Independent Component Analysis and Blind Source Separation - Biomedical Signal Analysis: Contemporary Methods and Applications - page 114

considering the necessary assumptions for successful algorithm applica-
tion. In [246] conditions are given for when to apply this algorithm, and
showed that points satisfying these conditions can indeed be found if the
sources contain at most one Gaussian component ([246], lemma 5). Lin
used a discrete approximation of the derivative operator to approximate
the Hessian; we suggested using kernel-based density estimation, which
can be directly differentiated. A similar algorithm based on Hessian di-
agonalization was proposed by Yeredor [291], using the character of a
random vector. However, the character is complex-valued, and additional
care has to be taken when applying a complex logarithm. Basically, this
is well-defined only locally at nonzeros. In algorithmic terms, the char-
acter can be easily approximated by samples. Yeredor suggested joint
diagonalization of the Hessian of the logarithmic character evaluated at
several points in order to avoid the locality of the algorithm. Instead of
joint diagonalization, we proposed to use a combined energy function
based on the previously defined separator. This also takes into account
global information, but does not have the drawback of being singular at
zeros of the density.

Complex generalization

Comon [59] showed separability of linear real BSS using the Darmois-
Skitovitch theorem (see theorem 4.3). He noted that his proof for the real
case can also be extended to the complex setting. However, a complex
version of the Darmois-Skitovitch theorem is needed. In [247], such a
theorem was derived as a corollary of a multivariate extension of the
Darmois-Skitovitch theorem, first noted by Skitovitch [234] and later
shown in [93]:

= i=1 α i x i Theorem 4.4 complex S-D theorem: Let s 1 and

s 2 = i=1 β i x i with x 1 ,...,x n independent complex random variables

and α j ,β j ∈ C for j =1 ,...,n .If s 1 and s 2 are independent, then all

x j with α j β j = 0 are Gaussian.

This theorem can be used to prove separability of complex BSS and
generalize this to the separation of dependent subspaces (see section 5.3).
Note that a simple complex-valued uniqueness proof [248], which does
not need the Darmois-Skitovitch theorem, can be derived similarly to the
case of real-valued random variables from above. Recently, additional