparse recovery (2009) CaiJ.-F.OsherS.ShenZ.Convergence of the linearized Bregman iteration for ℓ1-norm minimizationMath. Comput.200978212721361198.6510310.1090/S0025-5718-09-02242-X2521281 Goldstein, T., Osher, S.: The split Bregman method for l1 regularized problems. SIAM J. Imag. Sci., 2 (2009) HestenesM.R.Multiplier and gradient methodsJ. Optim. Theory Appl.196943033200174.2070510.1007/BF00927673271809 Candès, E.J., Recht, B.: Exact matrix completion via convex optimization. Found. Comput. Math. (2008, to appear) Tai, X.-C., Wu, C.: Augmented Lagrangian method, dual methods and split Bregman iteration for ROF model. CAM Report 09-05, UCLA, January 2009 Zhu, M., Chan, T.: An efficient primal-dual hybrid gradient algorithm for total variation image restoration. CAM Report 08-34, UCLA, May 2008 ArrowK.J.HurwiczL.UzawaH.Studies in Linear and Non-Linear Programming1958StanfordStanford University Press0091.16002 BrambleJ.H.PasciakJ.E.VassilevA.T.Analysis of the inexact Uzawa algorithm for saddle point problemsSIAM J. Numer. Anal.1997343107210920873.6503110.1137/S00361429942733431451114 LiY.OsherS.Coordinate descent optimization for ℓ1 minimization with application to compressed sensing; a greedy algorithmInverse Probl. Imaging2009334875031188.9019610.3934/ipi.2009.3.4872557916 Combettes, P.L., Wajs, V.R.: Signal recovery by proximal forward-backward splitting. Multiscale Model. Simul. 4(4) (2005) ChanT.F.GolubG.H.MuletP.A nonlinear primal-dual method for total variation-based image restorationMath. Oper. Res.19962192412521430131 Chen, G., Teboulle, M.: Convergence analysis of a proximal like minimization algorithm using Bregman functions. SIAM J. 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Theory2006521289130610.1109/TIT.2006.8715822241189 AvinashC.MalcolmS.Principles of Computerized Tomographic Imaging2001PhiladelphiaSociety for Industrial and Applied Mathematics Darbon, J., Osher, S.: Fast discrete optimization for sparse approximations and deconvolutions. Preprint, UCLA (2007) ChenG.TeboulleM.A proximal-based decomposition method for convex minimization problemsMath. Program.1994641811010823.9009710.1007/BF015825661274173 YinW.OsherS.GoldfarbD.DarbonJ.Bregman iterative algorithms for ℓ1 minimization with applications to compressed sensingSIAM J. Imag. Sci.200811431680525657310.1137/0707039832475828 Zhang, X., Burger, M., Bresson, X., Osher, S.: Bregmanized nonlocal regularization for deconvolution and sparse reconstruction. CAM Report 09-03, UCLA, January 2009 DaubechiesI.DefriseM.De MolC.An iterative thresholding algorithm for linear inverse problems with a sparsity constraintCommun. Pure Appl. 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A Math.1962255289728990118.10502144188 EcksteinJ.BertsekasD.P.On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operatorsMath. Program.1992552933180765.9007310.1007/BF015812041168183 Esser, E., Zhang, X., Chan, T.: A general framework for a class of first order primal-dual algorithms for TV minimization. CAM Report 09-67, UCLA, August 2009 Cai, J.-F., Osher, S., Shen, Z.: Split Bregman method and frame based image restoration. CAM Report 09-28, UCLA, March 2009 Lemaréchal, C., Sagastizábal, C.: Practical aspects of the Moreau-Yosida regularization, I: Theoretical properties (1994) 9408_CR47 9408_CR48 D.L. Donoho (9408_CR23) 2006; 52 9408_CR43 M.R. Hestenes (9408_CR35) 1969; 4 E.J. Candès (9408_CR13) 2006; 52 J. Douglas (9408_CR24) 1976; 1 Y. Li (9408_CR38) 2009; 3 K.J. Arrow (9408_CR1) 1958 S.S. Chen (9408_CR19) 1998; 20 W. Yin (9408_CR50) 2008; 1 S. Kim (9408_CR36) 2007; 1 9408_CR49 G. Chen (9408_CR18) 1994; 64 R.T. 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| Snippet | In this paper, we propose a unified primal-dual algorithm framework for two classes of problems that arise from various signal and image processing... |
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| SourceType | Open Access Repository Aggregation Database Enrichment Source Index Database Publisher |
| StartPage | 20 |
| SubjectTerms | Algorithms Computational Mathematics and Numerical Analysis Convergence Image processing Iterative methods Joints Mathematical and Computational Engineering Mathematical and Computational Physics Mathematical models Mathematics Mathematics and Statistics Optimization Rice Theoretical |
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| Title | A Unified Primal-Dual Algorithm Framework Based on Bregman Iteration |
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