Generic and typical ranks of three-way arrays

The concept of tensor rank, introduced in the twenties, has been popularized at the beginning of the seventies. This has allowed to carry out factor analysis on arrays with more than two indices. The generic rank may be seen as an upper bound to the number of factors that can be extracted from a given tensor, with certain uniqueness conditions. We explain how to obtain numerically the generic rank of tensors of arbitrary dimensions, and compare it with the rare algebraic results already known at order three. In particular, we examine the cases of symmetric tensors, tensors with symmetric matrix slices, or tensors with free entries. Related applications include antenna array processing.

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

[2]  Nikos D. Sidiropoulos,et al.  Blind PARAFAC receivers for DS-CDMA systems , 2000, IEEE Trans. Signal Process..

[3]  J. Berge,et al.  The typical rank of tall three-way arrays , 2000 .

[4]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

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

[6]  Michael Clausen,et al.  Algebraic complexity theory , 1997, Grundlehren der mathematischen Wissenschaften.

[7]  V. Strassen.  Rank and optimal computation of generic tensors , 1983 .

[8]  A. Smilde,et al.  Non‐triviality and identification of a constrained Tucker3 analysis , 2002 .

[9]  A. Stegeman,et al.  Symmetry transformations for square sliced three-way arrays, with applications to their typical rank , 2006 .

[10]  Lieven De Lathauwer,et al.  Blind Identification of Underdetermined Mixtures by Simultaneous Matrix Diagonalization , 2008, IEEE Transactions on Signal Processing.

[11]  R. A. Harshman,et al.  Data preprocessing and the extended PARAFAC model , 1984 .

[12]  T. Howell,et al.  Global properties of tensor rank , 1978 .

[13]  J. Berge,et al.  Partial uniqueness in CANDECOMP/PARAFAC , 2004 .

[14]  Volker Strassen,et al.  Algebraic Complexity Theory , 1991, Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity.

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

[16]  A. Stechow,et al.  Decomposition , 1902, The Indian medical gazette.

[17]  P ? ? ? ? ? ? ? % ? ? ? ? , 1991 .

[18]  Age K. Smilde,et al.  Modelling of spectroscopic batch process data using grey models to incorporate external information , 2001 .

[19]  J. Kruskal.  Rank, decomposition, and uniqueness for 3-way and n -way arrays , 1989 .

[20]  J. Berge,et al.  Typical rank and indscal dimensionality for symmetric three-way arrays of order I×2×2 or I×3×3 , 2004 .

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

[22]  J. Berge,et al.  Simplicity of core arrays in three-way principal component analysis and the typical rank of p×q×2 arrays , 1999 .

[23]  L. Tucker,et al.  Some mathematical notes on three-mode factor analysis , 1966, Psychometrika.

[24]  M. Atkinson,et al.  Bounds on the ranks of some 3-tensors , 1980 .

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