Biomedical Engineering Reference
In-Depth Information
Using the matrix inversion lemma in Eq. (C.92), we obtain
F
T
FF
T
I
−
1
y
x
=
+
ʾ
.
(2.40)
ThesolutioninEq.(
2.40
) is called the
L
2
-norm-regularizedminimum-normsolution,
or simply
L
2
-regularized minimum-norm solution.
Let us compute the noise influence term for the
L
2
-regularized minimum-norm
solution. Using Eq. (
2.31
), we have
F
T
FF
T
I
−
1
N
ʳ
j
j
+
ʾ
v
j
u
j
,
+
ʾ
=
(2.41)
2
ʳ
j
=
1
and the
L
2
-regularized minimum-norm solution is expressed as
F
T
FF
T
I
−
1
x
=
+
ʾ
(
Hx
+
ʵ
)
F
T
FF
T
I
−
1
N
ʳ
j
j
+
ʾ
v
j
u
j
ʵ
.
=
+
ʾ
+
Hx
(2.42)
2
ʳ
j
=
1
The second term, expressing the influence of noise, is
N
u
j
ʵ
)
ʳ
j
(
v
j
.
(2.43)
2
ʳ
j
+
ʾ
j
=
1
In the expression above, the denominator contains the positive constant
ʾ
, and it is
easy to see that this
ʾ
prevents the terms with smaller singular values from being
amplified.
One problem here is how to choose an appropriate value for
ʾ
. Our argument
ʾ
above only suggests that if the noise is large, we need a large
, but if small, a smaller
ʾ
can be used. However, the arguments above do not lead to the derivation of an
appropriate
. We will return to this problem in Sect.
2.10.2
where
L
2
-regularized
minimum-norm solution is re-derived based on a Bayesian formulation, in which
deriving the optimum
ʾ
ʾ
is embedded.
2.9
L
1
-Regularized Minimum-Norm Solution
2.9.1 L
1
-Norm Constraint
In the preceding section, we derived a solution that minimizes the
L
2
-norm of the
solution vector
x
. In this section, we argue for a solution that minimizes the
L
1
-norm
of
x
, which is defined in Eq. (C.64). The
L
1
-norm-regularized solution is obtained
