RESEARCH ARTICLE

Rank-r decomposition of symmetric tensors

  • Jie WEN ,
  • Qin NI ,
  • Wenhuan ZHU
Expand
  • College of Science, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China

Received date: 28 Sep 2016

Accepted date: 19 Jan 2017

Published date: 27 Nov 2017

Copyright

2017 Higher Education Press and Springer-Verlag GmbH Germany

Abstract

An algorithm is presented for decomposing a symmetric tensor into a sum of rank-1 symmetric tensors. For a given tensor, by using apolarity, catalecticant matrices and the condition that the mapping matrices are commutative, the rank of the tensor can be obtained by iteration. Then we can find the generating polynomials under a selected basis set. The decomposition can be constructed by the solutions of generating polynomials under the condition that the solutions are all distinct which can be guaranteed by the commutative property of the matrices. Numerical examples demonstrate the efficiency and accuracy of the proposed method.

Cite this article

Jie WEN , Qin NI , Wenhuan ZHU . Rank-r decomposition of symmetric tensors[J]. Frontiers of Mathematics in China, 2017 , 12(6) : 1339 -1355 . DOI: 10.1007/s11464-017-0632-5

1
AlexanderJ, HirschowitzA. Polynomial interpolation in several variables.J Algebraic Geom, 1995, 4: 201–222

2
BallicoE. On the typical rank of real bivariate polynomials.Linear Algebra Appl,2014, 452(1): 263–269

DOI

3
BallicoE, BernardiA. Decomposition of homogeneous polynomials with low rank.Math Z, 2012, 271(3): 1141–1149

DOI

4
BatselierK, WongN. Symmetric tensor decomposition by an iterative eigendecomposition algorithm.J Comput Appl Math, 2016, 308(15): 69–82

DOI

5
BernardiA, GimiglianoA, Id`aM. Computing symmetric rank for symmetric tensors.J Symbolic Comput, 2011, 46(1): 34–53

DOI

6
BrachatJ, ComonP, MourrainB, Tsigaridas E. Symmetric tensor decomposition.Linear Algebra Appl,2010, 433(11-12): 1851–1872

DOI

7
BuczynskaW, BuczynskiJ. Secant varieties to high degree Veronese reembeddings, catalecticant matrices and smoothable Gorenstein schemes.J Algebraic Geom, 2014, 23(1): 63–90

DOI

8
ComonP. Tensors: a brief introduction.IEEE Signal Processing Magazine, 2014, 31(3): 44–53

DOI

9
ComonP, GolubG, LimL H, MourrainB. Symmetric tensors and symmetric tensor rank.SIAM J Matrix Anal Appl,2008, 30(3): 1254–1279

DOI

10
FriedlandS. Remarks on the symmetric rank of symmetric tensors.SIAM J Matrix Anal Appl, 2016: 37(1): 320–337

DOI

11
HillarC, LimL H. Most tensor problems are NP-hard.J ACM,2013, 60(6): 45

DOI

12
JiangB, LiZ N. ZhangS Z. Characterizing real-valued multivariate complex polynomials and their symmetric tensor representations.SIAM J Matrix Anal Appl,2016, 37(1): 381–408

DOI

13
LandsbergJ M. Tensors: Geometry and Applications.Grad Stud Math, Vol 128. Providence: Amer Math Soc, 2012

14
NieJ. Generating polynomials and symmetric tensor decompositions.Found Comput Math,

DOI

15
OedingL, OttaavianiG. Eigenvectors of tensors and algorithms for Waring decomposition.J Symbolic Comput,2013, 54: 9–35

DOI

16
RobevaE. Orthogonal decomposition of symmetric tensor.SIAM J Matrix Anal Appl,2016, 37(1): 86–102

DOI

17
SturmfelsB. Solving Systems of Polynomial Equations.CBMS Regional Conference Series in Mathematics, No 97. Providence: Amer Math Soc, 2002

DOI

Outlines

/