The goal of affine matrix rank minimization problem is to reconstruct alow-rank or approximately low-rank matrix under linear constraints. In general,this problem is combinatorial and NP-hard. In this paper, a nonconvex fractionfunction is studied to approximate the rank of a matrix and translate thisNP-hard problem into a transformed affine matrix rank minimization problem. Theequivalence between these two problems is established, and we proved that theuniqueness of the global minimizer of transformed affine matrix rankminimization problem also solves affine matrix rank minimization problem ifsome conditions are satisfied. Moreover, we proved that the optimal solution tothe transformed affine matrix rank minimization problem can be approximatelyobtained by solving the regularization transformed affine matrix rankminimization problem for some proper smaller $\lambda>0$. Ultimately, the DCalgorithm is utilized to solve the regularization transformed affine matrixrank minimization problem and the numerical experiments on image inpaintingproblems show that our method performs effective in recovering low-rank images.
展开▼
