Rank-r decomposition of symmetric tensors

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.

Keywords

Symmetric tensor / symmetric rank / decomposition / generating polynomial / catalectieant matrix

References

[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

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

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

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

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

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

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

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

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

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

[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

[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,

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

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

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

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Higher Education Press and Springer-Verlag GmbH Germany