J.-F. Cardoso, Eigen-structure of the fourth-order cumulant tensor with application to the blind source separation problem, in Proc. ICASSP-90, Albuquerque, NM, 1990, pp. 2655{ 2658.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... that is less directly connected with the objective function framework, is the eigenmatrix decomposition of higher order cumulant tensors (de ned in Appendix A) Most solutions use the fourth order cumulant tensor, whose properties and relation to the estimation of ICA have been studied extensively [20, 21, 22, 29, 27, 36]. Related methods were also introduced in [106] The fourth order cumulant tensor can be de ned as the following linear operator T from the space of m Theta m matrices to itself: T (K) ij = X k;l cum (x i ; x j ; x k ; x l )K kl (21) where the subscript ij means the (i; j) th element of a ....

....; x j ; x k ; x l )K kl (21) where the subscript ij means the (i; j) th element of a matrix, and K is a m Theta m matrix. This is a linear operator, and thus has m 2 eigenvalues that correspond to eigenmatrices. Solving for the eigenvectors of such eigenmatrices, the ICA model can be estimated [20]. The advantage of this approach is that it requires no knowledge of the probability densities of the independent components. Moreover, cumulants can be used to approximate mutual information [36, 1] as shown above, though the approximation is often very crude. The main drawback of this approach ....

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J.-F. Cardoso. Eigen-structure of the fourth-order cumulant tensor with application to the blind source separation problem. In Proc. ICASSP'90, pages 26552658, Albuquerque, NM, USA, 1990.

....case. Wideband DOA is closely related to wideband beamforming, the problem of steering the sensor array to look only in a particular direction. Some references on the wideband DOA area are [83, 84, 63, 25, 28] Some references on the wideband beamformer are [35, 66, 1] ffl Separation of Sources [42, 21, 75, 91, 55, 2, 14, 16, 19, 18, 13, 48, 17, 18, 22, 7, 46, 57, 58, 61, 51, 53]. ffl Combined Separation and Equalization. The addition of multipath to the source separation problem or the addition of additional channels to the blind equalization 1 problem [7, 51] The subject of this dissertation) 1.2 Multichannel Source Separation and Equalization The full problem of ....

....broadband array. The general form of source separation studied herein, requires the adaptation of a matrix of FIR filters, which allows the description of each path between the sources and the sensors as in Figure 2.1. Eigenstructure based methods exist for source separation (see Cardoso et al. [13, 14]) as well as adaptive approaches such as the Herault Jutten (HJ) algorithm [42] the Weinstein et al. algorithms [86] EASI algorithms of Cardoso and Laheld [18] the information maximization algorithms of Bell and Sejnowski [7] and others. Our main contribution is in the consideration of ....

J.-F. Cardoso, "Eigen-structure of the fourth-order cumulant tensor with application to the blind source separation problem," in Int. Conf. on Acoustics Speech and Signal Processing, pp. 2655--2658, IEEE, 1990.

....whitened version 1 INTRODUCTION 1 1 Introduction The idea of ICA can be tracked back to 1983, when H#rault and Jutten made a pioneering work on a real time algorithm with a neuro mimetic architecture dedicated to BSS. On the theoretical side, the framework of high order moments allowed Cardoso [4, 5] to propose one of the rst direct solutions to a BSS problem. In 1991, H#rault and Jutten [21] published a complete presentation of their adaptive algorithm. Their article outline the transition from PCA to ICA (INCA as they call it) very clearly. More than 10 years later, a lot of work has been ....

J.-F. Cardoso. Eigen-structure of the fourth-order cumulant tensor with application to the blind source separation problem. In Proc. ICASSP'90, pages 26552658, Albuquerque, NM, USA, 1990.

....A by the same constant. For mathematical convenience, we de ne here that the independent components s i have unit variance. This makes the (non Gaussian) independent components unique, up to their signs. Note that no order is de ned between the independent components. In blind source separation (Cardoso, 1990; Jutten and Herault, 1991) the observed values of v correspond to a realization of an m dimensional discrete time signal v(t) t = 1; 2; Then the components s i (t) are called source signals, which are usually original, uncorrupted signals or noise sources. Another possible application of ....

....et al. 1996) Then the columns of A represent features, and s i signals the presence and the amplitude of the i th feature in the observed data v. The problem of estimating the matrix A in eq. 1) can be somewhat simpli ed by performing a preliminary sphering or prewhitening of the data v (Cardoso, 1990; Comon, 1994; Oja and Karhunen, 1995) The observed vector v is linearly transformed to a vector x = Mv such that its elements x i are mutually uncorrelated and all have unit variance. Thus the correlation matrix of x equals unity: Efxx T g = I. This transformation is always possible and can be ....

Cardoso, J.-F. (1990). Eigen-structure of the fourth-order cumulant tensor with application to the blind source separation problem. In Proc. ICASSP'90, pages 26552658, Albuquerque, NM, USA.

....large variances and are not tolerant to noise and numerical errors. Moreover, the algorithms may be computationally complicated. There is an extensive literature on cumulant based contrast functions for ICA both in neural network like solutions (see e.g. 1,6] and in signal processing (see e.g. [4,5,7,8]) and it is not the purpose of our paper to give a review of this mainstream of ICA research. Instead, we concentrate here on the bottom up approach in which the higher order statistics are implicitly embedded into the cost functions and algorithms by arbitrary nonlinear functions. In our ....

J.-F. Cardoso. Eigen-structure of the fourth-order cumulant tensor with application to the blind source separation problem. In Proc. ICASSP'90, pages 26552658, Albuquerque, NM, USA, 1990.

....of a contrast function of the output u. This function usually involves fourth order cumulants. Approaches can be divided into two categories, those that take some aspects of HOS into account implicitly through nonlinear functions of the output [9, 2, 14, 10, 4, 7] and those that do it explicitly [3, 8, 19, 13]. The former category includes many algorithms that are neural in nature. Another approach is to use generalized cross correlations [17, 12, 15] Bell and Sejnowski have recently showed that another possible criterion for BSS is mutual information [1] Combined with a nonlinear squashing ....

J.-F. Cardoso. Eigen-structure of the fourth-order cumulant tensor with application to the blind source separation problem. In Proc. ICASSP, pages 2655--2658, Albuquerque, NM, USA, April 3-6 1990.

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J.F. Cardoso, "Eigen-structure of the fourth-order cumulant tensor with application to the blind source separation problem", Proc. ICASSP'90, pp. 2655-2658, Albuquerque, 1990.

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J.F. Cardoso, "Eigen-Structure of the Fourth-Order Cumulant Tensor with Application to the Blind Source Separation Problem", Proc. ICASSP-90, pp. 2655-2658.

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J.F. Cardoso, "Eigen-structure of the fourth-order cumulant tensor with application to the blind source separation problem", Proc. ICASSP'90, pp. 2655-2658, Albuquerque, 1990.

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J.F. Cardoso, "Eigen-Structure of the Fourth-Order Cumulant Tensor with Application to the Blind Source Separation Problem", Proc. ICASSP-90, pp. 2655-2658.

....signaux qui empiriquement sont spatialement blancs. Ainsi pour r esoudre le probl eme de s eparation de sources par la m ethode des moments, il faut recourir a d autres moments que ceux du seul second ordre. Plusieurs estimateurs de A bas es sur la m ethode des moments ont et e publi es [7, 8, 9, 10, 11, 12], s appuyant essentiellement sur les cumulants d ordre 4. Voici un exemple simple, peut etre non publi e. Choisissons deux matrices quelconques N 1 et N 2 de taille n Theta n et d efinissons les matrices suivantes C i = Ef(x t g Gamma 2R x N i R x Gamma R x Trace (N i R x ) 8i = 1; 2: ....

Jean-Francois Cardoso. Eigen-structure of the fourth-order cumulant tensor with application to the blind source separation problem. In Proc. ICASSP, pages 2655--2658, 1990.

....F1 ; F m 2 of size m Theta m verifying Trace(F i F j ) ffi(i; j) 1 i; j m (14) Qx (F i ) i F i i 2 R 1 i m (15) Each F i is an eigen matrix and the real scalar i is the corresponding eigen value. In addition, the eigen matrices can be chosen to be hermitian. See [19] for a (simple) proof of these facts which hold for any cumulant tensor. In the case of source separation, where the cumulants have structure (7) we have seen that the range of Qx has a dimension equal to the number n of sources. Thus there are only n non zero eigenvalues of Qx out of m . We ....

J.-F. CARDOSO, "Eigen-structure of the fourth-order cumulant tensor with application to the blind source separation problem," in Proc. ICASSP, pp. 2655--2658, 1990.

....ces mthodes doivent exploiter l indpendance des signaux sources et ncessitent le recours aux statistiques d ordres suprieurs. Jutten [6] et Fety [4] ont propos des mthodes adaptatives utilisant des transformations non linaires tandis que les approches de Lacoume [5] Comon [3] et Cardoso [1] et [2] exploitent les cumulants du second et du quatrime ordre. Nous proposons ici une extension de la technique [1] et la comparons avec celle de [3] 2. MODELE CANONIQUE Les approches dcrites dans cet article utilisent l information du second ordre pour effectuer un blanchment du signal. Ceci ....

....hermitiennes orthonormes, sont les matrices propres de Q et l I ses valeurs propres. On peut remarquer que les quations (8) et (9) sont deux dcompositions propres de Q donc si les k p sont distincts alors les matrices propres M I et les a p a p concident : c est ce qui fonde la mthode [2]. Nous dfinissons le sous espace signal comme tant le sous espace engendr par les M premires matrices propres; le projecteur sur ce sous espace s crit alors : P S = M I M I (10) 5. CRITERES DE SEPARATION 5.1 Critre de Comon Aprs s tre ramen au modle canonique dfini au 2. Comon propose un ....

J.F. Cardoso, "Eigen-structure of the fourth-order cumulant tensor with application to the blind source separation problem", Proc. ICASSP'90, pp. 2655-2658, Albuquerque, 1990.

....ces mthodes doivent exploiter l indpendance des signaux sources et ncessitent le recours aux statistiques d ordres suprieurs. Jutten [6] et Fety [4] ont propos des mthodes adaptatives utilisant des transformations non linaires tandis que les approches de Lacoume [5] Comon [3] et Cardoso [1] et [2] exploitent les cumulants du second et du quatrime ordre. Nous proposons ici une extension de la technique [1] et la comparons avec celle de [3] 2. MODELE CANONIQUE Les approches dcrites dans cet article utilisent l information du second ordre pour effectuer un blanchment du signal. Ceci ....

....hermitiennes orthonormes, sont les matrices propres de Q et l I ses valeurs propres. On peut remarquer que les quations (8) et (9) sont deux dcompositions propres de Q donc si les k p sont distincts alors les matrices propres M I et les a p a p concident : c est ce qui fonde la mthode [2]. Nous dfinissons le sous espace signal comme tant le sous espace engendr par les M premires matrices propres; le projecteur sur ce sous espace s crit alors : P S = 1IM S M I M I (10) 5. CRITERES DE SEPARATION 5.1 Critre de Comon Aprs s tre ramen au modle canonique dfini au 2. Comon ....

J.F. Cardoso, "Eigen-structure of the fourth-order cumulant tensor with application to the blind source separation problem", Proc. ICASSP'90, pp. 2655-2658, Albuquerque, 1990.

....jl ik = e =1 S N 2 e e j i e l k (6) where e denotes the e th eigen value and e i j denotes the e th eigen matrix. If, in addition, the tensor satisfies the following horizontal symmetry : g jl ik = g lj ki , then the eigen matrices can be chosen hermitian : e l k = e l k [4]. 2.5 Delta symbols We denote by d i 1 . i p j 1 . j q a symbol which is equal to 1 if all indices are equal and zero otherwise. If in some orthonormal basis, a (1 1) tensor has coordinates equal to d i j , then it is the identity (1 1) tensor and its coordinates are equal to d i j in ....

J.F. Cardoso, "Eigen-structure of the fourth-order cumulant tensor with application to the blind source separation problem", Proc. ICASSP'90, pp. 2655-2658, Albuquerque, 1990.

....matrix to another hermitian matrix. ii) it is itself an hermitian operator in the (usual) sense that for any matrices M and N we have N Q (M) M Q(N) with the Euclidian scalar product M N = D Tr(NM H ) These are trivial consequences of cumulant symmetries. It follows [3] that the quadricovariance admits m 2 real eigenvalues, denoted , i =1,m 2 and m 2 corresponding orthonormal hermitian eigen matrices, denoted E i , i =1,m 2 , verifying : i =1,m 2 Q(E ) E i with E i = E i H , RR , Tr(E i E j ) d(i, j) As a simple consequence [3] of ....

.... follows [3] that the quadricovariance admits m 2 real eigenvalues, denoted , i =1,m 2 and m 2 corresponding orthonormal hermitian eigen matrices, denoted E i , i =1,m 2 , verifying : i =1,m 2 Q(E ) E i with E i = E i H , RR , Tr(E i E j ) d(i, j) As a simple consequence [3] of cumulant additivity and multilinearity, the quadricovariance of a linear mixture (1) of independent components takes the special form : 7) Q(M) p=1,n S k p aa p Maa p aa p aa p with no contribution from the additive noise (since it has been assumed Gaussian and independent of the ....

J.F. Cardoso, "Eigen-structure of the fourth-order cumulant tensor with application to the blind source separation problem", Proc. ICASSP'90, pp. 2655-2658, Albuquerque, 1990.

....processing using higer order statistics. We consider the problem of identifying directional vectors without knowing the array manifold (blind identification) by resorting only to 4th order cumulant statistics. Various cumulant based methods for blind identification have recently been reported [1 3]. They all exploit both second and fourth order information. In contrast, this paper considers the Fourth Order Only Blind Identification (FOOBI) problem. Restriction to 4th order offers two advantages. Firstly, while 2nd order cumulants are affected by additive Gaussian noise, 4th order ....

....Restriction to 4th order offers two advantages. Firstly, while 2nd order cumulants are affected by additive Gaussian noise, 4th order cumulants are not, so that no assumption about noise spatial structure is necessary. Secondly, more sources than sensors can be identified. Reported techniques [1 3] assume at most as many sources as sensors since they rely (explicitely or not) on a prior directional vector orthogonalization based on 2nd order information. In contrast, FOOBI approaches can identify more sources than sensors by turning the orthogonality constraint into a less stringent ....

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J.F. Cardoso, "Eigen-Structure of the Fourth-Order Cumulant Tensor with Application to the Blind Source Separation Problem", Proc. ICASSP-90, pp. 2655-2658.

....column, i.e. the signals are separated up to a phase shift and a possible permutation. Interestingly enough, this very criterion is independently arrived at in [11] via an approximate likelihood obtained by a 4th order GramCharlier expansion. 1 An algebraic approach to blind identification as in [13], yields closed form solutions in terms of eigenmatrices (see below) In order to deal with sample cumulants, algebraic solutions based on the structure of the true cumulants may be adapted into a criterionbased technique. It turns out that the notion of eigenmatrices yields a class of ....

....The above criterion (12) is based on the set (M r jr = 1; n) which is an orthonormal basis for the range of the matrix mapping Q. As a matter of fact, any matrix in this range, i.e. of the form Q(H) for some n 2 n matrix H, is exactly diagonalized by U when Q is defined from the exact cumulants [13]. Let then S be a set S = r ; H r jr = 1; 1 1 1 ; q) of q positive numbers r and q complex n2 n matrices C r and define for all r: C r def = Q(H r ) The function c S (V ) X r=1;q r jjdiag(V C r V H )jj 2 (13) is candidate to be an identification criterion and (12) obviously ....

J.-F. Cardoso, "Eigen-structure of the fourth-order cumulant tensor with application to the blind source separation problem," in Proc. ICASSP, pp. 2655-- 2658, 1990.

....; F m 2 of size m Theta m verifying Trace(F i F H j ) ffi(i; j) 1 i; j m 2 (14) Qx (F i ) i F i i 2 R 1 i m 2 (15) Each F i is an eigen matrix and the real scalar i is the corresponding eigen value. In addition, the eigen matrices can be chosen to be hermitian. See [19] for a (simple) proof of these facts which hold for any cumulant tensor. In the case of source separation, where the cumulants have structure (7) we have seen that the range of Qx has a dimension equal to the number n of sources. Thus there are only n non zero eigenvalues of Qx out of m 2 . We ....

J.-F. CARDOSO, "Eigen-structure of the fourth-order cumulant tensor with application to the blind source separation problem," in Proc. ICASSP, pp. 2655--2658, 1990.

No context found.

J.-F. Cardoso, Eigen-structure of the fourth-order cumulant tensor with application to the blind source separation problem, in Proc. ICASSP-90, Albuquerque, NM, 1990, pp. 2655{ 2658.

No context found.

J. Cardoso, "Eigen-structure of the fourth-order cumulant tensor with application to the blind source separation problem," in Proc. ICASSP, pp. 2655--2658, 1990.

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