Tài liệu Independent component analysis P11 ppt

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
xi
xj
xk
xl.

For an introduction to cumulants, see Section 2.7.

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.

11.1 DEFINITION OF CUMULANT TENSOR

We shall here consider only the fourth-order cumulant tensor, which we call for sim￾plicity 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
xi
xj
xk
xl,

where the indices i
j
k
l are from 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. (2.106).

In fact, all fourth-order cumulants of linear combinations of xi can be obtained

as linear combinations of the cumulants of xi . This can be seen using the additive

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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)