Independent Component Analysis and Blind Source Separation - Biomedical Signal Analysis: Contemporary Methods and Applications - page 120

1.3

1.2

1.1

1

0.9

0.8

0.7

0.6

0.5

0 20 40 60 80 100 120 140 160 180 200

Figure 4.8
Kurtosis maximization: absolute kurtosis versus angle. The function
α →| kurt((cos( α )sin( α )) z ) | is plotted with the uniform z from figure 4.4.

We have seen that prewhitening (i.e. PCA) is essential for this
algorithm — it reduces the search dimension by making the problem
easily accessible. The above equation can be interpreted as finding the
projection onto the line given by w such that z along this line is maximal
non Gaussian.
In figures 4.8 and 4.9, the absolute kurtosis is plotted for the uniform-
source example respectively the Laplacian example from above.

Gradient ascent kurtosis maximization
In practice local algorithms are often interesting. A differentiable func-
tion f :

n

can be maximized by local updates in the direction of
its gradient (which points to the direction of greatest ascent). Given a
suciently small learning rate η> 0andastartingpoint x (0)

R → R

n ,

∈ R

local maxima of f can be found by iterating

x ( t +1)= x ( t )+ η Δ x ( t )

with

f ( x ( t )) = ∂f Δ x ( t )=( Df )( x ( t )) =

∂ x ( x ( t ))
being the gradient of f at x ( t ). This algorithm is called gradient ascent .
Often, the learning rate η is chosen to be dependent on the time t ,and

∇