Iterative reweighted least squares for matrix rank minimization

Karthik Mohan, Maryam Fazel
2010 2010 48th Annual Allerton Conference on Communication, Control, and Computing (Allerton)  
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 1 norm subject to affine constraints. The Iterative Reweighted Least Squares (IRLSp) algorithm (0 < p ≤ 1), has been proposed as a method to solve the p (p ≤ 1) minimization problem with affine constraints. Recently Chartrand et al observed that IRLSp with p < 1 has better empirical performance than 1
more » ... on, 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.
doi:10.1109/allerton.2010.5706969 fatcat:zqqw4gbesndobge3maqrtxsvuy