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Independent Component Analysis - Chapter 17: Nonlinear ICA

This chapter deals with independent component analysis (ICA) for nonlinear mixing models. A fundamental difficulty in the nonlinear ICA problem is that it is highly nonunique without some extra constraints, which are often realized by using a suitable regularization. We also address the nonlinear blind source separation (BSS) problem. Contrary to the linear case, we consider it different from the respective nonlinear ICA problem. | Independent Component Analysis. Aapo Hyvarinen Juha Karhunen Erkki Oja Copyright 2001 John Wiley Sons Inc. ISBNs 0-471-40540-X Hardback 0-471-22131-7 Electronic 17 Nonlinear ICA This chapter deals with independent component analysis ICA for nonlinear mixing models. A fundamental difficulty in the nonlinear ICA problem is that it is highly nonunique without some extra constraints which are often realized by using a suitable regularization. We also address the nonlinear blind source separation BSS problem. Contrary to the linear case we consider it different from the respective nonlinear ICA problem. After considering these matters some methods introduced for solving the nonlinear ICA or BSS problems are discussed in more detail. Special emphasis is given to a Bayesian approach that applies ensemble learning to a flexible multilayer perceptron model for finding the sources and nonlinear mixing mapping that have most probably given rise to the observed mixed data. The efficiency of this method is demonstrated using both artificial and real-world data. At the end of the chapter other techniques proposed for solving the nonlinear ICA and BSS problems are reviewed. 17.1 NONLINEAR ICA AND BSS 17.1.1 The nonlinear ICA and BSS problems In many situations the basic linear ICA or BSS model n x As 17.1 j i is too simple for describing the observed data x adequately. Hence it is natural to consider extension of the linear model to nonlinear mixing models. For instantaneous 315 316 NONLINEAR ICA mixtures the nonlinear mixing model has the general form x f s 17.2 where x is the observed m-dimensional data mixture vector f is an unknown realvalued m-component mixing function and s is an n-vector whose elements are the n unknown independent components. Assume now for simplicity that the number of independent components n equals the number of mixtures m. The general nonlinear ICA problem then consists of finding a mapping h R R that gives components y h x 17.3 that are statistically