The Experts below are selected from a list of 11973 Experts worldwide ranked by ideXlab platform
Ahmad Karfoul - One of the best experts on this subject based on the ideXlab platform.
-
Line search and trust region strategies for Canonical Decomposition of semi-nonnegative semi-symmetric 3rd order tensors
Linear Algebra and its Applications, 2014Co-Authors: Julie Coloigner, Laurent Albera, Ahmad Karfoul, Pierre ComonAbstract:Numerical solutions are proposed to fit the CanDecomp/ParaFac (CP) model of real three-way arrays, when the latter are both nonnegative and symmetric in two modes. In other words, a semi- nonnegative INDSCAL analysis is performed. The nonnegativity constraint is circumvented by means of changes of variable into squares, leading to an unconstrained problem. In addition, two globalization strategies are studied, namely line search and trust region. Regarding the former, a global plane search scheme is considered. It consists in computing, for a given direction, one or two optimal stepsizes, depending on whether the same stepsize is used in various updating rules. Moreover, we provide a compact matrix form for the derivatives of the objective func- tion. This allows for a direct implementation of several iterative algorithms such as conjugate gradient, Levenberg-Marquardt and Newton-like methods, in matrix programming environments like MATLAB. Our numerical results show the advantage of our optimization strategies when combined with a priori information such as partial symmetry.
-
Iterative methods for the Canonical Decomposition of multi-way arrays: Application to blind underdetermined mixture identification
Signal Processing, 2011Co-Authors: Ahmad Karfoul, Laurent Albera, Lieven De LathauwerAbstract:Two main drawbacks can be stated in the alternating least square (ALS) algorithm used to fit the Canonical Decomposition (CAND) of multi-way arrays. First its slow convergence caused by the presence of collinearity between factors in the multi-way array it decomposes. Second its blindness to Hermitian symmetries of the considered arrays. Enhanced line search (ELS) scheme was found to be a good way to cope with the slow convergence of the ALS algorithm together with a partial use of the Hermitian symmetry. However, to our knowledge, required equations to perform the latter scheme are only given in the case of third and fifth order arrays. Therefore, our first contribution consists in generalizing the ELS procedure to the case of complex arrays of any order greater than three. Our second contribution is another improvement of the ALS scheme, able to profit from Hermitianity and positive semi-definiteness of the considered arrays. It consists in resorting to the CAND first of a third order array having one unitary loading matrix and second of several rank-1 arrays. An iterative algorithm is then proposed alternating between Procrustes problem solving and the computation of rank-one matrix approximations in order to achieve the CAND of the third order array.
-
Blind underdetermined mixture identification by joint Canonical Decomposition of HO cumulants
IEEE Transactions on Signal Processing, 2010Co-Authors: Ahmad Karfoul, Laurent Albera, Gwénaël BirotAbstract:A new family of cumulant-based algorithms is proposed in order to blindly identify potentially underdetermined mixtures of statistically independent sources. These algorithms perform a joint Canonical Decomposition (CAND) of several higher order cumulants through a CAND of a three-way array with special symmetries. These techniques are studied in terms of identifiability, performance and numerical complexity. From a signal processing viewpoint, the proposed methods are shown i) to have a better estimation resolution and ii) to be able to process more sources than the other classical cumulant-based techniques. Second, from a numerical analysis viewpoint, we deal with the convergence speed of several procedures for three-way array Decomposition, such as the ACDC scheme. We also show how to accelerate the iterative CAND algorithms by using differently the symmetries of the considered three-way array. Next, from a multilinear algebra viewpoint the paper aims at giving some insights on the uniqueness of a joint CAND of several Hermitian multiway arrays compared to the CAND of only one array. This allows us, as a result, to extend the concept of virtual array (VA) to the case of combination of several VAs.
W Van Paesschen - One of the best experts on this subject based on the ideXlab platform.
-
Canonical Decomposition of ictal scalp eeg reliably detects the seizure onset zone
NeuroImage, 2007Co-Authors: M De Vos, Anneleen Vergult, L De Lathauwer, W De Clercq, S Van Huffel, Patrick Dupont, Andre Palmini, W Van PaesschenAbstract:Long-term electroencephalographic (EEG) recordings are important in the presurgical evaluation of refractory partial epilepsy for the delineation of the irritative and ictal onset zones. In this paper we introduce a new algorithm for an automatic, fast and objective localizing of the ictal onset zone in ictal EEG recordings. We extracted the potential distribution of the ictal activity from EEG using the higher order Canonical Decomposition method, also referred to as the CP model. The CP model decomposes in a unique way a higher order tensor in a minimal sum of rank-1 'atoms'. We showed that only one atom is related to the seizure activity. Simulation experiments demonstrated that the method correctly extracted the potential distribution of the ictal activity even with low signal-to-noise ratios. In 37 ictal EEGs, the CP method correctly localized the seizure onset zone in 34 (92%) and visual assessment in 21 cases (57%) (p=0.00024). The CP method is a fast method to delineate the ictal onset zone in ictal EEGs and is more sensitive than visual interpretation of the ictal EEGs.
-
Canonical Decomposition of ictal scalp eeg and accurate source localisation principles and simulation study
Computational Intelligence and Neuroscience, 2007Co-Authors: M De Vos, L De Lathauwer, S Van Huffel, Bart Vanrumste, W Van PaesschenAbstract:Long-term electroencephalographic (EEG) recordings are important in the presurgical evaluation of refractory partial epilepsy for the delineation of the ictal onset zones. In this paper, we introduce a new concept for an automatic, fast, and objective localisation of the ictal onset zone in ictal EEG recordings. Canonical Decomposition of ictal EEG decomposes the EEG in atoms. One or more atoms are related to the seizure activity. A single dipole was then fitted to model the potential distribution of each epileptic atom. In this study, we performed a simulation study in order to estimate the dipole localisation error. Ictal dipole localisation was very accurate, even at low signal-to-noise ratios, was not affected by seizure activity frequency or frequency changes, and was minimally affected by the waveform and depth of the ictal onset zone location. Ictal dipole localisation error using 21 electrodes was around 10.0 mm and improved more than tenfold in the range of 0.5-1.0 mm using 148 channels. In conclusion, our simulation study of Canonical Decomposition of ictal scalp EEG allowed a robust and accurate localisation of the ictal onset zone.
Gwénaël Birot - One of the best experts on this subject based on the ideXlab platform.
-
Blind underdetermined mixture identification by joint Canonical Decomposition of HO cumulants
IEEE Transactions on Signal Processing, 2010Co-Authors: Ahmad Karfoul, Laurent Albera, Gwénaël BirotAbstract:A new family of cumulant-based algorithms is proposed in order to blindly identify potentially underdetermined mixtures of statistically independent sources. These algorithms perform a joint Canonical Decomposition (CAND) of several higher order cumulants through a CAND of a three-way array with special symmetries. These techniques are studied in terms of identifiability, performance and numerical complexity. From a signal processing viewpoint, the proposed methods are shown i) to have a better estimation resolution and ii) to be able to process more sources than the other classical cumulant-based techniques. Second, from a numerical analysis viewpoint, we deal with the convergence speed of several procedures for three-way array Decomposition, such as the ACDC scheme. We also show how to accelerate the iterative CAND algorithms by using differently the symmetries of the considered three-way array. Next, from a multilinear algebra viewpoint the paper aims at giving some insights on the uniqueness of a joint CAND of several Hermitian multiway arrays compared to the CAND of only one array. This allows us, as a result, to extend the concept of virtual array (VA) to the case of combination of several VAs.
Lieven De Lathauwer - One of the best experts on this subject based on the ideXlab platform.
-
Canonical POLYADIC Decomposition WITH A COLUMNWISE ORTHONORMAL FACTOR MATRIX
2013Co-Authors: Mikael Sørensen, Lieven De Lathauwer, Pierre ComonAbstract:Abstract. Canonical Polyadic Decomposition (CPD) of a higher-order tensor is an important tool in mathematical engineering. In many applications at least one of the matrix factors is constrained to be column-wise orthonormal. We first derive a relaxed condition that guarantees uniqueness of the CPD under this constraint. Second, we give a simple proof of the existence of the optimal low-rank approximation of a tensor in the case that a factor matrix is column-wise orthonormal. Third, we derive numerical algorithms for the computation of the constrained CPD. In particular, orthogonality-constrained versions of the CPD methods based on simultaneous matrix diagonalization and alternating least squares are presented. Numerical experiments are reported. Key words. higher-order tensor, polyadic Decomposition, Canonical Decomposition (CANDE-COMP), parallel factor (PARAFAC), simultaneous matrix diagonalization, alternating least squares
-
Iterative methods for the Canonical Decomposition of multi-way arrays: Application to blind underdetermined mixture identification
Signal Processing, 2011Co-Authors: Ahmad Karfoul, Laurent Albera, Lieven De LathauwerAbstract:Two main drawbacks can be stated in the alternating least square (ALS) algorithm used to fit the Canonical Decomposition (CAND) of multi-way arrays. First its slow convergence caused by the presence of collinearity between factors in the multi-way array it decomposes. Second its blindness to Hermitian symmetries of the considered arrays. Enhanced line search (ELS) scheme was found to be a good way to cope with the slow convergence of the ALS algorithm together with a partial use of the Hermitian symmetry. However, to our knowledge, required equations to perform the latter scheme are only given in the case of third and fifth order arrays. Therefore, our first contribution consists in generalizing the ELS procedure to the case of complex arrays of any order greater than three. Our second contribution is another improvement of the ALS scheme, able to profit from Hermitianity and positive semi-definiteness of the considered arrays. It consists in resorting to the CAND first of a third order array having one unitary loading matrix and second of several rank-1 arrays. An iterative algorithm is then proposed alternating between Procrustes problem solving and the computation of rank-one matrix approximations in order to achieve the CAND of the third order array.
-
a link between the Canonical Decomposition in multilinear algebra and simultaneous matrix diagonalization
SIAM Journal on Matrix Analysis and Applications, 2006Co-Authors: Lieven De LathauwerAbstract:Canonical Decomposition is a key concept in multilinear algebra. In this paper we consider the Decomposition of higher-order tensors which have the property that the rank is smaller than the greatest dimension. We derive a new and relatively weak deterministic sufficient condition for uniqueness. The proof is constructive. It shows that the Canonical components can be obtained from a simultaneous matrix diagonalization by congruence, yielding a new algorithm. From the deterministic condition we derive an easy-to-check dimensionality condition that guarantees generic uniqueness.
-
computation of the Canonical Decomposition by means of a simultaneous generalized schur Decomposition
SIAM Journal on Matrix Analysis and Applications, 2005Co-Authors: Lieven De Lathauwer, Bart De Moor, Joos VandewalleAbstract: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.
Laurent Albera - One of the best experts on this subject based on the ideXlab platform.
-
Line search and trust region strategies for Canonical Decomposition of semi-nonnegative semi-symmetric 3rd order tensors
Linear Algebra and its Applications, 2014Co-Authors: Julie Coloigner, Laurent Albera, Ahmad Karfoul, Pierre ComonAbstract:Numerical solutions are proposed to fit the CanDecomp/ParaFac (CP) model of real three-way arrays, when the latter are both nonnegative and symmetric in two modes. In other words, a semi- nonnegative INDSCAL analysis is performed. The nonnegativity constraint is circumvented by means of changes of variable into squares, leading to an unconstrained problem. In addition, two globalization strategies are studied, namely line search and trust region. Regarding the former, a global plane search scheme is considered. It consists in computing, for a given direction, one or two optimal stepsizes, depending on whether the same stepsize is used in various updating rules. Moreover, we provide a compact matrix form for the derivatives of the objective func- tion. This allows for a direct implementation of several iterative algorithms such as conjugate gradient, Levenberg-Marquardt and Newton-like methods, in matrix programming environments like MATLAB. Our numerical results show the advantage of our optimization strategies when combined with a priori information such as partial symmetry.
-
Iterative methods for the Canonical Decomposition of multi-way arrays: Application to blind underdetermined mixture identification
Signal Processing, 2011Co-Authors: Ahmad Karfoul, Laurent Albera, Lieven De LathauwerAbstract:Two main drawbacks can be stated in the alternating least square (ALS) algorithm used to fit the Canonical Decomposition (CAND) of multi-way arrays. First its slow convergence caused by the presence of collinearity between factors in the multi-way array it decomposes. Second its blindness to Hermitian symmetries of the considered arrays. Enhanced line search (ELS) scheme was found to be a good way to cope with the slow convergence of the ALS algorithm together with a partial use of the Hermitian symmetry. However, to our knowledge, required equations to perform the latter scheme are only given in the case of third and fifth order arrays. Therefore, our first contribution consists in generalizing the ELS procedure to the case of complex arrays of any order greater than three. Our second contribution is another improvement of the ALS scheme, able to profit from Hermitianity and positive semi-definiteness of the considered arrays. It consists in resorting to the CAND first of a third order array having one unitary loading matrix and second of several rank-1 arrays. An iterative algorithm is then proposed alternating between Procrustes problem solving and the computation of rank-one matrix approximations in order to achieve the CAND of the third order array.
-
Semi-algebraic Canonical Decomposition of multi-way arrays and joint eigenvalue Decomposition
2011Co-Authors: Xavier Luciani, Laurent AlberaAbstract:A semi-algebraic algorithm based on Joint EigenValue Decomposition (JEVD) is proposed to compute the CP Decomposition of multi-way arrays. The iterative part of the method is thus limited to the JEVD computation. In addition it involves less restrictive hypothesis than other recent semi-algebraic approaches. We also propose an original JEVD technique based on the LU factorization. Numerical examples highlight the main advantages of the proposed methods to solve both the JEVD and CP problems.
-
Blind underdetermined mixture identification by joint Canonical Decomposition of HO cumulants
IEEE Transactions on Signal Processing, 2010Co-Authors: Ahmad Karfoul, Laurent Albera, Gwénaël BirotAbstract:A new family of cumulant-based algorithms is proposed in order to blindly identify potentially underdetermined mixtures of statistically independent sources. These algorithms perform a joint Canonical Decomposition (CAND) of several higher order cumulants through a CAND of a three-way array with special symmetries. These techniques are studied in terms of identifiability, performance and numerical complexity. From a signal processing viewpoint, the proposed methods are shown i) to have a better estimation resolution and ii) to be able to process more sources than the other classical cumulant-based techniques. Second, from a numerical analysis viewpoint, we deal with the convergence speed of several procedures for three-way array Decomposition, such as the ACDC scheme. We also show how to accelerate the iterative CAND algorithms by using differently the symmetries of the considered three-way array. Next, from a multilinear algebra viewpoint the paper aims at giving some insights on the uniqueness of a joint CAND of several Hermitian multiway arrays compared to the CAND of only one array. This allows us, as a result, to extend the concept of virtual array (VA) to the case of combination of several VAs.