On the generic and typical ranks of 3-tensors

被引：22

作者：

Friedland, Shmuel ^{[1
]}

机构：

[1] Univ Illinois, Dept Math Stat & Comp Sci, Chicago, IL 60607 USA

来源：

LINEAR ALGEBRA AND ITS APPLICATIONS | 2012年 / 436卷 / 03期

关键词：

Rank of tensor; Generic rank; Maximal rank; Typical rank over reals; The generic rank conjecture; SECANT VARIETIES; 3-WAY ARRAYS; DECOMPOSITIONS; COMPUTATION; UNIQUENESS; COMPLEXITY;

D O I：

10.1016/j.laa.2011.05.008

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

We study the generic and typical ranks of 3-tensors of dimension l x m x n using results from matrices and algebraic geometry. We state a conjecture about the exact values of the generic rank of 3-tensors over the complex numbers, which is verified numerically for l, m, n <= 14. We also discuss the typical ranks over the real numbers, and give an example of an infinite family of 3-tensors of the form l = m, n = (m - 1)(2) + 1, m = 3, 4, ... , which have at least two typical ranks. (C) 2011 Elsevier Inc. All rights reserved.

引用

页码：478 / 497

页数：20

共 35 条

[1]

Abo H, 2009, T AM MATH SOC, V361, P767

[2]

[Anonymous], 1976, ALGEBRAIC GEOMETRY

[3] O(N2.7799) COMPLEXITY FOR N BY N APPROXIMATE MATRIX MULTIPLICATION [J].

BINI, D ;

CAPOVANI, M ;

ROMANI, F ;

LOTTI, G .

INFORMATION PROCESSING LETTERS, 1979, 8 (05) :234-235

[4]

Bochnak J., 1998, Ergebnisse der Mathematik und ihrer Grenzgebiete, DOI DOI 10.1007/978-3-662-03718-8

[5]

Burgisser P., 1997, ALGEBRAIC COMPLEXITY, V315

[6]

CARROLL J, 1970, PSYCHOMETRIKA, V9, P267

[7] Ranks of tensors, secant varieties of Segre varieties and fat points [J].

Catalisano, MV ;

Geramita, AV ;

Gimigliano, A .

LINEAR ALGEBRA AND ITS APPLICATIONS, 2002, 355 :263-285

[8] Generic and typical ranks of multi-way arrays [J].

Comon, P. ;

ten Berge, J. M. F. ;

De Lathauwer, L. ;

Castaing, J. .

LINEAR ALGEBRA AND ITS APPLICATIONS, 2009, 430 (11-12) :2997-3007

[9]

Coppi R., 1989, MULTIWAY DATA ANAL

[10] Computation of the canonical decomposition by means of a simultaneous generalized Schur decomposition [J].

De Lathauwer, L ;

De Moor, B ;

Vandewalle, J .

SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS, 2004, 26 (02) :295-327

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