GENERALIZATION OF LINEAR REGRESSION PROBLEMS - Neural-based Orthogonal Data Fitting: The EXIN Neural Networks - page 163

Thus, the general form of the barrier locus is given by

2

q j v j x bar ≈ ± (5.102)

λ − 2 γζ

j

The same considerations of the two-dimensional case can be applied here. In
particular, the following theorem results.

Theorem 106 (Barrier Asymptotic Behavior) The GeTLS convergence bar-
rier asymptotically follows the z directions ( i.e., the directions of the v j eigenvec-
tors for j = n ) .

5.3.8 Critical Loci: Center Trajectories

Equation (5.45) represents the locus of the centers of the equilevel hypersurfaces
in the x reference system; in the z reference system, rotated by the matrix V
around the origin, this equation becomes

ζ ) = − I n − 1 q

z c (γ , 2 γζ (5.103)

that is, for each component ( i = 1, ... , n ),

q i
λ i − 2 γζ

z ci = (5.104)

In each plane z j z i this locus is an equilateral hyperbola translated from the origin:

q i z j
λ i − λ j z j + q j z i = (5.105)

All these loci pass through the z origin, which coincides with the x origin. Its
asymptotes are given by

q j
λ

q i
λ

z j =− and z i = (5.106)

where λ = λ i − λ j .

Definition 107 (Asymptotes) The asymptote of the hyperbolas parallel to z i
( the coordinate along the direction of the eigenvector

v i associated to = σ 2

i

λ i )

is defined as the z i asymptote .

Assuming distinct singular values for A , these loci tend to degenerate into two
corresponding orthogonal straight lines for q j → 0or q i → 0.
The global locus (5.104) is parameterized by the product γζ . The parameter
t = 2 γζ will be used [see eq. (5.74)]. If a priori assumptions are not made, the