Affine matrix rank minimization problem via non-convex fraction function penalty

作者：Cui, Angang; Peng, Jigen^*; Li, Haiyang; Zhang, Chengyi; Yu, Yongchao

来源：Journal of Computational and Applied Mathematics, 2018, 336: 353-374.

DOI：10.1016/j.cam.2017.12.048

• Affine matrix rank minimization
• Low-rank
• Matrix completion
• Fraction function
• Iterative singular value thresholding algorithm
• Image inpainting

Affine matrix rank minimization problem is a fundamental problem in many important applications. It is well known that this problem is combinatorial and NP-hard in general. In this paper, a continuous promoting low rank non-convex fraction function is studied to replace the rank function in this NP-hard problem. An iterative singular value thresholding algorithm is proposed to solve the regularization transformed affine matrix rank minimization problem. With the change of the parameter in non-convex fraction function, we could get some much better results, which is one of the advantages for the iterative singular value thresholding algorithm compared with some state-of-art methods. Some convergence results are established. Moreover, we proved that the value of the regularization parameter lambda > 0 cannot be chosen too large. Indeed, there exists (lambda) over bar > 0 such that the optimal solution of the regularization transformed affine matrix rank minimization problem is equal to zero for any lambda > (lambda) over bar. Numerical experiments on matrix completion problems and image inpainting problems show that our method performs effective in finding a low-rank matrix compared with some state-of-art methods.

• 出版日期2018-7
• 单位西安交通大学; 广州大学; 西安工程大学