Ky Fan 2-k-Norm model for Low-Rank Matrix Recovery with ADMM

In order to recover a low-rank matrix, the nuclear norm minimization problem is generally used to instead of the rank function minimization problem. But it is difficult to satisfy the restricted isometry conditions of linear map. When the rank is large enough, this convex relaxation can fail to recover the matrix. To solve this problem, a new nonconvex model, Ky Fan 2-k-norm model, is proposed to replace the rank function. Extend the restricted isometry of vectors to the matrices, our model is more stable than the unclear norm model. The ADMM algorithm is used to transform the model into three subproblems, which is widely used in computer vision. To facilitate the update of X, we replace the model with a convex model of form -norm. Then we use the accelerated proximal gradient (APG) algorithm to calculate, and a closed form solution can be found by soft threshold operator. Extensive experiments on both synthetic data and real images demonstrate that the Ky Fan 2-k-norm model has better recovery ability than the nuclear norm model.