Nonnegative Compression for Semi-Nonnegative Independent Component Analysis

International audienceIn many Independent Component Analysis (ICA) problems the mixing matrix is nonnegative while the sources are unconstrained, giving rise to what we call hereafter the Semi-Nonnegative ICA (SN-ICA) problems. Exploiting the nonnegativity property can improve the ICA result. Besides, in some practical applications, the dimension of the observation space must be reduced. However, the classical dimension compression procedure, such as prewhitening, breaks the nonnegativity property of the compressed mixing matrix. In this paper, we introduce a new nonnegative compression method, which guarantees the nonnegativity of the compressed mixing matrix. Simulation results show its fast convergence property. An illustration of Blind Source Separation (BSS) of Magnetic Resonance Spectroscopy (MRS) data confirms the validity of the proposed method

A Fast Indscal Analysis for Blind Undetermined Mixture Identification from Several Even Higher Order Cumulants.

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Blind underdetermined mixture identification by joint canonical decomposition of HO cumulants

12 pages , in one part by the ANR DECOTES Contract (), in second part by the mv-EMD Contract() and in third part byInternational audienceA 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

A Unified Approach for Inverse Problem in EEG and Brain Connectivity with Application to Epilepsy. A Proof of Concept Study

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Canonical decomposition of even order Hermitian positive semi-definite arrays

International audienceMost of the algorithms today available to compute the canonical decomposition of higher order arrays are either computationally very heavy, or are not guaranteed to converge to the global optimum. The solution we propose in order to keep the numerical complexity moderate is~i) to stop the latter algorithms once the solution belongs to the convergence region of the global optimum, and~ii) to refine the solution with a mere gradient descent algorithm. The case of fourth order hermitian positive semi-definite arrays with complex entries is considered. In fact, the hermitian symmetry constraint is taken into account by optimizing a higher order multivariate polynomial criterion. A compact matrix form of the gradient is then computed based on an appropriate framework allowing for derivation in C whereas the cost function is not complex analytic. This compact expression is perfectly suitable for matrix-based programming environments such as MATLAB where loops are to be avoided at all costs. Eventually, computer results show a good performance of the proposed approach

Iterative methods for the canonical decomposition of multi-way arrays: Application to blind underdetermined mixture identification

14 pagesInternational audienceTwo 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

Canonical decomposition of even-order positive semi-definite hermitian arrays by procrustean rotation : application to ICA

Nous présentons ici une nouvelle famille de méthodes itératives, nommée CanDeP (multi-way array Canonical Decomposition based on Procrustes rotation), afin de calculer la décomposition canonique d'un tableau hermitien semi-défini positif d'ordre pair. Cette approche alterne jusqu'à convergence la résolution du problème de Procrustes et la décomposition de tableaux de rang 1. Appliquée aux cumulants d'ordre 2q, la méthode 2q-CanDeP permet de réaliser une analyse en composantes indépendantes performante, offrant i) un gain important en résolution, ii) la capacité d'identifier un mélange sous-déterminé de sources et iii) une robustesse à un bruit Gaussien de covariance spatiale inconnue

Décomposition canonique de tableaux hermitiens semi-définis positifs d'ordre pair par rotation procustéenne: application à l'ICA.

International audienceNous présentons ici une nouvelle famille de méthodes itératives, nommée CanDeP (multi-way array Canonical Decomposition based on Procrustes rotation), afin de calculer la décomposition canonique d'un tableau hermitien semi-défini positif d'ordre pair. Cette approche alterne jusqu'à convergence la résolution du problème de Procrustes et la décomposition de tableaux de rang 1. Appliquée aux cumulants d'ordre 2q, la méthode 2q-CanDeP permet de réaliser une analyse en composantes indépendantes performante, offrant i) un gain important en résolution, ii) la capacité d'identifier un mélange sous-déterminé de sources et iii) une robustesse à un bruit Gaussien de covariance spatiale inconnue

Line search and trust region strategies for canonical decomposition of semi-nonnegative semi-symmetric 3rd order tensors

International audienceNumerical 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

A novel ANN adaptive Riemannian-based kernel classification for motor imagery

International audienceMore recently, a number of studies show the interest of the use of the Riemannian geometry in EEG classification. The idea is to exploit the EEG covariance matrices, instead of the raw EEG data, and use the Riemannian geometry to directly classify these matrices. This paper presents a novel Artificial Neural Network approach based on an Adaptive Riemannian Kernel, named ARK-ANN, to classify Electroencephalographic (EEG) motor imaging signals in the context of Brain Computer Interface (BCI). A multilayer perceptron is used to classify the covariance matrices of Motor Imagery (MI) signals employing an adaptive optimization of the testing set. The contribution of a geodesic filter is also assessed for the ANN and the original method which uses an SVM classifier. The results demonstrate that the ARK-ANN performs better than the other methods and the geodesic filter gives slightly better results in the ARK-SVM, considered here as the reference method, in the case of inter-subject classification (accuracy of 87.4% and 86% for ARK-ANN and ARK-SVM, respectively). Regarding the cross-subject classification, the proposed method gives an accuracy of 77.3% and increases the precision by 8.2% in comparison to the SVM based method