Mathematics in Independent Component Analysis

Chapter 2. Neural Computation 16:1827-1850, 2004 69
1840 F. Theis
Furthermore, it follows from this theorem that linear BSS is a local problem
as proven already in Theis, Puntonet, and Lang (2003) using the restriction
of a random vector.
4.1 Example for Hessian Diagonalization BSS. In order to illustrate
the algorithm of local Hessian diagonalization, we give a two-dimensional
example. Let S be a random vector with densities
pS1 (s1) = 1
2 χ[−1,1](s1)
pS2 (s2) = 1
√ 2π exp
�
− 1
2 s2 2
�
where χ[−1,1] is one on [−1, 1] and zero everywhere else. The orthogonal
mixing matrix A is chosen to be
A = 1 √ 2
� �
1 1
.
−1 1
The mixture density pX of X := AS then is (det A = 1),
pX(x) = 1
2 √ 2π χ[−1,1]
� � �
1√2
(x1 − x2) exp − 1
4 (x1 + x2) 2
�
,
for x ∈ R 2 . pX is positive and C 2 in a neighborhood around 0. Then
∂1 ln pX(x) = ∂2 ln pX(x) =− 1
2 (x1 + x2)
∂ 2 1 ln pX(x) = ∂ 2 2 ln pX(x) = ∂1∂2 ln pX(x) =− 1
2
for x with |x| < 1
2 , and the Hessian of the logarithmic densities is
Hln p X (x) =− 1
2
� �
1 1
1 1
independent on x in a neighborhood around 0. Diagonalization of Hln p (0)
X
yields
� �
−1 0
,
0 0
and this equals AHln p X (0)A ⊤ , as stated in theorem 3.