A Unified Approach for Over and Under-determined Blind
Source Separation Based on Both Sparsity and Decorrelation
Fangchen Feng∗
∗ CNRS-SUPELEC-Univ Paris-Sud
Gif-sur-Yvette, France
Email: fangchen.feng@lss.supelec.fr
Matthieu Kowalski∗†
† Parietal project-team, INRIA, Saclay, France
Email: matthieu.kowalski@lss.supelec.fr
Abstract—Independent component analysis (ICA) has been a major
tool for blind sources separation (BSS). Both theoretical and practical
evaluations showed that the hypothesis of independence suit well for
audio and musical signals. In the last few years, optimization which
based on sparsity has emerged as an another efficient implement for
BSS. Our work starts from introducing a new BSS method that takes
advantages of both independence and sparsity using overcomplete Gabor
representation. It is shown theoretically and numerically that the method
works in both under-determined and determined case. Experimental
results are then given illustrating the good performances of this approach
and its robustness to noise.
I. INTRODUCTION
The instantaneous linear mixture model of blind source separation
assumes that :
x = As + e , (1)
where x ∈ RM×T
and s ∈ RN×T
are the matrices of mixture
channels and source signals respectively. A ∈ RM×N
is the mixing
matrix and e ∈ RM×T
models the background noise.
The ICA methods [7], [10] are often applied when M ≥ N
(over-determined or determined scenario). These methods try to
achieve separation by minimizing an independence criterion between
the components of the estimated sources. While two-steps methods
based on sparsity are largely applied in the under-determined case
(M < N) [8] : The mixing system is first estimated using the
clustering methods [1]. Then the source signals are estimated by
using optimization methods considering that the mixing system is
known [12].
In our work, we designed an algorithm that benefits from both
independence and sparsity which work in under-determined and
(over-)determined cases.
II. ON THE USE OF DECORRELATION AND SPARSITY
Motivated by [9] which shows that one should develops alternating
decomposition methods targeting decompositions into sparse compo-
nents rather than independent components, the idea of the proposed
approach is that we suppose that the sources are decorrelated which
is a direct consequence of the independence. This assumption can be
formulated as :
E(ssT
) = D (2)
where s has zero mean and D is a diagonal matrix with the energy
of each source signals on its diagonal. E is the expectation operator.
Under this assumption, along with the sparse property of signals [5]
[2], we obtain a constrained optimization :



arg min
A,α
1
2
||x − AαΦ∗
||2
2 + λ||α||1
s.t. αΦ∗
Φα∗
= D
(3)
where Φ is the Gabor transform operator and Φ∗
is the adjoint
operator of Φ, that is its Hermitian transpose. α are the synthesis
coefficients of s.
The minimization in both A and α makes the problem nonconvex
and we propose here an approximate algorithm in Algorithm 1,based
on the proximal alternating linearized minimization method [3].
Algorithm 1:
Initialization : α(0)
∈ CN×B
, A(0)
∈ RM×N
, k = 1;
repeat
1. α(k)
= prox λ
L
(α(k)
− f(k)
(α(k)
)
L(k) );
2. s = α(k)
Φ∗
;
3. D = diag(ssT
);
4. W = sqrtm((ssT
)−1
);
5. ˜s = sqrtm(D)Ws;
6. A(k+1)
= x˜sT
;
7. a
(k+1)
n = a
(k+1)
n /||a
(k+1)
n ||, n = 1, 2, . . . , N;
8. k = k + 1;
until convergence;
where f(k)
(α) = 1
2
||x − A(k)
αΦ∗
||2
2 and f(k)
= −A(k)T
(x −
A(k)
αΦ∗
)Φ. L(k)
is the Lipschitz constant of f(k)
(α) and ”sqrtm”
calculate the principal square root of a matrix. an is the n-th column
of matrix A, n = 1, 2, . . . , N.
Step 1 is one iteration of the sub-problem :
min
α
1
2
||x − AαΦ∗
||2
2 + λ||α||1 (4)
where the matrix A is supposed to be known. Step 2 to Step 5 are
designed to perform the projection onto the contraint, i.e. to enforce
that the estimated signals are decorrelated and to keep the energy of
each estimation unchanged at the same time. Step 6 and 7 aim to
find the appropriate A when given α with a direct application of the
decorrelation assumption (2) :
A(k+1)
= argmin
A
1
2
||x − AαΦ∗
||2
2 | αΦ∗
Φα = D, ||an|| = 1
Moreover, step 7 is also used to avoid the classical scale ambiguity
of source separation [4]. The algorithm is initialized with random
signals.
We demonstrate in Figure 1 that this algorithm (Opt-deco) works
better than DUET [12] in the under-determined setting. Figure 2
shows that it is more robust to noise than FastICA [6] in the
determined (high) and over-determined (low) setting. One can remark
that the proposed method outperforms FastICA in the low input SNR
condition. In the experiments, we use SDR (Signal Distorsion Ratio)
and SIR (Signal Interference Ratio) as the evaluation indicators [11].

5 10 15 20
−2
0
2
4
6
8
10
12
14
16
18
20
SDR
Input SNR(dB)
SDR(dB)
5 10 15 20
−2
0
2
4
6
8
10
12
14
16
18
20
Input SNR(dB)
SIR(dB)
SIR
Opt−deco
DUET
Opt−deco
DUET
Fig. 1. Performance evaluation in under-determined case with M = 2 and
N = 3
5 10 15 20
0
10
20
30
40
50
Input SNR(dB)
SDR(dB)
SDR
5 10 15 20
0
10
20
30
40
50
Input SNR(dB)
SIR(dB)
SIR
5 10 15 20
0
10
20
30
40
50
Input SNR(dB)
SDR(dB)
SDR
5 10 15 20
0
10
20
30
40
50
Input SNR(dB)
SDR(dB)
SIR
Opt−deco
FastICA
Opt−deco
FastICA
Opt−deco
FastICA
Opt−deco
FastICA
Fig. 2. High : Performance evaluation in determined case with M = 3 and
N = 3. Low : Performance evaluation in over-determined case with M = 3
and N = 5
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