Recognition of facial expressions - Knowledge Based Systems ...

defined as:
∆w
= −η
∇
ji
ji
∂E
E = −
∂w
ji
The gradient components can be expressed as follows:
∂E
∂w
ji
∂E
=
∂a
j
∂a
j
∂net
j
∂net
∂w
ji
j
The third partial derivative in the previous equation can be easily computed based on the
definition of
∂ net
j
∂net
∂w
ji
j
=
∂
∂w
ji
k
w
jk
a
k
=
k
∂w
jk
∂w
a
ji
k
Using the chain rule the previous relation can be written as:
∂net
∂w
ji
j
=
k
∂w
(
∂w
jk
ji
a
k
+ w
jk
∂a
∂w
k
ji
)
Examining the first partial derivative, it can be noticed that
∂w
∂w
jk
ji
is zero unless k = i .
Furthermore, examining the second partial derivative, if w
jk
is not zero, then there
∂ak
exists a connection from unit k to unit j which implies that must be zero
∂w
because otherwise the network would not be feed-forward and there would be a recurrent
connection.
Following the criteria,
ji
∂net
∂w
ji
j
= a
i
The middle partial derivative is
∂a
j
∂net
j
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