![](bg8.png)
tion that arises if two sources have ∆φ
ij
= 0 or π. In
that case, the problem becomes ill-posed, as was al-
ready the case in IPA (Almeida et al., 2011a). In fact,
using sources with ∆φ
ij
<
π
10
starts to deteriorate the
results of PLMF, even with zero noise.
One further aspect which warrants discussion is
PLMF’s identifiability. If we find two factorizations
such that X = M
1
A
1
⊙(z
1
f
T
1
)
= M
2
A
2
⊙(z
2
f
T
2
)
(i.e., two factorizations which perfectly describe the
same data X), does that imply that M
1
= M
2
, and
similar equalities for the other variables? It is quite
clear that the answer is negative: the usual indeter-
minancies of BSS apply to PLMF as well, namely
the indeterminancies of permutation, scaling, and sign
of the estimated sources. There is at least one fur-
ther indeterminancy: starting from a given solution
X = M
1
A
1
⊙(z
1
f
T
1
)
, one can always construct a
new one by defining z
2
≡ e
iψ
z
1
and f
2
≡ e
−iψ
f
1
,
while keeping M
2
≡ M
1
and A
2
≡ A
1
. Note that
S
1
= A
1
⊙(z
1
f
T
1
) = A
2
⊙(z
2
f
T
2
) = S
2
, thus the esti-
mated sources are exactly the same.
6 CONCLUSIONS
We presented an improved version of Phase Locked
Matrix Factorization (PLMF), an algorithm that di-
rectly tries to reconstruct a set of measured signals as
a linear mixing of phase-locked sources, by factoriz-
ing the data into a product of four variables: the mix-
ing matrix, the source amplitudes, their phase lags,
and a common oscillation.
PLMF is now able to estimate the sources even
when their common oscillation is unknown – an ad-
vantage which greatly increases the applicability of
the algorithm. Furthermore, the sub-problem for M
is now convex, and the sub-problems for z and f are
tackled in a more appropriate manner which should
find local minima. The results show good perfor-
mance for the noiseless case and good robustness to
small amounts of noise. The results show as well that
the proposed algorithm is accurate and can deal with
low amounts of noise, under the assumption that the
sources are fully phase-locked, even if the common
oscillation is unknown. This generalization brings us
considerably closer to being able to solve the Separa-
tion of Synchronous Sources (SSS) problem in real-
world data.
ACKNOWLEDGEMENTS
This work was partially funded by the DECA-Bio
project of the Institute of Telecommunications, and
by the Academy of Finland through its Centres of Ex-
cellence Program 2006-2011.
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