Information Technology Reference
In-Depth Information
Define,
∀
i
=
1, 2,
...
,
n
−
1,
f
i
(
t
)
f
n
(
t
)
ϕ
i
(
t
)
=
(2.87)
which gives
df
i
(
t
)
df
n
(
t
)
f
n
(
t
)
−
f
i
(
t
)
d
ϕ
i
(
t
)
dt
dt
dt
=
2
(
f
n
(
t
))
(
2
.
86
)
w (
t
)
−
2
2
=
(λ
n
−
λ
i
) ϕ
i
(
t
)
(2.88)
whose solution on [0,
∞
)
is
t
d
ν
w (ν)
ϕ
i
(
t
)
=
exp [
(λ
n
−
λ
i
)
]
∀
i
=
1, 2,
...
,
n
−
1
(2.89)
2
2
0
t
→∞
→
If
λ
n
is single, then
ϕ
i
(
t
)
0
∀
i
=
1, 2,
...
,
n
−
1andthen
lim
f
i
(
t
)
=
0
∀
i
=
1, 2,
...
,
n
−
1
t
→∞
which yields
lim
→∞
w (
t
)
=
w (
t
st
)
=
lim
f
n
(
t
)
z
n
(2.90)
t
t
→∞
Considering as a first approximation the constancy of the weight modulus, it
follows that
f
n
(
t
)
=±
w (
0
)
2
lim
t
→∞
t
→∞
w (
t
)
=
w (
t
st
)
lim
+
w (
0
)
2
z
n
T
z
n
>
0
∀
w/w
=
T
z
n
<
0
−
w (
0
)
2
z
n
∀
w/w
Remark 61 (Approximation by the ODE)
As will be apparent in the following
section
(
which deals mainly with the stochastic discrete MCA learning laws
)
,eq.
(
2.80
)
is only approximately valid in the first part of the time evolution of the MCA
learning laws (i.e., in approaching the minor component).
Theorem 62 (Time Constant)
If the initial weight vector has modulus less than
unity, the MCA EXIN weight vector reaches the minor component direction faster
than OJAn, which is faster than LUO. The contrary happens if the modulus of the
initial weight vector is more than unity. If the weight remains on the unit sphere,
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