## Computation of the canonical decomposition by means of a simultaneous generalized schur decomposition (2004)

Venue: | SIAM J. Matrix Anal. Appl |

Citations: | 41 - 9 self |

### BibTeX

@ARTICLE{Lathauwer04computationof,

author = {Lieven De Lathauwer and Bart De Moor and Joos Vandewalle},

title = {Computation of the canonical decomposition by means of a simultaneous generalized schur decomposition},

journal = {SIAM J. Matrix Anal. Appl},

year = {2004},

volume = {26},

pages = {295--327}

}

### Years of Citing Articles

### OpenURL

### Abstract

Abstract. The canonical decomposition of higher-order tensors is a key tool in multilinear algebra. First we review the state of the art. Then we show that, under certain conditions, the problem can be rephrased as the simultaneous diagonalization, by equivalence or congruence, of a set of matrices. Necessary and sufficient conditions for the uniqueness of these simultaneous matrix decompositions are derived. In a next step, the problem can be translated into a simultaneous generalized Schur decomposition, with orthogonal unknowns [A.-J. van der Veen and A. Paulraj, IEEE Trans. Signal Process., 44 (1996), pp. 1136–1155]. A first-order perturbation analysis of the simultaneous generalized Schur decomposition is carried out. We discuss some computational techniques (including a new Jacobi algorithm) and illustrate their behavior by means of a number of numerical experiments.

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Citation Context ...a minimal number of rank-1 terms: (2.1) A = R� r λr U (1) r ◦ U (2) r ◦···◦U (N) r The decomposition is visualized for third-order tensors in Figure 2.1. The terminology originates from psychometrics =-=[10]-=- and phonetics [26]. Later on, the decomposition model was also applied in chemometrics [1]. Recently, the decomposition drew the attention of researchers in signal processing [14, 16, 45, 46]. A good... |

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Citation Context ... rank-1 terms: (2.1) A = R� r λr U (1) r ◦ U (2) r ◦···◦U (N) r The decomposition is visualized for third-order tensors in Figure 2.1. The terminology originates from psychometrics [10] and phonetics =-=[26]-=-. Later on, the decomposition model was also applied in chemometrics [1]. Recently, the decomposition drew the attention of researchers in signal processing [14, 16, 45, 46]. A good tutorial of the cu... |

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Citation Context ...ations than an average random start. Whenever in this section we have used ALS iterations for the optimization of cost function f, we have also tried the general-purpose Levenberg–Marquardt algorithm =-=[39]-=- (we used the command lsqnonlin of the Optimization Toolbox 2.0 of MATLAB 5.3). In the last series of experiments, Levenberg–Marquardt gave consistently much less accurate results than ALS, even when ... |

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Citation Context ...ns the matrices in the intersection of the range of the cumulant tensor and the manifold of rank-1 matrices take the form of an outer product of a steering vector with itself; consequently MUSIC-like =-=[44]-=- algorithms are devised. In [6] the same author investigates the link between symmetry of the cumulant tensor and the rank-1 property of its components. The problem is subsequently reformulated in ter... |

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Citation Context ...ational cost is in line with results obtained for other simultaneous matrix decompositions. A Jacobi-rotation for a simultaneous real symmetric EVD can be computed by rooting a polynomial of degree 2 =-=[8, 9]-=-. For an SSD (Q = Z), polynomials are of degree 4 [25]. The Jacobi-result is an explicit solution for the CANDECOMP of rank-2 tensors. Apart from this result, a Jacobi-sweep is more expensive than an ... |

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Citation Context ...mposition—the SGSD. The reformulation in terms of orthogonal unknowns allows for the application of typical numerical procedures that involve orthogonal matrices. The SGSD as such already appeared in =-=[48]-=-. The difference is that in this paper it is applied to unsymmetric, instead of symmetric, matrices. This generalization may raise some confusion. It might, for instance, be tempting to consider also ... |

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Citation Context ...e that the two-sided Jacobi-algorithm cannot get stuck in a local optimum; local or global convergence is still an open problem for the computation of other simultaneous matrix decompositions as well =-=[4, 8, 9, 12, 23, 48]-=-. We have not observed convergence to a local optimum in any of our simulations for the unsymmetric CANDECOMP-problem. For the case where U (1) = U (2) , a meaningless result has been obtained for one... |

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Citation Context ...hows that a simultaneous EVD is much more robust than a single EVD. It is well known that, when eigenvalues are close, the eigenvectors in a single EVD may be strongly affected by small perturbations =-=[30]-=-. The reason is that for coinciding eigenvalues only the corresponding eigenspace is defined; different directions in this subspace will emerge as eigenvectors for different infinitesimal perturbation... |

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Citation Context ...calar product is zero. In [19] we discussed a possible multilinear generalization of the singular value decomposition (SVD). The different n-rank values can easily be read from this decomposition. In =-=[20]-=- we examined some techniques to compute the least-squares approximation of a given tensor by a tensor with prespecified n-ranks. On the other hand, in [19] we emphasized that the decomposition that wa... |

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Citation Context ... from psychometrics [10] and phonetics [26]. Later on, the decomposition model was also applied in chemometrics [1]. Recently, the decomposition drew the attention of researchers in signal processing =-=[14, 16, 45, 46]-=-. A good tutorial of the current state of the art in psychometrics and chemometrics is [3]. A λ1 λ2 λR U (2) 1 U (2) 2 = + + ...+ U (1) 1 U (3) 1 U (1) 2 U (3) 2 . U (1) R U (3) R Fig. 2.1. Visualizat... |

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Citation Context ..., however, equals 3, since in which A = E2 ◦ E1 ◦ E1 + E1 ◦ E2 ◦ E1 + E1 ◦ E1 ◦ E2, E1 = � 1 0 � � 0 , E2 = 1 is a decomposition in a minimal linear combination of rank-1 tensors (a proof is given in =-=[17]-=-). The scalar product 〈A, B〉 of two tensors A, B ∈ RI1×I2×...×IN is defined in a straightforward way as 〈A, B〉 def = � � ai1i2...iN bi1i2...iN . The Frobeniusi1 i2 ...� iN norm of a tensor A∈R I1×I2×.... |

66 | Non-orthogonal joint diagonalization in the leastsquares sense with application in blind source separation - Yeredor - 2002 |

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Citation Context ... algorithm. The reason is that the stationary points of the higher-order power iteration generally do not correspond to one of the terms in (2.4), and that the residue is in general not of rank R − 1 =-=[32]-=-. This even holds when the rank1 terms are mutually orthogonal [33]. Only when each of the matrices {U (n) } is column-wise orthonormal, the deflation approach will work, but in this special case, the... |

55 |
Rank-one approximation to high order tensors
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Citation Context ...t methods, given that a rigorous mathematical analysis of their convergence properties often proves to be extremely tough (as is witnessed by the fact that only very few related results are available =-=[4, 50]-=-). In a first series of experiments we will compare the accuracy of both techniques presented in section 7 and check whether an additional direct optimization of the cost function f, defined in (2.3),... |

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Citation Context ...d-order real-valued tensors is derived from a combinatorial algebraic perspective in [34]. The complex counterpart is concisely proved in [45]. The result is generalized to arbitrary tensor orders in =-=[47]-=-. In [6] complex fourth-order cumulant tensors are addressed. Here we will restrict ourselves to some remarks of a more general nature, that are of direct importance to this paper. From the CANDECOMP-... |

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Citation Context ...(with respect to their ideal values in an exact CANDECOMP) may have caused eigenvalues to cross each other. This is illustrated in the next example; a symmetric version of the example can be found in =-=[4]-=-.s308 L. DE LATHAUWER, B. DE MOOR, AND J. VANDEWALLE Example 6. Consider the following matrix pair: ⎛ ⎜ M1 = ⎜ ⎝ 1 − ɛ 0 0 0 1+ɛ 0 0 0 2 ⎞ 0 0 ⎟ 1 ⎠ 0 0 0 3 , M2 ⎛ ⎜ = ⎜ ⎝ 2 0 0 1 3 0 0 0 1−ɛ 0 0 0 ⎞ ... |

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Factor analysis of tongue shapes
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Citation Context ...epresenting the displacements of 13 points of the tongue of 5 test persons while pronouncing 10 vowels. A detailed description of these data and their analysis by means of a CANDECOMP can be found in =-=[29]-=-. The dataset can be downloaded from [21]. First, we observed that the two dominant 1-mode, 2-mode, and 3-mode singular values [19] explain 94.5%, 95.4%, and 96.0%, respectively, of the “energy” in th... |

37 |
The PARAFAC model for three-way factor analysis and multidimensional scaling
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Citation Context ...ank itself is usually determined by repeating the procedure for different values of R, and comparing the results. An alternative, also based on heuristics, is the evaluation of split-half experiments =-=[27]-=-.sA CANDECOMP ALGORITHM 301 ALS iterations can be very slow. In addition, it is sometimes observed that the algorithm moves through a “swamp”: the algorithm seems to converge, but then the convergence... |

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Citation Context ...ation schemes, at this moment only formulated in terms of (super-symmetric) cumulant tensors, have been developed as means to solve the problem of higher-order-only independent component analysis. In =-=[7]-=- Cardoso shows that under mild conditions the matrices in the intersection of the range of the cumulant tensor and the manifold of rank-1 matrices take the form of an outer product of a steering vecto... |

35 |
Rank, decomposition, and uniqueness for 3-way and n-way arrays
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Citation Context ... striking difference with the matrix case is that the rank of a real-valued tensor in the field of complex numbers is not necessarily equal to the rank of the same tensor in the field of real numbers =-=[35]-=-. Second, even if nonorthogonal rank-1 terms are allowed, the minimal number of terms is not bounded by min{I1, I2, ..., IN} in general (cf. Example 2); it is usually larger and depends also on the te... |

26 |
A decomposition for three-way arrays
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Citation Context ...ntains the following new contributions: • In the literature one finds that, in theory, the CANDECOMP can be computed by means of a matrix EVD (under the uniqueness assumptions specified in section 2) =-=[38, 43, 5, 42]-=-. We show that one can actually interpret the tensor decomposition as a simultaneous matrix decomposition. The simultaneous matrix decomposition is numerically more robust than a single EVD. • We show... |

24 |
A weighted non-negative least squares algorithm for three-way “PARAFAC” factor analysis
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Citation Context ...f acceleration methods have been proposed [26, 28, 31]. One could make use of a prediction technique, in which estimates of previous iteration steps are extrapolated to forecast new estimates [3]. In =-=[40]-=- a Gauss–Newton method is described, in which all the CANDECOMPfactors are updated simultaneously; in addition, the inherent indeterminacy of the decomposition has been fixed by adding a quadratic reg... |

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Citation Context ...nly increase again. The nature of swamps and how they can be avoided forms a topic of ongoing research [41, 36]. To cope with the slow convergence, a number of acceleration methods have been proposed =-=[26, 28, 31]-=-. One could make use of a prediction technique, in which estimates of previous iteration steps are extrapolated to forecast new estimates [3]. In [40] a Gauss–Newton method is described, in which all ... |

16 | A continuous Jacobi-like approach to the simultaneous reduction of real matrices
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Citation Context ...e that the two-sided Jacobi-algorithm cannot get stuck in a local optimum; local or global convergence is still an open problem for the computation of other simultaneous matrix decompositions as well =-=[4, 8, 9, 12, 23, 48]-=-. We have not observed convergence to a local optimum in any of our simulations for the unsymmetric CANDECOMP-problem. For the case where U (1) = U (2) , a meaningless result has been obtained for one... |

15 | DAISY: A database for the identification of systems - Moor, Gersem, et al. - 1997 |

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Citation Context ...ntains the following new contributions: • In the literature one finds that, in theory, the CANDECOMP can be computed by means of a matrix EVD (under the uniqueness assumptions specified in section 2) =-=[38, 43, 5, 42]-=-. We show that one can actually interpret the tensor decomposition as a simultaneous matrix decomposition. The simultaneous matrix decomposition is numerically more robust than a single EVD. • We show... |

14 | A counterexample to the possibility of an extension of the Eckart–Young low-rank approximation theorem for the orthogonal rank tensor decomposition
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Citation Context ...order power iteration generally do not correspond to one of the terms in (2.4), and that the residue is in general not of rank R − 1 [32]. This even holds when the rank1 terms are mutually orthogonal =-=[33]-=-. Only when each of the matrices {U (n) } is column-wise orthonormal, the deflation approach will work, but in this special case, the components can be obtained by means of a matrix SVD [19]. Because ... |

12 |
How 3-MFA data can cause degenerate Parafac solutions, among other relationships
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Citation Context ...eed drastically decreases and remains small for several iteration steps, after which it may suddenly increase again. The nature of swamps and how they can be avoided forms a topic of ongoing research =-=[41, 36]-=-. To cope with the slow convergence, a number of acceleration methods have been proposed [26, 28, 31]. One could make use of a prediction technique, in which estimates of previous iteration steps are ... |

11 |
Strategies for analyzing data from video fluorometric monitoring of liquid chromatographic effluents, Anal
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(Show Context)
Citation Context ...position is visualized for third-order tensors in Figure 2.1. The terminology originates from psychometrics [10] and phonetics [26]. Later on, the decomposition model was also applied in chemometrics =-=[1]-=-. Recently, the decomposition drew the attention of researchers in signal processing [14, 16, 45, 46]. A good tutorial of the current state of the art in psychometrics and chemometrics is [3]. A λ1 λ2... |

8 |
Continuous-time matrix algorithms, systolic algorithms and adaptive neural networks
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(Show Context)
Citation Context ...upp(Rk)R T k � � k R T k upp(Rk) in which skew(·) is the skew-symmetric and upp(·) the upper triangular part of a matrix. Proof. We will prove this result by resorting to the framework established in =-=[15, 22]-=-. The gradient of g with respect to Q can be determined by assuming that Q has a velocity ˙ Q on the manifold of orthogonal matrices and expressing the evolution of g: (6.10) ˙g = 〈∇Q g, ˙ Q〉 (see, e.... |

5 |
Resolution of multicomponent fluorescent mixtures by analysis of the excitation-emission-frequency .array
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(Show Context)
Citation Context ...ntains the following new contributions: • In the literature one finds that, in theory, the CANDECOMP can be computed by means of a matrix EVD (under the uniqueness assumptions specified in section 2) =-=[38, 43, 5, 42]-=-. We show that one can actually interpret the tensor decomposition as a simultaneous matrix decomposition. The simultaneous matrix decomposition is numerically more robust than a single EVD. • We show... |

4 |
An efficient algorithm for Parafac of three-way data with large numbers of observation units
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- 1991
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Citation Context ...nly increase again. The nature of swamps and how they can be avoided forms a topic of ongoing research [41, 36]. To cope with the slow convergence, a number of acceleration methods have been proposed =-=[26, 28, 31]-=-. One could make use of a prediction technique, in which estimates of previous iteration steps are extrapolated to forecast new estimates [3]. In [40] a Gauss–Newton method is described, in which all ... |

4 |
Component models for three-way data: An alternating least squares algorithm with optimal scaling features
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(Show Context)
Citation Context ...in risk that these eigenvalues have crossed each other. In Figure 9.6 we compare the mean value of �Û(1) − U (1) � obtained with a SGSD to the one obtained from the EVD of the matrix V2 · V −1 1 (cf. =-=[38, 43, 5, 42]-=-). It is clear that the SGSD is more accurate than a single EVD, because it takes all the matrices Vk into account. However, the technique is more sensitive to the condition number of U (1) . In the c... |

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(Show Context)
Citation Context ...simultaneous matrix decompositions. A Jacobi-rotation for a simultaneous real symmetric EVD can be computed by rooting a polynomial of degree 2 [8, 9]. For an SSD (Q = Z), polynomials are of degree 4 =-=[25]-=-. The Jacobi-result is an explicit solution for the CANDECOMP of rank-2 tensors. Apart from this result, a Jacobi-sweep is more expensive than an extended QZ-step if not min{R, K} ≫8. If the simultane... |

3 |
Two-factor degeneracies and a stabilization of
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(Show Context)
Citation Context ...eed drastically decreases and remains small for several iteration steps, after which it may suddenly increase again. The nature of swamps and how they can be avoided forms a topic of ongoing research =-=[41, 36]-=-. To cope with the slow convergence, a number of acceleration methods have been proposed [26, 28, 31]. One could make use of a prediction technique, in which estimates of previous iteration steps are ... |

1 |
An evaluation of five algorithms for generating an initial configuration for SINDSCAL
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- 1989
(Show Context)
Citation Context ...of the n-mode vectors in the CANDECOMP-model (1 � n � N) are close, then it seems unlikely that this configuration is found from a random start [14]. Some alternative initializations are discussed in =-=[11]-=-. The rank itself is usually determined by repeating the procedure for different values of R, and comparing the results. An alternative, also based on heuristics, is the evaluation of split-half exper... |

1 |
Jacobi-algorithm for simultaneous generalized Schur decomposition in higher-order-only ICA
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- 1998
(Show Context)
Citation Context .....,VK. 3. Compute orthogonal Q, Z and (approximately) upper triangular {Rk} (1�k�K) from the SGSD of (5.1)–(5.3): - extended QZ-iteration (section 7.1, [48]), or - Jacobi-type iteration (section 7.2, =-=[17, 18]-=-). 4. Compute U (1) and U (2) from {Rk} (1�k�K) and {Vk} (1�k�K) (and Q , Z). Compute U (3) from U (1) , U (2) and A. (Detailed outline in section 8.) (5. Minimize f( Â)=�A − Â�2 (section 2).) ai1i2i3... |

1 |
personal communication, Linguistics Dept
- Ladefoged
- 1996
(Show Context)
Citation Context ...nk estimates and different starting values, and cross-examining the results. On a SUN Ultra 2 Sparc and using MATLAB 4.2c, our computations took 0.2+0.04s of CPU-time, which was a drastic improvement =-=[37]-=-. 10. Conclusion. In this paper we have investigated the computation of the CANDECOMP, under the assumptions made in section 2. Currently, the calculation of the factors mostly takes the form of an AL... |