## Tensor Decompositions and Applications (2008)

### Cached

### Download Links

Citations: | 265 - 15 self |

### BibTeX

@MISC{Kolda08tensordecompositions,

author = {Tamara G. Kolda and Brett W. Bader},

title = {Tensor Decompositions and Applications },

year = {2008}

}

### OpenURL

### Abstract

This survey provides an overview of higher-order tensor decompositions, their applications, and available software. A tensor is a multidimensional or N-way array. Decompositions of higher-order tensors (i.e., N-way arrays with N ≥ 3) have applications in psychometrics, chemometrics, signal processing, numerical linear algebra, computer vision, numerical analysis, data mining, neuroscience, graph analysis, and elsewhere. Two particular tensor decompositions can be considered to be higher-order extensions of the matrix singular value decomposition: CANDECOMP/PARAFAC (CP) decomposes a tensor as a sum of rankone tensors, and the Tucker decomposition is a higher-order form of principal component analysis. There are many other tensor decompositions, including INDSCAL, PARAFAC2, CANDELINC, DEDICOM, and PARATUCK2 as well as nonnegative variants of all of the above. The N-way Toolbox, Tensor Toolbox, and Multilinear Engine are examples of software packages for working with tensors.

### Citations

1444 |
Independent component analysis, A new concept
- Comon
- 1994
(Show Context)
Citation Context ...s that optimizes a function that measures the “simplicity” of the core as measured by some objective [120]. Another is to use a Jacobi-type algorithm to maximize the magnitude of the diagonal entries =-=[46, 65, 162]-=-. Finally, the HOSVD generates an all-orthogonal core, as mentioned previously, which is yet another type of special core structure that might be useful. 4.4. Applications of Tucker. Several examples ... |

574 | Using linear algebra for intelligent information retrieval
- Berry, Dumais, et al.
- 1995
(Show Context)
Citation Context ... PARAFAC2 is that it can be used on the original data. Chew et al. [44] used PARAFAC2 for clustering documents across multiple languages. The idea is to extend the concept of latent semantic indexing =-=[75, 72, 21]-=- in a cross-language context by using multiple translations of the same collection of documents, i.e., a parallel corpus. In this case, Xk is a term-by-document matrix for the kth language in the para... |

362 |
Analysis of individual differences in multidimensional scaling via an n-way generalization of Eckart–Young decomposition
- Carroll, Chang
- 1970
(Show Context)
Citation Context ..., 106], and the idea of a multiway model is attributed to Cattell in 1944 [40, 41]. These concepts received scant attention until the work of Tucker in the 1960s [224, 225, 226] and Carroll and Chang =-=[38]-=- and Harshman [90] in 1970, all of which appeared in psychometrics literature. Appellof and Davidson [13] are generally credited as being the first to use tensor decompositions (in 1981) in chemometri... |

76 |
Polynomial interpolation in several variables
- Alexander, Hirschowitz
- 1995
(Show Context)
Citation Context ...) ⎤ ⎥ ,TENSOR DECOMPOSITIONS AND APPLICATIONS 467 except for when (N,I) ∈{(3, 5), (4, 3), (4, 4), (4, 5)}, in which case it should be increased by one. The result is due to Alexander and Hirschowitz =-=[7, 49]-=-. 3.2. Uniqueness. An interesting property of higher-order tensors is that their rank decompositions are often unique, whereas matrix decompositions are not. Sidiropoulos and Bro [199] and Ten Berge [... |

68 | Decomposition of quantics in sums of power of linear forms
- Comon, Mourrain
- 1996
(Show Context)
Citation Context ...the elements of G are zero, thereby eliminating interactions between corresponding components and improving uniqueness. Superdiagonalization of the core is impossible (even in the symmetric case; see =-=[50, 49]-=-), but it is possible to try to make as many elements zero or very small as possible. This was first observed by Tucker [226] and has been studied by several authors; see, e.g., [117, 103, 127, 172, 1... |

67 |
Tutorial and applications
- Bro, PARAFAC
- 1997
(Show Context)
Citation Context ... literature. Appellof and Davidson [13] are generally credited as being the first to use tensor decompositions (in 1981) in chemometrics, and tensors have since become extremely popular in that field =-=[103, 201, 27, 28, 31, 152, 241, 121, 12, 9, 29]-=-, even spawning a book in 2004 [200]. In parallel to the developments in psychometrics and chemometrics, there was a great deal of interest in decompositions of bilinear forms in the field of algebrai... |

64 |
Algorithm 862: Matlab tensor classes for fast algorithm prototyping
- Bader, Kolda
- 2006
(Show Context)
Citation Context ...ng CP and Tucker, with an emphasis toward data analysis in chemometrics. Like the N-way Toolbox, the PLS Toolbox can handle constraints and missing data. The MATLAB Tensor Toolbox, by Bader and Kolda =-=[16, 17, 18]-=-, is a generalpurpose set of classes that extends MATLAB’s core capabilities to support operations such as tensor multiplication and matricization. It comes with ALS-based algorithms for CP and Tucker... |

61 |
A fast non-negativityconstrained least squares algorithm
- Bro, Jong
- 1997
(Show Context)
Citation Context ...negative tensor factorization generically as NTF but fail to differentiate between CP and Tucker. Hereafter, we use the terminology NNCP (nonnegative CP) and NNT (nonnegative Tucker). Bro and De Jong =-=[32]-=- consider NNCP and NNT. They solve the subproblems in CP-ALS and Tucker-ALS with a specially adapted version of the NNLS method of Lawson and Hanson [150]. In the case of NNCP for a third-order tensor... |

59 |
The N-way toolbox for MATLAB
- Andersson, Bro
- 2000
(Show Context)
Citation Context ... 27, 28, 47, 129, 78, 48, 200, 69, 29, 6, 184], and a book has appeared very recently on multiway data analysis [139]. Moreover, there are several software packages available for working with tensors =-=[179, 11, 146, 85, 16, 17, 18, 239, 243]-=-. Wherever possible, the titles in the references section of this review are hyperlinked to either the publisher web page for the paper or the author’s version. Many older papers are now available onl... |

58 |
Tensorial extensions of independent component analysis for group fMRI data analysis
- Beckmann, Smith
- 2005
(Show Context)
Citation Context ...ion [229, 230, 231, 190, 236, 237, 193, 102, 235], numerical analysis [22, 108, 23, 89, 114, 88, 115], data mining [190, 4, 157, 211, 5, 209, 210, 44, 14], graph analysis [136, 135, 15], neuroscience =-=[20, 163, 165, 167, 170, 168, 169, 2, 3, 70, 71]-=-, and more. Several surveys have been written in other fields [138, 52, 104, 27, 28, 47, 129, 78, 48, 200, 69, 29, 6, 184], and a book has appeared very recently on multiway data analysis [139]. Moreo... |

50 |
A link between the canonical decomposition in multilinear algebra and simultaneous matrix diagonalization
- Lathauwer
(Show Context)
Citation Context ...A (N)) = R. They further observed that since rank(A ⊙ B) ≤ rank(A ⊗ B) ≤ rank(A) · rank(B), an even simpler necessary condition is ⎛ N∏ ⎜ min ⎝ rank(A (m) ⎞ ⎟ ) ⎠ ≥ R. n=1,...,N m=1 m̸=n De Lathauwer =-=[55]-=- has looked at methods to determine the rank of a tensor and the question of when a given CP decomposition is deterministically or generically (i.e., with probability one) unique. The CP decomposition... |

47 | Unsupervised multiway data analysis: A literature survey
- Acar, Yener
(Show Context)
Citation Context ...ning [190, 4, 157, 211, 5, 209, 210, 44, 14], graph analysis [136, 135, 15], neuroscience [20, 163, 165, 167, 170, 168, 169, 2, 3, 70, 71], and more. Several surveys have been written in other fields =-=[138, 52, 104, 27, 28, 47, 129, 78, 48, 200, 69, 29, 6, 184]-=-, and a book has appeared very recently on multiway data analysis [139]. Moreover, there are several software packages available for working with tensors [179, 11, 146, 85, 16, 17, 18, 239, 243]. Wher... |

45 | Numerical operator calculus in higher dimensions
- Beylkin, Mohlenkamp
(Show Context)
Citation Context ... signal processing [62, 196, 47, 43, 68, 80, 173, 60, 61], numerical linear algebra [87, 63, 64, 132, 244, 133, 149], computer vision [229, 230, 231, 190, 236, 237, 193, 102, 235], numerical analysis =-=[22, 108, 23, 89, 114, 88, 115]-=-, data mining [190, 4, 157, 211, 5, 209, 210, 44, 14], graph analysis [136, 135, 15], neuroscience [20, 163, 165, 167, 170, 168, 169, 2, 3, 70, 71], and more. Several surveys have been written in othe... |

41 | Symmetric tensor and symmetric tensor rank
- Comon, Golub, et al.
(Show Context)
Citation Context ...for all 1 ≤ in ≤ In. Figure 2.3 illustrates X = a ◦ b ◦ c, a third-order rank-one tensor. 2.2. Symmetry and Tensors. A tensor is called cubical if every mode is the same size, i.e., X ∈ R I×I×I×···×I =-=[49]-=-. A cubical tensor is called supersymmetric (thoughTENSOR DECOMPOSITIONS AND APPLICATIONS 459 Fig. 2.3 Rank-one third-order tensor, X = a ◦ b ◦ c. The (i, j, k) element of X is given by xijk = aibjck... |

31 |
A new efficient method for determining the number of components in PARAFAC models
- Bro, Kiers
(Show Context)
Citation Context ...rms of fit, and this does cause problems in practice [164, 186, 180]. When the data are noisy (as is frequently the case), then fit alone cannot determine the rank in any case; instead, Bro and Kiers =-=[34]-=- proposed a consistency diagnostic called CORCONDIA to compare different numbers of components. Assuming the number of components is fixed, there are many algorithms to compute a CP decomposition. Her... |

31 | Non-negative tensor factorization using alpha and beta divergences
- Cichocki, Zdunek, et al.
- 2007
(Show Context)
Citation Context ...onstraints can be imposed on the core tensor (to represent known interactions) and the factor matrices (per desired structure). They apply NNT with sparseness constraints to EEG data. Cichocki et al. =-=[45]-=- propose a nonnegative version of PARAFAC2 for EEG data. They develop a multiplicative update based on α- andβ-divergences as well as a regularized ALS algorithm and an alternating interior-point grad... |

26 |
Modeling and multiway analysis of chatroom tensors
- Acar, Camtepe, et al.
- 2005
(Show Context)
Citation Context ...173, 60, 61], numerical linear algebra [87, 63, 64, 132, 244, 133, 149], computer vision [229, 230, 231, 190, 236, 237, 193, 102, 235], numerical analysis [22, 108, 23, 89, 114, 88, 115], data mining =-=[190, 4, 157, 211, 5, 209, 210, 44, 14]-=-, graph analysis [136, 135, 15], neuroscience [20, 163, 165, 167, 170, 168, 169, 2, 3, 70, 71], and more. Several surveys have been written in other fields [138, 52, 104, 27, 28, 47, 129, 78, 48, 200,... |

22 |
Centering and scaling in component analysis
- Bro, Smilde
- 2003
(Show Context)
Citation Context ...ill be a need for libraries that take advantage of parallel processors and/or multicore architectures. Numerous datacentric issues exist as well, such as preparing the data for processing; see, e.g., =-=[36]-=-. Another problem is missing data, especially systematically missing data, which is an issue in chemometrics; see Tomasi and Bro [222] and references therein. For a general discussion on handling miss... |

21 |
Parallel Proportional Profiles" and other principles for determining the choice of factors by rotation
- Cattell
- 1944
(Show Context)
Citation Context ..., we wish to bring this research to the attention of SIAM readers. Tensor decompositions originated with Hitchcock in 1927[105, 106], and the idea of a multiway model is attributed to Cattell in 1944 =-=[40, 41]-=-. These concepts received scant attention until the work of Tucker in the 1960s [224, 225, 226] and Carroll and Chang [38] and Harshman [90] in 1970, all of which appeared in psychometrics literature.... |

19 |
Multiway analysis of epilepsy tensors
- Acar, Aykut-Bingol, et al.
- 2007
(Show Context)
Citation Context ...ion [229, 230, 231, 190, 236, 237, 193, 102, 235], numerical analysis [22, 108, 23, 89, 114, 88, 115], data mining [190, 4, 157, 211, 5, 209, 210, 44, 14], graph analysis [136, 135, 15], neuroscience =-=[20, 163, 165, 167, 170, 168, 169, 2, 3, 70, 71]-=-, and more. Several surveys have been written in other fields [138, 52, 104, 27, 28, 47, 129, 78, 48, 200, 69, 29, 6, 184], and a book has appeared very recently on multiway data analysis [139]. Moreo... |

18 |
Improving the speed of multi-way algorithms
- Andersson, Bro
- 1998
(Show Context)
Citation Context ...− 2〈 X ×1 A (1)T ···×N A (N)T , G 〉 + ‖G‖ 2 = ‖X‖ 2 − 2〈 G, G 〉 + ‖G‖ 2 = ‖X‖ 2 −‖G‖ 2 = ‖X‖ 2 −‖X ×1 A (1)T ×2 ···×N A (N)T ‖ 2 . The details of this transformation are readily available; see, e.g., =-=[10, 64, 134]-=-. Once again, we can use an ALS approach to solve (4.3). Because ‖X‖2 is constant, (4.3) can be recast as a series of subproblems involving the following maximization problem, which solves for the nth... |

16 |
Temporal analysis of semantic graphs using ASALSAN
- Bader, Harshman, et al.
(Show Context)
Citation Context ... 244, 133, 149], computer vision [229, 230, 231, 190, 236, 237, 193, 102, 235], numerical analysis [22, 108, 23, 89, 114, 88, 115], data mining [190, 4, 157, 211, 5, 209, 210, 44, 14], graph analysis =-=[136, 135, 15]-=-, neuroscience [20, 163, 165, 167, 170, 168, 169, 2, 3, 70, 71], and more. Several surveys have been written in other fields [138, 52, 104, 27, 28, 47, 129, 78, 48, 200, 69, 29, 6, 184], and a book ha... |

15 |
Discussion tracking in Enron email using PARAFAC
- Bader, Berry, et al.
- 2008
(Show Context)
Citation Context ...173, 60, 61], numerical linear algebra [87, 63, 64, 132, 244, 133, 149], computer vision [229, 230, 231, 190, 236, 237, 193, 102, 235], numerical analysis [22, 108, 23, 89, 114, 88, 115], data mining =-=[190, 4, 157, 211, 5, 209, 210, 44, 14]-=-, graph analysis [136, 135, 15], neuroscience [20, 163, 165, 167, 170, 168, 169, 2, 3, 70, 71], and more. Several surveys have been written in other fields [138, 52, 104, 27, 28, 47, 129, 78, 48, 200,... |

15 |
CANDELINC: a general approach to multidimensional analysis of many-way arrays with linear constraints on parameters
- Carroll, Pruzansky, et al.
- 1980
(Show Context)
Citation Context ...ual differences in scaling (INDSCAL) Carroll and Chang, 1970 [38] Parallel factors for cross products (PARAFAC2) Harshman, 1972 [92] CANDECOMP with linear constraints (CANDELINC) Carroll et al., 1980 =-=[39]-=- Decomposition into directional components (DEDICOM) Harshman, 1978 [93] PARAFAC and Tucker2 (PARATUCK2) Harshman and Lundy, 1996 [100] 5.1. INDSCAL. Individual differences in scaling (INDSCAL) is a s... |

14 |
Practical aspects of PARAFAC modeling of fluorescence excitationemission data
- Andersen, Bro
(Show Context)
Citation Context ... literature. Appellof and Davidson [13] are generally credited as being the first to use tensor decompositions (in 1981) in chemometrics, and tensors have since become extremely popular in that field =-=[103, 201, 27, 28, 31, 152, 241, 121, 12, 9, 29]-=-, even spawning a book in 2004 [200]. In parallel to the developments in psychometrics and chemometrics, there was a great deal of interest in decompositions of bilinear forms in the field of algebrai... |

14 | Cross-language information retrieval using parafac2
- Chew, Bader, et al.
- 2007
(Show Context)
Citation Context ...173, 60, 61], numerical linear algebra [87, 63, 64, 132, 244, 133, 149], computer vision [229, 230, 231, 190, 236, 237, 193, 102, 235], numerical analysis [22, 108, 23, 89, 114, 88, 115], data mining =-=[190, 4, 157, 211, 5, 209, 210, 44, 14]-=-, graph analysis [136, 135, 15], neuroscience [20, 163, 165, 167, 170, 168, 169, 2, 3, 70, 71], and more. Several surveys have been written in other fields [138, 52, 104, 27, 28, 47, 129, 78, 48, 200,... |

12 |
Selecting among three-mode principal component models of different types and complexities: A numerical convex hull based method
- Ceulemans, Kiers
- 2006
(Show Context)
Citation Context ... matrices to the identity matrix. There are obviously many choices for tensor decompositions, which may lead to confusion about which model to choose for a particular application. Ceulemans and Kiers =-=[42]-=- discuss methods for choosing between CP and the different Tucker models in the three-way case. 4.1. The n-Rank. Let X be an Nth-order tensor of size I1 × I2 ×···×IN .Then the n-rank of X, denoted ran... |

11 |
Strategies for analyzing data from video fluorometric monitoring of liquid chromatographic effluents, Anal
- Appellof, Davidson
- 1981
(Show Context)
Citation Context ...ed scant attention until the work of Tucker in the 1960s [224, 225, 226] and Carroll and Chang [38] and Harshman [90] in 1970, all of which appeared in psychometrics literature. Appellof and Davidson =-=[13]-=- are generally credited as being the first to use tensor decompositions (in 1981) in chemometrics, and tensors have since become extremely popular in that field [103, 201, 27, 28, 31, 152, 241, 121, 1... |

10 | Collective sampling and analysis of high order tensors for chatroom communications
- Acar, Çamtepe, et al.
- 2006
(Show Context)
Citation Context |

10 |
Maximum likelihood fitting using ordinary least squares algorithms
- Bro, Sidiropoulos, et al.
(Show Context)
Citation Context ...PCA that, in some sense, rotates the principal components so that they are statistically independent. Beckmann and Smith [20] extended CP to develop a probabilistic ICA. Bro, Sidiropoulos, and Smilde =-=[35]-=- and Vega-Montoto and Wentzell [234] formulated maximum-likelihood versions of CP. Acar and Yener [6] discuss several of these decompositions as well as some not covered here: shifted CP and Tucker [9... |

10 |
Tensor decomposition and approximation schemes for constraint satisfaction problems
- Vega, Kannan, et al.
(Show Context)
Citation Context ...less communications; see also [176]. Mahoney, Maggioni, and Drineas [161] extend the matrix CUR decomposition to tensors; others have done related work using sampling methods for tensor approximation =-=[74, 54]-=-. Vasilescu and Terzopoulos [233] explored higher-order versions of ICA, a variation of PCA that, in some sense, rotates the principal components so that they are statistically independent. Beckmann a... |

9 | O(n 2.7799 ) complexity for n∗n approximate matrix multiplication - Bini, Capovani, et al. - 1979 |

9 |
PARAFAC2-Part II. Modeling chromatographic data with retention time shifts
- Bro, Andersson, et al.
- 1999
(Show Context)
Citation Context ... literature. Appellof and Davidson [13] are generally credited as being the first to use tensor decompositions (in 1981) in chemometrics, and tensors have since become extremely popular in that field =-=[103, 201, 27, 28, 31, 152, 241, 121, 12, 9, 29]-=-, even spawning a book in 2004 [200]. In parallel to the developments in psychometrics and chemometrics, there was a great deal of interest in decompositions of bilinear forms in the field of algebrai... |

9 |
Modeling Multi-way Data with Linearly Dependent Loadings
- Bro, Harshman, et al.
- 2005
(Show Context)
Citation Context ...Ar, Br, Cr�. r=1 Here we assume G r is of size Mr×Nr×Pr, Ar is of size I×Mr, Br is of size J ×Nr, and Cr is of size K × Pr for r =1,...,R. Figure 5.4 shows an example. Bro, Harshman, and Sidiropoulos =-=[33, 6]-=- propose a version of this called the PARALIND model. De Almeida, Favier, and Mota [53] give an overview of some aspects of models of the form in (5.10) and their application to problems in blind beam... |

8 |
Lathauwer, Blind identification of convolutive MIM systems with 3 sources and 2
- Chen, Petropolu, et al.
- 2002
(Show Context)
Citation Context ...4 × 4 × 4 tensor to describe 2 × 2 matrix multiplication [208, 141, 147, 24]. In the last ten years, interest in tensor decompositions has expanded to other fields. Examples include signal processing =-=[62, 196, 47, 43, 68, 80, 173, 60, 61]-=-, numerical linear algebra [87, 63, 64, 132, 244, 133, 149], computer vision [229, 230, 231, 190, 236, 237, 193, 102, 235], numerical analysis [22, 108, 23, 89, 114, 88, 115], data mining [190, 4, 157... |

7 |
Structure-seeking multilinear methods for the analysis of fMRI data
- Andersen, Rayens
(Show Context)
Citation Context ...ther work on CP, discussed practical aspects of working with data (scaling, etc.), and gave examples illustrating the utility of the CP decomposition for event-related potentials. Andersen and Rayens =-=[8]-=- applied CP to fMRI data arranged as voxels by time by run and also as voxels by time by trial by run. Martínez-Montes et al. [163, 165] applied CP to a time-varying EEG spectrum arranged as a three-d... |

7 |
On the complexity of the multiplication of matrices of small formats
- Bläser
(Show Context)
Citation Context ...n be shown that the rank and border rank of the tensor are both equal to 7(see, e.g., [147]). The case of 3 × 3 matrix multiplication corresponds to a 9 × 9 × 9 tensor that has rank between 19 and 23 =-=[145, 26, 24]-=- and a border rank between 14 and 21 [192, 26, 24, 148].TENSOR DECOMPOSITIONS AND APPLICATIONS 471 procedure CP-ALS(X,R) initialize A (n) ∈ R In×R for n =1,...,N repeat for n =1,...,N do V ← A (1)T A... |

3 |
A general algorithm for obtaining simple structure of core arrays in N-way PCA with application to fluorimetric data
- Andersson, Henrion
- 1999
(Show Context)
Citation Context |

2 |
Robust tensor classifiers for color object recognition
- Bauckhage
- 2007
(Show Context)
Citation Context ...d Levin [194] applied CP to image compression and classification. Furukawa et al. [82] applied a CP model to bidirectional texture functions in order to build a compressed texture database. Bauckhage =-=[19]-=- extended discriminant analysis to higher-order data (color images, in this case) for classification. Beylkin and Mohlenkamp [22, 23] apply CP to operators and conjecture that the optimal separation r... |

2 |
The three basic factor-analytic research designs - their interrelations and derivatives
- Cattell
- 1952
(Show Context)
Citation Context ..., we wish to bring this research to the attention of SIAM readers. Tensor decompositions originated with Hitchcock in 1927[105, 106], and the idea of a multiway model is attributed to Cattell in 1944 =-=[40, 41]-=-. These concepts received scant attention until the work of Tucker in the 1960s [224, 225, 226] and Carroll and Chang [38] and Harshman [90] in 1970, all of which appeared in psychometrics literature.... |

2 | Canonical Tensor Decompositions, I3S report, RR-2004-17, CNRS-Laboratoire I3S, Université Nice-Sophia Antipolis - Comon - 2004 |

2 |
The Constrained Block-PARAFAC Decomposition. Presentation at TRICAP2006, Chania, Greece. Available online at http://www. telecom.tuc.gr/˜nikos/TRICAP2006main/TRICAP2006 Almeida.pdf
- Almeida, Favier, et al.
- 2006
(Show Context)
Citation Context ... J ×Nr, and Cr is of size K × Pr for r =1,...,R. Figure 5.4 shows an example. Bro, Harshman, and Sidiropoulos [33, 6] propose a version of this called the PARALIND model. De Almeida, Favier, and Mota =-=[53]-=- give an overview of some aspects of models of the form in (5.10) and their application to problems in blind beamforming and multiantenna coding. In a series of papers, De Lathauwer [56, 57] and De La... |

2 | of a Higher-Order Tensor in Block Terms – part I: Lemmas for Partitioned Matrices - “Decompositions |

1 |
MPEG video watermarking using tensor singular value decomposition
- Abdallah, Hamza, et al.
- 2007
(Show Context)
Citation Context ...ed the Tucker decomposition to the bidirectional texture function (BTF) for rendering texture in two-dimensional images. Many related applications exist, such as watermarking MPEG videos using Tucker =-=[1]-=-. Grigorascu and Regalia [87] consider extending ideas from structured matrices to tensors. They are motivated by the structure in higher-order cumulants and corresponding polyspectra in applications ... |

1 | recognition on epilepsy feature tensor - Seizure - 2007 |

1 | tensor toolbox, version 2.2. http://csmr.ca.sandia.gov/~tgkolda/ TensorToolbox - MATLAB - 2007 |

1 |
The Role of Tensor Rank in the Complexity Analysis of Bilinear Forms, presentation at ICIAM07
- Bini
- 2007
(Show Context)
Citation Context ...uth [130, sec. 4.6.4]. The most interesting example of this is Strassen matrix multiplication, which is an application of a decomposition of a 4 × 4 × 4 tensor to describe 2 × 2 matrix multiplication =-=[208, 141, 147, 24]-=-. In the last ten years, interest in tensor decompositions has expanded to other fields. Examples include signal processing [62, 196, 47, 43, 68, 80, 173, 60, 61], numerical linear algebra [87, 63, 64... |

1 | on multiway analysis in chemistry - Review |

1 |
Maximal rank of m× n× (mn− k) tensors
- Bshouty
- 1990
(Show Context)
Citation Context ...tchcock [105] first proposed this definition of rank in 1927, and Kruskal [141] did so independently 50 years later. For the perspective of tensor rank from an algebraic complexity point of view, see =-=[107, 37, 130, 24]-=- and references therein. An exact CP decomposition with R =rank(X) components is called the rank decomposition. The definition of tensor rank is an exact analogue to the definition of matrix rank, but... |

1 | Generic and typical ranks of multi-way arrays. Submitted for publication - Comon, Berge, et al. - 2007 |