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Noisy matrix completion: understanding statistical guarantees for convex relaxation via nonconvex optimization. (English) Zbl 07271856

90C25 Convex programming
90C26 Nonconvex programming, global optimization
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[1] E. Abbe, J. Fan, K. Wang, and Y. Zhong, Entrywise eigenvector analysis of random matrices with low expected rank, Ann. Statist., 48 (2020), pp. 1452-1474. · Zbl 1450.62066
[2] A. Ahmed, B. Recht, and J. Romberg, Blind deconvolution using convex programming, IEEE Trans. Inform. Theory, 60 (2014), pp. 1711-1732. · Zbl 1360.94057
[3] J. Bai and S. Ng, Confidence intervals for diffusion index forecasts and inference for factor-augmented regressions, Econometrica, 74 (2006), pp. 1133-1150. · Zbl 1152.91721
[4] A. I. Barvinok, Problems of distance geometry and convex properties of quadratic maps, Discrete Comput. Geom., 13 (1995), pp. 189-202. · Zbl 0829.05025
[5] A. Beck and M. Teboulle, A fast iterative shrinkage-thresholding algorithm for linear inverse problems, SIAM J. Imaging Sci., 2 (2009), pp. 183-202. · Zbl 1175.94009
[6] N. Boumal, V. Voroninski, and A. Bandeira, The non-convex Burer-Monteiro approach works on smooth semidefinite programs, in Advances in Neural Information Processing Systems, Curran Associates, Red Hook, NY, 2016, pp. 2757-2765.
[7] S. Burer and R. D. Monteiro, A nonlinear programming algorithm for solving semidefinite programs via low-rank factorization, Math. Program., 95 (2003), pp. 329-357. · Zbl 1030.90077
[8] J.-F. Cai, E. J. Candès, and Z. Shen, A singular value thresholding algorithm for matrix completion, SIAM J. Optim., 20 (2010), pp. 1956-1982. · Zbl 1201.90155
[9] J.-F. Cai, T. Wang, and K. Wei, Fast and provable algorithms for spectrally sparse signal reconstruction via low-rank Hankel matrix completion, Appl. Comput. Harmon. Anal., 46 (2019), pp. 94-121. · Zbl 1442.94017
[10] T. T. Cai and W.-X. Zhou, Matrix completion via max-norm constrained optimization, Electron. J. Stat., 10 (2016), pp. 1493-1525. · Zbl 1342.62091
[11] E. Candès, X. Li, Y. Ma, and J. Wright, Robust principal component analysis?, J. ACM, 58 (2011), 11. · Zbl 1327.62369
[12] E. Candès, X. Li, and M. Soltanolkotabi, Phase retrieval via Wirtinger flow: Theory and algorithms, IEEE Trans. Inform. Theory, 61 (2015), pp. 1985-2007. · Zbl 1359.94069
[13] E. Candès and Y. Plan, Matrix completion with noise, Proc. IEEE, 98 (2010), pp. 925 –936.
[14] E. Candès and B. Recht, Exact matrix completion via convex optimization, Found. Comput. Math., 9 (2009), pp. 717-772. · Zbl 1219.90124
[15] E. Candès and T. Tao, The power of convex relaxation: Near-optimal matrix completion, IEEE Trans. Inform. Theory, 56 (2010), pp. 2053 –2080. · Zbl 1366.15021
[16] Y. Cao and Y. Xie, Poisson matrix recovery and completion, IEEE Trans. Signal Process., 64 (2016), pp. 1609-1620. · Zbl 1412.94024
[17] V. Chandrasekaran, S. Sanghavi, P. A. Parrilo, and A. S. Willsky, Rank-sparsity incoherence for matrix decomposition, SIAM J. Optim., 21 (2011), pp. 572-596. · Zbl 1226.90067
[18] J. Chen and X. Li, Model-free nonconvex matrix completion: Local minima analysis and applications in memory-efficient Kernel PCA, J. Mach. Learn. Res., 20 (2010), 39. · Zbl 1441.62157
[19] J. Chen, D. Liu, and X. Li, Nonconvex rectangular matrix completion via gradient descent without \(\ell_{2,\infty}\) regularization, IEEE Trans. Inform. Theory, 66 (2020), pp. 5806-5841. · Zbl 1448.90078
[20] Y. Chen, Incoherence-optimal matrix completion, IEEE Trans. Inform. Theory, 61 (2015), pp. 2909-2923. · Zbl 1359.15022
[21] Y. Chen and E. Candès, The projected power method: An efficient algorithm for joint alignment from pairwise differences, Comm. Pure Appl. Math., 71 (2018), pp. 1648-1714. · Zbl 06919696
[22] Y. Chen and E. J. Candès, Solving random quadratic systems of equations is nearly as easy as solving linear systems, Comm. Pure Appl. Math., 70 (2017), pp. 822-883, https://doi.org/10.1002/cpa.21638. · Zbl 1379.90024
[23] Y. Chen, C. Cheng, and J. Fan, Asymmetry helps: Eigenvalue and eigenvector analyses of asymmetrically perturbed low-rank matrices, Ann. Statist., to appear.
[24] Y. Chen and Y. Chi, Robust spectral compressed sensing via structured matrix completion, IEEE Trans. Inform. Theory, 60 (2014), pp. 6576-6601. · Zbl 1360.94064
[25] Y. Chen and Y. Chi, Harnessing structures in big data via guaranteed low-rank matrix estimation: Recent theory and fast algorithms via convex and nonconvex optimization, IEEE Signal Process. Mag., 35 (2018), pp. 14-31, https://doi.org/10.1109/MSP.2018.2821706.
[26] Y. Chen, Y. Chi, J. Fan, and C. Ma, Gradient descent with random initialization: Fast global convergence for nonconvex phase retrieval, Math. Program., 176 (2019), pp. 5-37. · Zbl 1415.90086
[27] Y. Chen, J. Fan, C. Ma, and K. Wang, Spectral method and regularized MLE are both optimal for top-\(K\) ranking, Ann. Statist., 47 (2019), pp. 2204-2235. · Zbl 1425.62038
[28] Y. Chen, J. Fan, C. Ma, and Y. Yan, Inference and uncertainty quantification for noisy matrix completion, Proc. Natl. Acad. Sci. USA, 116 (2019), pp. 22931-22937. · Zbl 1431.90117
[29] Y. Chen, L. J. Guibas, and Q. Huang, Near-optimal joint optimal matching via convex relaxation, in International Conference on Machine Learning (ICML), ACM, New York, 2014, pp. 100-108.
[30] Y. Chen, A. Jalali, S. Sanghavi, and C. Caramanis, Low-rank matrix recovery from errors and erasures, IEEE Trans. Inform. Theory, 59 (2013), pp. 4324-4337.
[31] Y. Chen and M. J. Wainwright, Fast Low-Rank Estimation by Projected Gradient Descent: General Statistical and Algorithmic Guarantees, preprint, https://arxiv.org/abs/1509.03025, 2015.
[32] Y. Cheng and R. Ge, Non-convex matrix completion against a semi-random adversary, Proc. Mach. Learn. Res., 75 (2018), pp. 1362-1394.
[33] Y. Chi, Y. M. Lu, and Y. Chen, Nonconvex optimization meets low-rank matrix factorization: An overview, IEEE Trans. Signal Process., 67 (2019), pp. 5239-5269. · Zbl 07123429
[34] M. A. Davenport and J. Romberg, An overview of low-rank matrix recovery from incomplete observations, IEEE J. Sel. Topics Signal Process., 10 (2016), pp. 608-622.
[35] L. Ding and Y. Chen, The Leave-One-Out Approach for Matrix Completion: Primal and Dual Analysis, preprint, https://arxiv.org/abs/1803.07554, 2018.
[36] B. Efron, Correlation and large-scale simultaneous significance testing, J. Amer. Statist. Assoc., 102 (2007), pp. 93-103. · Zbl 1284.62340
[37] B. Efron, Correlated z-values and the accuracy of large-scale statistical estimates, J. Amer. Statist. Assoc., 105 (2010), pp. 1042-1055. · Zbl 1390.62139
[38] N. El Karoui, On the impact of predictor geometry on the performance on high-dimensional ridge-regularized generalized robust regression estimators, Probab. Theory Related Fields, 170 (2018), pp. 95-175. · Zbl 1407.62060
[39] J. Fan, X. Han, and W. Gu, Estimating false discovery proportion under arbitrary covariance dependence, J. Amer. Statist. Assoc., 107 (2012), pp. 1019-1035. · Zbl 1395.62219
[40] J. Fan, Y. Ke, Q. Sun, and W.-X. Zhou, Farmtest: Factor-adjusted robust multiple testing with approximate false discovery control, J. Amer. Statist. Assoc., 114 (2019), pp. 1880-1893. · Zbl 1428.62345
[41] J. Fan, Y. Ke, and K. Wang, Factor-adjusted regularized model selection, J. Econometrics, 216 (2020), pp. 71-85. · Zbl 1456.62114
[42] J. Fan, Y. Liao, and M. Mincheva, Large covariance estimation by thresholding principal orthogonal complements, J. R. Stat. Soc. Ser. B Stat. Methodol., 75 (2013), pp. 603-680. · Zbl 1411.62138
[43] J. Fan, W. Wang, and Y. Zhong, Robust covariance estimation for approximate factor models, J. Econometrics, 208 (2019), pp. 5-22. · Zbl 1452.62410
[44] J. Fan, L. Xue, and J. Yao, Sufficient forecasting using factor models, J. Econometrics, 201 (2017), pp. 292-306. · Zbl 1377.62185
[45] M. Fazel, Matrix Rank Minimization with Applications, PhD thesis, Stanford University, Stanford, CA, 2002.
[46] M. Fazel, H. Hindi, and S. Boyd, Rank minimization and applications in system theory, in American Control Conference, Vol. 4, IEEE, Piscataway, NJ, 2004, pp. 3273-3278.
[47] M. Fazel, H. Hindi, and S. P. Boyd, Log-det heuristic for matrix rank minimization with applications to Hankel and Euclidean distance matrices, in American Control Conference, IEEE, Piscataway, NJ, 2003, pp. 2156-2162.
[48] M. Fornasier, H. Rauhut, and R. Ward, Low-rank matrix recovery via iteratively reweighted least squares minimization, SIAM J. Optim., 21 (2011), pp. 1614-1640. · Zbl 1236.65044
[49] R. Ge, C. Jin, and Y. Zheng, No spurious local minima in nonconvex low rank problems: A unified geometric analysis, International Conference on Machine Learning, ACM, New York, 2017, pp. 1233-1242.
[50] R. Ge, J. D. Lee, and T. Ma, Matrix completion has no spurious local minimum, in Advances in Neural Information Processing Systems, 2016, Curran Associates, Red Hook, NY, pp. 2973-2981.
[51] D. Gross, Recovering low-rank matrices from few coefficients in any basis, IEEE Trans. Inform. Theory, 57 (2011), pp. 1548-1566. · Zbl 1366.94103
[52] S. Gunasekar, A. Acharya, N. Gaur, and J. Ghosh, Noisy matrix completion using alternating minimization, in Joint European Conference on Machine Learning and Knowledge Discovery in Databases, Springer, Berlin, 2013, pp. 194-209.
[53] M. Hardt, Understanding alternating minimization for matrix completion, in Foundations of Computer Science (FOCS), IEEE, Piscataway, NJ, 2014, pp. 651-660.
[54] Q.-X. Huang and L. Guibas, Consistent shape maps via semidefinite programming, Comput. Graph. Forum, 32 (2013), pp. 177-186.
[55] P. Jain, R. Meka, and I. S. Dhillon, Guaranteed rank minimization via singular value projection, in Advances in Neural Information Processing Systems, Curran Associates, Red Hook, NY, 2010, pp. 937-945.
[56] P. Jain, P. Netrapalli, and S. Sanghavi, Low-rank matrix completion using alternating minimization, in ACM Symposium on Theory of Computing, ACM, New York, 2013, pp. 665-674. · Zbl 1293.65073
[57] C. Jin, S. M. Kakade, and P. Netrapalli, Provable efficient online matrix completion via non-convex stochastic gradient descent, in Advances in Neural Information Processing Systems, Curran Associates, Red Hook, NY, 2016, pp. 4520-4528.
[58] I. T. Jolliffe, A note on the use of principal components in regression, J. R. Stat. Soc. Ser. C Appl. Stat., 31 (1982), pp. 300-303.
[59] P. Jung, F. Krahmer, and D. Stöger, Blind demixing and deconvolution at near-optimal rate, IEEE Trans. Inform. Theory, 64 (2017), pp. 704-727. · Zbl 1390.94235
[60] R. H. Keshavan, A. Montanari, and S. Oh, Matrix completion from a few entries, IEEE Trans. Inform. Theory, 56 (2010), 2980 –2998. · Zbl 1366.62111
[61] R. H. Keshavan, A. Montanari, and S. Oh, Matrix completion from noisy entries, J. Mach. Learn. Res., 11 (2010), pp. 2057-2078. · Zbl 1242.62069
[62] O. Klopp, Noisy low-rank matrix completion with general sampling distribution, Bernoulli, 20 (2014), pp. 282-303. · Zbl 1400.62115
[63] A. Kneip and P. Sarda, Factor models and variable selection in high-dimensional regression analysis, Ann. Statist., 39 (2011), pp. 2410-2447. · Zbl 1231.62131
[64] V. Koltchinskii, K. Lounici, and A. B. Tsybakov, Nuclear-norm penalization and optimal rates for noisy low-rank matrix completion, Ann. Statist., 39 (2011), pp. 2302-2329, https://doi.org/10.1214/11-AOS894. · Zbl 1231.62097
[65] M.-J. Lai, Y. Xu, and W. Yin, Improved iteratively reweighted least squares for unconstrained smoothed \(\ell_q\) minimization, SIAM J. Numer. Anal., 51 (2013), pp. 927-957. · Zbl 1268.49038
[66] Q. Li and G. Tang, Approximate support recovery of atomic line spectral estimation: A tale of resolution and precision, Appl. Comput. Harmon. Anal., 48 (2020), pp. 891-948. · Zbl 1436.62081
[67] Y. Li, C. Ma, Y. Chen, and Y. Chi, Nonconvex matrix factorization from rank-one measurements, Proc. Mach. Learn. Res., 89 (2019), pp. 1496-1505.
[68] S. Ling and T. Strohmer, Self-calibration and biconvex compressive sensing, Inverse Problems, 31 (2015), 115002. · Zbl 1327.93183
[69] S. Ling and T. Strohmer, Blind deconvolution meets blind demixing: Algorithms and performance bounds, IEEE Trans. Inform. Theory, 63 (2017), pp. 4497-4520. · Zbl 1370.94583
[70] Z. Liu and L. Vandenberghe, Interior-point method for nuclear norm approximation with application to system identification, SIAM J. Matrix Anal. Appl., 31 (2009), pp. 1235-1256. · Zbl 1201.90151
[71] C. Ma, K. Wang, Y. Chi, and Y. Chen, Implicit regularization in nonconvex statistical estimation: Gradient descent converges linearly for phase retrieval, matrix completion, and blind deconvolution, Found. Comput. Math., 20 (2020), pp. 451-632. · Zbl 1445.90089
[72] S. Ma, D. Goldfarb, and L. Chen, Fixed point and Bregman iterative methods for matrix rank minimization, Math. Program., 128 (2011), pp. 321-353. · Zbl 1221.65146
[73] R. Mazumder, T. Hastie, and R. Tibshirani, Spectral regularization algorithms for learning large incomplete matrices, J. Mach. Learn. Res., 11 (2010), pp. 2287-2322. · Zbl 1242.68237
[74] S. Negahban and M. J. Wainwright, Restricted strong convexity and weighted matrix completion: Optimal bounds with noise, J. Mach. Learn. Res., 13 (2012), pp. 1665-1697. · Zbl 1436.62204
[75] Y. Nesterov, How to make the gradients small, Optima, 88 (2012), pp. 10-11.
[76] N. Parikh and S. Boyd, Proximal algorithms, Found. Trends Optim., 1 (2014), pp. 127-239.
[77] D. Park, A. Kyrillidis, C. Carmanis, and S. Sanghavi, Non-square matrix sensing without spurious local minima via the Burer-Monteiro approach, Proc. Mach. Learn. Res., 54 (2017), pp. 65-74.
[78] D. Paul, E. Bair, T. Hastie, and R. Tibshirani, “Preconditioning” for feature selection and regression in high-dimensional problems, Ann. Statist., 36 (2008), pp. 1595-1618. · Zbl 1142.62022
[79] B. Recht, A simpler approach to matrix completion, J. Mach. Learn. Res., 12 (2011), pp. 3413-3430. · Zbl 1280.68141
[80] B. Recht, M. Fazel, and P. A. Parrilo, Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization, SIAM Rev., 52 (2010), pp. 471-501. · Zbl 1198.90321
[81] J. D. Rennie and N. Srebro, Fast maximum margin matrix factorization for collaborative prediction, in International conference on Machine Learning, ACM, New York, 2005, pp. 713-719.
[82] A. Rohde and A. B. Tsybakov, Estimation of high-dimensional low-rank matrices, Ann. Statist., 39 (2011), pp. 887-930. · Zbl 1215.62056
[83] A. Shapiro, Y. Xie, and R. Zhang, Matrix completion with deterministic pattern: A geometric perspective, IEEE Trans. Signal Process., 67 (2019), pp. 1088-1103. · Zbl 1414.15034
[84] A. Singer, Angular synchronization by eigenvectors and semidefinite programming, Appl. Comput. Harmon. Anal., 30 (2011), pp. 20-36. · Zbl 1206.90116
[85] A. M.-C. So and Y. Ye, Theory of semidefinite programming for sensor network localization, Math. Program., 109 (2007), pp. 367-384. · Zbl 1278.90482
[86] N. Srebro and A. Shraibman, Rank, trace-norm and max-norm, in International Conference on Computational Learning Theory, Springer, Berlin, 2005, pp. 545-560. · Zbl 1137.68563
[87] R. Sun and Z.-Q. Luo, Guaranteed matrix completion via non-convex factorization, IEEE Trans. Inform. Theory, 62 (2016), pp. 6535-6579. · Zbl 1359.94179
[88] P. Sur, Y. Chen, and E. J. Candès, The likelihood ratio test in high-dimensional logistic regression is asymptotically a rescaled \(c\) hi-square, Probab. Theory Related Fields, 175 (2019), pp. 487-558. · Zbl 1431.62319
[89] K.-C. Toh and S. Yun, An accelerated proximal gradient algorithm for nuclear norm regularized linear least squares problems, Pac. J. Optim., 6 (2010), pp. 615-640. · Zbl 1205.90218
[90] C. Tomasi and T. Kanade, Shape and motion from image streams under orthography: A factorization method, Int. J. Comput. Vis., 9 (1992), pp. 137-154.
[91] S. Tu, R. Boczar, M. Simchowitz, M. Soltanolkotabi, and B. Recht, Low-rank solutions of linear matrix equations via Procrustes flow, in International Conference on Machine Learning, ACM, New York, 2016, pp. 964-973.
[92] B. Vandereycken, Low-rank matrix completion by Riemannian optimization, SIAM J. Optim., 23 (2013), pp. 1214-1236. · Zbl 1277.15021
[93] R. Vershynin, Introduction to the non-asymptotic analysis of random matrices, in Compressed Sensing, Theory and Applications, Cambridge University Press, Cambridge, 2012, pp. 210-268.
[94] L. Wang, X. Zhang, and Q. Gu, A unified computational and statistical framework for nonconvex low-rank matrix estimation, Proc. Mach. Learn. Res., 54 (2017), pp. 981-990.
[95] K. Wei, J.-F. Cai, T. F. Chan, and S. Leung, Guarantees of Riemannian optimization for low rank matrix recovery, SIAM J. Matrix Anal. Appl., 37 (2016), pp. 1198-1222. · Zbl 1347.65109
[96] Z. Wen, W. Yin, and Y. Zhang, Solving a low-rank factorization model for matrix completion by a nonlinear successive over-relaxation algorithm, Math. Program. Comput., 4 (2012), pp. 333-361. · Zbl 1271.65083
[97] H. Zhang, Y. Zhou, Y. Liang, and Y. Chi, A nonconvex approach for phase retrieval: Reshaped Wirtinger flow and incremental algorithms, J. Mach. Learn. Res., 18 (2017), pp. 5164-5198.
[98] T. Zhang, J. M. Pauly, and I. R. Levesque, Accelerating parameter mapping with a locally low rank constraint, Magn. Resonance Med., 73 (2015), pp. 655-661.
[99] T. Zhao, Z. Wang, and H. Liu, A nonconvex optimization framework for low rank matrix estimation, in Advances in Neural Information Processing Systems, Curran Associates, Red Hook, NY, 2015, pp. 559-567.
[100] Q. Zheng and J. Lafferty, Convergence Analysis for Rectangular Matrix Completion using Burer-Monteiro Factorization and Gradient Descent, preprint, https://arxiv.org/abs/1605.07051, 2016.
[101] Y. Zhong and N. Boumal, Near-optimal bounds for phase synchronization, SIAM J. Optim., 28 (2018), pp. 989-1016. · Zbl 1396.90068
[102] Z. Zhou, X. Li, J. Wright, E. Candès, and Y. Ma, Stable principal component pursuit, in International Symposium on Information Theory, IEEE, Piscataway, NJ, 2010, pp. 1518-1522.
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