Low rank matrix completion: A smoothed l0-search

This paper focuses on algorithmic development for the low-rank matrix completion problem. It has been shown that in the l₀-search for low-rank matrix completion, the singular points in the objective function are the major reasons for failures. While different methods have been proposed to handle singular points, this paper rigorously analyzes them to show that there is a need for further improvement. To address the singularity issue, we propose a new objective function that is continuous everywhere. The new objective function is a good approximation of the original objective function in the sense that in the limit, the lower level sets of the new objective function are the closure of those of the original objective function. We formulate the matrix completion problem as the minimization of the new objective function and design a quasiNewton method to solve it. Simulations demonstrate that the new method achieves excellent numerical performance.