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## A multilinear singular value decomposition (2000)

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Venue: | SIAM J. Matrix Anal. Appl |

Citations: | 472 - 22 self |

### Citations

1851 |
Independent component analysis, a new concept?
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- 1994
(Show Context)
Citation Context ..., the interested reader is referred to the tutorial papers [34, 37, 38, 30] and the bibliography [41]; for the use of tensor decompositions as a basic tool in HOS-based signal processing, we refer to =-=[11, 12, 9, 19]-=-). Higher-order tensors do not merely play an important role in HOS. As a matter of fact they seem to be used in the most various disciplines, like chemometrics, psychometrics, econometrics, image pro... |

429 |
Some mathematical notes on three-mode factor analysis.
- Tucker
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Citation Context ...h person in a group of individuals is measured on each trait in a group of traits by each of a number of methods,” or “when individuals are measured by a battery of measures on a number of occasions” =-=[42, 43]-=-. For three-way data, the Tucker model consists of decomposing a real (I1 × I2 × I3)-tensor A according to (1) ai1i2i3 = I1� j1 I2� j2 I3� j3 sj1j2j3u (1) i1j1 u(2) i2j2 u(3) i3j3 , in which u (1) i1j... |

350 |
Introduction to Matrix Analysis,
- Bellman
- 1995
(Show Context)
Citation Context ...HOSVD can be obtained by unfolding A and S in model equation (5): A (n) = U (n) � · S (n) · U (n+1) ⊗ U (n+2) ⊗···⊗U (N) ⊗ U (1) ⊗ U (2) ⊗···⊗U (n−1)� T , (8) in which ⊗ denotes the Kronecker product =-=[2, 40]-=-. (The Kronecker product of two matrices F ∈ CI1×I2 and G ∈ CJ1×J2 is defined according to F ⊗ G def =(fi1i2 G) 1�i1�I1;1�i2�I2 .) Notice that the conditions (6) and (7) imply that S (n) has mutually ... |

257 | Multidimensional independent component analysis. - Cardoso - 1998 |

223 | Tensor Methods in Statistics. - McCullagh - 1987 |

138 |
Tutorial on higher-order statistics (spectra) in signal processing and system theory: theoretical results and some applications,
- Mendel
- 1991
(Show Context)
Citation Context ...lind identification of linear filters, without making assumptions on the minimum/nonminimum phase character, etc. (for general aspects of HOS, the interested reader is referred to the tutorial papers =-=[34, 37, 38, 30]-=- and the bibliography [41]; for the use of tensor decompositions as a basic tool in HOS-based signal processing, we refer to [11, 12, 9, 19]). Higher-order tensors do not merely play an important role... |

110 |
A Short Introduction to Perturbation Theory for Linear Operators,
- Kato
- 1982
(Show Context)
Citation Context ...· Y H . Then the m preferred n-mode singular vectors are given by the columns of U (n) 1 · X. (The m preferred right singular vectors in (31) are given by the columns of V (n) 1 · Y.) In analogy with =-=[27, 20]-=-, we state now the following perturbation theorem. Theorem 7 (first-order perturbation of the HOSVD). Consider a higher-order tensor A(ɛ), the elements of which are analytic functions of a real parame... |

107 |
Subspace Identification for Linear Systems”,
- Overschee, Moor
- 1996
(Show Context)
Citation Context ...s of a given tensor yield an orthonormal basis for its nmode vector space (and its orthogonal complement). For matrices, the property was the starting-point for the development of subspace algorithms =-=[45, 46, 47]-=-; Property 7 allows for an extension of this methodology in multilinear algebra. Property 8 (norm). Let the HOSVD of A be given as in Theorem 2; then the following holds. �A� 2 R1 � � � RN� 2 � �2 = =... |

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

89 | Decomposition of quantics in sums of power of linear forms.
- Comon, Mourrain
- 1996
(Show Context)
Citation Context ..., the interested reader is referred to the tutorial papers [34, 37, 38, 30] and the bibliography [41]; for the use of tensor decompositions as a basic tool in HOS-based signal processing, we refer to =-=[11, 12, 9, 19]-=-). Higher-order tensors do not merely play an important role in HOS. As a matter of fact they seem to be used in the most various disciplines, like chemometrics, psychometrics, econometrics, image pro... |

88 | Multilinear Algebra”, - Greub - 1967 |

87 |
Finite Dimensional Multilinear Algebra-Part I,
- Marcus
- 1973
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Citation Context ...gure 2 have proven to be very useful to gain insight in tensor techniques. The n-mode product of a tensor and a matrixis a special case of the inner product in multilinear algebra and tensor analysis =-=[32, 26]-=-. In the literature it often takes the form of an Einstein summation convention. Without going into details, this means that summations are written in full, but that the summation sign is dropped for ... |

84 |
Signal processing with higher-order spectra.
- Nikias, Mendel
- 1993
(Show Context)
Citation Context ...lind identification of linear filters, without making assumptions on the minimum/nonminimum phase character, etc. (for general aspects of HOS, the interested reader is referred to the tutorial papers =-=[34, 37, 38, 30]-=- and the bibliography [41]; for the use of tensor decompositions as a basic tool in HOS-based signal processing, we refer to [11, 12, 9, 19]). Higher-order tensors do not merely play an important role... |

76 |
The extension of factor analysis to three-dimensional matrices.
- Tucker
- 1964
(Show Context)
Citation Context ...h person in a group of individuals is measured on each trait in a group of traits by each of a number of methods,” or “when individuals are measured by a battery of measures on a number of occasions” =-=[42, 43]-=-. For three-way data, the Tucker model consists of decomposing a real (I1 × I2 × I3)-tensor A according to (1) ai1i2i3 = I1� j1 I2� j2 I3� j3 sj1j2j3u (1) i1j1 u(2) i2j2 u(3) i3j3 , in which u (1) i1j... |

67 | Super-symmetric decomposition of the fourth-order cumulant tensor. Blind identification of more sources than sensors. - Cardoso - 1991 |

63 |
The PARAFAC model for three-way factor analysis and multidimensional scaling, in:
- Harshman, Lundy
- 1984
(Show Context)
Citation Context ...t the HOSVD does not provide precise information about rank-related issues. With this respect, the “canonical decomposition” (CANDECOMP), or “parallel factors” model (PARAFAC) may be more informative =-=[10, 24, 1, 31, 39]-=-. Definition 11 (CANDECOMP). A canonical decomposition or parallel factors decomposition of a tensor A ∈ C I1×I2×···×IN is a decomposition of A as a linear combination of a minimal number of rank-1 te... |

54 |
A weighted non-negative least squares algorithm for three-way “PARAFAC” factor analysis
- Paatero
- 1997
(Show Context)
Citation Context ...t the HOSVD does not provide precise information about rank-related issues. With this respect, the “canonical decomposition” (CANDECOMP), or “parallel factors” model (PARAFAC) may be more informative =-=[10, 24, 1, 31, 39]-=-. Definition 11 (CANDECOMP). A canonical decomposition or parallel factors decomposition of a tensor A ∈ C I1×I2×···×IN is a decomposition of A as a linear combination of a minimal number of rank-1 te... |

42 | Decomposition, and Uniqueness for 3-Way and n-Way Arrays, - Kruskal, Rank - 1989 |

39 |
A decomposition for three-way arrays,
- Leurgans, Ross, et al.
- 1993
(Show Context)
Citation Context ...t the HOSVD does not provide precise information about rank-related issues. With this respect, the “canonical decomposition” (CANDECOMP), or “parallel factors” model (PARAFAC) may be more informative =-=[10, 24, 1, 31, 39]-=-. Definition 11 (CANDECOMP). A canonical decomposition or parallel factors decomposition of a tensor A ∈ C I1×I2×···×IN is a decomposition of A as a linear combination of a minimal number of rank-1 te... |

36 | P.: Independent component analysis, a survey of some algebraic methods
- Cardoso, Comon
- 1996
(Show Context)
Citation Context ..., the interested reader is referred to the tutorial papers [34, 37, 38, 30] and the bibliography [41]; for the use of tensor decompositions as a basic tool in HOS-based signal processing, we refer to =-=[11, 12, 9, 19]-=-). Higher-order tensors do not merely play an important role in HOS. As a matter of fact they seem to be used in the most various disciplines, like chemometrics, psychometrics, econometrics, image pro... |

33 | Eigen-structure of the fourth-order cumulant tensor with application to the blind source separation problem,” in - Cardoso - 1990 |

33 |
and uniqueness for 3-way and N-way arrays, in:
- Km&al, Rank
- 1989
(Show Context)
Citation Context ...ily be verified by checking some examples (see further). The rank of a higher-order tensor is usually defined in analogy with the fact that a rank-R matrixcan be decomposed as a sum of R rank-1 terms =-=[12, 29]-=-. Definition 3. An Nth-order tensor A has rank 1 when it equals the outer product of N vectors U (1) , U (2) , ..., U (N) , i.e., ai1i2...iN = u(1) i1 u(2) i2 ...u(N) iN , for all values of the indice... |

23 |
Three-mode principal component analysis.
- Kroonenberg
- 1983
(Show Context)
Citation Context ...ues are not significant (see also Property 6)), Â is still to be considered as a good approximation of A. The error is bounded as in (24). For procedures to enhance a given approximation, we refer to =-=[28, 17]-=-.s1268 L. DE LATHAUWER, B. DE MOOR, AND J. VANDEWALLE Fig. 6. Construction of H (1) for (a) a matrix and (b) a third-order tensor. Property 11 (link between HOSVD and matrixEVD). Let the HOSVD of A (n... |

21 | Independent component analysis based on higher-order statistics only. - Lathauwer, Moor, et al. - 1996 |

19 |
Kronecker products, unitary matrices and signal processing applications,
- Regalia, Mitra
- 1989
(Show Context)
Citation Context ...HOSVD can be obtained by unfolding A and S in model equation (5): A (n) = U (n) � · S (n) · U (n+1) ⊗ U (n+2) ⊗···⊗U (N) ⊗ U (1) ⊗ U (2) ⊗···⊗U (n−1)� T , (8) in which ⊗ denotes the Kronecker product =-=[2, 40]-=-. (The Kronecker product of two matrices F ∈ CI1×I2 and G ∈ CJ1×J2 is defined according to F ⊗ G def =(fi1i2 G) 1�i1�I1;1�i2�I2 .) Notice that the conditions (6) and (7) imply that S (n) has mutually ... |

17 |
Strategies for analyzing data from video fluorometric monitoring of liquid chromatographic effluents, Anal
- Appellof, Davidson
- 1981
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Citation Context |

13 | Higher-order power method— Application in independent component analysis - LATHAUWER, COMON, et al. - 1995 |

11 | Fourth-order cumulant structure forcing, Application to blind array processing - CARDOSO - 1992 |

11 |
Higherorder power method–Application in independent component analysis
- Lathauwer, Comon, et al.
- 1995
(Show Context)
Citation Context ...HOSVD has already proved its value. In [14] we showed that the decomposition is fundamentally related to the problem of blind source separation, also known as independent component analysis (ICA). In =-=[18]-=- we used the decomposition to compute an initial value for a tensorial equivalent of the power method, aiming at the computation of the best rank-1 approximation of a given tensor; a high-performant I... |

11 | editors. Multiway data analysis - Coppi, Bolasco - 1989 |

9 |
Blind source separation by simultaneous third-order tensor diagonalization
- Lathauwer, Moor, et al.
(Show Context)
Citation Context ...LUE DECOMPOSITION 1275 higher computational cost) than the computation of the HOSVD, in which the deviation from diagonality is simply considered as a perturbation effect. For algorithms, we refer to =-=[11, 16]-=-, exploring Jacobi-type procedures. 7.3. Canonical decomposition. A major difference between matrices and higher-order tensors is that the HOSVD does not provide precise information about rank-related... |

8 | Subspace Identi for Linear Systems: Theory{Implementation{Applications - Overschee, Moor - 1996 |

7 |
Blind sources separation by higher-order singular value decomposition
- DeLATHAUWER, DeMOOR, et al.
- 1994
(Show Context)
Citation Context ...he matrixSVD does lead to the definition of different (formally less striking) multilinear generalizations, as we will explain later on. In our own research the HOSVD has already proved its value. In =-=[14]-=- we showed that the decomposition is fundamentally related to the problem of blind source separation, also known as independent component analysis (ICA). In [18] we used the decomposition to compute a... |

7 |
Theory and Problems of Tensor Calculus
- Kay
- 1988
(Show Context)
Citation Context ...gure 2 have proven to be very useful to gain insight in tensor techniques. The n-mode product of a tensor and a matrixis a special case of the inner product in multilinear algebra and tensor analysis =-=[32, 26]-=-. In the literature it often takes the form of an Einstein summation convention. Without going into details, this means that summations are written in full, but that the summation sign is dropped for ... |

6 |
Dimensionality reduction in higherorder-only
- Lathauwer, Moor, et al.
- 1997
(Show Context)
Citation Context ...riance matrixof X, the number of skew or kurtic components might as well be estimated as the number of significant n-mode singular values of the third-order, resp., fourth-order, cumulant tensor of X =-=[17]-=-. Finally, we remember from section 2.2 that knowledge of the n-rank values of a given tensor does not allow us in general to make precise statements about the rank of that tensor. With this respect, ... |

4 |
The Candecomp–Candelinc family of models and methods for multidimensional data analysis
- Carroll, Pruzansky
- 1984
(Show Context)
Citation Context |

4 |
On the use of the singular value decomposition in identification and signal processing, in Numerical linear algebra, digital signal processing and parallel algorithms
- Vandewalle, Moor
- 1988
(Show Context)
Citation Context ...mation of the number of sources in the source separation problem, the estimation of filter lengths in identification, the estimation of the number of harmonics in the harmonic retrieval problem, etc. =-=[45, 46]-=-. Property 6 may play a similar role in multilinear algebra. Let us illustrate this by means of a small example. Consider the most elementary relationship in multivariate statistics, in which an I-dim... |

3 | Introduction to Vectors and Tensors. Linear and Multilinear Algebra - Bowen, Wang - 1980 |

3 |
Analytic properties of singular values and vectors
- Moor, Boyd
- 1989
(Show Context)
Citation Context ... one can prove that analyticity can be guaranteed by dropping the sign and ordering convention. With this goal we define, in analogy to the “unordered-unsigned singular value decomposition” (USVD) of =-=[20]-=- an unordered-unsigned higher-order singular value decomposition (UHOSVD). Theorem 5 (UHOSVD). If the elements of A∈CI1×I2×...×IN are analytic functions of a real parameter p, then there exist real an... |

2 |
Singular value decomposition: a powerful concept and tool
- Vandewalle, Callaerts
- 1990
(Show Context)
Citation Context ...mation of the number of sources in the source separation problem, the estimation of filter lengths in identification, the estimation of the number of harmonics in the harmonic retrieval problem, etc. =-=[45, 46]-=-. Property 6 may play a similar role in multilinear algebra. Let us illustrate this by means of a small example. Consider the most elementary relationship in multivariate statistics, in which an I-dim... |

2 | component analysis based on higher-order statistics only - “Independent - 1996 |

2 | source separation by simultaneous third-order tensor diagonalization - “Blind - 1996 |

1 |
From order 2 to HOS: new tools and applications
- Lacoume, Gaeta, et al.
- 1992
(Show Context)
Citation Context ...lind identification of linear filters, without making assumptions on the minimum/nonminimum phase character, etc. (for general aspects of HOS, the interested reader is referred to the tutorial papers =-=[34, 37, 38, 30]-=- and the bibliography [41]; for the use of tensor decompositions as a basic tool in HOS-based signal processing, we refer to [11, 12, 9, 19]). Higher-order tensors do not merely play an important role... |

1 | Multilinear Algebra - Nathcott - 1984 |

1 |
Higher order spectra in signal processing
- Nikias
- 1990
(Show Context)
Citation Context |

1 | Strategies for analyzing data from video monitoring of liquid chromatographic euents - Appellof, Davidson - 1981 |

1 | cumulant structure forcing. Application to blind array processing - Fourth-order |

1 | component analysis, a survey of some algebraic methods - Independent - 1996 |