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Estimation of high-dimensional low-rank matrices. (English) Zbl 1215.62056

Summary: Suppose that we observe entries or, more generally, linear combinations of entries of an unknown \(m\times T\)-matrix \(A\) corrupted by noise. We are particularly interested in the high-dimensional setting where the number \(mT\) of unknown entries can be much larger than the sample size \(N\). Motivated by several applications, we consider estimation of a matrix \(A\) under the assumption that it has small rank. This can be viewed as dimension reduction or sparsity assumption. In order to shrink toward a low-rank representation, we investigate penalized least squares estimators with a Schatten-\(p\) quasi-norm penalty term, \(p\leq 1\). We study these estimators under two possible assumptions-a modified version of the restricted isometry condition and a uniform bound on the ratio “empirical norm induced by the sampling operator/Frobenius norm.” The main results are stated as non-asymptotic upper bounds on the prediction risk and on the Schatten-\(q\) risk of the estimators, where \(q\in [p, 2]\). The rates that we obtain for the prediction risk are of the form \(rm/N\) (for \(m=T)\), up to logarithmic factors, where \(r\) is the rank of \(A\). The particular examples of multi-task learning and matrix completion are worked out in detail. The proofs are based on tools from the theory of empirical processes. As a by-product, we derive bounds for the \(k\) th entropy numbers of the quasi-convex Schatten class embeddings \(S_{p}^{M}\hookrightarrow S_{2}^{M},\, p<1\), which are of independent interest.

MSC:

62H12 Estimation in multivariate analysis
62G05 Nonparametric estimation
62F10 Point estimation
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[1] Abernethy, J., Bach, F., Evgeniou, T. and Vert, J.-P. (2009). A new approach to collaborative filtering: Operator estimation with spectral regularization. J. Mach. Learn. Res. 10 803-826. · Zbl 1235.68122
[2] Amini, A. and Wainwright, M. (2009). High-dimensional analysis of semidefinite relaxations for sparse principal components. Ann. Statist. 37 2877-2921. · Zbl 1173.62049
[3] Argyriou, A., Evgeniou, T. and Pontil, M. (2008). Convex multi-task feature learning. Mach. Learn. 73 243-272.
[4] Argyriou, A., Micchelli, C. A. and Pontil, M. (2010). On spectral learning. J. Mach. Learn. Res. · Zbl 1242.68201
[5] Argyriou, A., Micchelli, C. A., Pontil, M. and Ying, Y. (2008). A spectral regularization framework for multi-task structure learning. In Advances in Neural Information Processing Systems 20 (J.C. Platt, et al., eds.) 25-32. MIT Press, Cambridge, MA.
[6] Bach, F. R. (2008). Consistency of trace norm minimization. J. Mach. Learn. Res. 9 1019-1048. · Zbl 1225.68146
[7] Bickel, P. J. and Levina, E. (2008). Regularized estimation of large covariance matrices. Ann. Statist. 36 199-227. · Zbl 1132.62040
[8] Bickel, P. J., Ritov, Y. and Tsybakov, A. B. (2009). Simultaneous analysis of lasso and Dantzig selector. Ann. Statist. 37 1705-1732. · Zbl 1173.62022
[9] Bunea, F., She, Y. and Wegkamp, M. H. (2010). Optimal selection of reduced rank estimators of high-dimensional matrices. Available at . · Zbl 1216.62086
[10] Bunea, F., Tsybakov, A. B. and Wegkamp, M. H. (2007). Aggregation for Gaussian regression. Ann. Statist. 35 1674-1697. · Zbl 1209.62065
[11] Cai, T., Zhang, C.-H. and Zhou, H. H. (2010). Optimal rates of convergence for covariance matrix estimation. Ann. Statist. 38 2118-2144. · Zbl 1202.62073
[12] Candès, E. J. and Plan, Y. (2010a). Matrix completion with noise. Proc. IEEE 98 925-936.
[13] Candès, E. J. and Plan, Y. (2010b). Tight oracle bounds for low-rank matrix recovery from a mininal number of noisy random measurements. Available at . · Zbl 1366.90160
[14] Candès, E. J. and Recht, B. (2009). Exact matrix completion via convex optimization. Found. Comput. Math. 9 717-772. · Zbl 1219.90124
[15] Candès, E. J. and Tao, T. (2005). Decoding by linear programming. IEEE Trans. Inform. Theory 51 4203-4215. · Zbl 1264.94121
[16] Candès, E. J. and Tao, T. (2009). The power of convex relaxation: Near-optimal matrix completion. Unpublished manuscript. · Zbl 1366.15021
[17] McCarthy, C. A. (1967). C p . Israel J. Math. 5 249-272. · Zbl 0156.37902
[18] Dalalyan, A. and Tsybakov, A. (2008). Aggregation by exponential weighting, sharp oracle inequalities and sparsity. Mach. Learn. 72 39-61.
[19] Donoho, D. L., Johnstone, I. M., Hoch, J. C. and Stern, A. S. (1992). Maximum entropy and the nearly black object. J. Roy. Statist. Soc. Ser. B 54 41-81. JSTOR: · Zbl 0788.62103
[20] Edmunds, D. E. and Triebel, H. (1996). Function Spaces, Entropy Numbers, Differential Operators . Cambridge Univ. Press, Cambridge. · Zbl 0865.46020
[21] Edmunds, D. E. and Triebel, H. (1989). Entropy numbers and approximation numbers in function spaces. Proc. London Math. Soc. 58 137-152. · Zbl 0629.46034
[22] Foster, D. P. and George, E. I. (1994). The risk inflation criterion for multiple regression. Ann. Statist. 22 1947-1975. · Zbl 0829.62066
[23] Guédon, O. and Litvak, A. E. (2000). Euclidean projections of a p -convex body. In Geometric Aspects of Functional Analysis, Israel Seminar (GAFA) 1996-2000 (V. D. Milman and G. Schechtman, eds.). Lecutre Notes in Mathematics 1745 95-108. Springer, Berlin. · Zbl 0988.52010
[24] Gross, D. (2009). Recovering low-rank matrices from few coefficients in any basis. Available at . · Zbl 1366.94103
[25] Keshavan, R. H., Montanari, A. and Oh, S. (2009). Matrix completion from noisy entries. Available at . · Zbl 1242.62069
[26] Kolmogorov, A. N. and Tikhomirov, V. M. (1959). The \epsilon -entropy and \epsilon -capacity of sets in function spaces. Uspekhi Matem. Nauk 14 3-86. · Zbl 0090.33503
[27] Koltchinskii, V. (2008). Oracle inequalities in empirical risk minimization and sparse recovery problems. Ecole d’Eté de Probabilités de Saint-Flour, Lecture Notes .
[28] Lounici, K., Pontil, M., Tsybakov, A. B. and van de Geer, S. (2009). Taking advantage of sparsity in multi-task learning. In Proceedings of COLT-2009 .
[29] Lounici, K., Pontil, M., Tsybakov, A. B. and van de Geer, S. (2010). Oracle inequalities and optimal inference under group sparsity. Available at . · Zbl 1306.62156
[30] Meinshausen, N. and Bühlmann, P. (2006). High-dimensional graphs and variable selection with the lasso. Ann. Statist. 34 1436-1462. · Zbl 1113.62082
[31] Mendelson, S., Pajor, A. and Tomczak-Jaegermann, N. (2007). Reconstruction and subgaussian operators in asymptotic geometric analysis. Geom. Funct. Anal. 17 1248-1282. · Zbl 1163.46008
[32] Negahban, S., Ravikumar, P., Wainwright, M. J. and Yu, B. (2009). A unified framework for high-dimensional analysis of M -estimators with decomposable regularizers. In Advances in Neural Information Processing Systems, NIPS-2009 . · Zbl 1331.62350
[33] Negahban, S. and Wainwright, M. J. (2011). Estimation of (near) low rank matrices with noise and high-dimensional scaling. Ann. Statist. To appear. Available at . · Zbl 1216.62090
[34] Nemirovski, A. (2004). Regular Banach spaces and large deviations of random sums. Unpublished manuscript. · Zbl 1106.90059
[35] Pajor, A. (1998). Metric entropy of the Grassmann manifold. Convex Geom. Anal. 34 181-188. · Zbl 0942.46013
[36] Paulsen, V. I. (1986). Completely Bounded Maps and Dilations . In Pitman Research Notes in Mathematics 146 . Longman, New York. · Zbl 0614.47006
[37] Pietsch, A. (1980). Operator Ideals . Elsevier, Amsterdam. · Zbl 0434.47030
[38] Pinelis, I. F. and Sakhanenko, A. I. (1985). Remarks on inequalities for the probabilities of large deviations. Theory Probab. Appl. 30 143-148. · Zbl 0583.60023
[39] Ravikumar, P., Wainwright, M., Raskutti, G. and Yu, B. (2008). High-dimensional covariance estimation by minimizing \ell 1 -penalized log-determinant divergence. Unpublished manuscript. · Zbl 1274.62190
[40] Recht, B. (2009). A simpler approach to matrix completion. Available at . · Zbl 1280.68141
[41] Recht, B., Fazel, M. and Parrilo, P. A. (2010). Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization. SIAM Rev. 52 471-501. · Zbl 1198.90321
[42] Rigollet, P. and Tsybakov, A. B. (2010). Exponential screening and optimal rates of sparse estimation. Available at . · Zbl 1215.62043
[43] Rotfeld, S. Y. (1969). The singular numbers of the sum of completely continuous operators. In Topics in Mathematical Physics (M. S. Birman, ed.). Spectral Theory 3 73-78. English version published by Consultants Bureau, New York. · Zbl 0195.13701
[44] Srebro, N., Rennie, J. and Jaakkola, T. (2005). Maximum margin matrix factorization. In Advances in Neural Information Processing Systems 17 (L. Saul, Y. Weiss and L. Bottou, eds.) 1329-1336. MIT Press, Cambridge, MA.
[45] Srebro, N. and Shraibman, A. (2005). Rank, trace-norm and max-norm. In Learning Theory, Proceedings of COLT-2005. Lecture Notes in Comput. Sci. 3559 545-560. Springer, Berlin. · Zbl 1137.68563
[46] Tropp, J. A. (2010). User-friendly tail bounds for sums of random matrices. Available at . · Zbl 1259.60008
[47] Tsybakov, A. (2009). Introduction to Nonparametric Estimation . Springer, New York. · Zbl 1176.62032
[48] Tsybakov, A. and van de Geer, S. (2005). Square root penalty: Adaptation to the margin in classification and in edge estimation. Ann. Statist. 33 1203-1224. · Zbl 1080.62047
[49] van de Geer, S. (2000). Empirical Processes in M-estimation . Cambridge Univ. Press, Cambridge. · Zbl 1179.62073
[50] Vershynin, R. (2007). Some problems in asymptotic convex geometry and random matrices motivated by numerical algorithms. In Banach Spaces and Their Applications in Analysis (B. Randrianantoanina and N. Randrianantoanina, eds.) 209-218. de Gruyter, Berlin. · Zbl 1173.90476
[51] Zhao, P. and Yu, B. (2006). On model selection consistency of lasso. J. Mach. Learn. Res. 7 2541-2563. · Zbl 1222.62008
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