GeMCA EXIN THEORY - Neural-based Orthogonal Data Fitting: The EXIN Neural Networks - page 215

Then it follows that

= H + d α i
d

u i

d u i
d

+ α i D − 1 d D
d

(6.50)

ζ ζ ζ

r
u i

d D
d ζ

u i

d u i
d

I n + 1 + D − 1 d D
d

= H + α i − u i

ζ

u i Du i

ζ

ρ

= H + α i

u i

− rI n + 1 + D − 1 d D
d

= H + α i ρ u i (6.51)

ζ

Assuming that the eigenvalue decomposition of D − 1 R is equal to UD α U T , it can
be deduced that

H + = UD α U T

1 1

α n + 1

U T

= U diag , ... , 0 , ... ,

i th position

α − α i − α i

1

Equation (6.47) yields

diag I n
ζ

D − 1 d D
d ζ

1
1 − ζ

2 D − 1 J = = , −

Hence,

ρ =− rI n + 1 + diag I n
ζ

= diag 1
ζ

− r I n , − − r

1
1 − ζ

1
1 − ζ

, −

Then eq. (6.50) becomes

= α i UD α U T diag 1
ζ

− r I n , − − r u i

d u i
d ζ

1
1 − ζ

= α i UD α

(6.52)

To simplify eq. (6.52), the normalization of the eigenvectors must be taken into
account. Indeed, the constraint of belonging to the TLS hyperplane determines