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where
V
V
T
is the eigendecomposition of
A
T
A
,
q
≡
V
T
A
T
b
,and
x
crit
is the
value of
x
corresponding to
.
The first
n
equations in (6.2) yield the GeTLS EXIN critical value corres-
ponding to
γ
, which is one of the critical values for
E
GeTLS
(ζ
,
x
)
γ
:
x
crit
=
A
T
A
−
2
γζ
I
n
−
1
A
T
b
(6.4)
Equations (6.3) and (6.4) are the basis of the GeTLS EXIN theory. In this theory
the TLS hyperplane is defined as the locus whose equation is
u
n
+
1
=−
1. It
contains the GeTLS solutions for each value of
ζ
[24,31,35,156].
In conclusion, the GeTLS EXIN approach can be analyzed as a generalized
eigenvalue problem because of the form of the error function (5.6). This inter-
pretation is important from both a theoretical point of view (new theorems and
inequalities have been deduced) and a numerical point of view (other algorithms
can be used besides the GeTLS EXIN algorithm (neuron) [24,31,35], such as
Wilkinson's algorithm [35]).
In the next sections we analyze this symmetric positive definite (S/PD) gen-
eralized eigenvalue problem in detail.
6.1.1 Particular Cases
6.1.1.1 Case ζ
= 0
(OLS)
If
ζ
=
0, it follows that
2
0
n
0
D
=
(6.5)
0
T
1
Define as
u
⊥
each vector
u
belonging to the
n
-dimensional hyperplane passing
through the origin and whose normal is given by the vector
e
n
+
1
∈
n
+
1
,allof
whose components are null except the last. It follows that
1
γ
1
γ
u
⊥
⊥
e
n
+
1
⇒
Du
⊥
=
0
⇒
Ru
⊥
=
0
⇒
=
0
⇒
γ
=∞
(6.6)
because
R
is nonsingular. Hence,
is an eigenpair with an infinite eigen-
value whose algebraic multiplicity is
n
and
u
⊥
belongs to the subspace orthogonal
to
e
n
+
1
. It means that this eigenspace is parallel to the TLS hyperplane. It implies
that all critical points except one of the GeTLS EXIN cost functions disappear.
The following analysis estimates the remaining eigenpair and confirms the fore-
going considerations. From eq. (6.2), by defining
u
(
∞
,
u
⊥
)
=
u
1
,
u
n
+
1
T
with
u
1
n
∈
and
u
n
+
1
∈
, we can derive
A
T
AA
T
b
b
T
Ab
T
b
u
1
u
n
+
1
0
n
u
1
u
n
+
1
0
=
2
γ
0
T
1
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