Exact minimum rank approximation via Schatten pp-norm minimization

Minimizing the rank of a matrix with a given system of affine constraints is to recover the lowest-rank matrix with many important applications in engineering and science. A convex relaxation of the rank minimization problem by minimizing the nuclear norm instead of the rank of the matrix has recently been proposed. Recht and Fazel concluded that nuclear norm minimization subject to affine constraints is equivalent to rank minimization under a certain condition in terms of the rank-restricted isometry property. In this paper, we extend some recent results from nuclear norm minimization to Schatten pp-norm minimization and present a theoretical guarantee for the Schatten pp-norm minimization if a certain restricted isometry property holds for the linear affine transform. Our results improve on the previous works where recovery is used for nuclear norm minimization. An algorithm based on the Majorization Minimization algorithm has been proposed to solve Schatten pp-norm minimization. By using an approximate singular value decomposition procedure, we obtain a fast and robust algorithm. The numerical results on the matrix completion problem with noisy measurements indicate that our algorithm gives a more accurate reconstruction and takes less time compared with some other existing algorithms.