Digital Signal Processing Reference
In-Depth Information
information criterion (AIC) (Akaike 1970) and the Bayesian information criterion
(BIC) (Schwarz 1978), would provide a balance between the complexity of the
model (here the number of sources) and its ability to faithfully represent the data. It
would amount to adding a penalty term to equation (9.19). This penalty term would
merely prevent a high number of sources. But a sparsity-based method to estimate
N
s
within the GMCA framework can be designed.
For a fixed number of sources
n
s
<
N
s
, the sparse BSS problem (9.19) can be
written in the constrained form
n
s
Y
T
F
≤
σ.
p
(
P
n
s
,σ
): min
A
,
α
|
rank(
A
)
=
n
s
1
α
i
s
.
t
.
−
A
α
(9.31)
i
=
T
,
A
, and the number of sources, the problem we would
To jointly estimate
S
=
α
like to tackle is then
n
s
Y
T
F
≤
σ
p
p
min
n
s
∈{
min
1
α
i
s
.
t
.
−
A
α
.
1
,...,
N
c
}
A
,
α
|
rank(
A
)
=
n
s
i
=
σ<σ
(
n
s
), (
P
n
s
,σ
) has
no feasible solution in
A
that satisfies the rank condition. For a fixed
n
s
<
σ
(
n
s
) such that if
If
n
s
<
N
s
, there exists a minimal value
N
s
,this
σ
(
n
s
) is the approximation error between
Y
and its projection in
the subspace spanned by its singular vectors corresponding to the
n
s
largest singu-
lar values. Furthermore, in the noiseless case, for
n
s
<
minimal value
σ
(
n
s
) is always strictly
positive as the data lie in a subspace whose dimension is exactly
N
s
. When
n
s
N
s
,
=
N
s
,
σ
=
σ
(
N
s
)
the problem (
P
N
s
,σ
0.
This discussion suggests a constructive approach to jointly estimate the number
of sources
N
s
,
S
) has at least one solution for
=
T
, and
A
. This selection procedure uses GMCA to solve a
sequence of problems (
P
n
s
,σ
(
n
s
)
) for each constraint radius
=
α
σ
(
n
s
) with increasing
n
s
,
1
≤
n
s
≤
N
c
. This is summarized in Algorithm 36 (Bobin et al. 2008b).
Algorithm 36
GMCA-Based Selection of the Number of Sources
α
Task:
Jointly estimate the number of sources, source coefficients
, and mixing
matrix
A
.
Parameters:
The data
Y
, the dictionary
=
[
···
K
], number of iterations
1
N
iter
, stopping threshold
λ
min
, threshold update schedule.
Main iteration:
for
n
s
=
1 to
N
c
do
1. Add a new column to
A
.
2. Solve (
P
n
s
,σ
(
n
s
)
) with the GMCA algorithm using (
,
N
iter
,
n
s
,
N
c
,λ
min
)as
its parameters.
T
F
≤
σ
(
N
s
)
then
stop.
Output:
Estimated number of sources
n
s
.
if
Y
−
A
α
