Research On Algorithm And Application Of Low Rank Matrices Restoration

Recovering a large matrix from limited measurements is a challenging task arising in many real applications, such as image inpainting, compressive sensing, and medical imaging, and these kinds of problems are mostly formulated as low-rank matrix approximation problems. Due to the rank operator being nonconvex and discontinuous,most of the recent theoretical studies use the nuclear norm as a convex relaxation and the low-rank matrix recovery problem is solved through minimization of the nuclear norm regularized problem. However, a major limitation of nuclear norm minimization is that all the singular values are simultaneously minimized and the rank may not be well approximated(Hu et al., 2013).Correspondingly, in this paper, we propose a new multistage algorithm, which makes use of the concept of Truncated Nuclear Norm Regularization(TNNR) proposed by Hu et al., 2013, and iterative support detection(ISD) proposed by Wang and Yin,2010, to promote and extend the previous algorithms, and make the low-rank matrix remodeling recovery algorithm more generalized. In summary, the contributions of this paper mainly include the following. First, the new multistage algorithm overcomes the above limitation of nuclear norm, which only minimizes the smallest singular values;Simultaneously, the new algorithm also solves the rank estimate of low-rank matrices in Hu et al., 2013, namely, how to search the truncated position of the smallest singular values;Besides matrix completion problems considered by Hu et al., 2013, the proposed method can be also extended to the general low-rank matrix recovery problems.Extensive experiments well validate the superiority of our new algorithms over other state-of-the-art methods.