Accelerated projected gradient method for linear inverse problems with sparsity constraints

Journal of Fourier Analysis and Applications

Volumn 14, Issue 5-6, 2008, Pages 764-792

  Daubechies, Ingrid ^a   Fornasier, Massimo ^a   Loris, Ignace ^b  

^a PRINCETON UNIVERSITY   (United States)

^b VRIJE UNIVERSITEIT BRUSSEL   (Belgium)

Author keywords

Linear inverse problems; Projected gradient method; Sparse recovery

Indexed keywords

EID: 57349127864     PISSN: 10695869     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00041-008-9039-8     Document Type: Article

References (46)

1. Y.I. Alber A.N. Iusem M.V. Solodov 1998 On the projected subgradient method for nonsmooth convex optimization in a Hilbert space Math. Program. Ser. A 81 1 23 35
2. Anthoine, S.: Different Wavelet-based Approaches for the Separation of Noisy and Blurred Mixtures of Components. Application to Astrophysical Data. Ph.D. Thesis, Princeton University, 2005
3. L.M. Brègman 1965 Finding the common point of convex sets by the method of successive projection Dokl. Akad. Nauk SSSR 162 487 490
4. 2 singularities Commun. Pure Appl. Math. 57 2 219 266
5. Candès, E.J., Romberg, J.: Practical signal recovery from random projections. In: Wavelet Applications in Signal and Image Processing XI. Proc. SPIE Conf., vol. 5914 (2004)
6. E.J. Candès T. Tao 2006 Near-optimal signal recovery from random projections: universal encoding strategies? IEEE Trans. Inf. Theory 52 12 5406 5425
7. E. Candès J. Romberg T. Tao 2006 Stable signal recovery from incomplete and inaccurate measurements Commun. Pure Appl. Math. 59 8 1207 1223
8. E.J. Candès J. Romberg T. Tao 2006 Exact signal reconstruction from highly incomplete frequency information IEEE Trans. Inf. Theory 52 2 489 509
9. C. Canuto K. Urban 2004 Adaptive optimization of convex functionals in Banach spaces SIAM J. Numer. Anal. 42 5 2043 2075
10. A. Chambolle R.A. DeVore N.-Y. Lee B.J. Lucier 1998 Nonlinear wavelet image processing: variational problems, compression, and noise removal through wavelet shrinkage IEEE Trans. Image Process. 7 3 319 335
11. Christensen, O.: An Introduction to Frames and Riesz Bases. Birkhäuser, Boston (2003)
12. Cohen, A.: Numerical Analysis of Wavelet Methods. Studies in Mathematics and Its Applications, vol. 32. North-Holland, Amsterdam (2003)
13. A. Cohen W. Dahmen R. DeVore 2001 Adaptive wavelet methods for elliptic operator equations-Convergence rates Math. Comput. 70 27 75
14. A. Cohen W. Dahmen R. DeVore 2002 Adaptive wavelet methods II: Beyond the elliptic case Found. Comput. Math. 2 3 203 245
15. A. Cohen M. Hoffmann M. Reiss 2004 Adaptive wavelet Galerkin methods for linear inverse problems SIAM J. Numer. Anal. 42 4 1479 1501
16. P.L. Combettes V.R. Wajs 2005 Signal recovery by proximal forward-backward splitting Multiscale Model. Simul. 4 4 1168 1200
17. Dahlke, S., Maass, P.: An outline of adaptive wavelet Galerkin methods for Tikhonov regularization of inverse parabolic problems. In: Hon, J.-C., et al. (eds.) Recent Development in Theories and Numerics. Proceedings of the International Conference on Inverse Problems, Hong Kong, China, January 9-12 2002, pp. 56-66. World Scientific, Singapore (2003)
18. S. Dahlke M. Fornasier T. Raasch 2007 Adaptive frame methods for elliptic operator equations Adv. Comput. Math. 27 1 27 63
19. S. Dahlke M. Fornasier T. Raasch R. Stevenson M. Werner 2007 Adaptive frame methods for elliptic operator equations: The steepest descent approach IMA J. Numer. Anal. 27 717 740
20. Daubechies, I.: Ten Lectures on Wavelets. SIAM, Philadelphia (1992)
21. I. Daubechies G. Teschke 2005 Variational image restoration by means of wavelets: Simultaneous decomposition, deblurring, and denoising Appl. Comput. Harmon. Anal. 19 1 1 16
22. I. Daubechies M. Defrise C. DeMol 2004 An iterative thresholding algorithm for linear inverse problems Commun. Pure Appl. Math. 57 11 1413 1457
23. D.L. Donoho 1992 Superresolution via sparsity constraints SIAM J. Math. Anal. 23 5 1309 1331
24. D.L. Donoho 1995 De-noising by soft-thresholding IEEE Trans. Inf. Theory 41 3 613 627
25. D.L. Donoho 1995 Nonlinear solution of linear inverse problems by wavelet-vaguelette decomposition Appl. Comput. Harmon. Anal. 2 2 101 126
26. D.L. Donoho 2006 Compressed sensing IEEE Trans. Inf. Theory 52 4 1289 1306
27. D.L. Donoho B.F. Logan 1992 Signal recovery and the large sieve SIAM J. Appl. Math. 52 577 591
28. D.L. Donoho P. Starck 1989 Uncertainty principles and signal recovery SIAM J. Appl. Math. 49 906 931
29. B. Efron T. Hastie I. Johnstone R. Tibshirani 2004 Least angle regression Ann. Stat. 32 2 407 499
30. Engl, H.W., Hanke, M., Neubauer, A.: Regularization of Inverse Problems. Mathematics and Its Applications, vol. 375. Kluwer Academic, Dordrecht (1996)
31. M.A.T. Figueiredo R.D. Nowak 2003 An EM algorithm for wavelet-based image restoration IEEE Trans. Image Proc. 12 8 906 916
32. M.A.T. Figueiredo R.D. Nowak S.J. Wright 2007 Gradient projection for sparse reconstruction: Application to compressed sensing and other inverse problems IEEE J. Sel. Top. Signal Process. 1 4 586 597
33. M. Fornasier 2007 Domain decomposition methods for linear inverse problems with sparsity constraints Inverse Probl. 23 2505 2526
34. M. Fornasier F. Pitolli 2008 Adaptive iterative thresholding algorithms for magnetoenceophalography (MEG) J. Comput. Appl. Math. 10.1016/j.cam.2007.10. 048
35. M. Fornasier H. Rauhut 2008 Iterative thrsholding algorithms Appl. Comput. Harmon. Anal. 25 2 187 208
36. M. Fornasier H. Rauhut 2008 Recovery algorithms for vector valued data with joint sparsity constraints SIAM J. Numer. Anal. 46 2 577 613
37. Logan, B.F.: Properties of High-Pass Signals. Ph.D. Thesis, Columbia University, 1965
38. 1-norm regularization of wavelet coefficients Geophys. J. Int. 170 1 359 370
39. Mallat, S.: A Wavelet Tour of Signal Processing, 2nd edn. Academic Press, San Diego (1999)
40. R. Ramlau G. Teschke 2005 Tikhonov replacement functionals for iteratively solving nonlinear operator equations Inverse Probl. 21 5 1571 1592
41. H. Rauhut 2007 Random sampling of sparse trigonometric polynomials Appl. Comput. Harmon. Anal. 22 1 16 42
42. J.-L. Starck E.J. Candès D.L. Donoho 2003 Astronomical image representation by curvelet transform Astron. Astrophys. 298 785 800
43. J.-L. Starck M.K. Nguyen F. Murtagh 2003 Wavelets and curvelets for image deconvolution: a combined approach Signal Process. 83 2279 2283
44. G. Teschke 2007 Multi-frame representations in linear inverse problems with mixed multi-constraints Appl. Comput. Harmon. Anal. 22 1 43 60
45. van den Berg, E., Friedlander, M.P.: In pursuit of a root. Preprint (2007)
46. Wolfram, S.: The Mathematica Book, 5th edn. Wolfram Media/Cambridge University Press, Cambridge (2003)