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 inJournal of scientific computing Vol. 46; no. 1; pp. 20 - 46
Main Authors Zhang, Xiaoqun, Burger, Martin, Osher, Stanley
Format Journal Article
LanguageEnglish
Published Boston Springer US 01.01.2011
Springer Nature B.V
Subjects
Online AccessGet full text
ISSN0885-7474
1573-7691
1573-7691
DOI10.1007/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
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  organization: Department of Mathematics, UCLA
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Issue 1
Keywords Inexact Uzawa methods
Bregman iteration
minimization
Proximal point iteration
Saddle point
Language English
License cc-by-nc
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References FigueiredoM.A.T.NowakR.D.WrightS.J.Gradient projection for sparse reconstruction: Application to compressed sensing and other inverse problemsIEEE J. Sel. Top. Signal Proces.20071458659710.1109/JSTSP.2007.910281
GlowinskiR.Le TallecP.Augmented Lagrangian and Operator Splitting Methods in Nonlinear Mechanics1989PhiladelphiaSIAM0698.73001
Guo, X., Li, F., Michael, K.Ng.: A fast L1-TV algorithm for image restoration. ICM Technical Report 08-13, Hong Kong Baptist University, November 2008
CandèsE.J.RombergJ.TaoT.Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency informationIEEE Trans. Inf. Theory200652248950910.1109/TIT.2005.862083
RudinL.I.OsherS.FatemiE.Nonlinear total variation based noise removal algorithmsPhysica D1992602592680780.49028
Yin, W.: Analysis and generalizations of the linearized Bregman method. 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. Optim., 3 (1993)
KimS.KohK.LustigM.BoydS.GorinevskyD.An interior-point method for large-scale l1-regularized least squaresIEEE J. Sel. Top. Signal Proces.20071460661710.1109/JSTSP.2007.910971
Yang, J., Zhang, Y., Yin, W.: An efficient TV-L1 algorithm for deblurring multichannel images corrupted by impulsive noise. SIAM J. Sci. Comput. 31(4) (2009)
SetzerS.MorkenK.LysakerM.LieK.-A.TaiX.-C.Split Bregman algorithm, Douglas-Rachford splitting and frame shrinkageScale Space and Variational Methods in Computer Vision2009BerlinSpringer46447610.1007/978-3-642-02256-2_39
Beck, A., Teboulle, M.: A fast iterative shrinkage-thresholding algorithm with application to wavelet-based image deblurring. In: IEEE International Conference on Acoustics, Speech, and Signal Processing, pp. 693–696 (2009)
DonohoD.L.Compressed sensingIEEE. Trans. Inf. 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. Math.20045711141314571077.6505510.1002/cpa.20042
Powell, M.J.D.: A method for nonlinear constraints in minimization problems. Optimization (1972)
DouglasJ.RachfordH.H.On the numerical solutions of heat conduction problems in two and three space variablesTrans. Am. Math. Soc.1976197116
BregmanL.The relaxation method of finding the common points of convex sets and its application to the solution of problems in convex programmingUSSR Comput. Math. Math Phys.1967720021710.1016/0041-5553(67)90040-7
CensorY.LentA.An iterative row-action method for interval convex programmingJ. Optim. Theory Appl.19813433213530431.4904210.1007/BF00934676628201
GabayD.Applications of the method of multipliers to variational inequalitiesAugmented Lagrangian Methods: Applications to the Solution of Boundary-Value Problems1983AmsterdamNorth-Holland
OsherS.BurgerM.GoldfarbD.XuJ.YinW.An iterative regularization method for total variation based image restorationMultiscale Model. Simul.2005424604891090.9400310.1137/0406054122162864
Esser, E.: Applications of Lagrangian-based alternating direction methods and connections to split Bregman. CAM Report 09-31, UCLA, April 2009
Goldstein, T., Bresson, X., Osher, S.: Geometric applications of the split Bregman method: segmentation and surface reconstruction. UCLA CAM Report, 09-06, February (2009)
Ma, S., Goldfarb, D., Chen, L.: Fixed point and Bregman iterative methods for matrix rank minimization. Technical Report, Columbia University (2008)
HaleE.T.YinW.ZhangY.A fixed-point continuation method for l1-regularized minimization with applications to compressed sensingSIAM J. Optim.200819110711301180.6507610.1137/0706989202460734
RockafellarR.T.Augmented Lagrangians and applications of the proximal point algorithm in convex programmingMath. Oper. Res.195682421439
Borghi, A., Darbon, J., Peyronnet, S., Chan, T.F., Osher, S.: A simple compressive sensing algorithm for parallel many-core architectures. CAM Report 08-64, UCLA (2008)
ChambolleA.An algorithm for total variation minimization and applicationsJ. Math. Imag. Vis.200420899710.1023/B:JMIV.0000011321.19549.882049783
OsherS.MaoY.DongB.YinW.Fast linearized Bregman iteration for compressive sensing and sparse denoisingCommun. Math. Sci.201081931111190.490402655902
CaiJ.-F.OsherS.ShenZ.Linearized Bregman iterations for frame-based image deblurringSIAM J. Imag. Sci.2009212262521175.9401010.1137/0807333712486529
ChenS.S.DonohoDavid L.SaundersM.A.Atomic decomposition by basis pursuitSIAM J. Sci. Comput.1998201336110.1137/S10648275963040101639094
CaiJ.-F.CandèsE.J.ShenZ.A singular value thresholding algorithm for matrix completionSIAM J. Optim.2010204195619821201.9015510.1137/0807389702600248
MoreauJ.-J.Fonctions convexes duales et points proximaux dans un espace hilbertienC. R. Acad. Sci. Paris Ser. 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)
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References_xml – reference: HestenesM.R.Multiplier and gradient methodsJ. Optim. Theory Appl.196943033200174.2070510.1007/BF00927673271809
– reference: SetzerS.MorkenK.LysakerM.LieK.-A.TaiX.-C.Split Bregman algorithm, Douglas-Rachford splitting and frame shrinkageScale Space and Variational Methods in Computer Vision2009BerlinSpringer46447610.1007/978-3-642-02256-2_39
– reference: KimS.KohK.LustigM.BoydS.GorinevskyD.An interior-point method for large-scale l1-regularized least squaresIEEE J. Sel. Top. Signal Proces.20071460661710.1109/JSTSP.2007.910971
– reference: Cai, J.-F., Osher, S., Shen, Z.: Split Bregman method and frame based image restoration. CAM Report 09-28, UCLA, March 2009
– reference: Yin, W.: Analysis and generalizations of the linearized Bregman method. CAAM Report, Rice University (2009)
– reference: Esser, E.: Applications of Lagrangian-based alternating direction methods and connections to split Bregman. CAM Report 09-31, UCLA, April 2009
– reference: ArrowK.J.HurwiczL.UzawaH.Studies in Linear and Non-Linear Programming1958StanfordStanford University Press0091.16002
– reference: FigueiredoM.A.T.NowakR.D.WrightS.J.Gradient projection for sparse reconstruction: Application to compressed sensing and other inverse problemsIEEE J. Sel. Top. Signal Proces.20071458659710.1109/JSTSP.2007.910281
– reference: Guo, X., Li, F., Michael, K.Ng.: A fast L1-TV algorithm for image restoration. ICM Technical Report 08-13, Hong Kong Baptist University, November 2008
– reference: ChenS.S.DonohoDavid L.SaundersM.A.Atomic decomposition by basis pursuitSIAM J. Sci. Comput.1998201336110.1137/S10648275963040101639094
– reference: YinW.OsherS.GoldfarbD.DarbonJ.Bregman iterative algorithms for ℓ1 minimization with applications to compressed sensingSIAM J. Imag. Sci.200811431680525657310.1137/0707039832475828
– reference: GabayD.Applications of the method of multipliers to variational inequalitiesAugmented Lagrangian Methods: Applications to the Solution of Boundary-Value Problems1983AmsterdamNorth-Holland
– reference: Lemaréchal, C., Sagastizábal, C.: Practical aspects of the Moreau-Yosida regularization, I: Theoretical properties (1994)
– reference: Becker, S., Bobin, J., Candès, E.: Nesta: a fast and accurate first-order method for sparse recovery (2009)
– reference: EcksteinJ.BertsekasD.P.On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operatorsMath. Program.1992552933180765.9007310.1007/BF015812041168183
– reference: OsherS.BurgerM.GoldfarbD.XuJ.YinW.An iterative regularization method for total variation based image restorationMultiscale Model. Simul.2005424604891090.9400310.1137/0406054122162864
– reference: CensorY.LentA.An iterative row-action method for interval convex programmingJ. Optim. Theory Appl.19813433213530431.4904210.1007/BF00934676628201
– reference: AvinashC.MalcolmS.Principles of Computerized Tomographic Imaging2001PhiladelphiaSociety for Industrial and Applied Mathematics
– reference: DonohoD.L.Compressed sensingIEEE. Trans. Inf. Theory2006521289130610.1109/TIT.2006.8715822241189
– reference: 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
– reference: CandèsE.J.RombergJ.TaoT.Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency informationIEEE Trans. Inf. Theory200652248950910.1109/TIT.2005.862083
– reference: Goldstein, T., Bresson, X., Osher, S.: Geometric applications of the split Bregman method: segmentation and surface reconstruction. UCLA CAM Report, 09-06, February (2009)
– reference: Combettes, P.L., Wajs, V.R.: Signal recovery by proximal forward-backward splitting. Multiscale Model. Simul. 4(4) (2005)
– reference: Borghi, A., Darbon, J., Peyronnet, S., Chan, T.F., Osher, S.: A simple compressive sensing algorithm for parallel many-core architectures. CAM Report 08-64, UCLA (2008)
– reference: Powell, M.J.D.: A method for nonlinear constraints in minimization problems. Optimization (1972)
– reference: OsherS.MaoY.DongB.YinW.Fast linearized Bregman iteration for compressive sensing and sparse denoisingCommun. Math. Sci.201081931111190.490402655902
– reference: BregmanL.The relaxation method of finding the common points of convex sets and its application to the solution of problems in convex programmingUSSR Comput. Math. Math Phys.1967720021710.1016/0041-5553(67)90040-7
– reference: Beck, A., Teboulle, M.: A fast iterative shrinkage-thresholding algorithm with application to wavelet-based image deblurring. In: IEEE International Conference on Acoustics, Speech, and Signal Processing, pp. 693–696 (2009)
– reference: DaubechiesI.DefriseM.De MolC.An iterative thresholding algorithm for linear inverse problems with a sparsity constraintCommun. Pure Appl. Math.20045711141314571077.6505510.1002/cpa.20042
– reference: HaleE.T.YinW.ZhangY.A fixed-point continuation method for l1-regularized minimization with applications to compressed sensingSIAM J. Optim.200819110711301180.6507610.1137/0706989202460734
– reference: ChenG.TeboulleM.A proximal-based decomposition method for convex minimization problemsMath. Program.1994641811010823.9009710.1007/BF015825661274173
– reference: Candès, E.J., Recht, B.: Exact matrix completion via convex optimization. Found. Comput. Math. (2008, to appear)
– reference: GlowinskiR.Le TallecP.Augmented Lagrangian and Operator Splitting Methods in Nonlinear Mechanics1989PhiladelphiaSIAM0698.73001
– reference: DouglasJ.RachfordH.H.On the numerical solutions of heat conduction problems in two and three space variablesTrans. Am. Math. Soc.1976197116
– reference: Ma, S., Goldfarb, D., Chen, L.: Fixed point and Bregman iterative methods for matrix rank minimization. Technical Report, Columbia University (2008)
– reference: CaiJ.-F.CandèsE.J.ShenZ.A singular value thresholding algorithm for matrix completionSIAM J. Optim.2010204195619821201.9015510.1137/0807389702600248
– reference: RudinL.I.OsherS.FatemiE.Nonlinear total variation based noise removal algorithmsPhysica D1992602592680780.49028
– reference: Tai, X.-C., Wu, C.: Augmented Lagrangian method, dual methods and split Bregman iteration for ROF model. CAM Report 09-05, UCLA, January 2009
– reference: CaiJ.-F.OsherS.ShenZ.Linearized Bregman iterations for frame-based image deblurringSIAM J. Imag. Sci.2009212262521175.9401010.1137/0807333712486529
– reference: ChanT.F.GolubG.H.MuletP.A nonlinear primal-dual method for total variation-based image restorationMath. Oper. Res.19962192412521430131
– reference: Chen, G., Teboulle, M.: Convergence analysis of a proximal like minimization algorithm using Bregman functions. SIAM J. Optim., 3 (1993)
– reference: LiY.OsherS.Coordinate descent optimization for ℓ1 minimization with application to compressed sensing; a greedy algorithmInverse Probl. Imaging2009334875031188.9019610.3934/ipi.2009.3.4872557916
– reference: Zhu, M., Chan, T.: An efficient primal-dual hybrid gradient algorithm for total variation image restoration. CAM Report 08-34, UCLA, May 2008
– reference: CaiJ.-F.OsherS.ShenZ.Convergence of the linearized Bregman iteration for ℓ1-norm minimizationMath. Comput.200978212721361198.6510310.1090/S0025-5718-09-02242-X2521281
– reference: MoreauJ.-J.Fonctions convexes duales et points proximaux dans un espace hilbertienC. R. Acad. Sci. Paris Ser. A Math.1962255289728990118.10502144188
– reference: Goldstein, T., Osher, S.: The split Bregman method for l1 regularized problems. SIAM J. Imag. Sci., 2 (2009)
– reference: RockafellarR.T.Augmented Lagrangians and applications of the proximal point algorithm in convex programmingMath. Oper. Res.195682421439
– reference: ChambolleA.An algorithm for total variation minimization and applicationsJ. Math. Imag. Vis.200420899710.1023/B:JMIV.0000011321.19549.882049783
– reference: Yang, J., Zhang, Y., Yin, W.: An efficient TV-L1 algorithm for deblurring multichannel images corrupted by impulsive noise. SIAM J. Sci. Comput. 31(4) (2009)
– reference: BrambleJ.H.PasciakJ.E.VassilevA.T.Analysis of the inexact Uzawa algorithm for saddle point problemsSIAM J. Numer. Anal.1997343107210920873.6503110.1137/S00361429942733431451114
– reference: Darbon, J., Osher, S.: Fast discrete optimization for sparse approximations and deconvolutions. Preprint, UCLA (2007)
– reference: Zhang, X., Burger, M., Bresson, X., Osher, S.: Bregmanized nonlocal regularization for deconvolution and sparse reconstruction. CAM Report 09-03, UCLA, January 2009
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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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