Preprocessing of the data

Some ICA algorithms require a preliminary sphering or whitening of the data ${\bf x}$, and even those algorithms that do not necessarily need sphering, often converge better with sphered data. (Recall that the data has also been assumed to be centered, i.e., made zero-mean.) Sphering means that the observed variable ${\bf x}$ of Eq. (11) is linearly transformed to a variable ${\bf v}$

$${\bf v}={\bf Q}{\bf x}$$ (30)

such that the covariance matrix of ${\bf v}$ equals unity: $E\{{\bf v}{\bf v} ^T\}={\bf I}$. This transformation is always possible. Indeed, it can be accomplished by classical PCA [36,112,110,114]. In addition to sphering, PCA may allow us to determine the number of independent components (if m>n): if noise level is low, the energy of ${\bf x}$ is essentially concentrated on the subspace spanned by the nfirst principal components, with n the number of independent components in the model (11). Several methods exist for estimating the number of signals (here, independent components), see [146,94,144,11]. Thus this reduction of dimension partially justifies the assumption m=n that was made in Section 3, and will be retained here. After sphering we have from (11) and (30):

$${\bf v}={\bf B}{\bf s}$$ (31)

where ${\bf B}={\bf Q}{\bf A}$ is an orthogonal matrix, because

$$E\{{\bf v}{\bf v}^T\}={\bf B}E\{{\bf s}{\bf s}^T\} {\bf B}^T = {\bf B}{\bf B}^T={\bf I}$$

Recall that we assumed that the independent components s_i have unit variance. We have thus reduced the problem of finding an arbitrary matrix ${\bf A}$in model (11) to the simpler problem of finding an orthogonal matrix ${\bf B}$. Once ${\bf B}$ is found, Eq. (31) is used to solve the independent components from the observed ${\bf v}$ by

$$\hat{{\bf s}} = {\bf B}^T{\bf v}$$ (32)

It is also worthwhile to reflect why sphering alone does not solve the separation problem. This is because sphering is only defined up to an additional rotation: if ${\bf Q}_1$ is a sphering matrix, then ${\bf Q}_2={\bf U}{\bf Q}_1$ is also a sphering matrix if and only if ${\bf U}$ is an orthogonal matrix. Therefore, we have to find the correct sphering matrix that equally separates the independent components. This is done by first finding any sphering matrix ${\bf Q}$, and later determining the appropriate orthogonal transformation from a suitable non-quadratic criterion.

In the following, we shall thus assume in certain sections that the data is sphered. For simplicity, the sphered data will be denoted by ${\bf x}$, and the transformed mixing matrix by ${\bf A}$, as in the definitions of Section 3. If an algorithm needs preliminary sphering, this is mentioned in the corresponding section. If no mention of sphering is made, none is needed.