Iterative reweighted least squares for matrix rank minimization

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 ℓ_p (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.