# American Institute of Mathematical Sciences

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

 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

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.
Citation: Hui Zhang, Jian-Feng Cai, Lizhi Cheng, Jubo Zhu. Strongly convex programming for exact matrix completion and robust principal component analysis. Inverse Problems and 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), 1956-1982. doi: 10.1137/080738970. [3] J.-F. Cai, S. Osher and Z. Shen, Linearized Bregman iterations for compressed sensing, Math. Comp., 78 (2009), 1515-1536. 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), 2127-2136. 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), 226-252. [6] E. J. Candès and B. Recht, Exact matrix completion via convexoptimization, Foundations of Computational Mathematics, 9 (2009), 717-772. [7] E.J. Candès and T. Tao, The power of convex relaxation: Near-optimal matrix completion, IEEE Transactions on InformationTheory, 56 (2010), 2053-2086. doi: 10.1109/TIT.2010.2044061. [8] E. J. Candès and Y. Plan, Matrix completion with noise, Proceeding of the IEEE, 98 (2010), 925-936. 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), 1-37. [10] E. J. Candès and T. Tao, Decoding by linear programming, IEEE Transactions on Information Theory, 51 (2006), 4203-4215. [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), 489-509. 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), 572-596. 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), 33-61. doi: 10.1137/S1064827596304010. [14] D. L. Donoho, Compressed sensing, IEEE Trans. Inform. Theory, 52 (2006), 1289-1306. doi: 10.1109/TIT.2006.871582. [15] M. Fazel, "Matrix Rank Minimization with Applications," Ph.D. thesis, Stanford University, 2002. [16] M. P. Friedlander and P. Tseng, Exact regularization of convex programs, SIAM J. Optim., 18 (2007), 1326-1350. 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), 3181-3198. [18] D. Gross, Recovering low-rank matrices from few coefficients in any basis, IEEE Trans. Inform. Theory, 57 (2011), 1548-1566. 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), 1122-1142. [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), San Francisco, (2010), 1791-1798. [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), 1235-1256. 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. [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. [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), 269-278. [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), 3114-3117. [26] S. Ma, D. Goldfarb and L. Chen, Fixed point and Bregman iterativemethods for matrix rank minimization, Mathematical Programming Series A, 128 (2011), 321-353. 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), 239-243. doi: 10.1109/9.554402. [28] Y. Nesterov, Smooth minimization of non-smooth functions, Math. Program. Serie A, 103 (2005), 127-152. [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), 93-111. [30] B. Recht, M. Fazel and P. Parrilo, Guaranteed minimum rank solutionsof matrix equations via nuclear norm minimization, SIAM Review, 52 (2010), 471-501. 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), 3065-3070. [32] B. Recht, W. Xu and B. Hassibi, Null space conditions and thresholds for rank minimization, Mathematics Programming, 127 (2011), 175-202. doi: 10.1007/s10107-010-0422-2. [33] B. Recht, A simpler approach to matrix completion, J. Machine Learning Res., 12 (2011), 3413-3430. [34] R. T. Rockafellar, "Convex Analysis," Princeton University Press, Princeton, NJ, 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), 615-640. [36] C. Tomasi and T. Kanade, Shape and motion from image streams under orthography: A factorization method, International Journal of Computer Vision, 9 (1992), 137-154. [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, arXiv:0905.0233. [38] W. Yin, Analysis and generalizations of the linearized Bregman method, SIAM Journal on Image Science, 3 (2010), 856-877. [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), 143-168. [40] Z. D. Zhang, X. Liang, A. Ganesh and Y. Ma, TILT: Transform invariant low-rank textures, Computer Vision-ACCV, (2010), 314-328. [41] H. Zhang and L. Z. Cheng, Projected Landweber iteration for matrix completion, Journal of Computational and Applied Mathematics, 235 (2010), 593-601. 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), 454-459. 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), 115009, 15 pp. 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), 1956-1982. doi: 10.1137/080738970. [3] J.-F. Cai, S. Osher and Z. Shen, Linearized Bregman iterations for compressed sensing, Math. Comp., 78 (2009), 1515-1536. 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), 2127-2136. 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), 226-252. [6] E. J. Candès and B. Recht, Exact matrix completion via convexoptimization, Foundations of Computational Mathematics, 9 (2009), 717-772. [7] E.J. Candès and T. Tao, The power of convex relaxation: Near-optimal matrix completion, IEEE Transactions on InformationTheory, 56 (2010), 2053-2086. doi: 10.1109/TIT.2010.2044061. [8] E. J. Candès and Y. Plan, Matrix completion with noise, Proceeding of the IEEE, 98 (2010), 925-936. 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), 1-37. [10] E. J. Candès and T. Tao, Decoding by linear programming, IEEE Transactions on Information Theory, 51 (2006), 4203-4215. [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), 489-509. 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), 572-596. 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), 33-61. doi: 10.1137/S1064827596304010. [14] D. L. Donoho, Compressed sensing, IEEE Trans. Inform. Theory, 52 (2006), 1289-1306. doi: 10.1109/TIT.2006.871582. [15] M. Fazel, "Matrix Rank Minimization with Applications," Ph.D. thesis, Stanford University, 2002. [16] M. P. Friedlander and P. Tseng, Exact regularization of convex programs, SIAM J. Optim., 18 (2007), 1326-1350. 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), 3181-3198. [18] D. Gross, Recovering low-rank matrices from few coefficients in any basis, IEEE Trans. Inform. Theory, 57 (2011), 1548-1566. 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), 1122-1142. [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), San Francisco, (2010), 1791-1798. [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), 1235-1256. 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. [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. [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), 269-278. [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), 3114-3117. [26] S. Ma, D. Goldfarb and L. Chen, Fixed point and Bregman iterativemethods for matrix rank minimization, Mathematical Programming Series A, 128 (2011), 321-353. 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), 239-243. doi: 10.1109/9.554402. [28] Y. Nesterov, Smooth minimization of non-smooth functions, Math. Program. Serie A, 103 (2005), 127-152. [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), 93-111. [30] B. Recht, M. Fazel and P. Parrilo, Guaranteed minimum rank solutionsof matrix equations via nuclear norm minimization, SIAM Review, 52 (2010), 471-501. 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), 3065-3070. [32] B. Recht, W. Xu and B. Hassibi, Null space conditions and thresholds for rank minimization, Mathematics Programming, 127 (2011), 175-202. doi: 10.1007/s10107-010-0422-2. [33] B. Recht, A simpler approach to matrix completion, J. Machine Learning Res., 12 (2011), 3413-3430. [34] R. T. Rockafellar, "Convex Analysis," Princeton University Press, Princeton, NJ, 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), 615-640. [36] C. Tomasi and T. Kanade, Shape and motion from image streams under orthography: A factorization method, International Journal of Computer Vision, 9 (1992), 137-154. [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, arXiv:0905.0233. [38] W. Yin, Analysis and generalizations of the linearized Bregman method, SIAM Journal on Image Science, 3 (2010), 856-877. [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), 143-168. [40] Z. D. Zhang, X. Liang, A. Ganesh and Y. Ma, TILT: Transform invariant low-rank textures, Computer Vision-ACCV, (2010), 314-328. [41] H. Zhang and L. Z. Cheng, Projected Landweber iteration for matrix completion, Journal of Computational and Applied Mathematics, 235 (2010), 593-601. 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), 454-459. 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), 115009, 15 pp. doi: 10.1088/0266-5611/26/11/115009.
 [1] Qingshan You, Qun Wan, Yipeng Liu. A short note on strongly convex programming for exact matrix completion and robust principal component analysis. Inverse Problems and Imaging, 2013, 7 (1) : 305-306. doi: 10.3934/ipi.2013.7.305 [2] Yangyang Xu, Ruru Hao, Wotao Yin, Zhixun Su. Parallel matrix factorization for low-rank tensor completion. Inverse Problems and Imaging, 2015, 9 (2) : 601-624. doi: 10.3934/ipi.2015.9.601 [3] Yitong Guo, Bingo Wing-Kuen Ling. Principal component analysis with drop rank covariance matrix. Journal of Industrial and Management Optimization, 2021, 17 (5) : 2345-2366. doi: 10.3934/jimo.2020072 [4] Ke Wei, Jian-Feng Cai, Tony F. Chan, Shingyu Leung. Guarantees of riemannian optimization for low rank matrix completion. Inverse Problems and Imaging, 2020, 14 (2) : 233-265. doi: 10.3934/ipi.2020011 [5] Dan Zhu, Rosemary A. Renaut, Hongwei Li, Tianyou Liu. Fast non-convex low-rank matrix decomposition for separation of potential field data using minimal memory. Inverse Problems and Imaging, 2021, 15 (1) : 159-183. doi: 10.3934/ipi.2020076 [6] Tao Wu, Yu Lei, Jiao Shi, Maoguo Gong. An evolutionary multiobjective method for low-rank and sparse matrix decomposition. Big Data & Information Analytics, 2017, 2 (1) : 23-37. doi: 10.3934/bdia.2017006 [7] Ningyu Sha, Lei Shi, Ming Yan. Fast algorithms for robust principal component analysis with an upper bound on the rank. Inverse Problems and Imaging, 2021, 15 (1) : 109-128. doi: 10.3934/ipi.2020067 [8] Yun Cai, Song Li. Convergence and stability of iteratively reweighted least squares for low-rank matrix recovery. Inverse Problems and Imaging, 2017, 11 (4) : 643-661. doi: 10.3934/ipi.2017030 [9] Zhouchen Lin. A review on low-rank models in data analysis. Big Data & Information Analytics, 2016, 1 (2&3) : 139-161. doi: 10.3934/bdia.2016001 [10] Kaixin Gao, Zheng-Hai Huang, Xiaolei Liu. Enhancing low-rank tensor completion via first-order and second-order total variation regularizations. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022136 [11] Henry Adams, Lara Kassab, Deanna Needell. An adaptation for iterative structured matrix completion. Foundations of Data Science, 2021, 3 (4) : 769-791. doi: 10.3934/fods.2021028 [12] Paul Skerritt, Cornelia Vizman. Dual pairs for matrix groups. Journal of Geometric Mechanics, 2019, 11 (2) : 255-275. doi: 10.3934/jgm.2019014 [13] Huiyuan Guo, Quan Yu, Xinzhen Zhang, Lulu Cheng. Low rank matrix minimization with a truncated difference of nuclear norm and Frobenius norm regularization. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022045 [14] Yuhong Dai, Nobuo Yamashita. Convergence analysis of sparse quasi-Newton updates with positive definite matrix completion for two-dimensional functions. Numerical Algebra, Control and Optimization, 2011, 1 (1) : 61-69. doi: 10.3934/naco.2011.1.61 [15] Yuan Shen, Xin Liu. An alternating minimization method for matrix completion problems. Discrete and Continuous Dynamical Systems - S, 2020, 13 (6) : 1757-1772. doi: 10.3934/dcdss.2020103 [16] Yi Yang, Jianwei Ma, Stanley Osher. Seismic data reconstruction via matrix completion. Inverse Problems and Imaging, 2013, 7 (4) : 1379-1392. doi: 10.3934/ipi.2013.7.1379 [17] Simon Foucart, Richard G. Lynch. Recovering low-rank matrices from binary measurements. Inverse Problems and Imaging, 2019, 13 (4) : 703-720. doi: 10.3934/ipi.2019032 [18] Simon Arridge, Pascal Fernsel, Andreas Hauptmann. Joint reconstruction and low-rank decomposition for dynamic inverse problems. Inverse Problems and Imaging, 2022, 16 (3) : 483-523. doi: 10.3934/ipi.2021059 [19] Kevin N. Webster. Low-rank kernel approximation of Lyapunov functions using neural networks. Journal of Computational Dynamics, 2022  doi: 10.3934/jcd.2022026 [20] Zhengshan Dong, Jianli Chen, Wenxing Zhu. Homotopy method for matrix rank minimization based on the matrix hard thresholding method. Numerical Algebra, Control and Optimization, 2019, 9 (2) : 211-224. doi: 10.3934/naco.2019015

2021 Impact Factor: 1.483