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The straight line in the
i
0
direction yields
A
=
0, and it is evident that this
second derivative and all higher derivatives are null, just giving a constant
E
,
which agrees with the previous analysis.
ω
i
0
=
0
=
2
E
∂ω
∂
2
A
j
=
i
0
ω
j
3
2
i
0
2
j
λ
i
0
−
λ
j
j
=
i
0
ω
2
j
=
i
0
ω
j
=
i
0
ω
2
j
=
(2.74)
j
2
2
By the sign analysis of this formula [for
i
0
representing the index of a saddle
direction, the sign analysis of eq. (2.60) with reversed sign holds], the following
proposition holds.
Proposition 58 (Hypercrests/Valleys)
The hyperplane through the origin and
perpendicular to the minimum direction is made up of points which are maxima
along the minimum direction (hypercrest along the minimum direction). The hyper-
plane through the origin and perpendicular to the maximum direction is made up
of points which are that along the maximum direction
(
hypervalley along the
maximum direction
)
. The hyperplane through the origin and perpendicular to a
saddle direction is made up of two types of points: points given by the intersection
with the saddle cone of repulsion, which are minima along the saddle direction
(hypervalley along the saddle direction) and points given by the intersection with
the saddle cone of attraction (i.e., all the other points), which are maxima along
the saddle direction
(
hypercrest along the saddle direction
)
.
Figure 2.1 shows the hypercrests and hypervalleys for the three-dimensional
case.
2.5.2 Gradient Flows and Equilibria
The MCA EXIN ODE performs a gradient descent on the RQ cost function in
weight space. In discrete form, it yields
T
α (
t
)
)
+
w
(
t
)
R
w (
t
)
w (
t
+
1
)
=
w (
t
)
+
−
R
w (
t
w (
t
)
(2.75)
w
T
(
t
) w (
t
)
w
T
(
t
) w (
t
)
w
(
t
)
=
i
=
1
ω
i
(
t
)
z
i
yields
Expressing the weight vector as
#
j
%
(
t
)
λ
j
−
λ
i
α (
t
)
w (
$
1
+
2
j
&
ω
i
(
t
+
1
)
=
ω
i
(
t
)
i
ω
(2.76)
4
2
t
)
=
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