Symmetric Tensors and Symmetric Tensor Rank

A symmetric tensor is a higher order generalization of a symmetric matrix. In this paper, we study various properties of symmetric tensors in relation to a decomposition into a symmetric sum of outer product of vectors. A rank-1 order-$k$ tensor is the outer product of $k$ nonzero vectors. Any symmetric tensor can be decomposed into a linear combination of rank-1 tensors, each of which is symmetric or not. The rank of a symmetric tensor is the minimal number of rank-1 tensors that is necessary to reconstruct it. The symmetric rank is obtained when the constituting rank-1 tensors are imposed to be themselves symmetric. It is shown that rank and symmetric rank are equal in a number of cases and that they always exist in an algebraically closed field. We will discuss the notion of the generic symmetric rank, which, due to the work of Alexander and Hirschowitz [J. Algebraic Geom., 4 (1995), pp. 201-222], is now known for any values of dimension and order. We will also show that the set of symmetric tensors of symmetric rank at most $r$ is not closed unless $r=1$.

[1]  F. L. Hitchcock.  The Expression of a Tensor or a Polyadic as a Sum of Products , 1927 .

[2]  F. L. Hitchcock.  Multiple Invariants and Generalized Rank of a P‐Way Matrix or Tensor , 1928 .

[3]  R. Tennant.  Algebra , 1941, Nature.

[4]  L. Tucker,et al.  Some mathematical notes on three-mode factor analysis , 1966, Psychometrika.

[5]  J. Chang,et al.  Analysis of individual differences in multidimensional scaling via an n-way generalization of “Eckart-Young” decomposition , 1970 .

[6]  Richard A. Harshman,et al.  Foundations of the PARAFAC procedure: Models and conditions for an "explanatory" multi-model factor analysis , 1970 .

[7]  M. Marcus.  Finite dimensional multilinear algebra , 1973 .

[8]  I. Shafarevich.  Basic algebraic geometry , 1974 .

[9]  J. Ord,et al.  Characterization Problems in Mathematical Statistics , 1975 .

[10]  J. Kruskal.  Three-way arrays: rank and uniqueness of trilinear decompositions, with application to arithmetic complexity and statistics , 1977 .

[11]  Joe Brewer,et al.  Kronecker products and matrix calculus in system theory , 1978 .

[12]  T. Howell,et al.  Global properties of tensor rank , 1978 .

[13]  A. Mukherjea,et al.  Real and Functional Analysis , 1978 .

[14]  J. Jaja.  An Addendum to Kronecker’s Theory of Pencils , 1979 .

[15]  Grazia Lotti,et al.  O(n2.7799) Complexity for n*n Approximate Matrix Multiplication , 1979, Inf. Process. Lett..

[16]  M. Atkinson,et al.  Bounds on the ranks of some 3-tensors , 1980 .

[17]  K. Brown,et al.  Graduate Texts in Mathematics , 1982 .

[18]  H. Nicolai.  Introduction to Supersymmetry and Supergravity , 1983 .

[19]  V. Strassen.  Rank and optimal computation of generic tensors , 1983 .

[20]  G. Rota,et al.  The invariant theory of binary forms , 1984 .

[21]  D. Weinberg.  Canonical forms for symmetric tensors , 1984 .

[22]  P. McCullagh.  Tensor Methods in Statistics , 1987 .

[23]  N. Bourbaki.  Algebra I: Chapters 1-3 , 1989 .

[24]  J. Kruskal.  Rank, decomposition, and uniqueness for 3-way and n -way arrays , 1989 .

[25]  Volker Strassen,et al.  Algebraic Complexity Theory , 1991, Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity.

[26]  Jean-Francois Cardoso,et al.  Super-symmetric decomposition of the fourth-order cumulant tensor. Blind identification of more sources than sensors , 1991, [Proceedings] ICASSP 91: 1991 International Conference on Acoustics, Speech, and Signal Processing.

[27]  J. Alexander,et al.  La méthode d'Horace éclatée: application à l'interpolation en degré quatre , 1992 .

[28]  A. Brini,et al.  The umbral symbolic method for supersymmetric tensors , 1992 .

[29]  T. Yokonuma.  Tensor Spaces and Exterior Algebra , 1992 .

[30]  B. Reznick.  Sums of Even Powers of Real Linear Forms , 1992 .

[31]  J. Berge,et al.  Computational solutions for the problem of negative saliences and nonsymmetry in INDSCAL , 1993 .

[32]  Gian-Carlo Rota,et al.  Apolarity and Canonical Forms for Homogeneous Polynomials , 1993, Eur. J. Comb..

[33]  F. Zak.  Tangents and Secants of Algebraic Varieties , 1993 .

[34]  C. D. Boor,et al.  Polynomial interpolation in several variables , 1994 .

[35]  Joe W. Harris,et al.  Algebraic Geometry: A First Course , 1995 .

[36]  Pierre Comon,et al.  Decomposition of quantics in sums of powers of linear forms , 1996, Signal Process..

[37]  Michael Clausen,et al.  Algebraic complexity theory , 1997, Grundlehren der mathematischen Wissenschaften.

[38]  David A. Cox,et al.  Using Algebraic Geometry , 1998 .

[39]  D. Freed.  Five Lectures on Supersymmetry , 1999 .

[40]  J. Berge,et al.  Simplicity of core arrays in three-way principal component analysis and the typical rank of p×q×2 arrays , 1999 .

[41]  Joos Vandewalle,et al.  A Multilinear Singular Value Decomposition , 2000, SIAM J. Matrix Anal. Appl..

[42]  Tomas Sauer,et al.  Polynomial interpolation in several variables , 2000, Adv. Comput. Math..

[43]  J. Berge,et al.  The typical rank of tall three-way arrays , 2000 .

[44]  Nikos D. Sidiropoulos,et al.  Parallel factor analysis in sensor array processing , 2000, IEEE Trans. Signal Process..

[45]  D. Brillinger.  Time series - data analysis and theory , 1981, Classics in applied mathematics.

[46]  K. Chandler.  Linear Systems of Cubics Singular at General Points of Projective Space , 2002, Compositio Mathematica.

[47]  Phillip A. Regalia,et al.  On the Best Rank-1 Approximation of Higher-Order Supersymmetric Tensors , 2001, SIAM J. Matrix Anal. Appl..

[48]  Marc Moreno Maza,et al.  Computation of canonical forms for ternary cubics , 2002, ISSAC '02.

[49]  J. Berge,et al.  Partial uniqueness in CANDECOMP/PARAFAC , 2004 .

[50]  V. Varadarajan.  Supersymmetry for Mathematicians: An Introduction , 2004 .

[51]  Pierre Comon,et al.  Blind identification and source separation in 2×3 under-determined mixtures , 2004, IEEE Trans. Signal Process..

[52]  J. Berge.  Simplicity and typical rank of three-way arrays, with applications to Tucker-3 analysis with simple cores , 2004 .

[53]  J. Berge,et al.  Typical rank and indscal dimensionality for symmetric three-way arrays of order I×2×2 or I×3×3 , 2004 .

[54]  C. Wampler,et al.  Basic Algebraic Geometry , 2005 .

[55]  Pierre Comon,et al.  Blind identification of under-determined mixtures based on the characteristic function , 2005, Proceedings. (ICASSP '05). IEEE International Conference on Acoustics, Speech, and Signal Processing, 2005..

[56]  P. Comon,et al.  Optimal solutions to non-negative PARAFAC/multilinear NMF always exist , 2005 .

[57]  D. Eisenbud,et al.  Lectures on the Geometry of Syzygies , 2005 .

[58]  Liqun Qi,et al.  Eigenvalues of a real supersymmetric tensor , 2005, J. Symb. Comput..

[59]  Lieven De Lathauwer,et al.  A Link between the Canonical Decomposition in Multilinear Algebra and Simultaneous Matrix Diagonalization , 2006, SIAM J. Matrix Anal. Appl..

[60]  Vin de Silva,et al.  Tensor rank and the ill-posedness of the best low-rank approximation problem , 2006, math/0607647.

[61]  Berkant Savas,et al.  A Newton-Grassmann Method for Computing the Best Multilinear Rank-(r1, r2, r3) Approximation of a Tensor , 2009, SIAM J. Matrix Anal. Appl..