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Guarantees of riemannian optimization for low rank matrix completion
1. | School of Data Science, Fudan University, Shanghai, China |
2. | Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong SAR, China |
3. | Office of the President, King Abdullah University of Science and Technology, Thuwal 23955-6900, Kingdom of Saudi Arabia |
$ m $ |
$ n\times n $ |
$ r $ |
$ \begin{align*} m\geq C_\kappa n^{1.5}r\log^{1.5}(n), \end{align*} $ |
$ C_\kappa $ |
$ \begin{align*} m\geq C_\kappa nr^2\log^{2}(n) \end{align*} $ |
References:
[1] |
Y. Amit, M. Fink, N. Srebro and S. Ullman, Uncovering shared structures in multiclass classification, in: Proceedings of the 24th International Conference on Machine Learning, 2007, 17–24.
doi: 10.1145/1273496.1273499. |
[2] |
A. Argyriou, T. Evgeniou and M. Pontil, Multi-task feature learning, Advances in Neural Information Processing Systems, 2007. Google Scholar |
[3] |
P.-A. Absil, R. Mahony and R. Sepulchre, Optimization Algorithms on Matrix Manifolds,, Princeton University Press, 2008.
doi: 10.1515/9781400830244.![]() ![]() |
[4] |
A. Ahmed, B. Recht and J. Romberg,
Blind deconvolution using convex programming, IEEE Transactions on Information Theory, 60 (2014), 1711-1732.
doi: 10.1109/TIT.2013.2294644. |
[5] |
R. Ahlswede and A. Winter,
Strong converse for identification via quantum channels, IEEE Transactions on Information Theory, 48 (2002), 569-579.
doi: 10.1109/18.985947. |
[6] |
S. Bhojanapalli, A. Kyrillidis and S. Sanghavi, Dropping convexity for faster semi-definite optimization, JMLR: Workshop and Conference Proceedings, 49 (2016), 1-53. Google Scholar |
[7] |
J. Blanchard, J. Tanner and K. Wei,
CGIHT: Conjugate gradient iterative hard thresholding for compressed sensing and matrix completion, Information and Inference, 4 (2015), 289-327.
doi: 10.1093/imaiai/iav011. |
[8] |
N. Boumal and P.-A. Absil, RTRMC: A Riemannian trust-region method for low-rank matrix completion, in: Advances in Neural Information Processing Systems, 2011. Google Scholar |
[9] |
E. J. Candès and B. Recht,
Exact matrix completion via convex optimization, Foundations of Computational Mathematics, 9 (2009), 717-772.
doi: 10.1007/s10208-009-9045-5. |
[10] |
E. J. Candès and Y. Plan,
Tight oracle bounds for low-rank matrix recovery from a minimal number of random measurements, IEEE Transactions on Information Theory, 57 (2011), 2342-2359.
doi: 10.1109/TIT.2011.2111771. |
[11] |
Y. Chen and M. J. Wainwright, Fast low-rank estimation by projected gradient descent: General statistical and algorithmic guarantees, arXiv: 1509.03025, 2015. Google Scholar |
[12] |
E. J. Candès and T. Tao,
The power of convex relaxation: Near-optimal matrix completion, IEEE Transactions on Information Theory, 56 (2009), 2053-2080.
doi: 10.1109/TIT.2010.2044061. |
[13] |
Y. Chen,
Incoherence-optimal matrix completion, IEEE Transactions on Information Theory, 61 (2015), 2909-2923.
doi: 10.1109/TIT.2015.2415195. |
[14] |
J.-F. Cai, E. J. Candès and Z. Shen,
A singular value thresholding algorithm for matrix completion, SIAM Journal on Optimization, 20 (2010), 1956-1982.
doi: 10.1137/080738970. |
[15] |
E. J. Candès, Y. Eldar, T. Strohmer and V. Voroninski,
Phase retrieval via matrix completion, SIAM J. on Imaging Sciences, 6 (2013), 199-225.
doi: 10.1137/110848074. |
[16] |
E. J. Candès, X. Li and M. Soltanolkotabi,
Phase retrieval via Wirtinger flow: Theory and algorithms, IEEE Transactions on Information Theory, 61 (2015), 1985-2007.
doi: 10.1109/TIT.2015.2399924. |
[17] |
Y. Chen and E. Candès,
Solving random quadratic systems of equations is nearly as easy as solving linear systems, Communications on Pure and Applied Mathematics, 70 (2017), 822-883.
doi: 10.1002/cpa.21638. |
[18] |
L. Eldén, Matrix Methods in Data Mining and Pattern Recogonization, SIAM, 2007.
doi: 10.1137/1.9780898718867. |
[19] |
M. Fazel, Matrix rank minimization with applications, ph. D. dissertation, Stanford University, 2002. Google Scholar |
[20] |
D. Gross,
Recovering low-rank matrices from few coefficients in any basis, IEEE Transactions on Information Theory, 57 (2011), 1548-1566.
doi: 10.1109/TIT.2011.2104999. |
[21] |
D. Goldfarb and S. Ma,
Convergence of fixed-point continuation algorithms for matrix rank minimization, Foundations of Computational Mathematics, 11 (2011), 183-210.
doi: 10.1007/s10208-011-9084-6. |
[22] |
R. W. Gerchberg and W. O. Saxton, A practical algorithm for the determination of the phase from image and diffraction plane pictures, Optik, 35 (1972), 237-246. Google Scholar |
[23] |
N. J. A. Harvey, D. R. Karger and S. Yekhanin, The complexity of matrix completion, in: Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithms, 2006, 1103–1111.
doi: 10.1145/1109557.1109679. |
[24] |
J. P. Haldar and D. Hernando,
Rank-constrained solutions to linear matrix equations using PowerFactorization, IEEE Signal Processing Letters, 16 (2009), 584-587.
doi: 10.1109/LSP.2009.2018223. |
[25] |
P. Jain, R. Meka and I. Dhillon, Guaranteed rank minimization via singular value projection, in: NIPS, 2010. Google Scholar |
[26] |
P. Jain, P. Netrapalli and S. Sanghavi, Low-rank matrix completion using alternating minimization, in: Proceedings of the Forty-Fifth Annual ACM Symposium on Theory of Computing, 2013,665–674.
doi: 10.1145/2488608.2488693. |
[27] |
P. Jain and P. Netrapalli, Fast exact matrix completion with finite samples, JMLR: Workshop and Conference Proceedings, 40 (2015), 1-28. Google Scholar |
[28] |
A. Kyrillidis and V. Cevher,
Matrix recipes for hard thresholding methods, Journal of Mathematical Imaging and Vision, 48 (2014), 235-265.
doi: 10.1007/s10851-013-0434-7. |
[29] |
R. H. Keshavan, Efficient algorithms for collaborative filtering, ph. D. dissertation, Stanford University, 2012. Google Scholar |
[30] |
R. H. Keshavan, A. Montanari and S. Oh,
Matrix completion from a few entries, IEEE Transactions on Information Theory, 56 (2010), 2980-2998.
doi: 10.1109/TIT.2010.2046205. |
[31] |
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. |
[32] |
K. Lee and Y. Bresler,
ADMiRA: Atomic decomposition for minimum rank approximation, IEEE Transactions on Information Theory, 128 (2010), 4402-4416.
doi: 10.1109/TIT.2010.2054251. |
[33] |
S. Ling and T. Strohmer,
Blind deconvolution meets blind demixing: Algorithms and performance bounds, IEEE Transactions on Information Theory, 63 (2017), 4497-4520.
doi: 10.1109/TIT.2017.2701342. |
[34] |
B. Mishra, K. A. Apuroop and R. Sepulchre, A Riemannian geometry for low-rank matrix completion, arXiv 1306.2672, 2013. Google Scholar |
[35] |
B. Mishra, G. Meyer, S. Bonnabel and R. Sepulchre,
Fixed-rank matrix factorizations and Riemannian low-rank optimization, Computational Statistics, 29 (2014), 591-621.
doi: 10.1007/s00180-013-0464-z. |
[36] |
T. Ngo and Y. Saad, Scaled Gradients on Grassmann Manifolds for Matrix Completion, in: Advances in Neural Information Processing Systems, 2012. Google Scholar |
[37] |
J. Nocedal and S. J. Wright, Numerical Optimization, Springer, 2006. |
[38] |
B. Recht, M. Fazel and P. A. Parrilo,
Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization, SIAM Review, 52 (2010), 471-501.
doi: 10.1137/070697835. |
[39] |
B. Recht,
A simpler approach to matrix completion, The Journal of Machine Learning Research, 12 (2011), 3413-3430.
|
[40] |
C. D. Sa, K. Olukotun, C. Ré, Global convergence of stochastic gradient descent for some nonconvex matrix problems, in: ICML, 2015. Google Scholar |
[41] |
R. Sun and Z. Luo, Guaranteed matrix completion via non-convex factorization, 2015 IEEE 56th Annual Symposium on Foundations of Computer Science–FOCS 2015, IEEE Computer Soc., Los Alamitos, CA, 2015,270–289. |
[42] |
J. Sun, Q. Qu and J. Wright,
A geometric analysis of phase retrieval, Foundations of Computational Mathematics, 18 (2018), 1131-1198.
doi: 10.1007/s10208-017-9365-9. |
[43] |
J. Tanner and K. Wei, Normalized iterative hard thresholding for matrix completion, SIAM Journal on Scientific Computing, 35 (2013), S104–S125.
doi: 10.1137/120876459. |
[44] |
S. Tu, R. Boczar, M. Simchowitz, M. Soltanolkotabi and B. Recht, Low-rank solutions of linear matrix equations via Procrustes flow, in: ICML, 2016. Google Scholar |
[45] |
J. Tanner and K. Wei,
Low rank matrix completion by alternating steepest descent methods, Applied and Computational Harmonic Analysis, 40 (2016), 417-429.
doi: 10.1016/j.acha.2015.08.003. |
[46] |
L. Vandenberghe and S. Boyd,
Semidefinite programming, SIAM Review, 38 (1996), 49-95.
doi: 10.1137/1038003. |
[47] |
B. Vandereycken,
Low rank matrix completion by Riemannian optimization, SIAM Journal on Optimization, 23 (2013), 1214-1236.
doi: 10.1137/110845768. |
[48] |
Z. Wen, W. Yin and Y. Zhang,
Solving a low-rank factorization model for matrix completion by a non-linear successive over-relaxation algorithm, Mathematical Programming Computation, 4 (2012), 333-361.
doi: 10.1007/s12532-012-0044-1. |
[49] |
C. D. White, S. Sanghavi and R. Ward,
The local convexity of solving systems of quadratic equations, Results in Mathematics, 71 (2017), 569-608.
doi: 10.1007/s00025-016-0564-5. |
[50] |
K. Wei, J.-F. Cai, T. F. Chan and S. Leung,
Guarantees of Riemannian optimization for low rank matrix recovery, SIAM Journal on Matrix Analysis nad Applications, 37 (2016), 1198-1222.
doi: 10.1137/15M1050525. |
[51] |
K. Wei, Solving systems of phaseless equations via Kaczmarz methods: A proof of concept study, Inverse Problems, 31 (2015), 125008, 23pp.
doi: 10.1088/0266-5611/31/12/125008. |
[52] |
Q. Zheng and J. Lafferty, A Convergent Gradient Descent Algorithm for Rank Minimization and Semidefinite Programming from Random Linear Measurements, NIPS, 2015. Google Scholar |
[53] |
T. Zhao, Z. Wang and H. Liu, Nonconvex low rank matrix factorization via inexact first order oracle, NIPS, 2015. Google Scholar |
[54] |
Q. Zheng and J. Lafferty, Convergence analysis for rectangular matrix completion using Burer-Monteiro factorization and gradient descent, 2016, arXiv: 1605.0705. Google Scholar |
show all references
References:
[1] |
Y. Amit, M. Fink, N. Srebro and S. Ullman, Uncovering shared structures in multiclass classification, in: Proceedings of the 24th International Conference on Machine Learning, 2007, 17–24.
doi: 10.1145/1273496.1273499. |
[2] |
A. Argyriou, T. Evgeniou and M. Pontil, Multi-task feature learning, Advances in Neural Information Processing Systems, 2007. Google Scholar |
[3] |
P.-A. Absil, R. Mahony and R. Sepulchre, Optimization Algorithms on Matrix Manifolds,, Princeton University Press, 2008.
doi: 10.1515/9781400830244.![]() ![]() |
[4] |
A. Ahmed, B. Recht and J. Romberg,
Blind deconvolution using convex programming, IEEE Transactions on Information Theory, 60 (2014), 1711-1732.
doi: 10.1109/TIT.2013.2294644. |
[5] |
R. Ahlswede and A. Winter,
Strong converse for identification via quantum channels, IEEE Transactions on Information Theory, 48 (2002), 569-579.
doi: 10.1109/18.985947. |
[6] |
S. Bhojanapalli, A. Kyrillidis and S. Sanghavi, Dropping convexity for faster semi-definite optimization, JMLR: Workshop and Conference Proceedings, 49 (2016), 1-53. Google Scholar |
[7] |
J. Blanchard, J. Tanner and K. Wei,
CGIHT: Conjugate gradient iterative hard thresholding for compressed sensing and matrix completion, Information and Inference, 4 (2015), 289-327.
doi: 10.1093/imaiai/iav011. |
[8] |
N. Boumal and P.-A. Absil, RTRMC: A Riemannian trust-region method for low-rank matrix completion, in: Advances in Neural Information Processing Systems, 2011. Google Scholar |
[9] |
E. J. Candès and B. Recht,
Exact matrix completion via convex optimization, Foundations of Computational Mathematics, 9 (2009), 717-772.
doi: 10.1007/s10208-009-9045-5. |
[10] |
E. J. Candès and Y. Plan,
Tight oracle bounds for low-rank matrix recovery from a minimal number of random measurements, IEEE Transactions on Information Theory, 57 (2011), 2342-2359.
doi: 10.1109/TIT.2011.2111771. |
[11] |
Y. Chen and M. J. Wainwright, Fast low-rank estimation by projected gradient descent: General statistical and algorithmic guarantees, arXiv: 1509.03025, 2015. Google Scholar |
[12] |
E. J. Candès and T. Tao,
The power of convex relaxation: Near-optimal matrix completion, IEEE Transactions on Information Theory, 56 (2009), 2053-2080.
doi: 10.1109/TIT.2010.2044061. |
[13] |
Y. Chen,
Incoherence-optimal matrix completion, IEEE Transactions on Information Theory, 61 (2015), 2909-2923.
doi: 10.1109/TIT.2015.2415195. |
[14] |
J.-F. Cai, E. J. Candès and Z. Shen,
A singular value thresholding algorithm for matrix completion, SIAM Journal on Optimization, 20 (2010), 1956-1982.
doi: 10.1137/080738970. |
[15] |
E. J. Candès, Y. Eldar, T. Strohmer and V. Voroninski,
Phase retrieval via matrix completion, SIAM J. on Imaging Sciences, 6 (2013), 199-225.
doi: 10.1137/110848074. |
[16] |
E. J. Candès, X. Li and M. Soltanolkotabi,
Phase retrieval via Wirtinger flow: Theory and algorithms, IEEE Transactions on Information Theory, 61 (2015), 1985-2007.
doi: 10.1109/TIT.2015.2399924. |
[17] |
Y. Chen and E. Candès,
Solving random quadratic systems of equations is nearly as easy as solving linear systems, Communications on Pure and Applied Mathematics, 70 (2017), 822-883.
doi: 10.1002/cpa.21638. |
[18] |
L. Eldén, Matrix Methods in Data Mining and Pattern Recogonization, SIAM, 2007.
doi: 10.1137/1.9780898718867. |
[19] |
M. Fazel, Matrix rank minimization with applications, ph. D. dissertation, Stanford University, 2002. Google Scholar |
[20] |
D. Gross,
Recovering low-rank matrices from few coefficients in any basis, IEEE Transactions on Information Theory, 57 (2011), 1548-1566.
doi: 10.1109/TIT.2011.2104999. |
[21] |
D. Goldfarb and S. Ma,
Convergence of fixed-point continuation algorithms for matrix rank minimization, Foundations of Computational Mathematics, 11 (2011), 183-210.
doi: 10.1007/s10208-011-9084-6. |
[22] |
R. W. Gerchberg and W. O. Saxton, A practical algorithm for the determination of the phase from image and diffraction plane pictures, Optik, 35 (1972), 237-246. Google Scholar |
[23] |
N. J. A. Harvey, D. R. Karger and S. Yekhanin, The complexity of matrix completion, in: Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithms, 2006, 1103–1111.
doi: 10.1145/1109557.1109679. |
[24] |
J. P. Haldar and D. Hernando,
Rank-constrained solutions to linear matrix equations using PowerFactorization, IEEE Signal Processing Letters, 16 (2009), 584-587.
doi: 10.1109/LSP.2009.2018223. |
[25] |
P. Jain, R. Meka and I. Dhillon, Guaranteed rank minimization via singular value projection, in: NIPS, 2010. Google Scholar |
[26] |
P. Jain, P. Netrapalli and S. Sanghavi, Low-rank matrix completion using alternating minimization, in: Proceedings of the Forty-Fifth Annual ACM Symposium on Theory of Computing, 2013,665–674.
doi: 10.1145/2488608.2488693. |
[27] |
P. Jain and P. Netrapalli, Fast exact matrix completion with finite samples, JMLR: Workshop and Conference Proceedings, 40 (2015), 1-28. Google Scholar |
[28] |
A. Kyrillidis and V. Cevher,
Matrix recipes for hard thresholding methods, Journal of Mathematical Imaging and Vision, 48 (2014), 235-265.
doi: 10.1007/s10851-013-0434-7. |
[29] |
R. H. Keshavan, Efficient algorithms for collaborative filtering, ph. D. dissertation, Stanford University, 2012. Google Scholar |
[30] |
R. H. Keshavan, A. Montanari and S. Oh,
Matrix completion from a few entries, IEEE Transactions on Information Theory, 56 (2010), 2980-2998.
doi: 10.1109/TIT.2010.2046205. |
[31] |
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. |
[32] |
K. Lee and Y. Bresler,
ADMiRA: Atomic decomposition for minimum rank approximation, IEEE Transactions on Information Theory, 128 (2010), 4402-4416.
doi: 10.1109/TIT.2010.2054251. |
[33] |
S. Ling and T. Strohmer,
Blind deconvolution meets blind demixing: Algorithms and performance bounds, IEEE Transactions on Information Theory, 63 (2017), 4497-4520.
doi: 10.1109/TIT.2017.2701342. |
[34] |
B. Mishra, K. A. Apuroop and R. Sepulchre, A Riemannian geometry for low-rank matrix completion, arXiv 1306.2672, 2013. Google Scholar |
[35] |
B. Mishra, G. Meyer, S. Bonnabel and R. Sepulchre,
Fixed-rank matrix factorizations and Riemannian low-rank optimization, Computational Statistics, 29 (2014), 591-621.
doi: 10.1007/s00180-013-0464-z. |
[36] |
T. Ngo and Y. Saad, Scaled Gradients on Grassmann Manifolds for Matrix Completion, in: Advances in Neural Information Processing Systems, 2012. Google Scholar |
[37] |
J. Nocedal and S. J. Wright, Numerical Optimization, Springer, 2006. |
[38] |
B. Recht, M. Fazel and P. A. Parrilo,
Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization, SIAM Review, 52 (2010), 471-501.
doi: 10.1137/070697835. |
[39] |
B. Recht,
A simpler approach to matrix completion, The Journal of Machine Learning Research, 12 (2011), 3413-3430.
|
[40] |
C. D. Sa, K. Olukotun, C. Ré, Global convergence of stochastic gradient descent for some nonconvex matrix problems, in: ICML, 2015. Google Scholar |
[41] |
R. Sun and Z. Luo, Guaranteed matrix completion via non-convex factorization, 2015 IEEE 56th Annual Symposium on Foundations of Computer Science–FOCS 2015, IEEE Computer Soc., Los Alamitos, CA, 2015,270–289. |
[42] |
J. Sun, Q. Qu and J. Wright,
A geometric analysis of phase retrieval, Foundations of Computational Mathematics, 18 (2018), 1131-1198.
doi: 10.1007/s10208-017-9365-9. |
[43] |
J. Tanner and K. Wei, Normalized iterative hard thresholding for matrix completion, SIAM Journal on Scientific Computing, 35 (2013), S104–S125.
doi: 10.1137/120876459. |
[44] |
S. Tu, R. Boczar, M. Simchowitz, M. Soltanolkotabi and B. Recht, Low-rank solutions of linear matrix equations via Procrustes flow, in: ICML, 2016. Google Scholar |
[45] |
J. Tanner and K. Wei,
Low rank matrix completion by alternating steepest descent methods, Applied and Computational Harmonic Analysis, 40 (2016), 417-429.
doi: 10.1016/j.acha.2015.08.003. |
[46] |
L. Vandenberghe and S. Boyd,
Semidefinite programming, SIAM Review, 38 (1996), 49-95.
doi: 10.1137/1038003. |
[47] |
B. Vandereycken,
Low rank matrix completion by Riemannian optimization, SIAM Journal on Optimization, 23 (2013), 1214-1236.
doi: 10.1137/110845768. |
[48] |
Z. Wen, W. Yin and Y. Zhang,
Solving a low-rank factorization model for matrix completion by a non-linear successive over-relaxation algorithm, Mathematical Programming Computation, 4 (2012), 333-361.
doi: 10.1007/s12532-012-0044-1. |
[49] |
C. D. White, S. Sanghavi and R. Ward,
The local convexity of solving systems of quadratic equations, Results in Mathematics, 71 (2017), 569-608.
doi: 10.1007/s00025-016-0564-5. |
[50] |
K. Wei, J.-F. Cai, T. F. Chan and S. Leung,
Guarantees of Riemannian optimization for low rank matrix recovery, SIAM Journal on Matrix Analysis nad Applications, 37 (2016), 1198-1222.
doi: 10.1137/15M1050525. |
[51] |
K. Wei, Solving systems of phaseless equations via Kaczmarz methods: A proof of concept study, Inverse Problems, 31 (2015), 125008, 23pp.
doi: 10.1088/0266-5611/31/12/125008. |
[52] |
Q. Zheng and J. Lafferty, A Convergent Gradient Descent Algorithm for Rank Minimization and Semidefinite Programming from Random Linear Measurements, NIPS, 2015. Google Scholar |
[53] |
T. Zhao, Z. Wang and H. Liu, Nonconvex low rank matrix factorization via inexact first order oracle, NIPS, 2015. Google Scholar |
[54] |
Q. Zheng and J. Lafferty, Convergence analysis for rectangular matrix completion using Burer-Monteiro factorization and gradient descent, 2016, arXiv: 1605.0705. Google Scholar |




Algorithm | SC | RA | PICC | LCR |
RGrad, RCG (I) | | | ||
RGrad, RCG (II) | | | ||
GD [54] | | | |
Algorithm | SC | RA | PICC | LCR |
RGrad, RCG (I) | | | ||
RGrad, RCG (II) | | | ||
GD [54] | | | |
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