文献类型：期刊文献

中文题名：The bounds of restricted isometry constants for low rank matrices recovery

英文题名：The bounds of restricted isometry constants for low rank matrices recovery

作者：Wang HuiMin[1,2];Li Song[2]

机构：[1]Shaoxing Univ, Dept Math, Shaoxing 312000, Peoples R China;[2]Zhejiang Univ, Dept Math, Hangzhou 310027, Zhejiang, Peoples R China

年份：2013

卷号：0

期号：6

起止页码：1117

中文期刊名：中国科学：数学英文版

外文期刊名：SCIENCE CHINA-MATHEMATICS

收录：SCI-EXPANDED(收录号：WOS:000320365400002)、、Scopus(收录号：2-s2.0-84878786681)、WOS、PubMed

基金：This work was supported by National Natural Science Foundation of China (Grant Nos. 91130009, 11171299 and 11041005) and National Natural Science Foundation of Zhejiang Province in China (Grant Nos. Y6090091 and Y6090641).

语种：英文

中文关键词：矩阵;常数;等距;Schatten;最小化问题;回收;线性系统;线性映射

外文关键词：restricted isometry constants; low-rank matrix recovery; Schatten-p norm; nuclear norm; compressed sensing; convex optimization

中文摘要：This paper discusses conditions under which the solution of linear system with minimal Schatten-p norm, 0 < p 1, is also the lowest-rank solution of this linear system. To study this problem, an important tool is the restricted isometry constant (RIC). Some papers provided the upper bounds of RIC to guarantee that the nuclear-norm minimization stably recovers a low-rank matrix. For example, Fazel improved the upper bounds to δA4r < 0.558 and δA3r < 0.4721, respectively. Recently, the upper bounds of RIC can be improved to δA2r < 0.307. In fact, by using some methods, the upper bounds of RIC can be improved to δA2r < 0.4931 and δ A r < 0.309. In this paper, we focus on the lower bounds of RIC, we show that there exists linear maps A with δA2r > 1/ 2^(1/2) or δAr > 1/3 for which nuclear norm recovery fail on some matrix with rank at most r. These results indicate that there is only a little limited room for improving the upper bounds for δA2r and δAr . Furthermore, we also discuss the upper bound of restricted isometry constant associated with linear maps A for Schatten p (0 < p < 1) quasi norm minimization problem.

外文摘要：This paper discusses conditions under which the solution of linear system with minimal Schatten-p norm, 0 < p a (c) 1/2 1, is also the lowest-rank solution of this linear system. To study this problem, an important tool is the restricted isometry constant (RIC). Some papers provided the upper bounds of RIC to guarantee that the nuclear-norm minimization stably recovers a low-rank matrix. For example, Fazel improved the upper bounds to delta (4r) (A) < 0.558 and delta (3r) (A) < 0.4721, respectively. Recently, the upper bounds of RIC can be improved to delta (2r) (A) < 0.307. In fact, by using some methods, the upper bounds of RIC can be improved to delta (2r) (A) < 0.4931 and delta (2r) (A) < 0.309. In this paper, we focus on the lower bounds of RIC, we show that there exists linear maps A with delta (2r) (A) > 1/ae2 or delta (r) (A) > 1/3 for which nuclear norm recovery fail on some matrix with rank at most r. These results indicate that there is only a little limited room for improving the upper bounds for delta (2r) (A) and delta (r) (A) . Furthermore, we also discuss the upper bound of restricted isometry constant associated with linear maps A for Schatten p (0 < p < 1) quasi norm minimization problem.