On lq Optimization and Matrix Completion

Rank minimization problems, which consist of finding a matrix of minimum rank subject to linear constraints, have been proposed in many areas of engineering and science. A specific problem is the matrix completion problem in which a low rank data matrix can be recovered from incomplete samples of its entries by solving a rank penalized least squares problem. The rank penalty is in fact the l(0) norm of the matrix singular values. A recent convex relaxation of this penalty is the commonly used l(1) norm of the matrix singular values. In this paper, we bridge the gap between these two penalties and propose the l(q), 0 < q < 1 penalized least squares problem for matrix completion. An iterative algorithm is developed by solving a non-standard optimization problem and a non-trivial convergence result is proved. We illustrate with simulations comparing the reconstruction quality of the three matrix singular value penalty functions: l(0), l(1), and l(q), 0 < q < 1.