http://dx.doi.org/10.1137/100781894">
 

Document Type

Journal Article

Department/Unit

Department of Mathematics

Title

Recovering low-rank and sparse components of matrices from incomplete and noisy observations

Language

English

Abstract

Many problems can be characterized by the task of recovering the low-rank and sparse components of a given matrix. Recently, it was discovered that this nondeterministic polynomial-time hard (NP-hard) task can be well accomplished, both theoretically and numerically, via heuristically solving a convex relaxation problem where the widely acknowledged nuclear norm and l1 norm are utilized to induce low-rank and sparsity. This paper studies the recovery task in the general settings that only a fraction of entries of the matrix can be observed and the observation is corrupted by both impulsive and Gaussian noise. We show that the resulting model falls into the applicable scope of the classical augmented Lagrangian method. Moreover, the separable structure of the new model enables us to solve the involved subproblems more efficiently by splitting the augmented Lagrangian function. Hence, some splitting numerical algorithms are developed for solving the new recovery model. Some preliminary numerical experiments verify that these augmented-Lagrangianbased splitting algorithms are easily implementable and surprisingly efficient for tackling the new recovery model. © 2011 Society for Industrial and Applied Mathematics.

Keywords

Alternating direction method, Augmented Lagrangian method, Low-rank, Matrix recovery, Principal component analysis, Sparse

Publication Date

2011

Source Publication Title

SIAM Journal on Optimization

Volume

21

Issue

1

Start Page

57

End Page

81

Publisher

Society for Industrial and Applied Mathematics

ISSN (print)

10526234

ISSN (electronic)

10957189

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