# Independent component analysis P11

## ICA by Tensorial Methods One approach for estimation of independent component analysis (ICA) consists of using higher-order cumulant tensor. Tensors can be considered as generalization of matrices, or linear operators. Cumulant tensors are then generalizations of the covariance matrix. The covariance matrix is the second-order cumulant tensor, and the fourth order tensor is deﬁned by the fourth-order cumulants cum(xi xj xk xl ). For an introduction to cumulants, see Section . As explained in Chapter 6, we can use the eigenvalue decomposition of the covariance matrix to whiten the data. This means that we transform the data so that second-order correlations. | 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 11 ICA by Tensorial Methods One approach for estimation of independent component analysis ICA consists of using higher-order cumulant tensor. Tensors can be considered as generalization of matrices or linear operators. Cumulant tensors are then generalizations of the covariance matrix. The covariance matrix is the second-order cumulant tensor and the fourth order tensor is defined by the fourth-order cumulants cum x Xj Xk xi . For an introduction to cumulants see Section . As explained in Chapter 6 we can use the eigenvalue decomposition of the covariance matrix to whiten the data. This means that we transform the data so that second-order correlations are zero. As a generalization of this principle we can use the fourth-order cumulant tensor to make the fourth-order cumulants zero or at least as small as possible. This kind of approximative higher-order decorrelation gives one class of methods for ICA estimation. DEFINITION OF CUMULANT TENSOR We shall here consider only the fourth-order cumulant tensor which we call for simplicity the cumulant tensor. The cumulant tensor is a four-dimensional array whose entries are given by the fourth-order cross-cumulants of the data cum xj Xj Xk xi where the indices are from 1 to n. This can be considered as a four- dimensional matrix since it has four different indices instead of the usual two. For a definition of cross-cumulants see Eq. . In fact all fourth-order cumulants of linear combinations of Xi can be obtained as linear combinations of the cumulants of a . This can be seen using the additive 229 230 ICA BY TENSO RIAL METHODS properties of the cumulants as discussed in Section . The kurtosis of a linear combination is given by kurt cum 2 WjXi WjXj Wk k wiXi i i j k I w w w wfcum xi Xj Xk xi ijkl Thus the fourth-order cumulants contain all .

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