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These three expressions are respectively equivalent to
m
,
n
ma
mn
=
1
.
2
(49)
6
Numerical Optimization
2
The goal here to minimize (33) under the derived condition (49).
|
F
(
u
,
v
)
|
in (33)
is replaced with its ensemble average
P
(
u
,
v
)
in a similar manner to that reported by
Ando as follows
2
P
(
u
,
v
)=
E
[
|
F
(
u
,
v
)
|
]
.
(50)
That is, Equation (33), repeated below as
2
dudv
D
Ψ
(
u
,
v
)
|
F
(
u
,
v
)
|
,
(51)
is rewritten as
k
,
l
m
,
n
a
kl
a
mn
R
kl
,
mn
D
Ψ
(
u
,
v
)
P
(
u
,
v
)
dudv
=
16
(52)
where
a
kl
mn
b
kl
mn
c
kl
mn
R
kl
,
mn
≡
D
(
τ
τ
+
τ
τ
+
τ
τ
)
P
(
u
,
v
)
dudv
.
(53)
Our objective is now to minimize
k
,
l
m
,
n
a
kl
a
mn
R
kl
,
mn
J
0
≡
16
(54)
under the condition (49).
The optimal values of
a
mn
are computed by a traditional gradient-descent opti-
mization as follows. Taking i as the number of steps of optimization, gives
J
(
i
)
0
k
,
l m
,
n
a
(
i
)
kl
a
(
i
mn
R
kl
,
mn
.
≡
16
(55)
Differentiating this expression, gives
∂
J
(
i
)
0
k
,
l
a
(
i
)
=
16
kl
R
kl
,
mn
.
(56)
a
(
i
mn
∂
To satisfy condition (49), the values are updated at each iteration as follows:
∂
a
(
i
mn
−
α
J
(
i
)
0
a
tmp
mn
=
,
(57)
a
(
i
mn
∂
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