Recovery of Low-Rank Matrices Under Affine Constraints via a Smoothed Rank Function

• Published in: IEEE transactions on signal processing Vol. 62; no. 4; pp. 981 - 992
• Main Authors: Malek-Mohammadi, Mohammadreza, Babaie-Zadeh, Massoud, Amini, Arash, Jutten, Christian
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 15.02.2014; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: affine rank minimization; affine rank minimization (ARM); Algorithms; Applied sciences; Approximation algorithms; Approximation methods; Compressed sensing; compressive sensing; Computer Science; Detection, estimation, filtering, equalization, prediction; Engineering Sciences; Exact sciences and technology; Information, signal and communications theory; matrix completion; matrix completion (MC); Minimization; nuclear norm minimization; nuclear norm minimization (NNM); rank approximation; Sampling, quantization; Sensors; Signal and communications theory; Signal and Image Processing; Signal processing algorithms; Signal, noise; spherical section property; spherical section property (SSP); Telecommunications and information theory; Vectors; State of the art; nuclear norm minimization (NNM); rank approximation; spherical section property (SSP); Algorithm; Convex programming; Optimization; Accuracy; matrix completion (MC); affine rank minimization (ARM); Signal processing; Compressed sensing; compressive sensing; af fi ne rank minimization (ARM)
• ISSN: 1053-587X; 1941-0476; 1941-0476
• DOI: 10.1109/TSP.2013.2295557

In this paper, the problem of matrix rank minimization under affine constraints is addressed. The state-of-the-art algorithms can recover matrices with a rank much less than what is sufficient for the uniqueness of the solution of this optimization problem. We propose an algorithm based on a smooth approximation of the rank function, which practically improves recovery limits on the rank of the solution. This approximation leads to a non-convex program; thus, to avoid getting trapped in local solutions, we use the following scheme. Initially, a rough approximation of the rank function subject to the affine constraints is optimized. As the algorithm proceeds, finer approximations of the rank are optimized and the solver is initialized with the solution of the previous approximation until reaching the desired accuracy. On the theoretical side, benefiting from the spherical section property, we will show that the sequence of the solutions of the approximating programs converges to the minimum rank solution. On the experimental side, it will be shown that the proposed algorithm, termed SRF standing for smoothed rank function, can recover matrices, which are unique solutions of the rank minimization problem and yet not recoverable by nuclear norm minimization. Furthermore, it will be demonstrated that, in completing partially observed matrices, the accuracy of SRF is considerably and consistently better than some famous algorithms when the number of revealed entries is close to the minimum number of parameters that uniquely represent a low-rank matrix.