# Independent component analysis P12

## ICA by Nonlinear Decorrelation and Nonlinear PCA This chapter starts by reviewing some of the early research efforts in independent component analysis (ICA), especially the technique based on nonlinear decorrelation, that was successfully used by Jutten, H´ rault, and Ans to solve the ﬁrst ICA problems. e Today, this work is mainly of historical interest, because there exist several more efﬁcient algorithms for ICA. Nonlinear decorrelation can be seen as an extension of second-order methods such as whitening and principal component analysis (PCA). These methods give components that are uncorrelated linear combinations of input variables, as explained in Chapter 6. We. | 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 12 ICA by Nonlinear Decorrelation and Nonlinear PCA This chapter starts by reviewing some of the early research efforts in independent component analysis ICA especially the technique based on nonlinear decorrelation that was successfully used by Jutten Herault and Ans to solve the first ICA problems. Today this work is mainly of historical interest because there exist several more efficient algorithms for ICA. Nonlinear decorrelation can be seen as an extension of second-order methods such as whitening and principal component analysis PCA . These methods give components that are uncorrelated linear combinations of input variables as explained in Chapter 6. We will show that independent components can in some cases be found as nonlinearly uncorrelated linear combinations. The nonlinear functions used in this approach introduce higher order statistics into the solution method making ICA possible. We then show how the work on nonlinear decorrelation eventually lead to the Cichocki-Unbehauen algorithm which is essentially the same as the algorithm that we derived in Chapter 9 using the natural gradient. Next the criterion of nonlinear decorrelation is extended and formalized to the theory of estimating functions and the closely related EASI algorithm is reviewed. Another approach to ICA that is related to PCA is the so-called nonlinear PCA. A nonlinear representation is sought for the input data that minimizes a least meansquare error criterion. For the linear case it was shown in Chapter 6 that principal components are obtained. It turns out that in some cases the nonlinear PCA approach gives independent components instead. We review the nonlinear PCA criterion and show its equivalence to other criteria like maximum likelihood ML . Then two typical learning rules introduced by the authors are reviewed of which

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