GeMCA EXIN THEORY - Neural-based Orthogonal Data Fitting: The EXIN Neural Networks - page 214

u T R d u
d ζ

= u T d d ζ Du + α u T d D

d ζ

u + α u T D d u
d ζ

u T R − α u T D d u
d

= u T d d Du + α u T d D

d

u (6.46)

ζ ζ ζ

= u T d

d ζ

Du + α u T d D
d ζ

0 u

Hence,

u T d d ζ ( d Du / d ζ) u

=− α

u T Du

Considering that

= 2 I n

d D
d ζ

0

= 2 J (6.47)

0 T

− 1

and

u T Du = 2 ( 1 − ζ ) + 2 ζ x T x

then

x T x d d ζ 1

( 1 − ζ ) + ζ x T x

−

=− α (6.48)

This equation can be rewritten as

d d

ζ +

=− α

1 =− αφ ( , ζ ) ζ

6.3.2 Eigenvector Derivative

From eq. (6.46) it follows for the i th eigenvector u i associate with the eigen-
value α i :

α i
d ζ

d u i
d ζ

d

Du i + α i d D
d ζ

( R − α i D ) = u i (6.49)

and

D − 1 R − α i I n + 1 d u i
d

d α i
d

u i + α i D − 1 d D
d

= u i

ζ ζ ζ

Define

D − 1 R

H = − α i I n + 1