Open Access
April 2012 Noisy matrix decomposition via convex relaxation: Optimal rates in high dimensions
Alekh Agarwal, Sahand Negahban, Martin J. Wainwright
Ann. Statist. 40(2): 1171-1197 (April 2012). DOI: 10.1214/12-AOS1000


We analyze a class of estimators based on convex relaxation for solving high-dimensional matrix decomposition problems. The observations are noisy realizations of a linear transformation $\mathfrak{X}$ of the sum of an (approximately) low rank matrix $\Theta^{\star}$ with a second matrix $\Gamma^{\star}$ endowed with a complementary form of low-dimensional structure; this set-up includes many statistical models of interest, including factor analysis, multi-task regression and robust covariance estimation. We derive a general theorem that bounds the Frobenius norm error for an estimate of the pair $(\Theta^{\star},\Gamma^{\star})$ obtained by solving a convex optimization problem that combines the nuclear norm with a general decomposable regularizer. Our results use a “spikiness” condition that is related to, but milder than, singular vector incoherence. We specialize our general result to two cases that have been studied in past work: low rank plus an entrywise sparse matrix, and low rank plus a columnwise sparse matrix. For both models, our theory yields nonasymptotic Frobenius error bounds for both deterministic and stochastic noise matrices, and applies to matrices $\Theta^{\star}$ that can be exactly or approximately low rank, and matrices $\Gamma^{\star}$ that can be exactly or approximately sparse. Moreover, for the case of stochastic noise matrices and the identity observation operator, we establish matching lower bounds on the minimax error. The sharpness of our nonasymptotic predictions is confirmed by numerical simulations.


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Alekh Agarwal. Sahand Negahban. Martin J. Wainwright. "Noisy matrix decomposition via convex relaxation: Optimal rates in high dimensions." Ann. Statist. 40 (2) 1171 - 1197, April 2012.


Published: April 2012
First available in Project Euclid: 18 July 2012

zbMATH: 1274.62219
MathSciNet: MR2985947
Digital Object Identifier: 10.1214/12-AOS1000

Primary: 62F30 , 62F30
Secondary: 62H12

Keywords: composite regularizers , high-dimensional inference , nuclear norm

Rights: Copyright © 2012 Institute of Mathematical Statistics

Vol.40 • No. 2 • April 2012
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