Foundations of Computational Mathematics

, Volume 17, Issue 2, pp 423–465 | Cite as

Generating Polynomials and Symmetric Tensor Decompositions

  • Jiawang Nie


This paper studies symmetric tensor decompositions. For symmetric tensors, there exist linear relations of recursive patterns among their entries. Such a relation can be represented by a polynomial, which is called a generating polynomial. The homogenization of a generating polynomial belongs to the apolar ideal of the tensor. A symmetric tensor decomposition can be determined by a set of generating polynomials, which can be represented by a matrix. We call it a generating matrix. Generally, a symmetric tensor decomposition can be determined by a generating matrix satisfying certain conditions. We characterize the sets of such generating matrices and investigate their properties (e.g., the existence, dimensions, nondefectiveness). Using these properties, we propose methods for computing symmetric tensor decompositions. Extensive examples are shown to demonstrate the efficiency of proposed methods.


Symmetric tensor Tensor rank Generating polynomial Generating matrix Symmetric tensor decomposition Polynomial system 

Mathematics Subject Classification

15A69 65F99 



The author would like to thank Lek-Heng Lim, Luke Oeding, and two anonymous referees for the useful comments. The research was partially supported by the NSF Grants DMS-0844775 and DMS-1417985.


  1. 1.
    J. Alexander and A. Hirschowitz. Polynomial interpolation in several variables. J. Algebraic Geom.  4(1995), pp. 201-22.MathSciNetzbMATHGoogle Scholar
  2. 2.
    E. Balllico and A. Bernardi. Decomposition of homogeneous polynomials with low rank. Math. Z.   271,   1141-1149,   2012.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    A. Bernardi, A. Gimigliano and M. Idà. Computing symmetric rank for symmetric tensors. Journal of Symbolic Computation   46, (2011), 34-53.MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    J. Brachat, P. Comon, B. Mourrain and E. Tsigaridas. Symmetric tensor decomposition. Linear Algebra Appl.   433, no. 11-12, 1851-1872, 2010.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    W. Buczyńska and J. Buczyński. Secant varieties to high degree Veronese reembeddings, catalecticant matrices and smoothable Gorenstein schemes. Journal of Algebraic Geometry, to appear. arXiv:1012.3563
  6. 6.
    G. Comas and M. Seiguer. On the rank of a binary form. Foundations of Computational Mathematics, Vol. 11, No. 1, pp. 65-78, 2011.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    P. Comon. Tensor decompositions - state of the art and applications. Keynote address in IMA Conf. in signal processing, Warwick, UK, 2000.Google Scholar
  8. 8.
    P. Comon, G. Golub, L.-H. Lim and B. Mourrain. Symmetric tensors and symmetric tensor rank. SIAM Journal on Matrix Analysis and Applications, 30, no. 3, 1254-1279, 2008.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    P. Comon and B. Mourrain. Decomposition of quantics in sums of powers of linear forms. Signal Processing   53(2), 93-107, 1996. Special issue on high-order statistics.CrossRefzbMATHGoogle Scholar
  10. 10.
    D. Cox, J. Little and D. O’Shea. Ideals, varieties, and algorithms: an introduction to computational algebraic geometry and commutative algebra, Springer, 2007.Google Scholar
  11. 11.
    R. M. Corless, P. M. Gianni and B. M. Trager. A reordered Schur factorization method for zero-dimensional polynomial systems with multiple roots. Proc. ACM Int. Symp. Symbolic and Algebraic Computation, 133-140, Maui, Hawaii, 1997.Google Scholar
  12. 12.
    R. Curto and L. Fialkow. Solution of the truncated complex moment problem for flat data. Memoirs of the American Mathematical Society, 119(1996), No. 568, Amer. Math. Soc., Providence, RI, 1996.Google Scholar
  13. 13.
    R. Curto and L. Fialkow. Flat extensions of positive moment matrices: recursively generated relations. Memoirs of the American Mathematical Society, no. 648, American Mathematical Society, Providence, 1998.Google Scholar
  14. 14.
    R. Curto and L. Fialkow. Truncated K-moment problems in several variables. Journal of Operator Theory, 54(2005), pp. 189-226.MathSciNetzbMATHGoogle Scholar
  15. 15.
    J. E. Dennis and R. B. Schnabel. Numerical methods for unconstrained optimization and nonlinear equations, Prentice-Hall, Englewood Cliffs, NJ, 1983.zbMATHGoogle Scholar
  16. 16.
    J.B. Lasserre, M. Laurent, B. Mourrain, P. Rostalski, and P. Trebuchet. Moment matrices, border bases and real radical computation. J. Symbolic Computation  51, pp. 63–85.Google Scholar
  17. 17.
    J. Harris. Algebraic geometry: a first course. Graduate Textbooks in Mathematics, Springer, 1992.Google Scholar
  18. 18.
    C. Hillar and L.-H. Lim. Most tensor problems are NP-hard. Journal of the ACM, 60 (2013), no. 6, Art. 45.Google Scholar
  19. 19.
    A. Iarrobino and V. Kanev. Power Sums, Gorenstein algebras, and determinantal varieties. Lecture Notes in Mathematics #1721, Springer, 1999.Google Scholar
  20. 20.
    C. T. Kelley. Iterative methods for linear and nonlinear equations, Frontiers in Applied Mathematics 16, SIAM, Philadelphia, 1995.Google Scholar
  21. 21.
    J.M. Landsberg. Tensors: geometry and applications. Graduate Studies in Mathematics, 128. American Mathematical Society, Providence, RI, 2012.Google Scholar
  22. 22.
    L.-H. Lim. Tensors and hypermatrices, in: L. Hogben (Ed.), Handbook of Linear Algebra, 2nd Ed., CRC Press, Boca Raton, FL, 2013.Google Scholar
  23. 23.
    J. J. More. The Levenberg-Marquardt algorithm: implementation and theory, in: G. A. Watson, ed., Lecture Notes in Mathematics 630: Numerical Analysis, Springer-Verlag, Berlin, 1978, 105–116.Google Scholar
  24. 24.
    L. Oeding and G. Ottaviani. Eigenvectors of tensors and algorithms for Waring decomposition. J. Symbolic Comput.  54, 9-35, 2013.MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    I. Shafarevich. Basic algebraic geometry 1: varieties in projective space, 2nd edition, Springer, Berlin 1994.CrossRefzbMATHGoogle Scholar
  26. 26.
    B. Sturmfels. Solving systems of polynomial equations. CBMS Regional Conference Series in Mathematics, 97. American Mathematical Society, Providence, RI, 2002.Google Scholar
  27. 27.
    Y. X. Yuan. Recent advances in numerical methods for nonlinear equations and nonlinear least squares, Numerical Algebra Control and Optimization, 1 (2011), 15–34.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© SFoCM 2015

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of California San DiegoLa JollaUSA

Personalised recommendations