Semi-algebraic canonical decomposition of multi-way arrays and Joint Eigenvalue Decomposition

A semi-algebraic algorithm based on Joint EigenValue Decomposition (JEVD) is proposed to compute the CP decomposition of multi-way arrays. The iterative part of the method is thus limited to the JEVD computation. In addition it involves less restrictive hypothesis than other recent semi-algebraic approaches. We also propose an original JEVD technique based on the LU factorization. Numerical examples highlight the main advantages of the proposed methods to solve both the JEVD and CP problems.

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

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

[3]  Joos Vandewalle,et al.  Computation of the Canonical Decomposition by Means of a Simultaneous Generalized Schur Decomposition , 2005, SIAM J. Matrix Anal. Appl..

[4]  P. Comon,et al.  Tensor decompositions, alternating least squares and other tales , 2009 .

[5]  Laurent Albera,et al.  Joint Eigenvalue Decomposition Using Polar Matrix Factorization , 2010, LVA/ICA.

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

[7]  Ed F. Deprettere,et al.  Azimuth and elevation computation in high resolution DOA estimation , 1992, IEEE Trans. Signal Process..

[8]  Karim Abed-Meraim,et al.  A new Jacobi-like method for joint diagonalization of arbitrary non-defective matrices , 2009, Appl. Math. Comput..

[9]  Florian Roemer,et al.  A closed-form solution for Parallel Factor (PARAFAC) Analysis , 2008, 2008 IEEE International Conference on Acoustics, Speech and Signal Processing.

[10]  Laurent Albera,et al.  Iterative methods for the canonical decomposition of multi-way arrays: Application to blind underdetermined mixture identification , 2011, Signal Process..

[11]  R. Bro.  PARAFAC. Tutorial and applications , 1997 .

[12]  Laurent Albera,et al.  Multiway space-time-wave-vector analysis for source localization and extraction , 2010, 2010 18th European Signal Processing Conference.

[13]  Pierre Comon,et al.  Enhanced Line Search: A Novel Method to Accelerate PARAFAC , 2008, SIAM J. Matrix Anal. Appl..

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

[15]  Xiqi Gao,et al.  Simultaneous Diagonalization With Similarity Transformation for Non-Defective Matrices , 2006, 2006 IEEE International Conference on Acoustics Speech and Signal Processing Proceedings.

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

[17]  M. Haardt,et al.  A closed-form solution for multilinear PARAFAC decompositions , 2008, 2008 5th IEEE Sensor Array and Multichannel Signal Processing Workshop.

[18]  Josef A. Nossek,et al.  Simultaneous Schur decomposition of several nonsymmetric matrices to achieve automatic pairing in multidimensional harmonic retrieval problems , 1998, IEEE Trans. Signal Process..

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