INTRODUCTION TO THE TLS EXIN NEURON - Neural-based Orthogonal Data Fitting: The EXIN Neural Networks - page 136

The family of the equilevel hypersurfaces

E TLS ( x ) = γ (4.47)

is introduced, where plays the role of the family parameter. According to eq.

(4.46), these loci are coincident with the sets of vectors x solving the parametric
equation

γ

2 b T Ax + b T b − γ 1 + x T x = Q N ( x ) − γ Q D ( x ) = x T A T Ax − 0 (4.48)

So the equilevel hypersurfaces constitute a family of hyperconics with γ as a
parameter, because any hypersurface is given by a quadratic form equal to zero.
Equation (4.48) is recast in the form
x T A T A − γ I n x − 2 b T Ax + b T b − γ = 0

(4.49)

where I n is the n × n identity matrix. In (4.49) the unknown vector x is substi-
tuted by y + x c (translation), where
x c (γ ) = A T A − γ I n − 1 A T b

(4.50)

coincides with the center of the hyperconic with level γ . Then a parametric

n follows without first-order terms,
y T A T A − γ I n y − b T A A T A − γ I n − 1 A T b + b T b − γ = 0

equation in y ∈

(4.51)

Given the assumption of rank ( A ) = n , the right singular vectors of A are given

by the columns of the orthogonal matrix V ∈

n × n , given by the EVD of A T A :

A T A = V V T (4.52)

[see eq. (1.2)], being λ n <λ n − 1 ≤
··· ≤ λ 1 ( λ i = σ i ,where σ i is the i th singular value of A : see Section 1.3).
Replacing (4.52) into (4.51) yields
y T V ( − γ I n ) V T y − b T AV − γ I n − 1 V T A T b + b T b − γ =

) = T where = diag (λ 1 , λ 2 , ... , λ n

0 (4.53)

A rotation z = V T y can be performed so that
z T − γ I n z − b T AV − γ I n − 1 V T A T b + b T b − γ =

0 (4.54)

By introducing

b T AV − γ I n − 1 V T A T b b T b

g (γ ) = − + γ (4.55)