On l q Optimization and Matrix Completion

• Published in: IEEE transactions on signal processing Vol. 60; no. 11; pp. 5714 - 5724
• Main Authors: Marjanovic, Goran, Solo, Victor
• Format: Journal Article
• Language: English
• Published: 01.11.2012
• Subjects: Computer simulation; Convergence; Least squares method; Norms; Optimization; Reconstruction; Signal processing; Transaction processing
• ISSN: 1053-587X; 1941-0476
• DOI: 10.1109/TSP.2012.2212015

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 .