Mathematics in Independent Component Analysis

236 Chapter 16. Proc. ICA 2006, pages 917-925
where the fourth-order cumulant tensor is defined as cum(Xi, X j, Xk, Xl) := E(XiX jXkXl)−
E(XiX j)E(XkXl)−E(XiXk)E(X jXl)−E(XiXl)E(X jXK). The deviation (ii) from Gaussianity
of XG can simply be measured by kurtosis, which in the case of white X means
δG(X) :=
1
(n−d)
d�
j=n+1
�
�
�E(X 4 i )−3� � �.
Altogether, we can therefore define a total model deviation as the weighted sum of the
above indices; the weight in the following was chosen experimentally to approximately
yield even contributions of the two measures:
δ(X)=10n(d−n)δI(X)+δG(X)
For numerical tests, we generate two different non-Gaussian source data sets, see
figure 1(d) and also [4], figure 1. The first source set (I) is an n-dimensional dependent
sub-Gaussian random vector given by an isotropic uniform density within the unit
disc, and source set (II) a 2-dimensional dependent super- and sub-Gaussian, given by
p(s1, s2)∝exp(−|s1|)1[c(s1),c(s1)+1], where c(s1)=0 if|s1|≤ln 2 and c(s1)=−1 otherwise.
Normalization was chosen to guarantee Cov(S N)=Iin advance.
In order to test for model violations, we have to find two representations X=AS
and X=A ′ S ′ of the same mixtures. After multiplication by A −1 we may as before
assume that a single representation X=AS is given with X and S both fulfilling the
dimension reduction model (1), and we have to show that ANG = AGN = 0 if the
decomposition is minimal (corollary 1). The latter can be tested numerically by using
the so-called normalized crosserror
E(A) :=
1
2n(d−n)
� �ANG� 2 F +�AGN� 2 F
where�.�F is some matrix norm, in our case the Frobenius norm.
In order to reduce the d 2 -dimensional search space, after whitening we may assume
that A∈O(d), so only d(d− 1)/2 dimensions have to be searched. O(d) can be uniformly
sampled for example by choosing B with Gaussian i.i.d. coefficients and orthogonalizing
A := (BB ⊤ ) −1/2 B. We perform 10 4 Monte-Carlo runs with random A∈O(d).
Sources have been generated with T= 10 4 samples, n-dimensional non-Gaussian part
(a) and (b) from above, and (d−n)-dimensional i.i.d. Gaussians. We measure model
deviationδ(AS) and compare it with the deviation E(A) from block-diagonality.
The results for varying parameters are given in figure 1(a-c). In all three cases we
observe that the smaller the model deviation, the smaller also the crosserror. This gives
an asymptotic confirmation of corollary 1, indicating that by random sampling no nonuniqueness
realizations have been found.
3 Conclusion
By minimality of the decomposition (1), we gave a necessary condition for the uniqueness
of non-Gaussian subspace analysis. Together with the assumption of existing covariance,
this was already sufficient to guarantee model uniqueness. Our result allows
NGSA algorithms to find the unknown, unique signal space within a noisy highdimensional
data set [6].
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