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Theorem 97 (DLS EXIN Convergence)
The DLS EXIN learning law, express-
ing the minimization of the error cost
(
5.21
)
converges, under the usual stochastic
approximation assumptions, to the DLS solution
[
i.e., eq.
(
1.55
)]
.
Proof.
Express eq. (5.24) as
x
T
x
2
Q
x
,
R
,
r
,
dx
dt
=
1
(5.25)
where
Q
x
,
R
,
r
,
=−
Rx
−
r
x
T
x
+
x
T
Rx
x
−
2
x
T
r
x
+
x
(5.26)
The critical points are given by
Q
x
,
R
,
r
,
=
0, which, under the transforma-
tion
=
r
T
v
−
1
x
v
(5.27)
becomes
Q
v
,
R
,
r
,
=
0
(5.28)
which implies that
R
T
−
3
R
T
−
2
3
v
v
R
2
v
v
R
T
T
v
v
−
v
R
T
−
3
T
R
v
v
−
2
R
T
−
2
R
T
R
−
1
v
v
T
R
v
+
3
2
2
=
v
v
v
After some algebra,
T
R
R
−
RR
T
RR
T
v
−
/
v
v
=
v
(5.29)
v
T
v
R
ayleigh quotient
v
c
are the eigenvectors of the matrix
R
−
RR
T
Hence, the critical points
.
Following the proof of Theorem 60, it is easy to show that in the transformed
space, under the usual assumptions, the DLS EXIN learning law converges to
the eigenvector
/
min
associated with the minimum eigenvalue. Using eq. (5.14),
it holds that, on average,
v
A
T
b
b
T
A
b
T
b
=
A
T
I
−
b
b
T
b
−
1
b
T
A
RR
T
→
A
T
A
−
R
−
(5.30)
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