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5.3.2 GeTLS Hyperconic Family
Recast the GeTLS hyperconic family (5.49) as
n
z T 2 γζ I n z
z i
k i = 1
= g
, ζ )
(5.52)
i
=
1
where k i
and z i is the i th component of the vector z and is
the coordinate along the direction of the eigenvector
=
g
,
ζ ) /λ i
2
γζ
v i associated with
= σ 2
i
λ i
V T y .Asseeninthelasttwosections,
[see eq. (1.2)], because of the rotation z
=
γ 0,
the k i s are all negative and the family is composed of imaginary
γ
)
min
ellipsoids.
Case Here, k i > 0 i , and therefore the fam-
ily is composed of hyperellipsoids [( n 1)-dimensional ellipsoids] with axes
parallel to the A eigenvectors. As γ increases, the hyperellipsoid becomes larger
(for γ = γ min , its center, a unique point, represents the minimum point) and its
semiaxis parallel to
5.3.2.1 The
γ min γ λ n /
2
ζ
v n grows much more than the others, giving ellipsoids more
and more stretched in this direction. The density of these equilevel hypersurfaces
is proportional to the gap between
γ min and
λ n /
2
ζ
(e.g., for a small gap, the cost
v n has a value of
landscape is flat). The infinity in the direction of
λ n /
2
ζ
for the
cost function:
E GeTLS EXIN x 0 + s v n =
λ n
2 ζ
n
lim
s →±∞
x 0
(5.53)
ζ .For
γ λ n /
ζ =
.
because the hyperellipsoids tend to infinity as
2
0
5 (TLS),
2
n
γ min = σ
1 .
+
5.3.2.2 The
λ n /
2
ζ γ λ n −1 /
2
ζ
Case Figure 4.15 shows the existence
( γ ), which is a zero for g , ζ ) .
of one value of γ
ζ γ γ . Here, k n
1.
λ
/
2
0and k i
0
i
=
n . The family is composed
n
of two-sheeted hyperboloids (for n
2) consisting of two parts located
v n direction).
in the half-spaces z n
k n and z n
≤−
k n (i.e., along the
The hyperplanes z n =
-dimensional
ellipsoids, the remaining hyperplanes z i = const ( i = 1, ... , n 1) along
two-sheeted hyperboloids. If γ = γ , the hyperconic is made up of only
one point. As
const ( | z n | > k n ) intersect in
(
n 2
)
γ λ n / 2 ζ + , the hyperboloids tend to stretch on the
v n
direction.
γ λ n 1 / 2 ζ .Herethe k i s change sign. The family is now com-
posed of one-sheeted hyperboloids along the v n direction. All hyperplanes
z n = const intersect it along ( n 2 ) -dimensional ellipsoids, the remaining
hyperplanes z i = const ( i = 1, ... , n 1) along hyperboloids or cones.
These one-sheeted hyperboloids tend to stretch on the
2.
v n 1 direction as
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