Taleb, A. & Jutten, C. (1997), "Nonlinear source separation: The post-nonlinear Mixtures", Proc. ESANN97, 279-284.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the validity of the generalization. 1. INTRODUCTION Recently three classes of methods have been proposed for problems in which nonlinear mixtures of independent sources are to be separated, i.e, nonlinear ICA. The first class of methods adds nonlinear mixing model to the linear model [1] 2] [3] [4] These methods resemble linear ICA methods very much with the only drastic difference being the introduction of unknown scaling and slope parameters to the nonlinear transfer function(recall that in ICA, this nonlinear transfer function is the same for all mixtures, e.g, logistic function ....

A. Taleb and C. Jutten, "Nonlinear source separation: The post-nonlinear mixtures," ESANN, pp. 279--284, 1997.

....using a priori knowledge of the source distribution. This has been applied by Lee, Lewicki, Girolami, and Sejnowski (in press b) to separate three sources from two sensors. Second, researchers have recently tackled the problem of nonlinear mixing phenomena. Yang, Amari, and Cichocki (1997) Taleb and Jutten (1997), and Lee, Koehler, and Orglmeister (1997) propose extensions when linear mixing is combined with certain nonlinear mixing models. Other approaches use self organizing feature maps to identify nonlinear features in the data (Lin Cowan, 1997; Pajunen Karhunen, 1997) Hochreiter and 436 Te Won ....

Taleb, A., & Jutten, C. (1997). Nonlinear source separation: The post-nonlinear mixtures. In ESANN (pp. 279--284).

....mixing process is performed in two stages: a linear mixing followed by a nonlinear transfer function. A parametric sigmoidal nonlinearity and nonlinearities approximated by higher order polynomials are suggested to solve the post nonlinear problem. A similar approach was independently proposed in [12]. They approximated the inverse transfer function by multilayer perceptrons (MLP) that were trained in an unsupervised manner. Those models may be justified for several biomedical signal analysis problems such as fMRI and EEG data analysis. It may also be used to account for intrinsic ....

....For several situations the linear assumption may lead to incorrect solutions. Therefore the goal in this paper is to formulate an ICA framework that is able to separate nonlinear mixing models. Researchers have very recently started addressing the ICA formulation to nonlinear mixing models [21 25,12,26] The proposed nonlinear ICA methods can be roughly divided into two classes of approaches. The first class of methods is an obvious extension to the linear ICA model where nonlinear mixing models are added to the linear model and the task is to find the inverse of the linear model as well as the ....

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A. Taleb and C. Jutten. Nonlinear source separation: The post-nonlinear mixtures. In ESANN, pages 279--284, 1997.

.... of the estimation of the independent components in the general non linear model (1) To our knowledge, these problems have not been treated in the literature, although some algorithms for non linear ICA have already been proposed [2, 7, 8] In a special case, the problem has been treated in [9]. The questions of existence and uniqueness of solutions are, however, of fundamental importance even in the construction of algorithms for non linear ICA. We present two results in this paper. In Section 2, we show explicitly how to construct a function g from R n to R n so that the ....

A. Taleb and C. Jutten. Nonlinear source separation: The post-nonlinear mixtures. In Proc. European Symposium on Artiøcial Neural Networks (ESANN97), pages 279284, Bruges, Belgium, April 1997.

....applications in the signal processing area which may benefit from ICA as a preprocessing analysis. Nevertheless, the linear mixing model may not be appropriate for some real environment experiments. Therefore, researchers have recently started addressing the ICA problem to nonlinear mixing models [3, 8, 11, 13, 14, 15]. In [8, 11, 13] the nonlinear components are extracted using self organizingfeature maps (SOFM) However, due to the limited number of neurons that map the underlying distribution the derived components have a quantization error that increases with increasing distance to the neighboring ....

....functions. In particular, we assume that the mixing is performed in two stages: a linear mixing followed by a nonlinear transfer function. We focus on a parametric sigmoidal nonlinearity and on higher order polynomials. A similar approach has been independently studied by Taleb and Jutten [14]. They approximate the inverse transfer function by multilayer perceptrons (MLP) that are trained in an unsupervised manner. This kind of model may be justified for several biomedical signal analysis problems such as brain blood moving analysis in magnetic resonance imaging (MRI) and EEG analysis. ....

A. Taleb and C. Jutten. Nonlinear source separation: The post-nonlinear mixtures. In ESANN'97 . In press.

....probability density function and obtain a more reliable estimation of the mutual information. A short version of this paper appeared in [17] The non linear mixture model considered there and this paper contains crossing non linearities in the mixture. This model is more general than that in [10, 14] where the non linear mixture is obtained by operating non linear functions componentwise on the linear mixture. 2 Mixture models and de mixing systems Let us consider unknown source signals s i (t) i = 1; Delta Delta Delta ; n which are mutually independent and stationary. It is assumed that ....

....0:1434 Gamma0:5017 0:4779 0:3734 3 7 7 7 7 7 5 where the elements of A 1 are randomly chosen in [ Gamma0:5; 0:5] and those of A 2 are randomly chosen in [ Gamma1; 1] The SOM algorithms in [9, 11, 12] fail to extract sources from the non linear mixture in this example. The algorithms in [10, 14] are not applicable to the mixture x(t) which contains crossing non linearities. We applied the non linear MMI algorithm (18) 16) to extract source signals in the observed non linear mixture using the sigmoid function g(x) tanh(flx) with a gain fl 0 for each hidden unit in the demixing ....

A. Taleb and C. Jutten. Nonlinear source separation: the post-nonlinear mixtures. In ESANN'97 , pages 279--284, 1997.

.... treated in the literature, although some algorithms for nonlinear ICA have already been proposed (Burel, 1992; Deco and Brauer, 1995; Deco and Obradovic, 1995; Lee et al. 1997; Pajunen et al. 1996; Pajunen and Karhunen, 1997; Yang et al. 1998) In a special case, the problem has been treated in (Taleb and Jutten, 1997). The questions of existence and uniqueness of solutions are, however, of fundamental importance even in the construction of algorithms for nonlinear ICA. We present two results in this paper. In Section 2, we show explicitly how to construct a function g from R n to R n so that the components ....

....(Ahlfors, 1979) For example, any simply connected domain can be mapped onto another one by a conformal mapping. This result shows that it is possible to obtain uniqueness in nonlinear ICA by restricting the mixing function f to a certain class, thus complementing the uniqueness results in (Taleb and Jutten, 1997). ....

Taleb, A. and Jutten, C. (1997). Nonlinear source separation: The postnonlinear mixtures. In Proc. European Symposium on Artiøcial Neural Networks (ESANN97), pages 279284, Bruges, Belgium.

....to some extent extract the independent components using a priori knowledge of the source distribution. This has been applied by Lee et al. 1998b) to separate three sources from two sensors. Second, researchers have recently tackled the problem of nonlinear mixing phenomena. Yang et al. 1997) Taleb and Jutten (1997) and Lee et al. 1997) propose extensions when linear mixing is combined with certain nonlinear mixing models. Other approaches use self organizing feature maps to identify nonlinear features in the data (Lin and Cowan, 1997; Pajunen and Karhunen, 1997) More recently, Hochreiter and Schmidhuber ....

Taleb, A. and Jutten, C. (1997). Nonlinear source separation: The post-nonlinear mixtures. In ESANN, pages 279--284.

....IST 1999 14190) and by the French project Statistiques Avancees et Signal (SASI) s y x 1 x N e N w 1 w N A g 1 f 1 f N g N B Fig. 1. PNL mixtures: mixing and separating systems for all , but requires ) for all and all . Only a few researchers [9, 10, 11, 12, 13, 14, 15, 16] addressed source separation in nonlinear mixtures, whose observations are , The problem consists in restoring the sources by estimating a nonlinear separating system tem 54547 . However, generally, it can be deduced [15] from the Darmois s theorem [17] that the nonlinear mixtures ....

A. Taleb and C. Jutten, "Non-linear source separation: the post non-linear mixtures," in ESANN'97, Bruges, Belgium, April 1997, pp. 279--284.

....of the mixing matrix, and the inversion of the nonlinear functions. 1. INTRODUCTION TO POSTNONLINEAR MODELS Conventional source separation algorithms treat the case where the mixing model is linear instantaneous or convolutive. Conversely, there are only few contributions to the nonlinear case [1, 4, 10, 5, 8, 9]. In the general nonlinear case, it has been shown, about 50 years ago, by Darmois G. 2] that there exists an infinity of nontrivial nonlinear transforms,i.e. without diagonal Jacobian, able to map an independent component random vector into another one. This negative result can discourage ....

....F A s e Linear part Mixing matrix Componentwise distortion observations sources Figure 1: The PNL model part is a set of unknown nonlinear invertible functions f i which introduce a componentwise distortion on the linear mixture As. This model, called the postnonlinear (PNL) model [8], has interesting properties. First, its plausibility in many real world situations: for example, the distortions can model the unknown characteristics of sensors, and in general any static distortion. Another interesting characteristic of the PNL model is its separability property [7] Using only ....

A. Taleb and C. Jutten. Nonlinear source separation : The post-nonlinear mixtures. In ESANN 97, pages 279--284, Bruges (Belgium), April 1997.

....1 ; x 2 ; xn g and fy 1 ; y 2 ; y p g are independent, f and g bijective. We conjecture that if X and Y are not independent then f and g are necessarily linear in their domain of de nition (support of P n i=1 a i x i and P n i=1 b i x i ) We rst introduced the PNL model in [53], where MLPs are used to invert the nonlinear functions f i . We used the maximum likelihood as a cost function, where we replaced the unknown sources pdfs by a GramCharlier expansion up to the forth order. The obtained results were quite good when the nonlinearities were not too strong. In the ....

A. Taleb and C. Jutten. Non-linear source separation: the post-non-linear mixtures. In ESANN'97, pages 279-284, Bruges, Belgium, 1997.

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Taleb, A. & Jutten, C. (1997), "Nonlinear source separation: The post-nonlinear Mixtures", Proc. ESANN97, 279-284.

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A. Taleb and C. Jutten 1997, "Nonlinear source separation: The post-nonlinear mixtures," Proc. of ESANN'97, Bruges, Belgium, 279--284.

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Taleb, A. and Jutten, C. Nonlinear source separation: the post-nonlinear mixtures. Proc 1997 Symposium on Artificial Neural Networks. p. 279-284, 1997.

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Taleb, A. and Jutten, C. 1997. Nonlinear source separation: The post-nonlinear mixtures. In ESANN'97, 279--284.

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