On $l_q$ 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 l₀ “norm” of the matrix singular values. A recent 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 the lq, 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: l₀, l₁, and lq, 0 <; q <; 1.