GeMCA EXIN THEORY - Neural-based Orthogonal Data Fitting: The EXIN Neural Networks - page 217

α i
α 1 − α i

α i
α n + 1 − α i

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

i th position

− η
.

1
ζ

− r

2
2

u i − η

.
− η

α i

α i − α 1
0
.

η

= u 1 u n + 1

···

α i

η

α − α

i n + 1

So the following result has been proved:

d u i
d ζ

α i
α i − α 1 u 1 +···+

α i
α i − α i − 1 u i − 1 +

α i
α i − α i + 1 u i + 1 +···

= η

α i − α n + 1 u n + 1

α i

+

1
ζ ( 1 − ζ )

α i
α i − α j u j

= (6.53)

j = i

Defining τ f , e as

#
$

α g
α g − α f

for g = f

τ f , g = (6.54)

%

0 otherwise

eq. (6.53) can be written in a more compact form as

τ 1, i
0
.
τ n + 1, i

d u i
d ζ

= η U