Exact minimum rank approximation via Schatten p-norm minimization
Exact minimum rank approximation via Schatten p-norm minimization
复制标题
通过 Schatten p 范数最小化的精确最小秩近似
DOI：
10.1016/j.cam.2014.02.015
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发表时间：
2014-09-01
影响因子：
2.4
通讯作者：
Chen, Di-Rong
中科院分区：
文献类型：
--
作者：
Liu, Lu;Huang, Wei;Chen, Di-Rong
最小化具有给定仿射约束系统的矩阵的秩是为了恢复最低秩矩阵，在工程和科学中有许多重要应用。最近提出了一个凸松弛的秩最小化问题，最小化核范数，而不是秩的矩阵。Recht和Fazel的结论是，核范数最小化受仿射约束，是等价的秩最小化在一定条件下的秩限制等距性质。本文将核范数极小化的一些最新结果推广到Schatten p-范数极小化，并在线性仿射变换满足一定的限制等距性的条件下，给出了Schatten p-范数极小化的理论保证.我们的研究结果改善了以前的工作，其中恢复用于核范数最小化。提出了一种基于优化极小化算法的Schatten p-范数极小化问题的求解算法。通过使用近似奇异值分解过程，我们得到了一个快速和鲁棒的算法。对含噪声测量的矩阵完备化问题的数值结果表明，与已有的算法相比，该算法具有更高的重构精度和更短的重构时间. (C)2014由Elsevier B. V.出版
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 p-norm minimization and present a theoretical guarantee for the Schatten p-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 p-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. (C) 2014 Published by Elsevier B.V.