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The FastICA algorithm [5] can be interpreted as performing the above described
noise removal.
In FastICA the requirement of whitening the sources is also made and therefore
**R**_{ss}=**I** and
(**A**^{T}**A**)^{-1}=**I**.
Then, the sources can be found one by one and we can consider a single column **a**of the mixing matrix **A**.
To derive the FastICA algorithm from the modified EM-algorithm, it is sufficient to note that
the term
**X**_{G}**F**(**s**_{0G})^{T}/*M*=**as**_{0G}*F*(**s**_{0G})^{T}/*M*is
*C*_{f}**a** where *C*_{f} is a constant that depends only on the nonlinear function .
Then the update rule is

which is the FastICA algorithm, where the constant *C*_{f} is the expectation
.

The choice of fixed nonlinearity
is implicitly connected to the distribution of the
sources *s*. The derivation of the EM-algorithm required that

However, we see that
has certain degrees of freedom
due to taking the difference
**XF**(**s**_{0}^{T})-**X**_{G}**F**(**s**_{0G}^{T}).
Expanding *f* polynomially we obtain
where
*g*'(*s*)=*f*(*s*) and *g*(*s*) contains
all the powers of *f* higher than two and possibly lower moments too.
This representation follows since in the update rule constants and
linear terms of
will cancel out. Therefore they will appear
in the distribution *p*(*s*) in the exponent with the power raised by
one due to integration. Since *p*(*s*) must be a probability density,
the constant *a* will be fixed by the requirement
.
Mean and variance of *s* will determine the constants *b* and *c*,
since the sources are required to be zero-mean and whitened (variance
is fixed to unity). There is one free parameter *d* left, which means
that there is not only one distribution corresponding to but a family of *p*(*s*). Typically the family includes both super- and
sub-Gaussian densities, which is why the same
can be used
for both cases.

** Next:** Application to General ICA
** Up:** Fast Algorithms for Bayesian
** Previous:** Fast EM-algorithm by Filtering
*Harri Lappalainen*

*2000-03-09*