GeMCA EXIN THEORY - Neural-based Orthogonal Data Fitting: The EXIN Neural Networks - page 218

Using the following notation:

d u 1
d ζ

d U
d ζ

d u n + 1
d ζ

= ···

0 τ ··· τ 1,2 i , n + 1

τ 0 ··· τ 2,1 2, n + 1

T =

. .

. . .

0

τ n + 1,1 τ n + 1,2 ··· 0

the following main result is obtained:

d U
d

= η UT (6.55)

ζ

6.3.2.1 Analysis of Matrix T From definition (6.54) it is deduced that

τ f , g =− α g

α f τ g , f

(6.56)

6.4 RANK ONE ANALYSIS AROUND THE TLS SOLUTION

b ]givenby USW T

Consider the SVD of matrix [ A ; and the corresponding

diag σ

1 .

W T ,where

2 2

n

decomposition of matrix R given by W =

1 ,

... , σ

+

It follows that

D − 1 Ru D − 1 W W T u = γ u ⇒ = γ u (6.57)

Defining n = W T u , we obtain

W T D − 1 W n = γ n (6.58)

that is, n = γ n ,where

= W T D − 1 W = W T D − 1 RW (6.59)

Decomposing the orthonormal matrix W as W 1
w

,where

T
n

w

1 is its last

T
n

+

+ 1

row, after some simple mathematics we obtain

I n + 1 −

xy T

2 1 − 2 ζ

1

= (6.60)

ζ − ζ