D.-T. Pham. Mutual information approach to blind separation of stationary sources. IEEE Transactions on Information Theory, 48(7):1935.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....(multipath effect and reverberation are not considered) ICA performs blind source separation. For the case where sensor signals are convolutive mixtures of sources, mutual information minimization is still an attractive tool. More detailed theoretical justification of this approach is found in [22, 6]. The approachwe take here is based on the minimization of mutual information in order to force the network output signals, fy i (t)g are spatially independent (spatial independence includes both equal time independence and differenttime independence) The direct application of mutual information ....

D. T. Pham. Mutual information approach to blind separation of stationary sources. In First International Workshop on Independent Component Analysis and Signal Separation, pages 215--220, Aussois, France, 19,ce

....transform is found by minimizing a contrast function based on mutual information that directly extends the contrast function used for classical ICA. We also derive a contrast function in the Gaussian stationary case that is based on spectral densities and generalizes the contrast function of Pham [22] to richer classes of dependency. 1. INTRODUCTION Given a multivariate random variable x in R , independent component analysis (ICA) consists in finding a linear transform W such that the resulting components of s = Wx = s 1 , s m ) # are as independent as possible (see, e.g. 13, ....

.... the semiparametric approach of [3] to the Gaussian stationary case, making use of the notion of graphical models for time series [16] Not surprisingly, the contrast function that we obtain is a linear combination of entropy rate terms that directly extends the contrast function presented by Pham [22]. In Section 3, we derive the contrast function for our semiparametric model and review techniques to estimate it in the temporally independent case, while in Section 4, we extend these results to the Gaussian stationary case. In Section 5, we give a precise description of our algorithms, and we ....

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D. T. Pham. Mutual information approach to blind separation of stationary sources. IEEE Inf. Theory, 48(7):1935.

....that B = A or B = l] A A where l] is a permutation matrix and A a diagonal scaling matrix. Many source separation algorithms have been recently proposed based on like lihood [1, 2, 3, 4, 5, 6, 7, 8] contrast function [9, 10, 11, 12] estimating function [13, 14, 15, 16] information theory [17, 4, 18, 19], and more generally on principle component analysis (PCA) 20] Independent factor analysis (IFA) 21, 22, 23] and independent component analysis (ICA) 16, 24, 25] All these methods assume that the mixing matrix A is invertible and mainly search for a separating matrix Presented at the 19th ....

D.-T. Pham, "Mutual information approach to blind separation of stationary sources," in Proc. First International Conference on Independent Component Analysis and Blind Source Separation ICA '99, (Aussois, France), pp. 215-220, January 11-15, 1999.

....he sketched the Maximum Likelihood solution and emphasized on the relevance of score functions. Finally, the nonlinear functions used in our rst algorithm correspond to a heuristic choice (fortunately robust) optimal for particular distributions. Since this date, he bring nice contributions [45, 46, 44] to the problem. 3. SOURCE SEPARATION IN NONLINEAR MIXTURES When linear models fail, nonlinear models, because of their better approximation capabilities, appear to be powerfull tools for modeling practical situations. Examples of actual nonlinear systems include digital satellite and microwave ....

D.T. Pham. Mutual information approach to blind separation of stationary sources. In Proceedings of ICA'99, pages 215-220, Aussois, France, January 1999.

....(multipath effect and reverberation are not considered) ICA performs blind source separation. For the case where sensor signals are convolutive mixtures of sources, mutual information minimization is still an attractive tool. More detailed theoretical justification of this approach is found in [22, 6]. The approach we take here is based on the minimization of mutual information in order to force the network output signals, fy i (t)g are spatially independent (spatial independence includes both equal time independence and differenttime independence) The direct application of mutual information ....

D. T. Pham. Mutual information approach to blind separation of stationary sources. In First International Workshop on Independent Component Analysis and Signal Separation, pages 215--220, Aussois, France, 1999.

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D. T. Pham, "Mutual information approach to blind separation of stationary sourcs," IEEE Trans. Inform. Theory, vol. 48, pp. 1935.

....and the conditional entropy h[Y (m)j Y (1 : m 1) in examples 1 and 4 (using the notation of example 4) can be viewed as approximations to the entropy rate [5] denoted by h[Y ( since they converge to h[Y ( as m 1. In this respect conditional entropy may be prefered since, as shown in [11] h[Y (1 : m) m h[Y (m)jY (1 : m 1) h[Y ( 13) and further h[Y (m)jY (1 : m 1) already attains its limit if the process Y ( is Markovian of order m 1. Note also that h[Y (m)jY (1 : m 1) h[Y (1 : m) h[Y (1 : m 1) hence equals h[Y (1 : m) h[Y (2 : m) by stationarity and the last ....

....if this matrix is diagonal. Thus minimizing the above contrast amounts to trying to nd a matrix B which jointly approximately diagonalizes the matrices f X (1) f X (m) relative to the above measure of deviation from diagonality. The contrast (8) and (12) have been introduced in Pham [11]. Their Gaussian analogues and the contrast (14) or (15) have been introduced in Pham [13, 14] Pham [15] also provide an ecient algorithm to solve the associate problem of joint approximate diagonalization of several matrices. 3 Contrast for convolutive mixtures Consider now the case where the ....

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Pham, D. T. \Mutual Information Approach to Blind Separation of Sources. In Proceeding of ICA'99 workshop", 215 - 220, Aussois, France, January 1999.

....as a criterion for separation of sources and the closely ( 1] 5] 7] However, such works only consider the mutual information or entropy associated with the marginal instantaneous distribution of the observed signals, thus ignoring their temporal dependence. The recent works of Pham [8] extends the notion of mutual information to stationary processes. Some basic results are here recalled. Let Y 1 , YK be a set of K random vectors with joint density f Y 1 ; Y K , the mutual information between them is de ned as I(Y 1 ; YK ) Z log Q K i=1 f Y k (y k ) f ....

....the product of their marginal densities which would arise if these random variables are independent. Thus, it is non negative and can be zero only if the random variables are independent (see for ex. 4] 7] The concept of mutual information has been generalized to stationary process by Pham [8]. For convenience, for a sequence of variables or vectors fZ(t) t 2 ZZg, the notation Z(s : t) will denote the vector obtained by stacking the (components of) Z(s) Z(t) Let Y ( fY (t) t 2 ZZg be a (vector or scalar) stationary process, we de ne the conditional entropy of Y (T ) ....

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D. T. Pham \Mutual information approach to blind separation for stationary sources." In Proceeding of ICA'99 Conference, 215 - 220. Aussois, January 1999.

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D.-T. Pham. Mutual information approach to blind separation of stationary sources. IEEE Transactions on Information Theory, 48(7):1935.

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Dinh Tuan Pham. Mutual Information Approach to Blind Separation of Stationary Sources. IEEE Transactions on Information Theory, 48(7):1935-1946, July 2002.

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