Jacobi-type algorithms for homogeneous polynomial optimization on Stiefel manifolds with applications to tensor approximations
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- by Zhou Sheng, Jianze Li and Qin Ni HTML | PDF
- Math. Comp. 92 (2023), 2217-2245 Request permission
Abstract:
This paper mainly studies the gradient-based Jacobi-type algorithms to maximize two classes of homogeneous polynomials with orthogonality constraints, and establish their convergence properties. For the first class of homogeneous polynomials subject to a constraint on a Stiefel manifold, we reformulate it as an optimization problem on a unitary group, which makes it possible to apply the gradient-based Jacobi-type (Jacobi-G) algorithm. Then, if the subproblem can always be represented as a quadratic form, we establish the global convergence of Jacobi-G under any one of three conditions. The convergence result for the first condition is an easy extension of the result by Usevich, Li, and Comon [SIAM J. Optim. 30 (2020), pp. 2998–3028], while other two conditions are new ones. This algorithm and the convergence properties apply to the well-known joint approximate symmetric tensor diagonalization. For the second class of homogeneous polynomials subject to constraints on the product of Stiefel manifolds, we reformulate it as an optimization problem on the product of unitary groups, and then develop a new gradient-based multiblock Jacobi-type (Jacobi-MG) algorithm to solve it. We establish the global convergence of Jacobi-MG under any one of the above three conditions, if the subproblem can always be represented as a quadratic form. This algorithm and the convergence properties are suitable to the well-known joint approximate tensor diagonalization. As the proximal variants of Jacobi-G and Jacobi-MG, we also propose the Jacobi-GP and Jacobi-MGP algorithms, and establish their global convergence without any further condition. Some numerical results are provided indicating the efficiency of the proposed algorithms.References
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Additional Information
- Zhou Sheng
- Affiliation: Department of Data Science, Anhui University of Technology, Maanshan, Anhui, People’s Republic of China
- ORCID: 0000-0001-9295-5872
- Email: szhou03@live.com
- Jianze Li
- Affiliation: Shenzhen Research Institute of Big Data, The Chinese University of Hong Kong, Shenzhen, Guangdong, People’s Republic of China
- MR Author ID: 956204
- ORCID: 0000-0002-0760-7994
- Email: lijianze@gmail.com
- Qin Ni
- Affiliation: Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, Jiangsu, People’s Republic of China
- Email: niqfs@nuaa.edu.cn
- Received by editor(s): April 2, 2022
- Received by editor(s) in revised form: December 1, 2022
- Published electronically: April 5, 2023
- Additional Notes: The first author was supported in part by the Anhui Provincial Natural Science Foundation (No. 2208085QA07) and the Youth Foundation of Anhui University of Technology (No. QZ202114). The second author was supported in part by the National Natural Science Foundation of China (No. 11601371) and the GuangDong Basic and Applied Basic Research Foundation (No. 2021A1515010232). The third author was supported by the National Natural Science Foundation of China (No. 11771210).
The second author is the corresponding author. - © Copyright 2023 American Mathematical Society
- Journal: Math. Comp. 92 (2023), 2217-2245
- MSC (2020): Primary 15A69, 90C23; Secondary 65F99, 90C30
- DOI: https://doi.org/10.1090/mcom/3834