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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
(γ
,
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
(γ
,
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
(γ
,
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
∗
<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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