GENERALIZATION OF LINEAR REGRESSION PROBLEMS - Neural-based Orthogonal Data Fitting: The EXIN Neural Networks - page 158

Introducing the parameter

t = 2 ζγ (5.74)

which will be shown later to be very important, it follows that

t DLS ≥ t TLS ≥ t OLS = 0 (5.75)

where the equal sign is valid only for compatible systems ( γ min = 0). In the x
reference system, the position of the solution is given by eq. (5.45) for γ = γ min :

V − I n − 1 q x sol (γ min , ζ ) = 2 γ ζ (5.76) min

which in the rotated z reference system (no translation) becomes
z sol (γ min , ζ ) = − 2 γ min ζ I n − 1 q

(5.77)

Thus,

$

z n

q n
λ n

q n
λ n − γ TLS

( z n )

z sol n ≈ (5.78)

%

b T b
q n

z n

where γ TLS = k / b T b + λ n > 0 and therefore | z n | ≥ z n .

Proposition 105 (Relative Positions) ∀ i = 1, ... , n,

z i ≥ | z i | ≥ z i

(5.79)

which implies, in the x reference system, the same inequalities for the modules:
x 2 ≥

2 ≥ x 2

x (5.80)

where the equal sign is valid only for compatible systems.

Proof. From eq. (5.78),

q n

z n ≈ = Hq n (5.81)

2 λ n b T b

q n
b T b + λ n

1 −

λ n −

where

b T b + λ n
λ n b T b + 2 λ

H = 2 (5.82)

n

+ q n