NDLI: Iterative reweighted least squares for matrix rank minimization

Content Provider	IEEE Xplore Digital Library
Author	Mohan, K. Fazel, M.
Copyright Year	2010
Description	Author affiliation: Electrical Engineering Department, University of Washington, Seattle, USA (Mohan, K.; Fazel, M.)
Abstract	The classical compressed sensing problem is to find the sparsest solution to an underdetermined system of linear equations. A good convex approximation to this problem is to minimize the ℓ1 norm subject to affine constraints. The Iterative Reweighted Least Squares (IRLSp) algorithm (0 < p ≤ 1), has been proposed as a method to solve the ℓp (p ≤ 1) minimization problem with affine constraints. Recently Chartrand et al observed that IRLS-p with p < 1 has better empirical performance than ℓ1 minimization, and Daubechies et al gave ‘local’ linear and super-linear convergence results for IRLS-p with p = 1 and p < 1 respectively. In this paper we extend IRLS-p as a family of algorithms for the matrix rank minimization problem and we also present a related family of algorithms, sIRLS-p. We present guarantees on recovery of low-rank matrices for IRLS-1 under the Null Space Property (NSP). We also establish that the difference between the successive iterates of IRLS-p and sIRLS-p converges to zero and that the IRLS-0 algorithm converges to the stationary point of a non-convex rank-surrogate minimization problem. On the numerical side, we give a few efficient implementations for IRLS-0 and demonstrate that both sIRLS-0 and IRLS-0 perform better than algorithms such as Singular Value Thresholding (SVT) on a range of ‘hard’ problems (where the ratio of number of degrees of freedom in the variable to the number of measurements is large). We also observe that sIRLS-0 performs better than Iterative Hard Thresholding algorithm (IHT) when there is no apriori information on the low rank solution.
Starting Page	653
Ending Page	661
File Size	472723
Page Count	9
File Format	PDF
ISBN	9781424482153
e-ISBN	9781424482160
DOI	10.1109/ALLERTON.2010.5706969
Language	English
Publisher	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Publisher Date	2010-09-29
Publisher Place	USA
Access Restriction	Subscribed
Rights Holder	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subject Keyword	Null space Clustering algorithms Minimization Approximation algorithms Projection algorithms Compressed sensing Convergence
Content Type	Text
Resource Type	Article

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