Orthogonal Decomposition of Symmetric Tensors

A real symmetric tensor is orthogonally decomposable (or odeco) if it can be written as a linear combination of symmetric powers of $n$ vectors which form an orthonormal basis of $\mathbb R^n$. Motivated by the spectral theorem for real symmetric matrices, we study the properties of odeco tensors. We give a formula for all of the eigenvectors of an odeco tensor. Moreover, we formulate a set of polynomial equations that vanish on the odeco variety and we conjecture that these polynomials generate its prime ideal. We prove this conjecture in some cases and give strong evidence for its overall correctness.

[1]  B. Sturmfels,et al.  Binomial Ideals , 1994, alg-geom/9401001.

[2]  B. Sturmfels.  Gröbner bases and convex polytopes , 1995 .

[3]  Tamara G. Kolda,et al.  Orthogonal Tensor Decompositions , 2000, SIAM J. Matrix Anal. Appl..

[4]  M. SIAMJ..  A COUNTEREXAMPLE TO THE POSSIBILITY OF AN EXTENSION OF THE ECKART – YOUNG LOW-RANK APPROXIMATION THEOREM FOR THE ORTHOGONAL RANK TENSOR DECOMPOSITION , 2003 .

[5]  Lek-Heng Lim,et al.  Singular values and eigenvalues of tensors: a variational approach , 2005, 1st IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing, 2005..

[6]  Gene H. Golub,et al.  Symmetric Tensors and Symmetric Tensor Rank , 2008, SIAM J. Matrix Anal. Appl..

[7]  Pierre Comon,et al.  Symmetric tensor decomposition , 2009, 2009 17th European Signal Processing Conference.

[8]  Claudiu Raicu.  Secant varieties of Segre–Veronese varieties , 2010, 1011.5867.

[9]  J. Landsberg.  Tensors: Geometry and Applications , 2011 .

[10]  J. Landsberg,et al.  Equations for secant varieties of Veronese and other varieties , 2011, 1111.4567.

[11]  Anima Anandkumar,et al.  A Method of Moments for Mixture Models and Hidden Markov Models , 2012, COLT.

[12]  B. Sturmfels,et al.  The number of eigenvalues of a tensor , 2010, 1004.4953.

[13]  Christopher J. Hillar,et al.  Most Tensor Problems Are NP-Hard , 2009, JACM.

[14]  Luke Oeding,et al.  Eigenvectors of tensors and algorithms for Waring decomposition , 2011, J. Symb. Comput..

[15]  Zhou Hui-xia.  Newton Method for Computing the Z-Eigenvalues of a Real Symmetric Tensor , 2014 .

[16]  Anima Anandkumar,et al.  Tensor decompositions for learning latent variable models , 2012, J. Mach. Learn. Res..