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(MADALINE, i.e., many ADALINES; see [191]) solve each subproblem
Ax
i
≈
b
i
,
i
=
1,
...
,
d separately
. The GeTLS EXIN MADALINE is made
up of a single layer of
d
GeTLS EXIN neurons, each having the same input
(row
a
i
n
,
i
=
1,
...
,
m
), a weight vector (
x
j
n
,
j
∈
∈
=
1,
...
,
d
)anda
m
×
d
,
i
=
1,
...
,
m
n
,
j
=
1,
...
,
d
). At convergence, the solution
is
X
=
[
x
1
,
x
2
,
...
,
x
d
]. The instantaneous cost function is
target (
b
ij
∈
a
i
x
j
−
b
ij
2
d
1
2
E
(
i
)
(
X
)
=
(5.36)
1
−
ζ )
+
ζ
x
j
2
2
(
j
=
1
and the global cost function is
m
E
(
i
)
(
E
GeTLS EXIN MAD
(
x
)
=
x
)
(5.37)
i
=
1
The GeTLS EXIN MADALINE discrete learning law is then
T
X
(
t
+
1
)
=
X
(
t
)
−
α (
t
)
a
i
κ
(
t
)
+
ζα(
t
)
X
(
t
)(
t
)
(5.38)
where
δ
j
(
t
)
=
a
i
x
j
(
t
)
−
b
ij
(5.39)
δ
j
(
t
)
(
1
−
ζ )
+
ζ
x
j
(
t
)
γ
j
(
t
)
=
(5.40)
2
2
κ(
t
)
=
γ
1
(
t
)
,
γ
2
(
t
)
,
...
,
γ
d
(
t
)
T
d
∈
(5.41)
diag
γ
(
t
)
∈
2
1
2
2
2
d
d
×
d
(
t
)
=
(
t
)
γ
(
t
)
...
γ
,
,
,
(5.42)
This network has independent weight vectors. As seen in Remark 20, a network
with
dependent
weight vectors is better, at least when all data are equally
perturbed and all subproblems
Ax
i
b
i
have the same degree of incompatibility.
≈
5.3 GeTLS STABILITY ANALYSIS
5.3.1 GeTLS Cost Landscape: Geometric Approach
For convenience, the GeTLS EXIN cost function (5.6) is repeated here:
T
2
(
Ax
−
b
)
(
Ax
−
b
)
(
1
−
ζ)
+
ζ
x
T
x
1
E
GeTLS EXIN
(
x
)
=
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