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Reconstruction of the singularities of a potential from backscattering data in 2D and 3D
Strongly convex programming for exact matrix completion and robust principal component analysis
1. | Department of Mathematics and Systems Science, National University of Defense Technology, Changsha, Hunan 410073, China, China, China |
2. | Department of Mathematics, University of Iowa, Iowa City, IA 52242, United States |
References:
[1] |
A. Argyriou, T. Evgeniou and M. Pontil, Multi-task feature learning,, Advances in Neural Information Processing Systems (NIPS), (2007). Google Scholar |
[2] |
J.-F. Cai, E.J. Candès and Z. Shen, A singular value thresholding algorithm for matrixcompletion,, SIAM J. on Optimization, 20 (2010), 1956.
doi: 10.1137/080738970. |
[3] |
J.-F. Cai, S. Osher and Z. Shen, Linearized Bregman iterations for compressed sensing,, Math. Comp., 78 (2009), 1515.
doi: 10.1090/S0025-5718-08-02189-3. |
[4] |
J.-F. Cai, S. Osher and Z. Shen, Convergence of the linearized Bregman iteration for ℓ$_1$-norm minimization,, Math. Comp., 78 (2009), 2127.
doi: 10.1090/S0025-5718-09-02242-X. |
[5] |
J.-F. Cai, S. Osher and Z. Shen, Linearized Bregman iterations for frame-based image deblurring,, SIAM J. Imaging Sciences, 2 (2009), 226.
|
[6] |
E. J. Candès and B. Recht, Exact matrix completion via convexoptimization,, Foundations of Computational Mathematics, 9 (2009), 717.
|
[7] |
E.J. Candès and T. Tao, The power of convex relaxation: Near-optimal matrix completion,, IEEE Transactions on InformationTheory, 56 (2010), 2053.
doi: 10.1109/TIT.2010.2044061. |
[8] |
E. J. Candès and Y. Plan, Matrix completion with noise,, Proceeding of the IEEE, 98 (2010), 925.
doi: 10.1109/JPROC.2009.2035722. |
[9] |
E. J. Candès, X. Li, Y. Ma and J. Wright, Robust principal component analysis?,, Journal of ACM, 58 (2011), 1.
|
[10] |
E. J. Candès and T. Tao, Decoding by linear programming,, IEEE Transactions on Information Theory, 51 (2006), 4203.
|
[11] |
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), 489.
doi: 10.1109/TIT.2005.862083. |
[12] |
V. Chandrasekharan, S. Sanghavi, P. Parillo and A. Wilsky, Rank-sparsity incoherence for matrix decomposition,, SIAM Journal on Optimization, 21 (2011), 572.
doi: 10.1137/090761793. |
[13] |
S. S. Chen, D. L. Donoho and M. A. Saunders, Atomic decomposition by basis pursuit,, SIAM J. Sci. Comput., 20 (1998), 33.
doi: 10.1137/S1064827596304010. |
[14] |
D. L. Donoho, Compressed sensing,, IEEE Trans. Inform. Theory, 52 (2006), 1289.
doi: 10.1109/TIT.2006.871582. |
[15] |
M. Fazel, "Matrix Rank Minimization with Applications,", Ph.D. thesis, (2002). Google Scholar |
[16] |
M. P. Friedlander and P. Tseng, Exact regularization of convex programs,, SIAM J. Optim., 18 (2007), 1326.
doi: 10.1137/060675320. |
[17] |
H. Gao, J.-F. Cai, Z. Shen and H. Zhao, Robust principalcomponent analysis-based four-dimensional computed tomography,, Physics in Medicine and Biology, 56 (2011), 3181. Google Scholar |
[18] |
D. Gross, Recovering low-rank matrices from few coefficients in any basis,, IEEE Trans. Inform. Theory, 57 (2011), 1548.
doi: 10.1109/TIT.2011.2104999. |
[19] |
H. Ji, S. Huang, Z. Shen and Y. Xu, Robust video restoration by joint sparse and low rank matrix approximation,, SIAM Journal on Imaging Sciences, 4 (2011), 1122.
|
[20] |
H. Ji, C. Liu, Z. W. Shen and Y. Xu, Robust video denoising using low rank matrix completion,, IEEE Conf. on Computer Vision and Pattern Recognition (CVPR), (2010), 1791. Google Scholar |
[21] |
Z. Liu and L. Vandenberghe, Interior-point method for nuclear norm approximation with application to system identification,, SIAM Journal on Matrix Analysis and Applications, 31 (2009), 1235.
doi: 10.1137/090755436. |
[22] |
Z. Lin, A. Ganesh, J. Wright, L. Wu, M. Chen and Y. Ma, Fastconvex optimization algorithms for exact recovery of a corrupted low-rank matrix,, Technical Report UILU-ENG-09-2214, (2010), 09. Google Scholar |
[23] |
Z. Lin, M. Chen, L. Wu and Y. Ma, The augmentedLagrange multiplier method for exact recovery of corrupted low-rankmatrices,, Technical Report UILU-ENG-09-2215, (2010), 09. Google Scholar |
[24] |
K. Min, Z. D. Zhang, J. Writht and Y. Ma, Decomposing background topic from keywords by principle component pursuit,, Proceeding of the 19th ACM Interational Conference on Information and Knowledge, (2010), 269. Google Scholar |
[25] |
J. Meng, W. Yin, E. Houssain and Z. Han, Collaborative spectrum sensing from sparse observations using matrix completion for cognitive radio networks,, Proceedings of ICASSP, (2010), 3114. Google Scholar |
[26] |
S. Ma, D. Goldfarb and L. Chen, Fixed point and Bregman iterativemethods for matrix rank minimization,, Mathematical Programming Series A, 128 (2011), 321.
doi: 10.1007/s10107-009-0306-5. |
[27] |
M. Mesbahi and G. P. Papavassilopoulos, On the rank minimization problem over a positive semidefinite linear matrix inequality,, IEEE Transactions on Automatic Control, 42 (1997), 239.
doi: 10.1109/9.554402. |
[28] |
Y. Nesterov, Smooth minimization of non-smooth functions,, Math. Program. Serie A, 103 (2005), 127.
|
[29] |
S. Osher, Y. Mao, B. Dong and W. Yin, Fast linearized Bregman iteration for compressive sensing and sparse denoising,, Comm. in Math. Sciences, 1 (2010), 93.
|
[30] |
B. Recht, M. Fazel and P. Parrilo, Guaranteed minimum rank solutionsof matrix equations via nuclear norm minimization,, SIAM Review, 52 (2010), 471.
doi: 10.1137/070697835. |
[31] |
B. Recht, W. Xu and B. Hassibi, Necessary and sufficient conditions forsuccess of the nuclear norm heuristic for rank minimization,, Proceedings of the 47th IEEE Conference on Decision and Control, (2008), 3065. Google Scholar |
[32] |
B. Recht, W. Xu and B. Hassibi, Null space conditions and thresholds for rank minimization,, Mathematics Programming, 127 (2011), 175.
doi: 10.1007/s10107-010-0422-2. |
[33] |
B. Recht, A simpler approach to matrix completion,, J. Machine Learning Res., 12 (2011), 3413. Google Scholar |
[34] |
R. T. Rockafellar, "Convex Analysis,", Princeton University Press, (1970).
|
[35] |
K. C. Toh and S. Yun, An accelerated proximal gradient algorithm fornuclear norm regularized least squares problems,, Pacific Journal of Optimization, 6 (2009), 615.
|
[36] |
C. Tomasi and T. Kanade, Shape and motion from image streams under orthography: A factorization method,, International Journal of Computer Vision, 9 (1992), 137. Google Scholar |
[37] |
J. Wright, A. Ganesh, S. Rao and Y. Ma, Robust principal component analysis: Exact recovery of corrupted low-rank matrices via convex optimization,, submitted to Journal of the ACM, (2009). Google Scholar |
[38] |
W. Yin, Analysis and generalizations of the linearized Bregman method,, SIAM Journal on Image Science, 3 (2010), 856.
|
[39] |
W. Yin, S. Osher, D. Goldfarb and J. Darbon, Bregman iterative algorithms for $l_1$ minimization with applications to compressed sensing,, SIAM J. Imaging Sciences, 1 (2008), 143.
|
[40] |
Z. D. Zhang, X. Liang, A. Ganesh and Y. Ma, TILT: Transform invariant low-rank textures,, Computer Vision-ACCV, (2010), 314. Google Scholar |
[41] |
H. Zhang and L. Z. Cheng, Projected Landweber iteration for matrix completion,, Journal of Computational and Applied Mathematics, 235 (2010), 593.
doi: 10.1016/j.cam.2010.06.010. |
[42] |
H. Zhang, L. Z. Cheng and W. Zhu, A lower bound guaranteeing exact matrix completion via singular value thresholding algorithm,, Applied and Computational Harmonic Analysis, 31 (2011), 454.
doi: 10.1016/j.acha.2011.04.004. |
[43] |
H. Zhang, L. Z. Cheng and W. Zhu, Nuclear norm regularization with low-rank constraint for matrix completion,, Inverse Problems, 26 (2010).
doi: 10.1088/0266-5611/26/11/115009. |
show all references
References:
[1] |
A. Argyriou, T. Evgeniou and M. Pontil, Multi-task feature learning,, Advances in Neural Information Processing Systems (NIPS), (2007). Google Scholar |
[2] |
J.-F. Cai, E.J. Candès and Z. Shen, A singular value thresholding algorithm for matrixcompletion,, SIAM J. on Optimization, 20 (2010), 1956.
doi: 10.1137/080738970. |
[3] |
J.-F. Cai, S. Osher and Z. Shen, Linearized Bregman iterations for compressed sensing,, Math. Comp., 78 (2009), 1515.
doi: 10.1090/S0025-5718-08-02189-3. |
[4] |
J.-F. Cai, S. Osher and Z. Shen, Convergence of the linearized Bregman iteration for ℓ$_1$-norm minimization,, Math. Comp., 78 (2009), 2127.
doi: 10.1090/S0025-5718-09-02242-X. |
[5] |
J.-F. Cai, S. Osher and Z. Shen, Linearized Bregman iterations for frame-based image deblurring,, SIAM J. Imaging Sciences, 2 (2009), 226.
|
[6] |
E. J. Candès and B. Recht, Exact matrix completion via convexoptimization,, Foundations of Computational Mathematics, 9 (2009), 717.
|
[7] |
E.J. Candès and T. Tao, The power of convex relaxation: Near-optimal matrix completion,, IEEE Transactions on InformationTheory, 56 (2010), 2053.
doi: 10.1109/TIT.2010.2044061. |
[8] |
E. J. Candès and Y. Plan, Matrix completion with noise,, Proceeding of the IEEE, 98 (2010), 925.
doi: 10.1109/JPROC.2009.2035722. |
[9] |
E. J. Candès, X. Li, Y. Ma and J. Wright, Robust principal component analysis?,, Journal of ACM, 58 (2011), 1.
|
[10] |
E. J. Candès and T. Tao, Decoding by linear programming,, IEEE Transactions on Information Theory, 51 (2006), 4203.
|
[11] |
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), 489.
doi: 10.1109/TIT.2005.862083. |
[12] |
V. Chandrasekharan, S. Sanghavi, P. Parillo and A. Wilsky, Rank-sparsity incoherence for matrix decomposition,, SIAM Journal on Optimization, 21 (2011), 572.
doi: 10.1137/090761793. |
[13] |
S. S. Chen, D. L. Donoho and M. A. Saunders, Atomic decomposition by basis pursuit,, SIAM J. Sci. Comput., 20 (1998), 33.
doi: 10.1137/S1064827596304010. |
[14] |
D. L. Donoho, Compressed sensing,, IEEE Trans. Inform. Theory, 52 (2006), 1289.
doi: 10.1109/TIT.2006.871582. |
[15] |
M. Fazel, "Matrix Rank Minimization with Applications,", Ph.D. thesis, (2002). Google Scholar |
[16] |
M. P. Friedlander and P. Tseng, Exact regularization of convex programs,, SIAM J. Optim., 18 (2007), 1326.
doi: 10.1137/060675320. |
[17] |
H. Gao, J.-F. Cai, Z. Shen and H. Zhao, Robust principalcomponent analysis-based four-dimensional computed tomography,, Physics in Medicine and Biology, 56 (2011), 3181. Google Scholar |
[18] |
D. Gross, Recovering low-rank matrices from few coefficients in any basis,, IEEE Trans. Inform. Theory, 57 (2011), 1548.
doi: 10.1109/TIT.2011.2104999. |
[19] |
H. Ji, S. Huang, Z. Shen and Y. Xu, Robust video restoration by joint sparse and low rank matrix approximation,, SIAM Journal on Imaging Sciences, 4 (2011), 1122.
|
[20] |
H. Ji, C. Liu, Z. W. Shen and Y. Xu, Robust video denoising using low rank matrix completion,, IEEE Conf. on Computer Vision and Pattern Recognition (CVPR), (2010), 1791. Google Scholar |
[21] |
Z. Liu and L. Vandenberghe, Interior-point method for nuclear norm approximation with application to system identification,, SIAM Journal on Matrix Analysis and Applications, 31 (2009), 1235.
doi: 10.1137/090755436. |
[22] |
Z. Lin, A. Ganesh, J. Wright, L. Wu, M. Chen and Y. Ma, Fastconvex optimization algorithms for exact recovery of a corrupted low-rank matrix,, Technical Report UILU-ENG-09-2214, (2010), 09. Google Scholar |
[23] |
Z. Lin, M. Chen, L. Wu and Y. Ma, The augmentedLagrange multiplier method for exact recovery of corrupted low-rankmatrices,, Technical Report UILU-ENG-09-2215, (2010), 09. Google Scholar |
[24] |
K. Min, Z. D. Zhang, J. Writht and Y. Ma, Decomposing background topic from keywords by principle component pursuit,, Proceeding of the 19th ACM Interational Conference on Information and Knowledge, (2010), 269. Google Scholar |
[25] |
J. Meng, W. Yin, E. Houssain and Z. Han, Collaborative spectrum sensing from sparse observations using matrix completion for cognitive radio networks,, Proceedings of ICASSP, (2010), 3114. Google Scholar |
[26] |
S. Ma, D. Goldfarb and L. Chen, Fixed point and Bregman iterativemethods for matrix rank minimization,, Mathematical Programming Series A, 128 (2011), 321.
doi: 10.1007/s10107-009-0306-5. |
[27] |
M. Mesbahi and G. P. Papavassilopoulos, On the rank minimization problem over a positive semidefinite linear matrix inequality,, IEEE Transactions on Automatic Control, 42 (1997), 239.
doi: 10.1109/9.554402. |
[28] |
Y. Nesterov, Smooth minimization of non-smooth functions,, Math. Program. Serie A, 103 (2005), 127.
|
[29] |
S. Osher, Y. Mao, B. Dong and W. Yin, Fast linearized Bregman iteration for compressive sensing and sparse denoising,, Comm. in Math. Sciences, 1 (2010), 93.
|
[30] |
B. Recht, M. Fazel and P. Parrilo, Guaranteed minimum rank solutionsof matrix equations via nuclear norm minimization,, SIAM Review, 52 (2010), 471.
doi: 10.1137/070697835. |
[31] |
B. Recht, W. Xu and B. Hassibi, Necessary and sufficient conditions forsuccess of the nuclear norm heuristic for rank minimization,, Proceedings of the 47th IEEE Conference on Decision and Control, (2008), 3065. Google Scholar |
[32] |
B. Recht, W. Xu and B. Hassibi, Null space conditions and thresholds for rank minimization,, Mathematics Programming, 127 (2011), 175.
doi: 10.1007/s10107-010-0422-2. |
[33] |
B. Recht, A simpler approach to matrix completion,, J. Machine Learning Res., 12 (2011), 3413. Google Scholar |
[34] |
R. T. Rockafellar, "Convex Analysis,", Princeton University Press, (1970).
|
[35] |
K. C. Toh and S. Yun, An accelerated proximal gradient algorithm fornuclear norm regularized least squares problems,, Pacific Journal of Optimization, 6 (2009), 615.
|
[36] |
C. Tomasi and T. Kanade, Shape and motion from image streams under orthography: A factorization method,, International Journal of Computer Vision, 9 (1992), 137. Google Scholar |
[37] |
J. Wright, A. Ganesh, S. Rao and Y. Ma, Robust principal component analysis: Exact recovery of corrupted low-rank matrices via convex optimization,, submitted to Journal of the ACM, (2009). Google Scholar |
[38] |
W. Yin, Analysis and generalizations of the linearized Bregman method,, SIAM Journal on Image Science, 3 (2010), 856.
|
[39] |
W. Yin, S. Osher, D. Goldfarb and J. Darbon, Bregman iterative algorithms for $l_1$ minimization with applications to compressed sensing,, SIAM J. Imaging Sciences, 1 (2008), 143.
|
[40] |
Z. D. Zhang, X. Liang, A. Ganesh and Y. Ma, TILT: Transform invariant low-rank textures,, Computer Vision-ACCV, (2010), 314. Google Scholar |
[41] |
H. Zhang and L. Z. Cheng, Projected Landweber iteration for matrix completion,, Journal of Computational and Applied Mathematics, 235 (2010), 593.
doi: 10.1016/j.cam.2010.06.010. |
[42] |
H. Zhang, L. Z. Cheng and W. Zhu, A lower bound guaranteeing exact matrix completion via singular value thresholding algorithm,, Applied and Computational Harmonic Analysis, 31 (2011), 454.
doi: 10.1016/j.acha.2011.04.004. |
[43] |
H. Zhang, L. Z. Cheng and W. Zhu, Nuclear norm regularization with low-rank constraint for matrix completion,, Inverse Problems, 26 (2010).
doi: 10.1088/0266-5611/26/11/115009. |
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