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have high residuals). In the following, robust nonlinear neurons are introduced
to overcome this problem.
1.11 BASIC DATA LEAST SQUARES PROBLEM
The TLS problem can be viewed as an unconstrained perturbation problem
because all columns of [ A ; b ] can have error perturbations. The OLS problem
constrains the columns of A to be errorless; the opposite case is the data least
squares (DLS) problem, because the error is assumed to lie only in the data
matrix A .
Definition 36 (Basic DLS Problem) Given the overdetermined set ( 1.4 ) ,the
data least squares ( DLS ) problem searches for
n A
A F
A )
min
A
subject to b
R
(
(1.53)
m
×
Once a minimizing A is found, then any x satisfying
A x = b
(1.54)
A =
A ) .
is called a DLS solution ( the corresponding DLS correction is
A
The DLS case is particularly appropriate for certain deconvolution problems,
such as those that may arise in system identification or channel equalization [51].
Theorem 37 The DLS problem ( 1.53 ) is solved by
b T b
b T A v min v min
b T A v min = 0
x =
(1.55)
where v min is the right singular vector corresponding to the smallest singular
value of the matrix P b A, where P b
I b b T b 1 b T is a projection matrix
that projects the column space of A into the orthogonal complement of b. If the
smallest singular vector is repeated, then the solution is not unique. The minimum
norm solution is given by
=
b T AV min V min A T b V min V min A T b
b T b
b T AV min V min A T b = 0
x =
(1.56)
where V min is the right singular space corresponding to the repeated smallest
singular value of P b A.
Proof. See [51], which derives its results from the constrained total least squares
(CTLS) [1,2].
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