An approximation theory of matrix rank minimization and its application to quadratic equations

作者：Zhao Yun Bin^*

来源：Linear Algebra and Its Applications, 2012, 437(1): 77-93.

DOI：10.1016/j.laa.2012.02.021

• Matrix rank minimization
• Singular values
• Matrix norms
• Semidefinite programming
• Duality theory
• Quadratic equations

Matrix rank minimization problems are gaining plenty of recent attention in both mathematical and engineering fields. This class of problems, arising in various and across-discipline applications, is known to be NP-hard in general. In this paper, we aim at providing an approximation theory for the rank minimization problem, and prove that a rank minimization problem can be approximated to any level of accuracy via continuous optimization (especially, linear and nonlinear semidefinite programming) problems. One of the main results in this paper shows that if the feasible set of the problem has a minimum rank element with the least Frobenius norm, then any accumulation point of solutions to the approximation problem, as the approximation parameter tends to zero, is a minimum rank solution of the original problem. The tractability under certain conditions and convex relaxation of the approximation problem are also discussed. An immediate application of this theory to the system of quadratic equations is presented in this paper. It turns out that the condition for such a system without a nonzero solution can be characterized by a rank minimization problem, and thus the proposed approximation theory can be used to establish some sufficient conditions for the system to possess only zero solution.

• 出版日期2012-7-1