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Local Convergence of the Alternating Least Squares Algorithm For Canonical Tensor Approximation
, 2011
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Detection of crossing white matter fibers with highorder tensors and
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Blind identification of underdetermined mixtures based on the hexacovariance and higherorder cyclostationarity
 Proc. SSP’09
, 2009
"... Abstract—Blind identification of underdetermined mixtures can be addressed efficiently by using the second ChAracteristic Function (CAF) of the observations. Our contribution is twofold. First, we propose the use of a LevenbergMarquardt algorithm, herein called LEMACAF, as an alternative to an Alte ..."
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Abstract—Blind identification of underdetermined mixtures can be addressed efficiently by using the second ChAracteristic Function (CAF) of the observations. Our contribution is twofold. First, we propose the use of a LevenbergMarquardt algorithm, herein called LEMACAF, as an alternative to an Alternating Least Squares algorithm known as ALESCAF, which has been used recently in the case of real mixtures of real sources. Second, we extend the CAF approach to the case of complex sources for which the previous algorithms are not suitable. We show that the complex case involves an appropriate tensor stowage, which is linked to a particular tensor decomposition. An extension of the LEMACAF algorithm, called LEMACAFC is then proposed to blindly estimate the mixing matrix by exploiting this tensor decomposition. In our simulation results, we first provide performance comparisons between third and fourth order versions of ALESCAF and LEMACAF in various situations involving BPSK sources. Then, a performance study of LEMACAFC is carried out considering 4QAM sources. These results show that the proposed algorithm provides satisfying estimations especially in the case of a large underdeterminacy level. Index Terms—Blind identification, blind source separation, characteristic function, complex sources, underdetermined mixtures, tensor decompositions I.
EEG extended source localization: tensorbased vs. conventional methods,” submitted to NeuroImage
, 2013
"... HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte p ..."
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HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et a ̀ la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. EEG extended source localization: tensorbased vs. conventional methods
GENERAL TENSOR DECOMPOSITION, MOMENT MATRICES AND APPLICATIONS
, 2011
"... The tensor decomposition addressed in this paper may be seen as a generalisation of Singular Value Decomposition of matrices. We consider general multilinear and multihomogeneous tensors. We show how to reduce the problem to a truncated moment matrix problem and give a new criterion for flat exten ..."
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The tensor decomposition addressed in this paper may be seen as a generalisation of Singular Value Decomposition of matrices. We consider general multilinear and multihomogeneous tensors. We show how to reduce the problem to a truncated moment matrix problem and give a new criterion for flat extension of QuasiHankel matrices. We connect this criterion to the commutation characterisation of border bases. A new algorithm is described. It applies for general multihomogeneous tensors, extending the approach of J.J. Sylvester to binary forms. An example illustrates the algebraic operations involved in this approach and how the decomposition can be recovered from eigenvector computation.
ICAbased eeg denoising: a comparative analysis of fifteen methods
 in Special Issue of the Bulletin of the Polish Academy of Sciences
, 2012
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Cited by 3 (2 self)
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HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et a ̀ la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
Blind source separation of underdetermined mixtures of eventrelated sources
 Signal Processing
, 2014
"... HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte p ..."
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HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et a ̀ la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. Blind source separation of underdetermined mixtures of eventrelated sources
ON THE GLOBAL CONVERGENCE OF THE HIGHORDER POWER METHOD FOR RANKONE TENSOR APPROXIMATION
, 2013
"... Tensor decomposition has consequential applications in various fields of disciplines, but it remains to be an extremely challenging task even to this date. A slightly more manageable endeavor has been to find a low rank approximation in place of the decomposition. Even for this less stringent task ..."
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Tensor decomposition has consequential applications in various fields of disciplines, but it remains to be an extremely challenging task even to this date. A slightly more manageable endeavor has been to find a low rank approximation in place of the decomposition. Even for this less stringent task, it is an established fact that tensors beyond matrices can fail to have best low rank approximations, with the notable exception that the best rankone approximation always exists for tensors of any order. Toward the latter, the most popular approach is via the notion of alternating projection. The specific numerical scheme appears as a variant of the power method. The so called proofs of convergence in the literature, however, have been only on the limiting behavior of the objective values. This is not enough, as the dynamics of the iterates itself should also be analyzed. The purpose of this paper is to fill the gap by presenting a rigorous convergence analysis of the power method for the rankone approximation. Also briefly discussed is a comparison of the power method with the global optimization.
WATER ANALYSIS WITH THE HELP OF TENSOR CANONICAL DECOMPOSITIONS
"... This study has been started two years ago by the laboratories of Radiochimie, Sciences Analytiques et Environnement (LRSAE) of the University of Nice SophiaAntipolis (UNS) and PROcessus de Transferts et d’Echanges dans l’Environ ..."
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This study has been started two years ago by the laboratories of Radiochimie, Sciences Analytiques et Environnement (LRSAE) of the University of Nice SophiaAntipolis (UNS) and PROcessus de Transferts et d’Echanges dans l’Environ
NONNEGATIVE 3WAY TENSOR FACTORIZATION TAKING INTO ACCOUNT POSSIBLE MISSING DATA
"... This paper deals with the problem of incomplete data i.e. data with missing, unknown or unreliable values, in the polyadic decomposition of a nonnegative threeway tensor. The main advantage of the nonnegativity constraint is that the approximation problem becomes well posed. To tackle simultaneou ..."
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This paper deals with the problem of incomplete data i.e. data with missing, unknown or unreliable values, in the polyadic decomposition of a nonnegative threeway tensor. The main advantage of the nonnegativity constraint is that the approximation problem becomes well posed. To tackle simultaneously these two problems, we suggest the use of a weighted least square cost function whose weights are gradually modified through the iterations. Moreover, the nonnegative nature of the loading matrices is taken into account directly in the problem parameterization. Then, the three gradient components can be explicitly derived allowing to efficiently implement the CP decomposition using standard optimization algorithms. In our case, we focus on the conjugate gradient and the BFGS algorithms. Finally, the good behaviour of the proposed approaches and their robustness versus possible model errors is illustrated through computer simulations in the context of data analysis.