Semidefinite Relaxations for Best Rank-1 Tensor Approximations

Jiawang Nie University of California, San Diego Li Wang University of Texas at Arlington

Optimization and Control mathscidoc:1904.25011

SIAM Journal on Matrix Analysis and Applications, 35, (3), 25, 2014
This paper studies the problem of finding best rank-1 approximations for both symmetric and nonsymmetric tensors. For symmetric tensors, this is equivalent to optimizing homogeneous polynomials over unit spheres; for nonsymmetric tensors, this is equivalent to optimizing multiquadratic forms over multispheres. We propose semidefinite relaxations, based on sum of squares representations, to solve these polynomial optimization problems. Their special properties and structures are studied. In applications, the resulting semidefinite programs are often large scale. The recent Newton-CG augmented Lagrangian method by Zhao, Sun, and Toh [SIAM J. Optim., 20 (2010), pp. 1737–1765] is suitable for solving these semidefinite relaxations. Extensive numerical experiments are presented to show that this approach is efficient in getting best rank-1 approximations.
polynomial, relaxation, rank-1 approximation, semidefinite program, sum of squares, tensor
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@inproceedings{jiawang2014semidefinite,
  title={Semidefinite Relaxations for Best Rank-1 Tensor Approximations},
  author={Jiawang Nie, and Li Wang},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20190427031742725931277},
  booktitle={SIAM Journal on Matrix Analysis and Applications},
  volume={35},
  number={3},
  pages={25},
  year={2014},
}
Jiawang Nie, and Li Wang. Semidefinite Relaxations for Best Rank-1 Tensor Approximations. 2014. Vol. 35. In SIAM Journal on Matrix Analysis and Applications. pp.25. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20190427031742725931277.
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