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## Generic and typical ranks of multi-way arrays

Venue: | Linear Algebra Appl |

Citations: | 28 - 5 self |

### Citations

591 |
Individual differences and multidimensional scaling. In:
- Carroll
- 1972
(Show Context)
Citation Context ...neric and typical ranks are discussed for various tensor structures. The typical rank of three-way arrays over the real field has been relevant for psychological data analysis since Carroll and Chang =-=[7]-=- and Harshman [8] independently proposed a method which they christened Candecomp and Parafac, respectively. Therefore, we shall subsequently refer to this decomposition with the acronym CP, as severa... |

540 |
Foundation of the PARAFAC procedure: model and conditions for an explanatory mutil-mode factor analysis,”
- Harshman
- 1972
(Show Context)
Citation Context ... ranks are discussed for various tensor structures. The typical rank of three-way arrays over the real field has been relevant for psychological data analysis since Carroll and Chang [7] and Harshman =-=[8]-=- independently proposed a method which they christened Candecomp and Parafac, respectively. Therefore, we shall subsequently refer to this decomposition with the acronym CP, as several other authors d... |

464 | A multilinear singular value decomposition,”
- Lathauwer, Moor, et al.
- 2000
(Show Context)
Citation Context ..., “Dynamical systems, control and optimization”, 2007–2011), (4) EU: ERNSI. This publication only reflects the authors’ views. 2 application of CP, chemometricians also use Tucker3 component analysis =-=[11]-=- [12] [13] quite often. This is another model aiming at decomposing a data array as weighted sum of rank-one arrays, the weights being collected in a socalled core array. Typically, the underlying che... |

415 |
Some mathematical notes on three-mode factor analysis
- Tucker
- 1966
(Show Context)
Citation Context ...al systems, control and optimization”, 2007–2011), (4) EU: ERNSI. This publication only reflects the authors’ views. 2 application of CP, chemometricians also use Tucker3 component analysis [11] [12] =-=[13]-=- quite often. This is another model aiming at decomposing a data array as weighted sum of rank-one arrays, the weights being collected in a socalled core array. Typically, the underlying chemometric m... |

233 | Algebraic Complexity Theory - Bürgisser, Clausen, et al. - 1997 |

141 |
Polynomial interpolation in several variables
- Alexander, Hirschowitz
- 1995
(Show Context)
Citation Context ...n studied for several decades [1] [2]. However, the value of the generic rank for arbitrary dimensions is not yet known in the unsymmetric case, and has been known in the symmetric case only recently =-=[3]-=- [4] [5]. The existence itself of the generic rank is not ensured in the real case, and there exist in general several typical ranks (see section 2.1 for definitions). The typical tensor rank of three... |

124 | Blind PARAFAC receivers for DS-CDMA systems
- Sidiropoulos, Giannakis, et al.
- 2000
(Show Context)
Citation Context ...tautologies. Besides Chemometrics, CP has found important applications in signal processing, especially in Independent Component Analysis [16] [17] and in multi-user access in wireless communications =-=[18]-=- [19] [20]. Moreover, the decomposition is finding its way to scientific computing, where it leads to a way around the Curse of Dimensionality [21, p. 125] [22] [23]. The paper is organized as follows... |

124 | Parallel factor analysis in sensor array processing,”
- Sidiropoulos, Bro, et al.
- 2000
(Show Context)
Citation Context ...logies. Besides Chemometrics, CP has found important applications in signal processing, especially in Independent Component Analysis [16] [17] and in multi-user access in wireless communications [18] =-=[19]-=- [20]. Moreover, the decomposition is finding its way to scientific computing, where it leads to a way around the Curse of Dimensionality [21, p. 125] [22] [23]. The paper is organized as follows. In ... |

115 |
Multi-Way Analysis
- Smilde, Bro, et al.
- 2004
(Show Context)
Citation Context ...study of typical rank of three-way arrays is of great theoretical importance for CP. Although CP was developed in a psychometric environment, its main area of applications has been Chemometrics, e.g. =-=[10]-=-. In addition to straightforward Email addresses: pcomon@unice.fr (P. Comon), J.M.F.ten.Berge@rug.nl (J.M.F. ten Berge), Lieven.DeLathauwer@kuleuven-kortrijk.be (L. De Lathauwer). 1 Part of this work ... |

96 | Symmetric tensors and symmetric tensor rank.
- Comon, Golub, et al.
- 2008
(Show Context)
Citation Context ...d for several decades [1] [2]. However, the value of the generic rank for arbitrary dimensions is not yet known in the unsymmetric case, and has been known in the symmetric case only recently [3] [4] =-=[5]-=-. The existence itself of the generic rank is not ensured in the real case, and there exist in general several typical ranks (see section 2.1 for definitions). The typical tensor rank of three-way arr... |

88 | Decomposition of quantics in sums of powers of linear forms
- Comon, Mourrain
- 1996
(Show Context)
Citation Context ...udied for several decades [1] [2]. However, the value of the generic rank for arbitrary dimensions is not yet known in the unsymmetric case, and has been known in the symmetric case only recently [3] =-=[4]-=- [5]. The existence itself of the generic rank is not ensured in the real case, and there exist in general several typical ranks (see section 2.1 for definitions). The typical tensor rank of three-way... |

86 | Algorithms for numerical analysis in high dimensions
- Beylkin, Mohlenkamp
(Show Context)
Citation Context ...-user access in wireless communications [18] [19] [20]. Moreover, the decomposition is finding its way to scientific computing, where it leads to a way around the Curse of Dimensionality [21, p. 125] =-=[22]-=- [23]. The paper is organized as follows. In section 2, definitions and historical remarks are provided. Next, a numerical algorithm is described in section 3, which is able to compute the generic ran... |

79 |
A link between the canonical decomposition in multilinear algebra and simultaneous matrix diagonalization
- Lathauwer
- 2006
(Show Context)
Citation Context ...always smaller than or equal to (but generally much smaller than) the generic rank. Other sufficient conditions have been derived in the literature under somewhat different assumptions, see e.g. [41] =-=[42]-=- for tensors enjoying Hermitean symmetries. We report values of the smallest typical/generic rank of 3-way arrays with equal dimensions in table 2. The values shown in table 2 can also be compared to ... |

75 |
and optimal computation of generic tensors
- Strassen, Rank
- 1983
(Show Context)
Citation Context ...A03 15A72 15A57 46M05 Preprint submitted to Linear Algebra Appl. vol.430, pp.2997–3007, June 2009 1 Introduction Generic ranks, defined in the complex field, have been studied for several decades [1] =-=[2]-=-. However, the value of the generic rank for arbitrary dimensions is not yet known in the unsymmetric case, and has been known in the symmetric case only recently [3] [4] [5]. The existence itself of ... |

71 |
The Extension of Factor Analysis to Three-Dimensional Matrices
- Tucker
- 1964
(Show Context)
Citation Context ...namical systems, control and optimization”, 2007–2011), (4) EU: ERNSI. This publication only reflects the authors’ views. 2 application of CP, chemometricians also use Tucker3 component analysis [11] =-=[12]-=- [13] quite often. This is another model aiming at decomposing a data array as weighted sum of rank-one arrays, the weights being collected in a socalled core array. Typically, the underlying chemomet... |

69 |
Multiple invariants and generalized rank of a p-way matrix or tensor
- Hitchcock
- 1927
(Show Context)
Citation Context ...fac, respectively. Therefore, we shall subsequently refer to this decomposition with the acronym CP, as several other authors did before, even if the CP decomposition had been introduced much earlier =-=[9]-=-. This CP decomposition generalizes Principal Component Analysis to three-way data, by seeking the best least squares approximation of a data array by the sum of a limited number of rank-one arrays. I... |

55 | Induction for secant varieties of Segre varieties
- Abo, Ottaviani, et al.
(Show Context)
Citation Context ...c rank of symmetric tensors is completely described by the Alexander-Hirschowitz (AH) theorem [3]. Recently, Abro and Ottaviani attempted to generalize the AH theorem to non symmetric complex tensors =-=[26]-=-, and provided an almost exhaustive list of exceptions. This recent contribution is the most significant step towards the complete characterization of the generic rank of unsymmetric tensors with free... |

40 | Canonical tensor decompositions
- Comon
- 2004
(Show Context)
Citation Context ...ations in distinguishing constrained Tucker3 models from tautologies. Besides Chemometrics, CP has found important applications in signal processing, especially in Independent Component Analysis [16] =-=[17]-=- and in multi-user access in wireless communications [18] [19] [20]. Moreover, the decomposition is finding its way to scientific computing, where it leads to a way around the Curse of Dimensionality ... |

38 | Data preprocessing and the extended Parafac model, in Research Methods for Multimode Data Analysis - Harshman, Lundy - 1984 |

32 |
decomposition, and uniqueness for 3-way and N-way arrays
- Rank
- 1989
(Show Context)
Citation Context ...(1) is exactly satisfied. This decomposition is referred to as the tensor Canonical or Parallel factor Decomposition (CP). A property is called typical if it holds true on a set of nonzero volume [2] =-=[24]-=- [25] [5]. This supposes that some topology has been defined on KN1×N2×...NL ; this can be the Zariski topology for instance, or an Euclidean topology. A property is said to be generic if it is true a... |

31 |
Typical tensorial rank
- Lickteig
- 1985
(Show Context)
Citation Context ...y a finite number of CP solutions. rank obtained for asymmetric tensors with equal dimensions, N , and order L, with an algorithm referred to as rangj(N,L). These results are consistent with those of =-=[43]-=-. We also indicate the dimensionality of the fiber of solutions. This number is simply defined as the difference: F (N,L) = R̄(N,L) (LN − L+ 1)−NL. For those values of dimension and order for which F ... |

26 | Khatri–Rao space-time codes
- Sidiropoulos, Budampati
- 2002
(Show Context)
Citation Context ...s. Besides Chemometrics, CP has found important applications in signal processing, especially in Independent Component Analysis [16] [17] and in multi-user access in wireless communications [18] [19] =-=[20]-=-. Moreover, the decomposition is finding its way to scientific computing, where it leads to a way around the Curse of Dimensionality [21, p. 125] [22] [23]. The paper is organized as follows. In secti... |

22 |
Simplicity of core arrays in threeway principal component analysis and the typical rank of p × q × 2 arrays
- Berge, Kiers
- 1999
(Show Context)
Citation Context ...s exactly satisfied. This decomposition is referred to as the tensor Canonical or Parallel factor Decomposition (CP). A property is called typical if it holds true on a set of nonzero volume [2] [24] =-=[25]-=- [5]. This supposes that some topology has been defined on KN1×N2×...NL ; this can be the Zariski topology for instance, or an Euclidean topology. A property is said to be generic if it is true almost... |

22 |
On Canonical Forms.
- Wakeford
- 1918
(Show Context)
Citation Context ...as the uncentered N1 × (N2 − 1)× (N3 − 1) array. 3 Computation of Generic Ranks The algorithm proposed is directly inspired by [4], which is in turn based on the so-called Terracini’s lemma [36] [37] =-=[38]-=-; note that the latter is often attributed to Lasker, and is hence almost one hundred years old. In a few works, the principle is based on the fact that the dimension of an irreducible variety is equa... |

22 | Blind identification of overcomplete mixtures of sources (BIOME
- Albera, Ferreol, et al.
- 2004
(Show Context)
Citation Context ...d is always smaller than or equal to (but generally much smaller than) the generic rank. Other sufficient conditions have been derived in the literature under somewhat different assumptions, see e.g. =-=[41]-=- [42] for tensors enjoying Hermitean symmetries. We report values of the smallest typical/generic rank of 3-way arrays with equal dimensions in table 2. The values shown in table 2 can also be compare... |

21 |
Tensor-product approximation to operators and functions in high dimensions
- HACKBUSCH, KHOROMSKIJ
(Show Context)
Citation Context ... access in wireless communications [18] [19] [20]. Moreover, the decomposition is finding its way to scientific computing, where it leads to a way around the Curse of Dimensionality [21, p. 125] [22] =-=[23]-=-. The paper is organized as follows. In section 2, definitions and historical remarks are provided. Next, a numerical algorithm is described in section 3, which is able to compute the generic rank of ... |

20 |
Global properties of tensor rank
- Howell
- 1978
(Show Context)
Citation Context ...9 15A03 15A72 15A57 46M05 Preprint submitted to Linear Algebra Appl. vol.430, pp.2997–3007, June 2009 1 Introduction Generic ranks, defined in the complex field, have been studied for several decades =-=[1]-=- [2]. However, the value of the generic rank for arbitrary dimensions is not yet known in the unsymmetric case, and has been known in the symmetric case only recently [3] [4] [5]. The existence itself... |

19 | Blind identification of underdetermined mixtures by simultaneous matrix diagonalization
- Lathauwer, Castaing
- 2008
(Show Context)
Citation Context ...pplications in distinguishing constrained Tucker3 models from tautologies. Besides Chemometrics, CP has found important applications in signal processing, especially in Independent Component Analysis =-=[16]-=- [17] and in multi-user access in wireless communications [18] [19] [20]. Moreover, the decomposition is finding its way to scientific computing, where it leads to a way around the Curse of Dimensiona... |

15 |
Etude Algébrique des Multitableaux: Apports de l’algèbre Tensorielle
- Franc
- 1992
(Show Context)
Citation Context ...the real field was initiated by Kruskal [27] [24], who noted that 2 × 2 × 2 arrays had both rank 2 and rank 3 with positive probability. Kruskal also added a few typical ranks for small arrays. Franc =-=[28]-=- discussed some more results, including bounds on typical rank. Ten Berge and Kiers [25] gave a first result of some generality, in solving the typical rank issue for all two-slice arrays (that is, ar... |

15 |
The typical rank of tall three-way arrays
- Berge
(Show Context)
Citation Context ...ge and Kiers [25] gave a first result of some generality, in solving the typical rank issue for all two-slice arrays (that is, arrays of format 2 × N2 × N3). These results were further generalized in =-=[29]-=-, to include all cases where, for N1 ≥ N2 ≥ N3, N1 > N2N3 − N2. Additional miscellaneous results can be found in [30] [29] [31] [32]. When Carroll and Chang developed Candecomp, the main applications ... |

14 |
Typical rank and INDSCAL dimensionality for symmetric three-way arrays of order Ix2x2 or Ix3x3
- Berge, Sidiropoulos, et al.
(Show Context)
Citation Context ...plications they had in mind (a scalar product fitting problem related to Indscal) involved three-way arrays with slices that are symmetric in two of the three modes. Ten Berge, Sidiropoulos and Rocci =-=[33]-=- noted that this form of symmetry would affect typical ranks, and examined a number of cases; also see [34]. Quite often, indeed, symmetry of slices appears to entail lower typical rank values. On the... |

10 |
Bounds on the ranks of some 3-tensors
- Atkinson, Lloyd
- 1980
(Show Context)
Citation Context ...(that is, arrays of format 2 × N2 × N3). These results were further generalized in [29], to include all cases where, for N1 ≥ N2 ≥ N3, N1 > N2N3 − N2. Additional miscellaneous results can be found in =-=[30]-=- [29] [31] [32]. When Carroll and Chang developed Candecomp, the main applications they had in mind (a scalar product fitting problem related to Indscal) involved three-way arrays with slices that are... |

9 |
ten Berge. Partial uniqueness in CANDECOMP/PARAFAC
- F
(Show Context)
Citation Context ...arrays of format 2 × N2 × N3). These results were further generalized in [29], to include all cases where, for N1 ≥ N2 ≥ N3, N1 > N2N3 − N2. Additional miscellaneous results can be found in [30] [29] =-=[31]-=- [32]. When Carroll and Chang developed Candecomp, the main applications they had in mind (a scalar product fitting problem related to Indscal) involved three-way arrays with slices that are symmetric... |

7 | Supervised classification, a probabilistic approach - Comon - 1995 |

7 |
rappresentazione delle forme quaternarie mediante somme di potenze di forme lineari. Atti della R. Acc. delle Scienze di
- Sulla
- 1916
(Show Context)
Citation Context ...ical rank as the uncentered N1 × (N2 − 1)× (N3 − 1) array. 3 Computation of Generic Ranks The algorithm proposed is directly inspired by [4], which is in turn based on the so-called Terracini’s lemma =-=[36]-=- [37] [38]; note that the latter is often attributed to Lasker, and is hence almost one hundred years old. In a few works, the principle is based on the fact that the dimension of an irreducible varie... |

6 |
Non-Triviality and Identification of a Constrained Tucker3
- Berge, Smilde
(Show Context)
Citation Context ... zero elements in arbitrary arrays, we need tools to tell models from tautologies. This is where the concept of typical rank has found another realm of application. For instance, Ten Berge and Smilde =-=[14]-=- have argued that a sparse core hypothesized by Gurden et al. [15] is indeed a model and not a tautology. Their hypothetical core was a 5×5× 3 array with only 5 nonzero entries, hence of rank 5 at mos... |

6 |
Statement of some current results about three-way arrays. Unpublished manuscript
- Kruskal
- 1983
(Show Context)
Citation Context ... product used in the next section. 4 2.2 Historical remarks Bounds on the typical rank over the complex field were given in [2]. The study of typical rank over the real field was initiated by Kruskal =-=[27]-=- [24], who noted that 2 × 2 × 2 arrays had both rank 2 and rank 3 with positive probability. Kruskal also added a few typical ranks for small arrays. Franc [28] discussed some more results, including ... |

6 |
Symmetry transformations for square sliced three way arrays, with applications to their typical rank
- BERGE, STEGEMAN
(Show Context)
Citation Context ...s of format 2 × N2 × N3). These results were further generalized in [29], to include all cases where, for N1 ≥ N2 ≥ N3, N1 > N2N3 − N2. Additional miscellaneous results can be found in [30] [29] [31] =-=[32]-=-. When Carroll and Chang developed Candecomp, the main applications they had in mind (a scalar product fitting problem related to Indscal) involved three-way arrays with slices that are symmetric in t... |

6 | Generic and typical ranks of three-way arrays
- COMON, BERGE
(Show Context)
Citation Context ...ric N2×N2 matrix slices, but assume in addition that every row and column in the latter matrix slices are zero-mean. In order to achieve this, it is sufficient to generate vectors b(r) with zero mean =-=[39]-=-; in other words, only N2 − 1 random numbers need to be drawn, the last entry of each vector b(r) being obtained via bN2 = − ∑N2−1 n2=1 bn2 . The Jacobian takes then the expression below: J = ... |

5 |
Modelling of spectroscopic batch process data using greymodels to incorporate external information
- Gurden, Westerhuis, et al.
- 2001
(Show Context)
Citation Context ...rom tautologies. This is where the concept of typical rank has found another realm of application. For instance, Ten Berge and Smilde [14] have argued that a sparse core hypothesized by Gurden et al. =-=[15]-=- is indeed a model and not a tautology. Their hypothetical core was a 5×5× 3 array with only 5 nonzero entries, hence of rank 5 at most. Because 5×5×3 arrays have a typical rank of at least 7, it is c... |

2 |
The Carroll & Chang conjecture of equal INDSCAL components when Candecomp/Parafac gives perfect fit. Linear Algebra Appl
- BERGE, STEGEMAN, et al.
- 2008
(Show Context)
Citation Context ... with slices that are symmetric in two of the three modes. Ten Berge, Sidiropoulos and Rocci [33] noted that this form of symmetry would affect typical ranks, and examined a number of cases; also see =-=[34]-=-. Quite often, indeed, symmetry of slices appears to entail lower typical rank values. On the other hand, there are also cases where symmetry of the slices does not affect the typical rank. A partial ... |

2 | Méthodes PARAFAC pour la Séparation de Signaux - CASTAING - 2006 |