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## Tensor Decompositions and Applications (2009)

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Venue: | SIAM REVIEW |

Citations: | 723 - 18 self |

### Citations

3779 | Indexing by latent semantic analysis.
- Deerwester, Dumais, et al.
- 1990
(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... |

1851 |
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 ... |

676 | Using Linear Algebra for Intelligent Information Retrieval,
- Berry
- 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... |

611 |
Analysis of individual differences in multidimensional scaling via an N-way generalization of and 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... |

555 |
Foundations of the PARAFAC procedure: Models and conditions for an "explanatory" multi-modal factor analysis. UCLA Working Papers in Phonetics,
- Harshman
- 1970
(Show Context)
Citation Context ...ea 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 chemometrics, and tensors ha... |

472 | A multilinear singular value decomposition.
- Lathauwer, Moor, et al.
- 2000
(Show Context)
Citation Context ...1, 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, 211, 5, 209, 210, 44, 14], graph analysis [136, 135, 15],... |

455 |
The approximation of one matrix by another of lower rank.
- Eckart, Young
- 1936
(Show Context)
Citation Context ...R has a CP decomposition that is generically unique if R ≤ L and R(R − 1) ≤ IJK(3IJK − IJ − IK − JK − I − J − K +3)/4. 3.3. Low-Rank Approximations and the Border Rank. For matrices, Eckart and Young =-=[76]-=- showed that a best rank-k approximation is given by the leading k factors of the SVD. In other words, let R be the rank of a matrix A and assume its SVD is given by A = R∑ r=1 σr ur ◦ vr with σ1 ≥ σ2... |

421 |
The varimax criterion for analytic rotation in factor analysis.
- Kaiser
- 1958
(Show Context)
Citation Context ...(AT)(T−1RT−T)(AT)T. Thus, the solution obtained in A is not unique [95]. Nevertheless, it is standard practice to apply some accepted rotation to “fix” A. A common choice is to adopt VARIMAX rotation =-=[112]-=- such that the variance across columns of A is maximized. A further practice in some problems is to ignore the diagonal entries of X in the residual calculation [93]. For many cases, this makes sense ... |

194 | Tensor rank and the ill-posedness of the best low-rank approximation problem.
- Silva, Lim
- 2007
(Show Context)
Citation Context ...coordinate system. This notion of tensors is not to be confused with tensors in physics and engineering (such as stress tensors) [175], which are generally referred to as tensor fields in mathematics =-=[69]-=-. A third-order tensor has three indices, as shown in Figure 1.1. A first-order tensor is a vector, a second-order tensor is a matrix, and tensors of order three or higher are called higher-order tens... |

143 |
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 [... |

143 | Using latent semantic analysis to improve access to textual information.
- Dumais, Furnas, et al.
- 1988
(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... |

124 |
Tensorial extensions of independent component analysis for multisubject FMRI analysis.
- CF, SM
- 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... |

121 |
Tutorial and a 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... |

107 |
The expression of a tensor or a polyadic as a sum of products
- Hitchcock
- 1927
(Show Context)
Citation Context ... A third-order tensor: X∈ RI×J×K . in applied mathematics journals. Therefore, 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] a... |

100 |
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... |

99 | Symmetric tensors and symmetric tensor rank,”
- Comon, Golub, et al.
- 2008
(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... |

95 | Tensor rank is NP-complete,” - Hastad - 1990 |

92 |
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... |

89 |
A fast non negativity-constrained 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... |

89 | 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... |

85 | Numerical operator calculus in higher dimensions,
- Beylkin, Mohlenkamp
- 2002
(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... |

82 | Unsupervised multiway data analysis: A literature survey.
- Acar, Yener
- 2007
(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... |

81 |
A link between the canonical decomposition in multilinear algebra and simultaneous matrix diagonalization.
- Lathauwer
- 2006
(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... |

61 |
A new efficient method for determining the number of components in PARAFAC models.
- Bro, HA
- 2003
(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... |

59 | Sparse image coding using a 3d non-negative tensor factorization.
- Hazan, Polak, et al.
- 2005
(Show Context)
Citation Context ...nsor 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, 211, 5, 209, 210, 44, 14], graph analysis [136, 135, 15], neuroscience [20, 163, 165, 167, 170, 168, 169, 2, 3, 70, 71],... |

55 | Computation of the canonical decomposition by means of a simultaneous generalized Schur decomposition.
- Lathauwer, Moor, et al.
- 2004
(Show Context)
Citation Context ...d for reducing the CP fitting problem in the three-way case to a generalized eigenvalue problem, which works when the first two factor matrices are of full rank. De Lathauwer, De Moor, and Vandewalle =-=[66]-=- cast CP as a simultaneous generalized Schur decomposition (SGSD) if, for a third-order tensor X ∈ R I×J×K ,rank(X) ≤ min{I,J} and the matrix rank of the unfolded tensor X (3) is greater than or equal... |

48 | Kruskal’s permutation lemma and the identification of CANDECOMP/PARAFAC and bilinear models.
- Jiang, Sidisopoulos
- 2004
(Show Context)
Citation Context ...queness for the CP decomposition in (3.4) is kA + kB + kC ≥ 2R +2.468 TAMARA G. KOLDA AND BRETT W. BADER Kruskal’s result is nontrivial and has been reproven and analyzed many times over; see, e.g., =-=[199, 218, 111, 206]-=-. Sidiropoulos and Bro [199] recently extended Kruskal’s result to N-way tensors. Let X be an N-way tensor with rank R and suppose that its CP decomposition is (3.5) X = R∑ r=1 a (1) r ◦ a(2) r ◦···◦a... |

47 | Appearance based object modeling using texture database: acquisition, compression and rendering.
- FURUKAWA, KAWASAKI, et al.
- 2002
(Show Context)
Citation Context ...e [14] used CP for automatic conversation detection in email over time using a term-byauthor-by-time array. Shashua and 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) ... |

44 | Decompositions of a higher-order tensor in block terms—Part III: Alternating least squares algorithms
- Lathauwer, Nion
(Show Context)
Citation Context ...ier, 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 Lathauwer and Nion [67] (and references therein) explore a general class of decompositions as in (5.10), looking C1 CR G 1 B1 G R BR X ≈ A1 + ···+ AR Fig. 5.4 Block decomposition of a third-or... |

43 |
Recent developments in CANDECOMP/PARAFAC algorithms: a critical review
- Faber, Bro, et al.
(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... |

42 | Hierarchical Kronecker tensor-product approximations.
- Hackbusch, Khoromskij, et al.
- 2005
(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... |

40 |
Multiway analysis of epilepsy tensors
- Acar, 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... |

40 |
Data preprocessing and the extended PARAFAC model, in:
- Harshman, Lundy
- 1984
(Show Context)
Citation Context ... uniqueness is due to Kruskal [141, 143] and depends on the concept of k-rank. The k-rank of a matrix A, denoted kA, is defined as the maximum value k such that any k columns are linearly independent =-=[141, 98]-=-. Kruskal’s result [141, 143] says that a sufficient condition for uniqueness for the CP decomposition in (3.4) is kA + kB + kC ≥ 2R+ 2. D ow nl oa de ds09 /0 6/ 12stos1 61 .5 3. 8. 10 . R ed ist rib ... |

39 |
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.... |

38 |
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... |

38 | Existence and computation of low Kronecker-rank approximations for large linear systems of tensor product structure.
- Grasedyck
- 2004
(Show Context)
Citation Context ... applications ranging from psychometrics and chemometrics to computer visualization and data mining, but many more applications of tensors are being developed. For instance, in mathematics, Grasedyck =-=[86]-=- uses the tensor structure in finite element computations to generate low-rank Kronecker product approximations. As another example, Lim [154] (see also Qi [183]) discusses higher-order extensions of ... |

37 | Dimensionality Reduction in Higher-Order Signal Processing and
- Lathauwer, Vandewalle
- 2004
(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... |

35 |
Modeling and multi-way analysis of chatroom tensors,”in ISI
- Acar, amtepe, 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,... |

35 | A Newton-Grassmann method for computing the best multilinear rank-(r1, r2, r3) approximation of a tensor.
- Eldén, Savas
- 2007
(Show Context)
Citation Context ...AND APPLICATIONS 479 methods for speeding up the HOOI algorithm, examining how to do the computations, how to initialize the method, and how to compute the singular vectors. Recently, Eldén and Savas =-=[77]-=- proposed a Newton–Grassmann optimization approach for computing a Tucker decomposition of a three-way tensor. The problem is cast as a nonlinear program with the factor matrices constrained to a Gras... |

34 | Nonnegative 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... |

33 |
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... |

32 | A randomized algorithm for a tensor-based generalization of the singular value decomposition, Linear Algebra and its
- DRINEAS, MAHONEY
(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... |

31 |
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... |

28 |
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... |

28 | Non-negative tensor factorisation for sound source separation
- FitzGerald, Cranitch, et al.
- 2005
(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... |

27 |
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,... |

25 |
Practical aspects of PARAFAC modeling of fluorescence excitation-emission data 1.
- Andersen, Bro
- 2003
(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... |

24 | Temporal analysis of semantic graphs using ASALSAN, in ICDM,
- Bader, Harshman, et al.
- 2007
(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... |

24 | Tensor decomposition and approximation schemes for constraint satisfaction problems
- Vega, Karpinski, et al.
- 2005
(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... |

23 |
Independent component analysis and (simultaneous) third-order tensor diagonalization
- Lathauwer, Moor, et al.
- 2001
(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 ... |

23 |
Tensor-product approximation to operators and functions in high dimensions,
- Hackbusch, Khoromskij
- 2007
(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... |

20 | 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,... |

20 |
Frommatrix to tensor: Multilinear algebra and signal processing
- Lathauwer, DeMoor
- 1998
(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... |

20 |
Topographic component (parallel factor) analysis of multichannel evoked potentials: practical issues in trilinear spatiotemporal decomposition.
- Field, Graupe
- 1991
(Show Context)
Citation Context ...red CP in the context of event-related potentials in brain imaging; this work also included results on uniqueness and remarks about how to choose the number of factors. Later work by Field and Graupe =-=[79]-=- put the work of Möcks in context with the other work on CP, discussed practical aspects of working with data (scaling, etc.), and gave examples illustrating the utility of the CP decomposition for ev... |

19 | Blind identification of underdetermined mixtures by simultaneous matrix diagonalization
- Lathauwer, Castaing
- 2008
(Show Context)
Citation Context ...ered the application of CP to sensor array processing. Other applications in telecommunications include [198, 197, 59]. CP also has important applications in independent component analysis (ICA); see =-=[58]-=- and references therein. Several authors have used CP decompositions in neuroscience. As mentioned previously, Möcks [166] independently discovered CP in the context of event-related potentials in bra... |

19 |
An approach to nmode component analysis,
- Kapteyn, Neudecker, et al.
- 1986
(Show Context)
Citation Context ...hree-way and some for N-way). Name Proposed by Three-mode factor analysis (3MFA/Tucker3) Tucker, 1966 [226] Three-mode PCA (3MPCA) Kroonenberg and De Leeuw, 1980 [140] N-mode PCA Kapteyn et al., 1986 =-=[113]-=- Higher-order SVD (HOSVD) De Lathauwer et al., 2000 [63] N-mode SVD Vasilescu and Terzopoulos, 2002 [229] The Tucker decomposition is a form of higher-order PCA. It decomposes a tensor into a core ten... |

17 |
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... |

17 |
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... |

17 | Modeling multi-way data with linearly dependent loadings
- Bro, Harshman, et al.
(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... |

17 |
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.... |

16 |
A model for the analysis of asymmetric data in marketing research
- Harshman, Green, et al.
- 1982
(Show Context)
Citation Context ...lumns of A may be scaled in a number of ways without affecting the solution. One choice is to have unit length in the 2-norm; other choices give rise to different benefits of interpreting the results =-=[95]-=-. Second, the matrix A can be transformed with any nonsingular matrix T with no loss of fit to the data because ARA T =(AT)(T −1 RT −T )(AT) T . Thus, the solution obtained in A is not unique [95]. Ne... |

16 | Uniqueness proof for a family of models sharing features of Tucker’s three-mode factor analysis and PARAFAC/CANDECOMP
- Harshman, Lundy
- 1996
(Show Context)
Citation Context ... the notion of n-rank, algorithms and computational issues, and applications. Section 5 covers other decompositions, including INDSCAL [38], PARAFAC2 [92], CANDELINC [39], DEDICOM [93], and PARATUCK2 =-=[100]-=-, and their applications. Section 6 provides information about software for tensor computations. We summarize our findings in section 7. 2. Notation and Preliminaries. In this review, we have tried to... |

16 |
Optimal evaluation of pairs of bilinear forms,”
- JaJa
- 1979
(Show Context)
Citation Context ...rd algorithm to determineTENSOR DECOMPOSITIONS AND APPLICATIONS 465 Table 3.2 Maximum ranks over R for three-way tensors. Tensor size Maximum rank Citation I × J × 2 min{I,J} +min{I,J,⌊max{I,J}/2}⌋} =-=[109, 143]-=- 3 × 3 × 3 5 [143] Table 3.3 Typical rank over R for three-way tensors. Tensor size Typical rank Citation 2 × 2 × 2 {2, 3} [143] 3 × 3 × 2 {3, 4} [142, 212] 5 × 3 × 3 {5, 6} [214] I × J × 2withI≥2J (v... |

15 |
Paesschen, W.: Canonical decomposition of ictal scalp EEG reliably detects the seizure onset zone. Neuroimage
- Vos, Vergult, et al.
- 2007
(Show Context)
Citation Context ...alled ERPWAVELAB for multichannel analysis of time-frequency transformed event-related activity of EEG474 TAMARA G. KOLDA AND BRETT W. BADER and MEG data. Recently, Acar et al. [2] and De Vos et al. =-=[70, 71]-=- used CP for analyzing epileptic seizures. Stegeman [203] explained the differences between a threeway extension of ICA and CP for multisubject fMRI data in terms of the higher-order statistical prope... |

13 | O(n2.7799) complexity for n×n approximate matrix multiplication - Bini, Capovani, et al. - 1979 |

12 | Collective sampling and analysis of high order tensors for chatroom communications
- Acar, Çamtepe, et al.
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Citation Context |

11 |
Structure-seeking multilinear methods for the analysis of fMRI data
- Andersen, Rayens
- 2004
(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... |

11 |
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... |

11 | Maximum Likelihood Fitting Using Ordinary Least Squares Algorithms
- Bro, Sidiropoulos, et al.
- 2002
(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... |

11 |
Regalia, Tensor displacement structures and polyspectral matching, in Fast Reliable Algorithms for Matrices with Structure
- Grigorascu, A
- 1999
(Show Context)
Citation Context ...1, 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, 211, 5, 209, 210, 44, 14], graph analysis [136, 135, 15],... |

9 | Baynast, “Blind deconvolution of DS-CDMA signals by means of decomposition in rank-(1,L,L) terms
- Lathauwer, de
- 2008
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Citation Context |

9 | A least-squares approximation of high-dimensional uncertain systems
- Doostan, Iaccarino, et al.
- 2007
(Show Context)
Citation Context ...erators such as the Newton potential; see Hackbusch, Khoromskij, and Tyrtyshnikov [89] and Hackbusch and Khoromskij [88]. Very recent work has focused on applying CP decompositions to stochastic PDEs =-=[242, 73]-=-. 4. Compression and the Tucker Decomposition. The Tucker decomposition was first introduced by Tucker in 1963 [224] and refined in subsequent articles by Levin [153] and Tucker [225, 226]. Tucker’s 1... |

9 |
Three-way DEDICOM: Analyzing multiple matrices of asymmetric relationships. Paper presented at the Annual Meeting of the North American Psychometric Society
- Harshman, Lundy
- 1992
(Show Context)
Citation Context ...ARATUCK2 model. 5.4.2. DEDICOM Applications. Most of the applications of DEDICOM in the literature have focused on two-way (matrix) data, but there are some three-way applications. Harshman and Lundy =-=[99]-=- analyzed asymmetric measures of yearly trade (import-export) among a set of nations over a period of 10 years. Lundy et al. [160] presented an application of three-way DEDICOM to skew-symmetric data ... |

9 |
Body diagonalization of core matrices in three-way principal components analysis: Theoretical bounds and simulation
- Henrion
- 1993
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8 |
MultiArray: a C++ library for generic programming with arrays.
- Garcia, Lumsdaine
- 2005
(Show Context)
Citation Context ...perations and internal and external contractions. The tensors are assumed to be dense, though symmetries are exploited to optimize storage. The Boost Multidimensional Array Library (Boost.MultiArray) =-=[83]-=- provides a C++ class template for multidimensional arrays that is efficient and convenient for expressing dense N-dimensional arrays. These arrays may be accessed using a familiar syntax of native C+... |

8 |
Three-Way Data Resolution by Alternating Slice-Wise Diagonalization (ASD
- Jiang, Wu, et al.
(Show Context)
Citation Context ...vey, Faber, Bro, and Hopke [78] compare ALS with six different methods, none of which is better than ALS in terms of quality of solution, though the alternating slicewise diagonalization (ASD) method =-=[110]-=- is acknowledged as a viable alternative when computation time is paramount. In another survey, Tomasi and Bro [223] compare ALS and ASD to four other methods plus three variants that apply Tucker-bas... |

7 |
Lathauwer, Blind identification of convolutive MIM systems with 3 sources and 2
- Chen, Petropolu, et al.
- 2002
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6 |
Application of the three-way decomposition for matrix compression
- Ibraghimov
- 2002
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5 |
A general algorithm for obtaining simple structure of core arrays in N -way PCA with application to fluorometric data
- Andersson, Henrion
- 1999
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4 |
Mpeg video watermarking using tensor singular value decomposition. Image Analysis and Recognition
- 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 ... |

4 | of a Higher-Order Tensor in Block Terms – part II: Definitions and Uniqueness - “Decompositions |

4 |
Tensor-based techniques for the blind separation
- Lathauwer, Castaing
(Show Context)
Citation Context ...modeling of fluorescence excitation-emission data. Sidiropoulos, Bro, and Giannakis [196] considered the application of CP to sensor array processing. Other applications in telecommunications include =-=[198, 197, 59]-=-. CP also has important applications in independent component analysis (ICA); see [58] and references therein. Several authors have used CP decompositions in neuroscience. As mentioned previously, Möc... |

4 |
Computing nonnegative tensor factorizations
- Friedlander, Hatz
(Show Context)
Citation Context ...case of NNCP for a third-order tensor, subproblem (3.7) requires a least-squares solution for A. The change here is to impose nonnegativity constraints on A. Similarly, for NNCP, Friedlander and Hatz =-=[81]-=- solve a bound constrained linear least-squares problem. Additionally, they impose sparseness constraints by regularizing the NNCP with an l1-norm penalty function. While this function is nondifferent... |

4 | vs ‘slice’ methods for representing three-way structure via matrix notation
- Harshman, Hong, et al.
- 2002
(Show Context)
Citation Context ...tions in the area of tensor decompositions. The notation used here is very similar to that proposed by Kiers [122]. Other standards have been proposed as well; see Harshman [94] and Harshman and Hong =-=[96]-=-. The order of a tensor is the number of dimensions, also known as ways or modes. 3 Vectors (tensors of order one) are denoted by boldface lowercase letters, e.g., a. Matrices (tensors of order two) a... |

4 |
Shifted factor analysis—Part I: Models and properties
- Harshman, Hong, et al.
(Show Context)
Citation Context ...5] 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 =-=[97]-=- and convoluted CP [171]. 6. Software for Tensors. The earliest consideration of tensor computation issues dates back to 1973, when Pereyra and Scherer [182] considered basic operations in tensor spac... |

4 | principal component analysis theory, algorithms and applications - N-way - 1994 |

4 | Global properties of tensor rank. Linear Algebra and its Applications, 22:9 – 23 - Howell - 1978 |

3 |
Robust tensor classifiers for color object recognition, in Image Analysis and 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... |

3 |
Fourth-order cumulant based blind identification of underdetermined mixtures
- Lathauwer, Castaing
- 2007
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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 |

2 |
CuBatch, a MATLAB interface for n-mode data analysis
- Gourvénec, Tomasi, et al.
(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... |

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 |

1 | of article to appear - Golub, Loan - 1996 |