lq 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 is recovered from incomplete samples of its entries by solving a rank penalized least squares problem. The rank penalty is in fact the l₀ norm of the matrix singular values. A convex relaxation of this penalty is the commonly used l₁ norm of the matrix singular values. In this paper we bridge the gap between these two penalties and propose a simple method for solving the l_q, q ∈ (0, 1), penalized least squares problem for matrix completion. We illustrate with simulations comparing our method to others in terms of solution quality.