Symmetric rank-1 approximation of symmetric high-order tensors

ABSTRACT

Finding the symmetric rank-1 approximation to a given symmetric tensor is an important problem due to its wide applications and its close relationship to the Z-eigenpair of a tensor. In this paper, we propose a method based on the proximal alternating linearized minimization to directly solve the optimization problem. Global convergence of our algorithm is established. Numerical experiments show that our algorithm is very competitive in speed, accuracy and robustness compared to other state-of-the-art methods.

KEYWORDS:

• Rank-1 approximation
• symmetric tensor
• proximal alternating linearized minimization
• Bose–Einstein condensate

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 65K05
• 90C22
• 90C26

Acknowledgements

The authors would like to thank Bo Peng for his efforts in an early investigation of this project. They are grateful to the associate editor and two anonymous referees for their valuable comments and suggestions.

Disclosure statement

No potential conflict of interest was reported by the author(s).

ORCID

Xin Liu http://orcid.org/0000-0002-5705-0805

Additional information

Funding

The research was supported in part by NSFC grants [grant numbers 11526096, 11601185, 11622112, 11971466, 11688101, 11421101 and 11831002]. The research was also supported by the National Center for Mathematics and Interdisciplinary Sciences, CAS, Youth Innovation Promotion Association of Chinese Academy of Sciences, and Key Research Program of Frontier Sciences QYZDJ-SSW-SYS010, CAS.

Notes on contributors

Leqin Wu

Dr Leqin Wu is an assistant professor at Jinan University. He graduated from Peking University in 2007 and got his bachelor degree, then received his PhD at Chinese Academy of Sciences in 2012 under the supervision of Professor Ya-xiang Yuan. He spent the following two years at the Department of Biological Statistics, University of Rochester, US, and afterwards works as an assistant professor at Jinan University (Guangzhou, China) until now. His current research interest is optimization algorithm design, especially for inverse problems in differential equations.

Xin Liu

Dr Xin Liu is an associate professor at Academy of Mathematics and Systems Science, Chinese Academy Sciences. He got his bachelor degree from the School of Mathematical Sciences, Peking University in 2004, and PhD from the University of Chinese Academy of Sciences in 2009, under the supervision of Professor Ya-xiang Yuan. His research interests include optimization problems with orthogonality constraints, linear and nonlinear eigenvalue problems, nonlinear least squares and distributed optimization. Dr Xin Liu is the principal investigator of five NSFC (National Science Foundation of China) grants including the Excellent Youth Grant. He has served as an associate editor of ‘Mathematical Programming Computation’ since 2015, and ‘Mathematic Numerica Sinica’ since 2017.

Zaiwen Wen

Dr Zaiwen Wen, associate professor at Peking University. He holds an MS in Computational Mathematics from Academy of Mathematics and Systems Science (AMSS) (2004) and a PhD in Operations Research from Columbia University (2009). His research interests include large-scale computational optimization and their applications in sciences and engineering. He was awarded the National Natural Science Foundation of China for Excellent Young Scholars in 2013, the Young top-notch talent in 2015 and the Science and Technology Award for Chinese Youth in 2016. He is an associate editor of Journal of the Operations Research Society of China, Journal of Computational Mathematics and a technical editor of Mathematical Programming Computation.