# Approximation of rank function and its application to the nearest low-rank correlation matrix

The rank function $\rank(\cdot)$ is neither continuous nor convex which brings much difficulty to the solution of rank minimization problems. In this paper, we provide a unified framework to construct the approximation functions of $\rank(\cdot)$, and study their favorable properties. Particularly, with two families of approximation functions, we propose a convex relaxation method for the rank minimization problems with positive semidefinite cone constraints, and illustrate its application by computing the nearest low-rank correlation matrix. Numerical comparisons with the convex relaxation method in \cite{LQ09} indicate that our method tends to yield a better local optimal solution.

## Citation

Department of Mathematics, South China University of Technology, Guangzhou City, China, July 10, 2011