Mathematics in Independent Component Analysis

Chapter 2. Neural Computation 16:1827-1850, 2004 71
1842 F. Theis
0.4
0.2
0
2
1
0
−1
−2 −2
−1 0
1
2
0.2
0.1
0
2
1
0
−1
−2 −2
−1 0
Figure 2: Independent Laplacian density p S (s) = 1
2 exp(−|x1|−|x2|): theoretic
(left) and approximated (right) densities. For the approximation, 1000 samples
and gaussian kernel approximation (see equation 4.3) with standard deviation
0.37 were used.
Rij[ ˆpX] can be calculated using lemma 2—here Rij[ϕ(x − x (k) )] ≡ 0—and
equation 4.4:
Rij[ ˆpX](x) = 1
�
ν�
Rij ϕ(x − x
ν2 k=1
(k) �
)
= 1
ν 2
�
ϕ
k�=l
− (∂iϕ)
= 4κ2
ν 2
= 4κ2
ν 2
�
k�=l
�
k