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Volume 36, Issue 3
SNIG Property of Matrix Low-Rank Factorization Model

Hong Wang, Xin Liu, Xiaojun Chen & Yaxiang Yuan

DOI: 10.4208/jcm.1707-m2016-0796

J. Comp. Math., 36 (2018), pp. 374-390.

Published online: 2018-06

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  • Abstract

Recently, the matrix factorization model attracts increasing attentions in handling large-scale rank minimization problems, which is essentially a nonconvex minimization problem. Specifically, it is a quadratic least squares problem and consequently a quartic polynomial optimization problem. In this paper, we introduce a concept of the SNIG ("Second-order Necessary optimality Implies Global optimality") condition which stands for the property that any second-order stationary point of the matrix factorization model must be a global minimizer. Some scenarios under which the SNIG condition holds are presented. Furthermore, we illustrate by an example when the SNIG condition may fail.

  • Keywords

Low rank factorization, Nonconvex optimization, Second-order optimality condition, Global minimizer.

  • AMS Subject Headings

15A18, 15A83, 65K05, 90C26

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

hong.wang@connect.polyu.hk (Hong Wang)

liuxin@lsec.cc.ac.cn (Xin Liu)

xiaojun.chen@polyu.edu.hk (Xiaojun Chen)

yyx@lsec.cc.ac.cn (Yaxiang Yuan)

  • BibTex
  • RIS
  • TXT
@Article{JCM-36-374, author = {Wang , HongLiu , XinChen , Xiaojun and Yuan , Yaxiang}, title = {SNIG Property of Matrix Low-Rank Factorization Model}, journal = {Journal of Computational Mathematics}, year = {2018}, volume = {36}, number = {3}, pages = {374--390}, abstract = {

Recently, the matrix factorization model attracts increasing attentions in handling large-scale rank minimization problems, which is essentially a nonconvex minimization problem. Specifically, it is a quadratic least squares problem and consequently a quartic polynomial optimization problem. In this paper, we introduce a concept of the SNIG ("Second-order Necessary optimality Implies Global optimality") condition which stands for the property that any second-order stationary point of the matrix factorization model must be a global minimizer. Some scenarios under which the SNIG condition holds are presented. Furthermore, we illustrate by an example when the SNIG condition may fail.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1707-m2016-0796}, url = {http://global-sci.org/intro/article_detail/jcm/12266.html} }
TY - JOUR T1 - SNIG Property of Matrix Low-Rank Factorization Model AU - Wang , Hong AU - Liu , Xin AU - Chen , Xiaojun AU - Yuan , Yaxiang JO - Journal of Computational Mathematics VL - 3 SP - 374 EP - 390 PY - 2018 DA - 2018/06 SN - 36 DO - http://doi.org/10.4208/jcm.1707-m2016-0796 UR - https://global-sci.org/intro/article_detail/jcm/12266.html KW - Low rank factorization, Nonconvex optimization, Second-order optimality condition, Global minimizer. AB -

Recently, the matrix factorization model attracts increasing attentions in handling large-scale rank minimization problems, which is essentially a nonconvex minimization problem. Specifically, it is a quadratic least squares problem and consequently a quartic polynomial optimization problem. In this paper, we introduce a concept of the SNIG ("Second-order Necessary optimality Implies Global optimality") condition which stands for the property that any second-order stationary point of the matrix factorization model must be a global minimizer. Some scenarios under which the SNIG condition holds are presented. Furthermore, we illustrate by an example when the SNIG condition may fail.

Wang , HongLiu , XinChen , Xiaojun and Yuan , Yaxiang. (2018). SNIG Property of Matrix Low-Rank Factorization Model. Journal of Computational Mathematics. 36 (3). 374-390. doi:10.4208/jcm.1707-m2016-0796
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