## The Annals of Statistics

- Ann. Statist.
- Volume 40, Number 2 (2012), 639-1284

### Noisy matrix decomposition via convex relaxation: Optimal rates in high dimensions

Alekh Agarwal, Sahand Negahban, and Martin J. Wainwright

#### Abstract

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.

#### Article information

**Source**

Ann. Statist. Volume 40, Number 2 (2012), 1171-1197.

**Dates**

First available: 18 July 2012

**Permanent link to this document**

http://projecteuclid.org/euclid.aos/1342625465

**Digital Object Identifier**

doi:10.1214/12-AOS1000

**Zentralblatt MATH identifier**

06073789

**Mathematical Reviews number (MathSciNet)**

MR2985947

**Subjects**

Primary: 62F30: Inference under constraints 62F30: Inference under constraints

Secondary: 62H12: Estimation

**Keywords**

High-dimensional inference nuclear norm composite regularizers

#### Citation

Agarwal, Alekh; Negahban, Sahand; Wainwright, Martin J. Noisy matrix decomposition via convex relaxation: Optimal rates in high dimensions. The Annals of Statistics 40 (2012), no. 2, 1171--1197. doi:10.1214/12-AOS1000. http://projecteuclid.org/euclid.aos/1342625465.

#### Supplemental materials

- Supplementary material: Simulations and proofs. This supplementary material contains numerical simulations that demonstrate excellent agreement between the theoretical predictions and the practical behavior of our estimators. We also provide proofs for our upper and lower bounds, including slightly sharpened versions of Corollaries 2 and 6.Digital Object Identifier: doi: 10.1214/12-AOS1000SUPPSupplemental files available for subscribers.