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n
domain
− {
0
}
. Among these, all the directions associated with the nonminor
components are saddle directions and correspond to ever-larger values of the
cost function as the corresponding eigenvalues increase. They are minima
in any direction of the space spanned by bigger eigenvectors
(
in the sense
of eigenvectors associated with bigger eigenvalues
)
but are maximaz in any
direction of the space spanned by smaller eigenvectors: While getting deeper
(i.e., toward the minimum) in the cost landscape, the number of dimensions of
escape from a saddle point becomes smaller. The critical direction associated
with the minimum eigenvalue is the only global minimum.
2.5.1.3 Hypercrests and Hypervalleys
Using eqs. (2.48) and (2.50), the
energy function can be recast as
j
=
1
λ
2
j
ω
(
t
)
j
E
=
j
=
1
ω
(2.69)
2
j
(
t
)
The critical loci of this cost function
along
a critical axis indexed with
i
0
are
found by differentiating the function along the corresponding coordinate. The
derivative is
λ
i
0
j
=
i
0
ω
E
∂ω
i
0
=
∂
ω
i
0
j
=
1
ω
2
2
j
2
j
j
=
i
0
ω
−
λ
j
(2.70)
j
2
2
A
j
λ
i
0
−
λ
j
=
A
=
j
=
i
0
ω
2
0 always because of the assumption on the spectrum
of
R
, except for
ω
j
=
0
∀
j
=
i
0
(i.e., the
i
0
direction). Thus, the critical loci are
given by
∂
E
∂ω
i
0
=
0
(2.71)
!
ω
=
0
hyperplane
⊥
i
0
direction
i
0
⇔
ω
=
0
(2.72)
j
i
0
direction
(
const
E
)
"
∀
j
=
i
0
(
ω
j
=
0
∀
j
, i.e., the origin, is out of the domain). The typology of these critical
loci can be studied by checking the second derivative:
j
2
E
∂ω
∂
2
A
j
ω
2
j
2
i
0
i
0
=
ω
−
4
ω
(2.73)
j
3
2
2
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