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4
INTRODUCTION TO THE
TLS EXIN NEURON
*
4.1 FROM MCA EXIN TO TLS EXIN
m
Section 1.5.1, given the system
Ax
=
b
,where
A
∈
×
n
and
As seen in
m
, the TLS solution mimimizes the following cost function:
b
∈
[
A
;
b
]
x
T
;−
1
T
2
T
E
TLS
(
x
)
=
(
Ax
−
b
)
(
Ax
−
b
)
1
+
x
T
x
2
=
(4.1)
x
T
;−
1
T
2
2
which is the Rayleigh quotient of [
A
;
b
]
T
[
A
;
b
] constrained to the TLS hyperplane
(i.e.,
x
n
+
1
1).
Hence, the TLS solution is parallel to the right singular vector (
∈
=−
n
+
1
)cor-
responding to the minimum singular value of [
A
;
b
]. Define
=
a
i
;
b
i
T
T
ξ
i
∨
=
ξ
i
y
i
(4.2)
with
a
i
the
i
th row of
A
,where
y
i
is the output of the MCA linear neuron
of weight vector
∈
1
, and learning law minimizing the
Rayleigh quotient of the autocorrelation matrix of the input data
R
,whichis
equivalent
to [
A
n
+
1
, input
ξ
i
n
+
∈
b
]
T
[
A
;
;
b
]
/
m
. Then, to find the TLS solution, the MCA solution
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