Partial orthogonal rank-one decomposition of complex symmetric tensors based on the Takagi factorization

Abstract This paper is devoted to the computation of rank-one decomposition of complex symmetric tensors. Based on the Takagi factorization of complex symmetric matrices, we derive algorithm for computing the partial orthogonal rank-one decomposition of complex symmetric tensors with an order being a power of two, denoted by CSTPOROD. We consider the properties of this decomposition. We design a strategy (tensor embedding) to computing the partial orthogonal rank-one decomposition of complex symmetric tensors, whose order is not the power of two. Similar to the case of complex symmetric tensors, we consider how to compute the partial orthogonal rank-one decomposition of general complex tensors. We illustrate our algorithms via numerical examples.

[1]  W. Gragg,et al.  Singular value decompositions of complex symmetric matrices , 1988 .

[2]  Pierre Comon,et al.  Independent component analysis, A new concept? , 1994, Signal Process..

[3]  Wei Xu,et al.  Block Lanczos tridiagonalization of complex symmetric matrices , 2005, SPIE Optics + Photonics.

[4]  Sabine Van Huffel,et al.  Exponential data fitting using multilinear algebra: the single‐channel and multi‐channel case , 2005, Numer. Linear Algebra Appl..

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

[6]  Jean-Franois Cardoso.  High-Order Contrasts for Independent Component Analysis , 1999, Neural Computation.

[7]  Raf Vandebril,et al.  A comparison between the complex symmetric based and classical computation of the singular value decomposition of normal matrices , 2014, Numerical Algorithms.

[8]  C. L. Nikias,et al.  Signal processing with higher-order spectra , 1993, IEEE Signal Processing Magazine.

[9]  Sabine Van Huffel,et al.  Enhanced resolution based on minimum variance estimation and exponential data modeling , 1993, Signal Process..

[10]  Andrzej Cichocki,et al.  Tensor Decompositions for Signal Processing Applications: From two-way to multiway component analysis , 2014, IEEE Signal Processing Magazine.

[11]  Wei Xu,et al.  A Divide-and-Conquer Method for the Takagi Factorization , 2008, SIAM J. Matrix Anal. Appl..

[12]  Pierre Comon,et al.  Tensor Decompositions, State of the Art and Applications , 2002 .

[13]  L. De Lathauwer,et al.  Line Search Computation of the Block Factor Model for Blind Multi-User Access in Wireless Communications , 2006, 2006 IEEE 7th Workshop on Signal Processing Advances in Wireless Communications.

[14]  Tamara G. Kolda,et al.  Tensor Decompositions and Applications , 2009, SIAM Rev..

[15]  Tamara G. Kolda,et al.  A Counterexample to the Possibility of an Extension of the Eckart-Young Low-Rank Approximation Theorem for the Orthogonal Rank Tensor Decomposition , 2002, SIAM J. Matrix Anal. Appl..

[16]  Ngai Wong,et al.  A Constructive Algorithm for Decomposing a Tensor into a Finite Sum of Orthonormal Rank-1 Terms , 2014, SIAM J. Matrix Anal. Appl..

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

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

[19]  Yimin Wei,et al.  Complex-valued neural networks for the Takagi vector of complex symmetric matrices , 2017, Neurocomputing.

[20]  Lieven De Lathauwer,et al.  On the Uniqueness of the Canonical Polyadic Decomposition of Third-Order Tensors - Part II: Uniqueness of the Overall Decomposition , 2013, SIAM J. Matrix Anal. Appl..

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

[22]  Wei Xu,et al.  A twisted factorization method for symmetric SVD of a complex symmetric tridiagonal matrix , 2009, Numer. Linear Algebra Appl..

[23]  Jan Swevers,et al.  Identification of nonlinear systems using Polynomial Nonlinear State Space models , 2010, Autom..

[24]  Ngai Wong,et al.  Symmetric tensor decomposition by an iterative eigendecomposition algorithm , 2014, J. Comput. Appl. Math..

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

[26]  Lieven De Lathauwer,et al.  A Block Factor Analysis Based Receiver for Blind Multi-User Access in Wireless Communications , 2006, 2006 IEEE International Conference on Acoustics Speech and Signal Processing Proceedings.

[27]  Raf Vandebril,et al.  On Generic Nonexistence of the Schmidt-Eckart-Young Decomposition for Complex Tensors , 2014, SIAM J. Matrix Anal. Appl..

[28]  Liqun Qi,et al.  New ALS Methods With Extrapolating Search Directions and Optimal Step Size for Complex-Valued Tensor Decompositions , 2011, IEEE Transactions on Signal Processing.