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Inverse Problems and Imaging (IPI)
 

Strongly convex programming for exact matrix completion and robust principal component analysis

Pages: 357 - 372, Volume 6, Issue 2, May 2012      doi:10.3934/ipi.2012.6.357

 
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Hui Zhang - Department of Mathematics and Systems Science, National University of Defense Technology, Changsha, Hunan 410073, China (email)
Jian-Feng Cai - Department of Mathematics, University of Iowa, Iowa City, IA 52242, United States (email)
Lizhi Cheng - Department of Mathematics and Systems Science, National University of Defense Technology, Changsha, Hunan 410073, China (email)
Jubo Zhu - Department of Mathematics and Systems Science, National University of Defense Technology, Changsha, Hunan 410073, China (email)

Abstract: The common task in matrix completion (MC) and robust principle component analysis (RPCA) is to recover a low-rank matrix from a given data matrix. These problems gained great attention from various areas in applied sciences recently, especially after the publication of the pioneering works of Cand├Ęs et al.. One fundamental result in MC and RPCA is that nuclear norm based convex optimizations lead to the exact low-rank matrix recovery under suitable conditions. In this paper, we extend this result by showing that strongly convex optimizations can guarantee the exact low-rank matrix recovery as well. The result in this paper not only provides sufficient conditions under which the strongly convex models lead to the exact low-rank matrix recovery, but also guides us on how to choose suitable parameters in practical algorithms.

Keywords:  Strongly convex programming, exact matrix completion, robust principal component analysis, low-rank matrix, dual certificate.
Mathematics Subject Classification:  Primary: 15B52, 90C25; Secondary: 60B20, 94A08, 94A12.

Received: July 2011;      Revised: January 2012;      Available Online: May 2012.

 References