The Best Rank-1 Approximation of a Symmetric Tensor and Related Spherical Optimization Problems

In this paper, we show that for a symmetric tensor, its best symmetric rank-$1$ approximation is its best rank-$1$ approximation. Based on this result, a positive lower bound for the best rank-$1$ approximation ratio of a symmetric tensor is given. Furthermore, a higher order polynomial spherical optimization problem can be reformulated as a multilinear spherical optimization problem. Then, we present a modified power algorithm for solving the homogeneous polynomial spherical optimization problem. Numerical results are presented, illustrating the effectiveness of the proposed algorithm.

[1]  Gerry Leversha.  Mathematics in signal processing V (based on proceedings of a conference at University of Warwick in December 2000) edited by J. G. McWhirter and I. K. Proudler. Pp. 360. £105.00. 2002. ISBN 0 19 850734 8 (Oxford University Press). , 2003, The Mathematical Gazette.

[2]  Jeong Whan Yoon,et al.  On the use of homogeneous polynomials to develop anisotropic yield functions with applications to sheet forming , 2008 .

[3]  Pierre Comon,et al.  Blind channel identification and extraction of more sources than sensors , 1998, Optics & Photonics.

[4]  Jorge J. Moré,et al.  Computing a Trust Region Step , 1983 .

[5]  Phillip A. Regalia,et al.  Monotonic convergence of fixed-point algorithms for ICA , 2003, IEEE Trans. Neural Networks.

[6]  Pablo A. Parrilo,et al.  Semidefinite programming relaxations for semialgebraic problems , 2003, Math. Program..

[7]  Tamara G. Kolda,et al.  Shifted Power Method for Computing Tensor Eigenpairs , 2010, SIAM J. Matrix Anal. Appl..

[8]  Pierre Comon,et al.  Robust Independent Component Analysis by Iterative Maximization of the Kurtosis Contrast With Algebraic Optimal Step Size , 2010, IEEE Transactions on Neural Networks.

[9]  Jiawang Nie.  Sum of squares methods for minimizing polynomial forms over spheres and hypersurfaces , 2012 .

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

[11]  Gene H. Golub,et al.  Rank-One Approximation to High Order Tensors , 2001, SIAM J. Matrix Anal. Appl..

[12]  P. Comon,et al.  A polynomial based approach to extract the maxima of an antipodally symmetric spherical function and its application to extract fiber directions from the Orientation Distribution Function in Diffusion MRI , 2008 .

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

[14]  Arogyaswami Paulraj,et al.  An analytical constant modulus algorithm , 1996, IEEE Trans. Signal Process..

[15]  Shuzhong Zhang,et al.  Approximation algorithms for homogeneous polynomial optimization with quadratic constraints , 2010, Math. Program..

[16]  S. Lang.  Real and Functional Analysis , 1983 .

[17]  Yiju Wang,et al.  On the best rank-1 approximation to higher-order symmetric tensors , 2007, Math. Comput. Model..

[18]  O. Taussky.  Sums of Squares , 1970 .

[19]  Anthony Man-Cho So,et al.  Deterministic approximation algorithms for sphere constrained homogeneous polynomial optimization problems , 2011, Math. Program..

[20]  Xi-Lin Li,et al.  A new gradient search interpretation of super-exponential algorithm , 2006, IEEE Signal Processing Letters.

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

[22]  L. Lathauwer,et al.  On the Best Rank-1 and Rank-( , 2004 .

[23]  A. Ivic.  Sums of squares , 2020, An Introduction to 𝑞-analysis.

[24]  L. Lathauwer.  First-order perturbation analysis of the best rank-(R1, R2, R3) approximation in multilinear algebra , 2004 .

[25]  Jean B. Lasserre,et al.  Global Optimization with Polynomials and the Problem of Moments , 2000, SIAM J. Optim..

[26]  Zhi-Quan Luo,et al.  A Semidefinite Relaxation Scheme for Multivariate Quartic Polynomial Optimization with Quadratic Constraints , 2010, SIAM J. Optim..

[27]  Pierre Comon,et al.  Analytical blind discrete source separation , 2000, 2000 10th European Signal Processing Conference.

[28]  Fei Wang,et al.  Z-eigenvalue methods for a global polynomial optimization problem , 2009, Math. Program..

[29]  Yinyu Ye,et al.  The cubic spherical optimization problems , 2012, Math. Comput..

[30]  T. Rao,et al.  Tensor Methods in Statistics , 1989 .

[31]  Y. Nesterov.  Random walk in a simplex and quadratic optimization over convex polytopes , 2003 .

[32]  Jean B. Lasserre,et al.  Polynomials nonnegative on a grid and discrete optimization , 2001 .

[33]  Liqun Qi,et al.  The Best Rank-One Approximation Ratio of a Tensor Space , 2011, SIAM J. Matrix Anal. Appl..

[34]  Zhong Wan,et al.  Global Minimization of Normal Quartic Polynomials Based on Global Descent Directions , 2004, SIAM J. Optim..

[35]  Liqun Qi,et al.  Extrema of a Real Polynomial , 2004, J. Glob. Optim..

[36]  Fei Wang,et al.  An Eigenvalue Method for Testing Positive Definiteness of a Multivariate Form , 2008, IEEE Transactions on Automatic Control.