GeMCA EXIN THEORY - Neural-based Orthogonal Data Fitting: The EXIN Neural Networks - page 220

6.5.1 Analysis of the Saddles

For each i such that 1 < i < n + 1, define the gaps

2
i
σ 2
i

σ

g i =

= σ 2

g i

i − 1

2

σ

i

2

which are greater than or equal to 1. By assuming that σ

i is an eigenvalue of R ,
the gaps depend on b T b . For instance, g i −→ 0for b −→ 0.
For ζ ≤ 0 . 5, the following strictest inequalities derived from the previous
theory are true for the intermediate eigenvalue α i (1 < i < n + 1) of matrix K :

2 2

i
2 ζ
u 1

ζ − ≤ ζ ≤

5

σ σ

i
2 ( 1 − ζ )
l 1

≤ α i ≤ 0 .

(6.66)

σ 2

2
i

0 ≤ ζ ≤ ζ −

σ

i
2

≤ α i ≤

ζ
l 2

ζ
u 1

2

ζ − is defined here as the left crossover and is given by The value

1

ζ − =

(6.67)

+ g i

1

It is positive and less than or equal to 0.5.
For ζ → 0 (OLS), the bounds l 2and u 1 tend to infinity; as seen before, all
these critical values are infinite.
For ζ ≥ 0 . 5, the strictest inequalities are given by

2
i

2
i

0 . 5 ≤ ζ ≤ ζ +

σ σ

≤ α i ≤

ζ
l 3

2 − ζ )

u 2

2 ( 1

(6.68)

≤ σ 2

2

ζ + ≤ ζ ≤ 1

σ

i i − 1

≤ α i

ζ
l 3

2 ζ

u 3

2

The value ζ + is here defined as the right crossover and is given by

g i
1 + g i

ζ + = (6.69)