Mathematics in Independent Component Analysis

Chapter 16. Proc. ICA 2006, pages 917-925 235
The basis choice from lemma 3 together with assumption (ii) can be used to prove
the following fact:
Lemma 4. The non-Gaussian transformation is invertible i.e. ANN∈ Gl(n).
The next lemma can be seen as modification of lemma 1, and indeed it can be shown
similarly.
Lemma 5. If ˆSN fulfills � ˆSNHˆSN −∇ˆSN(∇ˆSN) ⊤� e1+ ˆS 2 Nc≡0 for some constant vector
c∈R n , then the source component (SN)1 is Gaussian and independent of (SN)(2 : n).
Here more generally ei ∈ Rn denotes the i-th unit vector. Putting these lemmas
together, we can finally prove theorem 1: According to lemma 4, ANN is invertible, so
from the left yields
multiplying equation (5) from lemma 2 by B −1
N A−1
NN
�
ˆSNHˆSN −∇ˆSN(∇ˆSN) ⊤� B ⊤ NA⊤ GN + CˆS 2 N≡ 0 (6)
for any BN∈ Gl(n) and some fixed, real matrix C∈Mat(n×(d− n)).
We claim that AGN= 0. If not, then there exists v∈R d−n with�A⊤ GNv�=1. Choose
BN from (4) such that B−1 N SN is decorrelated. This is invariant under left-multiplication
by an orthogonal matrix, so we may moreover assume that B⊤ NA⊤ GNv=e1. Multiplying
equation (6) in turn by v from the right therefore shows that the vector function
�
ˆSNHˆSN −∇ˆSN(∇ˆSN) ⊤� e1+ cˆS 2 N≡ 0 (7)
is zero; here c := Cv∈R. This means that ˆSN fulfills the condition of lemma 5, which
implies that (SN)1 is Gaussian and independent of the rest. But this contradicts (ii) for
S, hence AGN= 0. Plugging this result into equation (5), evaluation at sN= 0 shows
that ANGA ⊤ GG = 0. Since AGN = 0 and A∈Gl(d), necessarily AGG ∈ Gl(d−n), so
ANG= 0 as was to prove.
2 Simulations
In this section, we will provide experimental validation of the uniqueness result of
corollary 1. In order to stay unbiased and not test a single algorithm, we have to uniformly
search the parameter space for possibly equivalent model representations. The
model assumptions (1) will not be perfectly fulfilled, so we introduce a measure of
model deviation based on 4-th order cumulants in the following.
Let the non-Gaussian dimension n and the total dimension d be fixed. Given a random
vector X=(XN, XG), we can without loss of generality assume that Cov(X)=I.
Any possible model deviation consists of (i) a deviation from the independence of XN
and XG and (ii) a deviation from the Gaussianity of XG. In the case of non-vanishing
kurtoses, the former can be approximated for example by
δI(X) :=
1
n(n−d)d 2
n�
d�
d�
i=1 j=n+1 k=1 l=1
d�
cum 2 (Xi, X j, Xk, Xl),