Biomedical Engineering Reference
In-Depth Information
using [
4
-
6
]
N
2
x
=
argmin
x
1
|
x
j
|
subject to
y
−
Fx
=
d
.
(2.44)
j
=
The only difference between the equation above and Eq. (
2.35
) is to minimize either
L
2
norm
1
=
j
|
2
in Eq. (
2.35
)or
L
1
norm,
in Eq. (
2.44
). Although it
may look as if there is no significant difference between the two methods, the results
of source estimation are significantly different. The
L
1
-norm regularization gives a
“so-called” sparse solution, in which only few
x
j
have nonzero values and a majority
of other
x
j
have values close to zero.
Using the method of Lagrange multipliers and following exactly the same argu-
ments as in Sect.
2.8
,the
L
1
-norm solution can be obtained by minimizing the cost
function
x
x
x
j
|
F
, i.e.,
N
2
x
=
argmin
x
F
:
F
=
y
−
Fx
+
ʾ
1
|
x
j
|
,
(2.45)
j
=
where again
is a positive constant that controls the balance between the first and the
second terms in the cost function above. Unfortunately, the minimization problem
in Eq. (
2.45
) does not have a closed-form solution, so numerical methods are used
here to obtain the solution
ʾ
x
.
2.9.2 Intuitive Explanation for Sparsity
Actually, it is not easy to provide an intuitive explanation regarding why the opti-
mizationinEq.(
2.44
)or(
2.45
) causes a sparse solution. The straightforward (and
intuitively clear) formulation to obtain a sparse solution should use the
L
0
-norm
minimization, such that
N
2
x
=
argmin
x
1
T (
x
j
)
subject to
y
−
Hx
=
d
,
(2.46)
j
=
where the function
T (
x
)
is defined in Eq. (C.65). In the above formulation, since
j
=
1
T (
x
j
)
indicates the number of nonzero
x
j
,
x
is the solution that has the smallest
2
number of nonzero
x
j
and still satisfies
d
. The optimization prob-
lem in Eq. (
2.46
) is known to require impractically long computational time. The
optimization for the
L
1
-norm cost function in Eq. (
2.44
) approximates this
L
0
-norm
optimization in Eq. (
2.46
) so as to obtain a sparse solution within a reasonable range
of computational time [
7
].
y
−
Hx
=
