On the generic and typical ranks of 3-tensors

Linear Algebra and Its Applications

Volumn 436, Issue 3, 2012, Pages 478-497

  Friedland, Shmuel ^a  

^a UNIVERSITY OF ILLINOIS AT CHICAGO   (United States)

Author keywords

Generic rank; Maximal rank; Rank of tensor; The generic rank conjecture; Typical rank over reals

Indexed keywords

ALGEBRAIC GEOMETRY; COMPLEX NUMBER; GENERIC RANKS; MAXIMAL RANK; RANK OF TENSOR; REAL NUMBER; TYPICAL RANK OVER REALS;

TENSORS;

EID: 80655149002     PISSN: 00243795     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.laa.2011.05.008     Document Type: Article

References (35)

1. H. Abo, G. Ottaviani, and C. Peterson Induction for secant varieties of Segre varieties Trans. Amer. Math. Soc. 361 2009 767 792
2. n 2.7799) complexity for matrix multiplication Inform. Process. Lett. 8 1979 234 235
3. J. Bochnak, M. Coste, and M.F. Roy Real Algebraic Geometry Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 1998 Springer-Verlag Berlin pp. 36
4. P. Bürgisser, M. Clausen, and M.A. Shokrollahi Algebraic Complexity Theory, with the Collaboration of Thomas Lickteig Grundlehren der Mathematischen Wissenschaften 315 1997 Springer-Verlag Berlin xxiv + 618 pp.
5. J. Carrol, and J. Chang Analysis of individual differences in multidimensional scaling via N-way generalization of Eckhart-Young decomposition Psychometrika 9 1970 267 283
6. M.V. Catalisano, A.V. Geramita, and A. Gimigliano Ranks of tensors, secant varieties of Segre varieties and fat points Linear Algebra Appl. 355 2002 263 285
7. P. Comon, J.M.F. Ten Berge, L. De Lathauwer, and J. Castaing Generic and typical ranks of multi-way arrays Linear Algebra Appl. 430 2009 2997 3007
8. R. Coppi, and S. Bolasco Multiway Data Analysis 1989 Elsevier Science Publishers (North-Holland)
9. V. De Silva, and L.-H. Lim Tensor rank and the ill-posedness of the best low-rank approximation problem SIAM J. Matrix Anal. Appl. 30 2008 1084 1127
10. S. Friedland, 3-Tensors: ranks and approximations, in: Workshop on Algorithms for Modern Massive Data Sets, Stanford-Yahoo, June 21-24, 2006. Available from: .
11. S. Friedland, and C. Krattenthaler 2-Adic valuations of certain ratios of products of factorials and applications Linear Algebra Appl. 426 2007 159 189 arXiv.math.NT/0508498v2 (Pubitemid 47176568)
12. S. Friedland, and R. Loewy Subspaces of symmetric matrices containing matrices with a multiple first eigenvalue Pacific J. Math. 62 1976 389 399
13. S. Friedland, V. Mehrmann, A. Miedlar, M. Nkengla, Fast low rank approximations of matrices and tensors, 2008. Available from: .
14. J. Harris Algebraic Geometry: A First Course 1992 Springer
15. J. Harris, and L.W. Tu On symmetric and skew-symmetric determinantal varieties Topology 23 1984 71 84
16. R. Harshman Foundations of the PARAFAC procedure: models and conditions for an explanatory multi-modal factor analysis UCLA Working Pap. Phonetics 16 1970 1 84
17. J. Ja'Ja' Optimal evaluation of a pair of bilinear forms SIAM J. Comput. 8 1979 281 293
18. T.G. Kolda, and B.W. Bader Tensor decompositions and applications SIAM Rev. 51 2009 455 500
19. J.B. Kruskal Three-way arrays: rank and uniqueness of trilinear decompositions, with applications to arithmetic complexity and statistics Linear Algebra Appl. 18 1977 95 138
20. J.B. Kruskal Rank, decomposition, and uniqueness for 3-way and N-way arrays Multi-way Data Analysis vol. 7-18 1989 North-Holland Amsterdam
21. L. de Lathauwer, B. de Moor, and J. Vandewalle Computation of the canonical decomposition by means of a simultaneous generalized Schur decomposition SIAM J. Matrix Anal. Appl. 26 2004 295 327 (Pubitemid 40541662)
22. T. Lickteig Typical tensorial rank Linear Algebra Appl. 69 1985 95 120
23. D. Mumford Algebraic Geometry I: Complex Projective Varieties 1976 Springer-Verlag Berlin
24. R. Rocci, Unpublished notes, 1993.
25. I.R. Shafarevich Basic Algebraic Geometry 1977 Springer-Verlag Berlin
26. A. Stegeman Degeneracy in CANDECOMP/PARAFAC explained for p × p × 2 arrays of rank p + 1 or higher Psychometrika 71 2006 483 501 (Pubitemid 44788544)
27. V. Strassen Rank and optimal computation of generic tensors Linear Algebra Appl. 52/53 1983 645 685
28. J.M. F ten Berge Kruskal's polynomial for 2 × 2 × 2 arrays and a generalization 2 × n × n arrays Psychometrika 56 1991 631 636
29. J.M.F. Ten Berge The typical rank of tall three-way arrays Psychometrika 65 2000 525- 532
30. J.M.F. Ten Berge Partial uniqueness in Candecomp/Parafac J. Chemometrics 18 2004 12 16
31. J.M.F. ten Berge, and HAL Kiers Simplicity of core arrays in three-way principal component analysis and the typical rank of p × q × 2 arrays Linear Algebra Appl. 294 1999 169 179 (Pubitemid 129563094)
32. J.M.F. ten Berge, and A. Stegeman Symmetry transformations for squared sliced three-way arrays, with applications to their typical rank Linear Algebra Appl. 418 2006 215 224 (Pubitemid 44286920)
33. A. Terracini, Sulla rappresentazione delle forme quaternarie mediante somme di potenze di forme lineari, Atti Rc. Accad. delle Scienze di Torino, vol. 51, 1915-16.
34. F.R. Gantmacher The Theory of Matrices vol. 2 1964 Chelsea Pub. Company New York
35. L.R. Tucker Some mathematical notes of three-mode factor analysis Psychometrika 31 1966 279 311