A Link between the Canonical Decomposition in Multilinear Algebra and Simultaneous Matrix Diagonalization

Canonical decomposition is a key concept in multilinear algebra. In this paper we consider the decomposition of higher-order tensors which have the property that the rank is smaller than the greatest dimension. We derive a new and relatively weak deterministic sufficient condition for uniqueness. The proof is constructive. It shows that the canonical components can be obtained from a simultaneous matrix diagonalization by congruence, yielding a new algorithm. From the deterministic condition we derive an easy-to-check dimensionality condition that guarantees generic uniqueness.

[1]  Martin J. Mohlenkamp,et al.  Numerical operator calculus in higher dimensions , 2002, Proceedings of the National Academy of Sciences of the United States of America.

[2]  Alle-Jan van der Veen,et al.  Joint diagonalization via subspace fitting techniques , 2001, ICASSP.

[3]  Eric Moulines,et al.  A blind source separation technique using second-order statistics , 1997, IEEE Trans. Signal Process..

[4]  H. Kiers.  Towards a standardized notation and terminology in multiway analysis , 2000 .

[5]  Gene H. Golub,et al.  Matrix computations (3rd ed.) , 1996 .

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

[7]  S. Leurgans,et al.  A Decomposition for Three-Way Arrays , 1993, SIAM J. Matrix Anal. Appl..

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

[9]  Arie Yeredor,et al.  Non-orthogonal joint diagonalization in the least-squares sense with application in blind source separation , 2002, IEEE Trans. Signal Process..

[10]  P. Paatero.  The Multilinear Engine—A Table-Driven, Least Squares Program for Solving Multilinear Problems, Including the n-Way Parallel Factor Analysis Model , 1999 .

[11]  R. D. Hill,et al.  On the cone of positive semidefinite matrices , 1987 .

[12]  Dinh Tuan Pham,et al.  Joint Approximate Diagonalization of Positive Definite Hermitian Matrices , 2000, SIAM J. Matrix Anal. Appl..

[13]  E. Oja,et al.  Independent Component Analysis , 2001 .

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

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

[16]  K. S. Banerjee.  Generalized Inverse of Matrices and Its Applications , 1973 .

[17]  J. Vandewalle,et al.  An introduction to independent component analysis , 2000 .

[18]  Nikos D. Sidiropoulos,et al.  Cramer-Rao lower bounds for low-rank decomposition of multidimensional arrays , 2001, IEEE Trans. Signal Process..

[19]  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.

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

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

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

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

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

[25]  J. Cardoso,et al.  Blind beamforming for non-gaussian signals , 1993 .

[26]  L. De Lathauwer,et al.  Parallel factor analysis by means of simultaneous matrix decompositions , 2005 .

[27]  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..

[28]  Moeness G. Amin,et al.  Blind source separation based on time-frequency signal representations , 1998, IEEE Trans. Signal Process..

[29]  N. Sidiropoulos,et al.  On the uniqueness of multilinear decomposition of N‐way arrays , 2000 .

[30]  Nikos D. Sidiropoulos,et al.  Kruskal's permutation lemma and the identification of CANDECOMP/PARAFAC and bilinear models with constant modulus constraints , 2004, IEEE Transactions on Signal Processing.

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

[32]  Dinh-Tuan Pham,et al.  Blind separation of instantaneous mixtures of nonstationary sources , 2001, IEEE Trans. Signal Process..

[33]  Lieven De Lathauwer,et al.  The canonical decomposition and blind identification with more inputs than outputs : Some algebraic results , 2003 .

[34]  B. Flury.  Common Principal Components and Related Multivariate Models , 1988 .

[35]  Martin J. Mohlenkamp,et al.  Algorithms for Numerical Analysis in High Dimensions , 2005, SIAM J. Sci. Comput..

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

[37]  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..