GeMCA EXIN THEORY - Neural-based Orthogonal Data Fitting: The EXIN Neural Networks - page 223

inequality is given by

2
n

2
n

σ

≤ α n + 1 ≤ σ

σ n
2

+ 1 + 1

≤ (6.81)

− ζ )
l 1

2 ( 1 ζ

u 1

2 ζ

u 2

From inequality (6.40) it follows that

2
2
2 ( 1 − ζ )
u 3

b ⊥ γ

OLS
1 − ζ

α n + 1 ≤ = (6.82)

For ζ → 0 (OLS), the bounds l 1and u 3 tend to finite values, while u 1and
u 2 tend to infinity. It follows that

2
2

2
2

2 ( 1 − ζ ) → b ⊥

b ⊥

α n + 1 ≤ = γ OLS (6.83)

2

From (6.81) we also derive

2
n

σ

+ 1

≤ γ OLS (6.84)

2

For ζ ≥ 0 . 5, the following strictest inequalities derived from the previous

theory are true:

2
n + 1
2 ζ
l 3

2
n + 1

σ σ 0 . 5 ≤ ζ ≤ ζ +

≤ α n + 1 ≤

1 − ζ )
u 2

2 (

(6.85)

2
n

σ

σ n
2

ζ + ≤ ζ ≤ 1

+ 1

≤ α n + 1 ≤

ζ
l 3

2 ζ

u 3

For ζ → 1 (DLS) the bounds l 3and u 3 tend to finite values. Hence, the bound
for the DLS smallest eigenvalue is given by

2
n

σ

≤ α n + 1 ≤ σ n
2

+ 1

(6.86)

2

which shows that the bound (2.117) is also valid for i = n + 1. In other words,
the bound (2.117) is valid for each finite eigenvalue of K .