GENERALIZATION OF LINEAR REGRESSION PROBLEMS - Neural-based Orthogonal Data Fitting: The EXIN Neural Networks - page 150

Repeating the same reasoning as that of Section 4.7, analogously to eq. (4.48),
search for the sets of vectors x solving the two-parameter equation in the n -
dimensional vector x :

x T A T Ax − 2 b T Ax + b T b − 2 γ( 1 − ζ) − 2 γζ x T x = 0 (5.43)

Similar to eq. (4.50), the center of the hyperconic x c is given by
x c (γ , ζ ) = A T A − 2 γζ I n − 1 A T b

(5.44)

which, using the EIV decomposition (4.52) of A T A , becomes

V − I n − 1 q x c (γ , ζ ) = 2 γζ (5.45)

As in eq. (4.51), the following two-parameter equation in the n -dimensional
vector y , without first-order terms, is obtained:
y T A T A − 2 γζ I n y − b T A A T A − 2 γζ I n − 1 A T b + b T b − 2 γ( 1 − ζ) = 0
(5.46)

By using eq. (4.52), eq. (5.46) becomes
y T V ( − V T y b T AV

− 2 γζ I n − 1 V T A T b + b T b − 2 γ( 1 − ζ) = 0

2 γζ I n ) −

(5.47)

By introducing
g (γ , ζ ) = b T AV − 2 γζ I n − 1 V T A T b − b T b + 2 γ( 1 − ζ) (5.48)

the equations corresponding to eqs. (4.56) and (4.58) are
z T − 2 γζ I n z = g (γ , ζ ) (5.49)

and

n

q i

− ζ) − b T b = q T g (γ , ζ ) =

λ i − 2 γζ +

2 γ( 1

i = 1

I n ) − 1 q b T b

( − 2 γζ + 2 γ( 1 − ζ) − (5.50)

respectively. Again, for any ζ , one and only one value γ min of γ exists in the
interval ( 0, λ n / 2 ζ ) such that g (γ min , ζ) = 0. This point corresponds to the unique
minimum of the cost function E GeTLS EXIN ( x ) , with position given by
x GeTLS sol = x c (γ min , ζ ) = A T A − 2 γ min ζ I n − 1 A T b (5.51)

Figure 4.15 is also valid for the GeTLS problem if σ 2 is replaced by λ i / 2 ζ .

i