Promoting low rank non convex fraction function penalty for affine matrix rank minimization
Affine matrix rank minimization problem is a fundamental problem with a lot of important applications in many fields. It is well known that this problem is combinatorial and NP hard in general. Therefore, it is important to choose the suitable substitution for this matrix rank minimization problem. In this paper, a promoting low rank non convex fraction function is studied to replace the rank function, and then affine matrix rank minimization problem can be translated into a transformed affine matrix rank minimization problem. An iterative singular value thresholding algorithm is proposed to solve the regularization transformed affine matrix rank minimization problem. Moreover, we proved that the optimal solution to the regularization transformed affine affine matrix rank minimization problem also solves the transformed affine matrix rank minimization problem if the certain condition is satisfied, which is similar to the strong duality theory in the MC/MC framework, and we also proved that the value of the regularization parameter \lambda>0$ can not be chosen too larger which avoid leading to a zero solution. Indeed, there exists \bar{\lambda}>0$ such that the optimal solution of the regularization transformed affine matrix rank minimization problem equals to zero for any \lambda>\bar{\lambda} Numerical experiments on completion of low rank random matrices show that our method performs powerful in finding a low rank matrix and the numerical experiments for the image inpainting problems show that our algorithm has better performances than some state of art methods.