Digital Signal Processing Reference
In-Depth Information
Algorithm 35
Fast GMCA Algorithm
Task:
Sparse blind source separation.
Parameters:
The data
Y
, the dictionary
1
···
K
], number of iterations
N
iter
, number of sources
N
s
and channels
N
c
, stopping threshold
=
[
λ
min
, threshold
update schedule.
Initialization:
(0)
0,
A
(0)
a random matrix.
α
=
Apply the MCA Algorithm 30 with
to each data channel
y
i
to get
β
.
max
i
,
l
β
l
]
.
Set threshold
λ
0
=
[
i
,
Main iteration:
for
t
=
1
to
N
iter
do
λ
t
A
(
t
)
+
β
.
(
t
+
1)
Update the coefficients
α
:
α
=
Thresh
1)
+
; normalize columns to a
Update the mixing matrix
A
:
A
(
t
+
1)
=
βα
(
t
+
unit
2
-norm.
Update the threshold
λ
t
according to the given schedule.
λ
t
≤
λ
min
then
stop.
Reconstruct the sources:
s
i
=
k
=
1
k
α
if
(
N
iter
)
i
,
k
,
i
=
1
,...,
N
s
.
Output:
Estimated sources
s
i
i
=
1
,...,
N
s
and mixing matrix
A
(
N
iter
)
.
wavelet transform), the algorithm becomes even faster because the MCA step is
replaced by application of the fast analysis operator to each channel.
9.4.4 Estimating the Number of Sources
In BSS, the number of sources
N
s
is assumed to be a fixed known parameter of the
problem. In practical situations, this is an exception rather than a rule, and estimat-
ing
N
s
from the data is a crucial and strenuous problem.
As we supposed
N
s
≤
N
c
, the number of sources is the dimension of the sub-
space of the whole
N
c
-dimensional space (recall that
N
c
is the number of channels)
in which the data lie. A misestimation of the number of sources
N
s
may entail two
difficulties:
Underestimation:
In the GMCA algorithm, underestimating the number of
sources will clearly lead to poor unmixed solutions that are made of linear com-
binations of true sources. The solution may then be suboptimal with respect to
the sparsity of the estimated sources.
Overestimation:
In such case, the GMCA algorithm may have to cope with a
mixing matrix estimate that becomes ill conditioned.
Relatively little work has focused on the estimation of the number of sources
N
s
.
One can think of using model selection criteria such as the minimum descrip-
tion length (MDL) devised by Balan (2007). Such criteria, including the Akaike
