Strongly convex programming for exact matrix completion and robust principal component analysis

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May 2012, 6(2): 357-372. doi: 10.3934/ipi.2012.6.357

Strongly convex programming for exact matrix completion and robust principal component analysis

Hui Zhang ^1, , Jian-Feng Cai ^2, , Lizhi Cheng ^1, and Jubo Zhu ^1,

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

Received July 2011 Revised January 2012 Published May 2012

Abstract
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The common task in matrix completion (MC) and robust principle component analysis (RPCA) is to recover a low-rank matrix from a given data matrix. These problems gained great attention from various areas in applied sciences recently, especially after the publication of the pioneering works of Candès et al.. One fundamental result in MC and RPCA is that nuclear norm based convex optimizations lead to the exact low-rank matrix recovery under suitable conditions. In this paper, we extend this result by showing that strongly convex optimizations can guarantee the exact low-rank matrix recovery as well. The result in this paper not only provides sufficient conditions under which the strongly convex models lead to the exact low-rank matrix recovery, but also guides us on how to choose suitable parameters in practical algorithms.

Keywords: Strongly convex programming, low-rank matrix, robust principal component analysis, exact matrix completion, dual certificate..

Mathematics Subject Classification: Primary: 15B52, 90C25; Secondary: 60B20, 94A08, 94A1.

Citation: Hui Zhang, Jian-Feng Cai, Lizhi Cheng, Jubo Zhu. Strongly convex programming for exact matrix completion and robust principal component analysis. Inverse Problems & Imaging, 2012, 6 (2) : 357-372. doi: 10.3934/ipi.2012.6.357

References:

[1]	A. Argyriou, T. Evgeniou and M. Pontil, Multi-task feature learning,, Advances in Neural Information Processing Systems (NIPS), (2007).
[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).
[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.
[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.
[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.
[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.
[24]	K. Min, Z. D. Zhang, J. Writht and Y. Ma, Decomposing background topic from keywords by principle component pursuit,, Proceeding of the 19^th ACM Interational Conference on Information and Knowledge, (2010), 269.
[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.
[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 47^th IEEE Conference on Decision and Control, (2008), 3065.
[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.
[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.
[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).
[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.
[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).
[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).
[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.
[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.
[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.
[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.
[24]	K. Min, Z. D. Zhang, J. Writht and Y. Ma, Decomposing background topic from keywords by principle component pursuit,, Proceeding of the 19^th ACM Interational Conference on Information and Knowledge, (2010), 269.
[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.
[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 47^th IEEE Conference on Decision and Control, (2008), 3065.
[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.
[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.
[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).
[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.
[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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