GeMCA EXIN THEORY - Neural-based Orthogonal Data Fitting: The EXIN Neural Networks - page 205

6

GeMCA EXIN THEORY

6.1 GeMCA APPROACH

The cost function (5.6) can be rewritten as a generalized Rayleigh quotient:

T

u T Ru
u T Du 2 ( Ax − b )

( Ax − b )
( 1 − ζ) + ζ x T x

1

E GeTLS ( x ) = = (6.1)

n times
2 ζ , ... ,2 ζ ,2 ( 1 − ζ ) where u = x T ;− 1 T , R = [ A ; b ] T [ A ; b ],and D = diag

.

For 0 <ζ < 1, the error (5.6) is the ratio of positive definite quadratic forms
and represents the generalized Rayleigh quotient of the symmetric positive
definite pencil ( R , D ) [178]. Its critical points solve the generalized eigenvalue
problem Ru = γ Du :
A T AA T b
b T Ab T b

u = 2 γ ζ I n u

0

(6.2)

0 T 1 − ζ

The last equation in (6.2) yields the GeTLS EXIN secular equation:

b T Ax crit b T b

0
⇒ q T − 2 γζ I n − 1 q + 2 γ( 1 − ζ) − b T b = 0

+ 2 γ( 1 − ζ) − =

(6.3)