Information Technology Reference
In-Depth Information
;−
1
T
[it follows from (1.12)
and (1.13)]. The TLS solution is parallel to the right singular vector corresponding
to the minimum singular value of [
A
;
b
], which can be expressed as a Rayleigh
quotient (see Section 2.1):
b
are
parallel
with the solution vector
x
T
A
;
1
T
b
]
x
T
2
i
=
1
a
i
x
−
b
i
[
A
;
;−
2
m
i
=
1
r
i
2
n
2
2
2
σ
=
=
=
(1.21)
x
T
;−
1
T
+
1
2
1
+
x
T
x
2
2
the square of the distance from
a
i
b
i
T
with
a
i
2
the
i
th row of
A
and
r
i
;
∈
n
+
1
to the nearest point in the subspace:
a
∈
a
a
b
˜
R
r
(
[
A
;
b
]
)
=
n
,
b
∈
,
b
=
x
T
Thus, the TLS solution
x
minimizes the sum of the squares of the orthogonal
distances (weighted squared residuals):
i
=
1
|
a
i
x
−
b
i
|
2
E
TLS
(
x
)
=
(1.22)
1
+
x
T
x
which is the Rayleigh quotient (see Section 2.1) of [
A
;
b
]
T
[
A
;
b
] constrained to
x
n
+
1
=−
1. It can also be rewritten as
T
E
TLS
(
x
)
=
(
Ax
−
b
)
(
Ax
−
b
)
1
+
x
T
x
(1.23)
This formulation is very important from the neural point of view because it
can be considered as the energy function to be minimized for the training of a
neural network whose final weights represent the TLS solution. This is the basic
idea of the TLS EXIN neuron.
1.6 MULTIDIMENSIONAL TLS PROBLEM
1.6.1 Unique Solution
Definition 16 (Multidimensional TLS Problem)
Given the overdetermined
set
(
1.1
)
, the total least squares
(
TLS
)
problem searches for
[
A
;
B
]
−
[
A
;
B
]
F
B
)
∈
R
(
A
)
min
subject to
R
(
(1.24)
[
A
;
B
]
∈
m
×
(
n
+
d
)
Once a minimizing
[
A
;
B
]
is found, then any
X satisfying
A X
B
=
(1.25)
Search WWH ::
Custom Search