Probing the pareto frontier for basis pursuit solutions

SIAM Journal on Scientific Computing

Volumn 31, Issue 2, 2008, Pages 890-912

  Van Ewout Berg, D E N ^a   Friedlander, Michael P ^a  

^a UNIVERSITY OF BRITISH COLUMBIA   (Canada)

Author keywords

Basis pursuit; Convex program; Duality; Finding; Newton's method; Norm regularization; One ; Projected gradient; Root ; Sparse solutions

Indexed keywords

CONVEX OPTIMIZATION; CURVE FITTING; ECONOMIC AND SOCIAL EFFECTS; LEAST SQUARES APPROXIMATIONS; NUMERICAL METHODS;

BASIS PURSUITS; CONVEX PROGRAMS; DUALITY; FINDING; NEWTON'S METHODS; NORM REGULARIZATION; ONE-; PROJECTED GRADIENT; REGULARISATION; ROOT-; SPARSE SOLUTIONS;

NEWTON-RAPHSON METHOD;

EID: 65649137930     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/080714488     Document Type: Article

References (48)

1. J. BARZILAI AND J. M. BORWEIN, Two-point step size gradient methods, IMA J. Numer. Anal., 8 (1988), pp. 141-148.
2. E. VAN DEN BERG, M. P. FRIEDLANDER, G. HENNENFENT, F. HERRMANN, R. SAAB, AND O. YLLMAZ, Algorithm 890: Sparco: A testing framework for sparse reconstruction,ACM Trans. Math. Software, to appear.
3. D. P. BERTSEKAS, Nonlinear Programming, 2nd ed., Athena Scientific, Belmont, MA, 1999.
4. D. P. BERTSEKAS, Convex Analysis and Optimization, Athena Scientific, Belmont, MA, 2003.
5. E. G. BIRGIN, J. M. MARTÍ;NEZ, AND M. RAYDAN, Nonmonotone spectral projected gradient methods on convex sets, SIAM J. Optim., 10 (2000), pp. 1196-1211.
6. E. G. BIRGIN, J. M. MARTINEZ, AND M. RAYDAN, Inexact spectral projected gradient methods on convex sets, IMA J. Numer. Anal., 23 (2003), pp. 1196-1211.
7. Å. BJÖRCK, Numerical Methods for Least Squares Problems, SIAM, Philadelphia, 1996.
8. S. BOYD AND l. VANDENBERGHE, Convex Optimization, Cambridge University Press, Cambridge, UK, 2004.
9. 1-magic, http://www.l1-magic.org/
10. E. J. CANDÈS, J. ROMBERG, AND T. TAO, Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information, IEEE Trans. Inform. Theory, 52 (2006), pp. 489-509.
11. E. J. CANDES, J. ROMBERG, AND T. TAO, Stable signal recovery from incomplete and inaccurate measurements, Comm. Pure Appl. Math., 59 (2006), pp. 1207-1223.
12. E. J. CANDÈS AND T. TAO, Near-optimal signal recovery from random projections: Universal encoding strategies?, IEEE Trans. Inform. Theory, 52 (2006), pp. 5406-5425.
13. A. CHAMBOLLE, R. De VORE, N.-Y. LEE, AND B. LUCIER, Nonlinear wavelet image processing: Variational problems, compression, and noise removal through wavelet shrinkage, IEEE Trans. Image Process., 7 (1998), pp. 319-335.
14. S. S. CHEN, D. L. DONOHO, AND M. A. SAUNDERS, Atomic decomposition by basis pursuit, SIAM J. Sci. Comput., 20 (1998), pp. 33-61.
15. S. S. CHEN, D. L. DONOHO, AND M. A. SAUNDERS, Atomic decomposition by basis pursuit, SIAM Rev., 43 (2001), pp. 129-159.
16. ILOG CPLEX. Mathematical Programming System, http: //www.cplex.com.
17. Y. DAI, W. W. Hager, K. SCHITTKOWSKI, AND H. ZHANG, The cyclic Barzilai-Borwein method for unconstrained optimization, IMA J. Numer. Anal., 7 (2006), pp. 604-627.
18. Y.-H. DAI AND R. FLETCHER, Projected Barzilai-Borwein methods for large-scale box-constrained quadratic programming, Numer. Math., 100 (2005), pp. 21-47.
19. I. DAS AND J. E. DENNIS, A closer look at drawbacks of minimizing weighted sums of objectives for Pareto set generation in multicriteria optimization problems, Struct. Optim., 14 (1997), pp. 63-69.
20. I. DAUBECHIES, M. FORNASIER, AND I. LORIS, Accelereated projected gradient method for linear inverse problems with sparsity constraints, J. Fourier Anal. Appl., 2007, to appear.
21. A. DAX, On regularized least norm problems, SIAM J. Optim., 2 (1992), pp. 602-618.
22. R. S. DEMBO, S. C. EISENSTAT, AND T. STEIHAUG, Inexact Newton methods,SIAM J. Numer. Anal., 19 (1982), pp. 400-408.
23. D. L. DONOHO, Compressed sensing, IEEE Trans. Inform. Theory, 52 (2006), pp. 1289-1306.
24. 1-norm solution is also the sparsest solution, Comm. Pure Appl. Math., 59 (2006), pp. 797-829.
25. D. L. DONOHO AND Y. TSAIG, Fast Solution of l1-norm Minimization Problems When the Solution May Be Sparse, http://www.stanford.edu/ t saig/research.html.
26. B. EFRON, T. HASTIE, I. JOHNSTONE, AND R. TIBSHIRANI, Least angle regression, Ann. Statist., 32 (2004), pp. 407-499.
27. M. FIGUEIREDO, R. NOWAK, AND S. J. WRIGHT, Gradient Projection for Sparse Reconstruction: Application to Compressed Sensing and Other Inverse Problems, in IEEE J. Selected Topics in Signal Process., IEEE Press, Piscataway, NJ, 2007, pp. 586-597.
28. M. P. FRIEDLANDER AND M. A. SAUNDERS, Discussion: The Dantzig selector: Statistical estimation when p is much larger than n, Ann. Statist., 35 (2007), pp. 2385-2391.
29. P. C. HANSEN, Rank-Deficient and Discrete Ill-Posed Problems, SIAM, Philadelphia, 1998.
30. P. C. HANSEN AND D. P. O'LEARY, The use of the L-curve in the regularization of discrete ill-posed problems, SIAM J. Sci. Comput., 14 (1993), pp. 1487-1503.
31. G. HENNENFENT AND F. J. HERRMANN, Sparseness-constrained data continuation with frames: Applications to missing traces and aliased signals in 2/3-D, in Proceedings of the SEG International Exposition and 75th Annual Meeting, Houston, TX, 2005.
32. G. HENNENFENT AND F. J. HERRMANN, Simply denoise: Wavefield reconstruction via jittered undersampling, Geophys., 73 (2008), pp. V19-V28.
33. F. HERRMANN AND S. ROSS-ROSS, SlimPy: A Python interface to Unix-like pipe based linear operators, http://slim.eos.ubc.ca/SLIMpy.
34. S.-J. KLM, K. KOH, M. LUSTIG, S. BOYD, AND D. GORINEVSKY, An interior-point method for large scale L\-regularized least squares, IEEE J. Trans. Sel. Top. Signal Process., 1 (2007), pp. 606-617.
35. S. LEYFFER, A note on multiobjective optimization and complementarity constraints,Preprint ANL/MCS-P1290-0905, Mathematics and Computer Science Division, Argonne National Laboratory, Illinois, Argonne, IL, 2005.
36. M. LUSTIG, D. L. DONOHO, AND J. M. PAULY, Sparse MRI: The application of compressed sensing for rapid MR imaging, Magn. Resonance Med., 58 (2007), pp. 1182-1195.
37. M. LUSTIG, D. L. DONOHO, J. M. SANTOS, AND J. M. PAULY, Compressed sensing MRI, IEEE Signal Process. Mag., 25 (2007), pp. 72-82.
38. MATH WORKS, MATLAB User's Guide, MathWorks, Natick, MA, 1992.
39. Mathematical Programming System, http://www.mosek.com.
40. J. NOCEDAL AND S. J. WRIGHT, Numerical Optimization, 2nd ed., Springer, New York, 2006.
41. M. R. OSBORNE, B. PRESNELL, AND B. A. TURLACH, A new approach to variable selection in least squares problems, IMA J. Numer. Anal., 20 (200 0), pp. 389-403.
42. M. R. OSBORNE, B. PRESNELL, AND B. A. TURLACH, On the LASSO and its dual, J. Comput. Graph. Statist., 9 (2000), pp. 319-337.
43. R. T. ROCKAFELLAR, Convex Analysis, Princeton University Press, Princeton, NJ, 1970.
44. M. A. SAUNDERS, PDCO. MATLAB Software for Convex Optimization, http://www.stanford.edu/group/SOL/software/pdco.html.
45. J. F. STURM, Using SeDuMi 1.02,A MATLAB Toolbox for Optimization Over Symmetric Cones (Updated for Version 1.05), Technical report, Depart ment of Econometrics, Tilburg University, Tilburg, The Netherlands, 1998.
46. R. TIBSHIRANI, Regression shrinkage and selection via the Lasso,J. Roy.Statist.Soc.Ser. B., 58 (1996), pp. 267-288.
47. A. N. TIKHONOV AND V. Y. ARSENIN, Solutions of Ill-Posed Problems, V.H.Winston and Sons, Washington, DC, 1977 (in Russian).
48. J. A. TROPP, Just relax: Convex programming methods for identifying sparse signals in noise, IEEE Trans. Inform. Theory, 52 (2006), pp. 1030-105 1.