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Independent component analysis (ICA) is a recently developed, useful extension of standard principal component analysis (PCA). The ICA model is utilized mainly in blind separation of unknown source signals from their linear mixtures. In this application only the source signals which correspond to the coefficients of the ICA expansion are of interest. In this paper, we propose neural structures related to multilayer feedforward networks for performing complete ICA. The basic ICA network consists of whitening, separation, and basis vector estimation layers. It can be used for both blind source separation and estimation of the basis vectors of ICA. We consider learning algorithms for each layer, and modify our previous nonlinear PCA type algorithms so that their separation capabilities are greatly improved. The proposed class of networks yields good results in test examples with both artificial and real-world data.

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A Bayesian ensemble learning method is introduced for unsupervised extraction of dynamic processes from noisy data. The data are assumed to be generated by an unknow n nonlinear mapping from unknow n factors. The dynamics of the factors are modeled using a nonlinear state-space model. The nonlinear mappings in the model are represented using multilayer perceptron networks. The proposed method is computationally demanding, but it allows the use of higher-dimensional nonlinear latent variable models than other existing approaches. Experiments with chaotic data show that the new method is able to blindly estimate the factors and the dynamic process that generated the data. It clearly outperforms currently available nonlinear prediction techniques in this very dif cult test problem.

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In this paper, we review recent advances in blind source separation (BSS) and independent component analysis (ICA) for nonlinear mixing models. After a general introduction to BSS and ICA, we discuss in more detail uniqueness and separability issues, presenting some new results. A fundamental difficulty in the nonlinear BSS problem and even more so in the nonlinear ICA problem is that they provide non-unique solutions without extra constraints, which are often implemented by using a suitable regularization. In this paper, we explore two possible approaches. The first one is based on structural constraints. Especially, post-nonlinear mixtures are an important special case, where a nonlinearity is applied to linear mixtures. For such mixtures, the ambiguities are essentially the same as for the linear ICA or BSS problems. The second approach uses Bayesian inference methods for estimating the best statistical parameters, under almost unconstrained models in which priors can be easily added. In the later part of this paper, various separation techniques proposed for post-nonlinear mixtures and general nonlinear mixtures are reviewed.

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