BAYESIAN MULTICHANNEL NONNEGATIVE MATRIX FACTORIZATIONFOR AUDIO SOURCE SEPARATION AND LOCALIZATION
Kousuke Itakura, Yoshiaki Bando, Eita Nakamura,
Katsutoshi Itoyama, Kazuyoshi Yoshii, Tatsuya Kawahara

Graduate School of Informatics, Kyoto University, Sakyo-ku, Kyoto 606-8501, Japan
ABSTRACT

Spatial correlation matrices Gjf corresponding to $ources are
given by weighted sum ofthose corresponding to directions

This paper presents a Bayesian extension of multichannel nonnegative matrix factorization (MNMF) that decomposes the complex
spectrograms of mixture signals recorded by a microphone array
into basis spectra, their temporal activations, and the spatial correlation matrices of sources (directions) in the time-frequency-channel
domain. Although the original MNMF can be used in a blind setting, prior knowledge of a microphone array is useful for improving
source separation. The impulse response (spatial correlation matrix)of each direction can be measured in an anechoic room, however,it differs from that in a real environment where the microphone array is used. To solve this, we propose a unified Bayesian model ofsource separation and localization by introducing a prior distribution
determined by an anechoic spatial correlation matrix on areal spatial correlation matrix with respect to each direction. This enables us
to adaptively estimate areal spatial correlation matrix and the direction of each source. Experimental results showed that our method
outperformed the original MNMF and the state-of-the-art methods
with prior knowledge in terms of signal-to-distortion ratio (SDR)
even when the method was used in an unknown environment withacoustic characteristics different from those of the anechoic room.

Index Terms- Source separation, source localization, multichannel nonnegative matrix factorization, Bayesian modeling
1. INTRODUCTION
Microphone array processing forms the basis of computational auditory scene analysis that aims to recognize individual auditory eventsin asound mixture. In multichannel source separation, phase differences between microphones playa key role. For ex am pIe, frequencydomain independent component analysis (ICA) [1] and independentvector analysis (lVA) [2, 3] can separate mixture sounds in such a
way that source signals are statistically independent from each other.
On the other hand, separation methods based on time-frequency (TF)
clustering [4, 5] assign each TF bin of mixture sounds exclusively to
one of the sources by focusing on the phase information.
The power spectrograms of sources as weil as phase differences
between microphones have recently been modeled for multichannel
source separation. Nonnegative matrix factorization (NMF) [6] is awell-known technique of single-channel source separation that approximates the power spectrogram of each source as a rank-l matrix. Various approaches have been proposed to separate sounds
by NMF [7- 9]. To deal with spatial correlation matrices over microphones, Sawada et al. [10] proposed a multichannel extensionof NMF (MNMF). Kitamura et al. [11] proposed a rank-l MNMF
that restricts the spatial correlation matrices to rank-l matrices andThis study was partially supported by Grant-in-Aid for Scientific Research Nos. 24220006 and 15K12063.

978-1-5090-4117-6/17/$31.00 ©2017 IEEE

551

Sldt

rL

W11h-.

Spatial correlation matrices GiJ
corresponding to directions

öD '
2

d

ILIBasiswll/
I

Activation hUt

f: frequency bint,:timeframe
Observed spectrogramsof a mixture sound xr!

m: microphone index
d: direction index
k: source index
l: basis index

Fig. 1. The generative story of multichannel NMF.
this model was shown to be an NMF-integrated version of IVA. TF
clustering-based method [5] can also be integrated with NMF [12].
Deep neural networks have been used to estimate the power densities
of sources [13]. In general, most of multichannel separation methods
have been designed so that they can be used in a blind setting.
In this paper we propose a Bayesian extension of MNMF that
can incorporate various kinds of prior knowledge in a principledmanner. When prior knowledge about an environment, microphones,
and/or sources (e.g., microphone array geometry, impulse responses
measured in an anechoic room, and the template spectra of sources)
are available, for ex am pIe, the parameters of MNMF can be converged to reasonable values. Furthermore, anonparametric Bayesian
extension of MNMF would be feasible to automatically estimate thenumber of sources according to observed data in a similar way to a
nonparametric Bayesian model of TF clustering [5].
As illustrated in Fig. 1, we design the generative process of thecomplex spectrograms of multi-channel mixture signals and then try
to solve the inverse problem. More specifically, the power spectrogram densities of each source are determined by the product of abasis matrix and an activation matrix. The spatial correlation matrices of each source, on the other hand, are given by the weighted sum
ofthose corresponding to different directions. Using these two typesof variables, the complex spectrogram of each source is stochastically generated. The mixture spectrograms are given by the sum of
the source spectrograms. The proposed method uses spatial correlation matrices (impulse responses) measured in an anechoic room
to determine prior distributions on real spatial correlation matrices.
Given the mixture spectrograms as observed data, we optimize all
the parameters iteratively using Gibbs sampling. Finally, the source
spectrograms are obtained by multichannel Wiener filtering and thesource directions are determined from the direction weights.

ICASSP2017

2. RELATED WORK

the complex spectrograms of source signals are defined as folIows:

A conventional approach to multichannel source separation is to esti mate a linear unmixing filter that decomposes the complex spectraof mixture signals into those of source signals in the frequency domain [1 - 3, 14]. Mixture signals are usually modeled as the sum ofsource signals convolved with the impulse responses of the corresponding source directions. This is equivalent to an instantaneous
mixing process in the frequency domain, Le., the complex spectraof mixture signals are the sum of source spectra multiplied by theimpulse-response spectra. Using such Iinearity between mixture andsource spectra, frequency-domain ICA can estimate a linear unmixing filter for each frequency bin [1]. The permutation of separated
source spectra, however, is not aligned between different frequency
bins. One way to resolve this permutation ambiguity is to focus on
the directions and inter-frequency correlations of the sources [14].
IVA [2, 3] is an extension of ICA that can jointly deal with all frequency components in a vectorial manner.
Another popular approach to multichannel source separation is
nonlinear time-frequency hard masking based on the sparseness (disjointness) of source spectrograms [4, 5, 12, 15- 17]. If each TF bin
is assigned to one of sources independently [16], the permutation
ambiguity arises as in ICA. To avoid this problem, Otsuka et al. [5]proposed a Bayesian mixture method inspired by latent Dirichlet allocation (LDA) in which each TF bin is exclusively assigned to oneof sources, eachof which is further exclusively assigned to one of directions. The impulse responses measured in an anechoic room canbe used as prior knowledge for joint source separation and localization. This method was extended to a BayesianJactor-mixture model
called NMF-LDA [12] that approximates the power spectrogram ofeach source as a low-rank matrix (weighted sum ofrank-l basis matri ces) for completing missing TF bins assigned to other sources.
An alternative extension is a Bayesian mixture-mixture model called
LDA-LDA [l7] that exclusively assigns each TF bin of the power
spectrogram of each source to one of bases.
MNMF [10, 18] i snon linear ti me-frequency soft masking method.
More specifically, MNMF decomposes the complex spectrograms of
mixture signals into basis spectra, temporal activations, and spatial
correlation matrices. This is aJactor-Jactor model because the multichannel mixture spectrum at each TF bin is modeled as a weighted
sum of source spectra (Le., each TF bin is not assigned to one ofsources), each of which is further modeled as a weighted sum ofbasis spectra. To reduce the initialization sensitivity ofMNMF, various restrictions on the spatial correlation matrix have been proposed.
Kitamura et al. [ll] restricted those to rank-l matrices. Nikunen
et al. [19] calculated those using the geometry of the microphonearray. The main contribution of this paper is to formulate a Bayesian
Jactor-Jactor model to incorporate prior knowledge of a microphonearray into the framework of MNMF as in [5] . A Bayesian model
enables prior knowledge to adapt to the recorded environment.

3. BAYESIAN MNMF
This section explains a Bayesian model of MNMF that decomposes
the complex spectrograms of mixture signals into basis spectra, temporal activations, and spatial correlation matrices in a statistical manner. Bayesian estimation of real spatial correlation matrices based on
those measured in an anechoic room leads to accurate joint source
separation and localization in an arbitrary environment.

Xtj = [Xtjl , ',XtjM]T E C M ,(1)

Ytj = [Ytjl,',YtjK]T E C K ,(2)where Xtjrn and Ytj k respectively denote the complex spectrum of
microphone m and that of source k at time frame t and frequency
f. As is often the case with related work, we assurne that the source
spectrum Ytj k is complex Gaussian distributed as folIows:
Ytjk

I Atjk

K

L

Xtj =

When K sources are observed with M microphones, each TF bin of
the complex spectrograms of observed mixture signals and that of

552

(4)

ajkYtjk,

k=lwhere a j k E C M is a steering vector of source k at frequency f.
We represent spatial directions with D discretized values indexed by
d = 1, ... , D. Note that a steering vector ajd depends on both
direction d and frequency fand it should theoretically be equivalent
to a steering vector a j k of a particular source k because the source
exists at one of the directions. Relaxing this constraint, the steeringvector of the source is represented as the weighted sum of steeringvectors of all directions as folIows:
D

ajk =

L

(5)

Ukdajd ,d= lwhere Ukd is the weight of direction d for source k and {U k d}.f= l is
expected to be sparse to associate source k with one direction.
Using Eqs. (3), (4), and (5), the observation spectrum Xtj isfound to be multivariate complex Gaussian distributed as folIows:
Xtf

I A, S, G

N e. (0,AtjkSkdGil ),(6)

where S k d = U%d and Gil = afda7d is a spatial correlation matrix
for direction d at frequency f.
If the power spectrogram densities {Atj k}of each source
k are assumed to have a low-rank structure, they are decomposed as
L

Atjk =

L

Wktfhklt,

(7)

t=l

where W k tt is the power spectrum density of basis I at frequency fand h k tt is the volume of basis I at time frame t .
Plugging Eq. (7) into Eq. (6), the Iikelihood of the unknown
parameters W , H , S ,and G for the observed data X is given by
Xt f

I W, H, S, G

N e. (0,tWktjhkttS k dG i l ) , (8)For mathematical convenience, the conjugate prior distributions are
put on those model parameters as folIows:
Wktj

Gamma(aü , bü ) ,

Skd
Gfd

(9)
(10)

h k tt3.1. Model rormulation(3)

N c(O, Atjk ) ,where Atjk is the power spectrum density of source k at time framet and frequency f.
Assuming an instantaneous mixing process in the frequency domain, the observation Xtj is represented using source spectra andsteering vectors as folIows:

Gamma(ag , M),

(ll)
(12)

where Wc is the complex Wishart distribution defined by
WC( X lv A) =,IX lv - M exp{ -tr( A - 1 X)}
IAlv 7r M(M - l )/2f(v - m) '_!....--'_ _(13)

where n t/ and <P t/kid are newly-introduced auxiliary variables. The
auxiliary function L is maximized, i.e., the eguality holds, when ntf
and <Ptf kld are given by
(18)

rc

where v :::: M is a degree of freedom and A >- 0 E MXM is ascale matrix. To use prior knowledge about a microphone array, thesteering vectors
f = l are measured for each direction d in an
anechoic room andis set as=+ EI) , whereE > 0 is a small number to makepositive definite.

<Pt /kid = Yt /kld

10gp(X IW , H ,S , G)

Wklf I X ,L (- log IYtfltf(14)

where Xtf = Xf}Xtf and Ytf = 2: kld Ytf kld . To derive a lower
bound L from Eg. (14), we use two inegualities used in [21]. First,
for a convex function j(Z) = - log IZ I (Z C: 0 E MXM ), we
0 by using a first-order
ca1culate a tangent plane at arbitrary n
Taylor expansion as folIows:c:rc- log IZ I :::: - log In l- tr(n - l Z) + M,

(15)

where the eguality holds when n = Z. Second, for a concave functi on g(Z) = -tr( Z-l A) with any matrix A C: 0 , we use thefollowing ineguality:-tr ( (2:{;=1Z k) -lA) :::: -Ktr( zkl<P kA<Pr) ,

(16)

.

(19)

The parameters Wklf , hklt , Skd, and G /d can be sampled fromthe following conditional distributions proportional to the product ofEgs. (9)-(12) and (17):3.2. Bayesian inference
Our goal is to ca1culate the posterior distribution p(W , H, S, G IX)
using the Bayes' theorem p(W , H, S, G IX) =and find optimal parameters that maximize the posterior in practice.
Since p(W, H, S , G IX) is analytically intractable, but a posterior
distribution of each parameter conditioned on the remaining parameters (e.g. , p(WIH , G , S , X) is tractable, we can use Gibbs sampling [20] that alternately and iteratively updates one of the parameters W , H, S, and G according to the conditional posterior distribution by fixing the other parameters.
To derive a tractable conditional posterior of each parameter, we
use a variational approach [21]. Although the conditional posterior
is proportional to the complete joint likelihood given by the productof Egs. (8)- (12), it is difficult to directIy get sampIes from the conditional posterior because of the complicated form of Eg. (8). Therefore, the log-likelihood function defined by Eg. (8) is lower bounded
by a tractable auxiliary function having auxiliary variables. The auxiliary function should become egual to the log-likelihood function
when it is maximized with respect to the auxiliary variables. Suchan auxiliary function can be used as a proxy for the log-likelihood
function. More specifically, letting Ytf kld = Wklf hkltS kd G j d , thelog-likelihood is given by- 1

GIG(a:f, Pk/f , Tklf) 'hklt I(20)(21)

Skd I X,

G fd I X,

fd

GIG(ag, Pkd, Tkd),(22)

MGIGc(vo, R fd, Ufd),

(23)

where
is a set of all parameters exc1uding *. GIG indicates thegeneralized inverse Gaussian distribution [22] and MGIGc indicatesthe complex matrix GIG distribution [23], defined by
2( I) - exp{ (-"y - 1) logx - px - T/X }p'I/
GIG X"( , p,T 2T'I/ 2 K 'I (2#)(24)

MGIGC( X b , R , U) cx: IX I'I - M exp{ -tr(RX + U X-ln , (25)where K 'I is the modified Bessel function of the second kind, "( is areal number, P > 0, T > 0, R >- 0, and U >- O. To draw sampIes
from the GIG and complex MGIG distributions , we use a rejection
sampling method [24] and a Metropolis-Hastings (MH) method [25] ,respectively. In the MH method, we use as a proposal distribution acomplex Wishart distribution whose mode eguals to that of a target
complex MGIG distribution (the mode of an MGIG distribution canbe calculated by using an algebraic Riccati eguation [23]). The conditional posterior parameters P:, T: , Rjd, and Ufd are given by
Pklj = b:f + L hkltSkdtr( G j dnZ/) ,td(G fd <Pt/kld X tf <Ptfkld) ,Tklj = L
td

(26)
(27)

pZlt = b3 + L Wklfskdtr(Gjdnz,/),
fd

(28)

Tk'lt = L Wkl}Skd tr (G /d<PtfkldXtj<Ptfkld) ,
jd

(29)

Pkd = bg + L Wklfhklttr(Gjdnz/),
tfl

(30)

(G /d <Ptfkld Xt / <P tfkld) ,Tkd = Ltfl
R fd =

+L

(31)

(32)

tkl

where {Z k C: o} t'= 1 is a set of arbitrary matrices, {<p k } t'= 1 is a setof auxiliary matrices that sum to the identity matrix (2: k <Pk = I) ,and the eguality holds when <P k = Z k( 2: k' Z k' )- 1.
Using Inegualities (15) and (16), the log-likelihood function givenby Eg. (14) is lower bounded by L as folIows:
10gp(X IW , H , S ,G)

2: L

(-tr

- log Intf l + M)

t/

-

L tr
tfkld

L

(17)

Ujd = L WklfhkltSkdn-;/.
tkl

3.3. Source separation and localization
The multichannel mixture spectrum Xtj over microphones at timeframe t and freguency bin j is decomposed into the sum of multichannel source spectra {Xt fk}L'= l using multichannel Wiener filtering [10] as folIows:
Xtfk =

where Ytj k = 2:ld Ytf kld.

553

(33)

(34)

The source spectrograms of the first channel are transformed
into time-domain source signals using the inverse short-time Fouriertrans form . The direction d( k) of each source k can be estimated byfinding the direction such that the weight S k d of direction d for thesource k is maximized as folIows:
(35)

t

O°

Mic.ll. . .,. 0 . ". 100 cm
M JC.
' 2
· . 4 0----.0
M JC\ 28 cmi\"'0"/
4. EVALUATION

Mic.3

This section reports comparative quantitative experiments conducted
for evaluating the proposed method.

Fig. 2. Positions of microphones and sources.

4.1. Experimental conditions
We synthesized convolutive mixture sounds as test data. Fig. 2 shows
the locations of microphones and sources. Three sources were convoluted using impulse responses measured with 4 microphones in aroom where the reverberation time RT60 was 400 ms. We used music signals (including guitar, bass, vocal, hi-hat, and piano sounds)
and speech signals selected from the SiSEC data set [26] and the
JNAS phonetically balanced Japanese utterances [27]. 30 mixture
signals were used for evaluation: 10 mixtures of music signals, 10
mixtures of speech signals, and 10 mixtures of music and speech
signals. The audio signals were sampled at 16 kHz and a shorttime Fourier transformation was carried out with a 5l2-pt Hanningwindow and a 256-pt shift size. Hyperparameters were determined
experimentally as folIows: L = 20, aü = a3 = aü = bü = b3 =bü = 1, Va = M + 1 ,and E = 0.01. The steering vectors
weremeasured for all directions with an angular interval of 5° (D = 72)
in an anechoic room and they were different from those used to generate the test data.
We compared our method with the standard and state-of-the-art
methods such as IVA [2], MNMF [10], NMF-LDA [12], and LDALDA [17]. The parameters of each method were updated or sampled
200 times. The signal-to-distortion ratio (SDR), signal-to-inferences
ratio (SIR) and signal-to-artifacts ratio (SAR) [28] were used to evaluate the separation performance. We compared the localization performance ofthe proposed method with those ofNMF-LDA [12] andLDA-LDA [17] . We evaluated the localization performance in termsof the average of the absolute localization errors.
4.2. Experimental results
The experimental results are listed in Tables 1, 2, and 3. With all
signal mixtures the SDR was highest for the proposed method, but
in terms of SIR the proposed method was inferior to NMF-LDA andLDA-LDA, which are methods based on TF clustering (hard masking of TF bins). In terms of SAR the proposed method was almost
the same as the conventional MNMF and was better than any of theother conventional methods. The proposed Bayesian extension was
found to work weil because the SDR and SIR for it were higher thanthose of MNMF and the SAR for it was almost the same as that for
MNMF.
The average absolute localization errors are listed in Table 4.
Although the localization errors for the proposed method were larger
than those for the conventional methods, the differences were lessthan 1 0 . We can thus say that the localization performance of the
proposed method is comparable to that of the conventional methods.
The results show that the proposed model could separate andlocalize mixture sounds in an environment where the effective steering vectors are different from those measured in an anechoic room.
This suggests that the proposed model can not only utilize the prior

554

Table 1. Evaluation on music signal mixtures.
SDR
SIR
SAR
3.2 dB
8.1 dB
7.5 dB
Bayesian MNMF
MNMF[lO]
1.0 dB
6.2 dB
6.7 dB
NMF-LDA [12]8.7 dB
3.2 dB
0.5 dB
LDA-LDA [17]0.7 dB
8.7 dB
3.3 dB
IVA [2]0.3 dB
4.9 dB
5.7 dB
Table 2. Evaluation on speech signal mixtures.
SDR
SIR
SAR
6.0 dB
12.6 dB
7.5 dB
Bayesian MNMF
MNMF[lO]
7.7 dB
4.8 dB
10.0 dB
NMF-LDA [12]4.2 dB
14.0 dB
5.2 dB
5.8 dB
17.0 dB
6.3 dB
LDA-LDA [17]
IVA [2]
3.4dB
7.5dB
7.1dB
Table 3. Evaluation on music and speech signal mixtures.
SDR
SIR
SAR
4.9 dB
13.0 dB
6.6 dB
Bayesian MNMF
MNMF[lO]
6.1 dB
1.8 dB
8.6 dB
NMF-LDA [12]1.1 dB
9.8 dB
4.1 dB
LDA-LDA [17]2.8 dB
14.2 dB
3.9 dB
IVA [2]0.1 dB
5.3 dB
5.3 dB
Table 4. Average absolute localization error (degree).
speechmusic
music+speech
Bayesian MNMF
2.00
2.50
2.50
NMF-LDA [12]
1.67
1.67
1.83
LDA-LDA [17]
1.67
1.83
2.00
knowledge on the steering vectors but also adapt them according to
the environment where mixture signals are observed.

5. CONCLUSION
This paper presented a Bayesian extension ofMNMF for audio source
separation and localization that can incorporate prior knowledge about
an environment, microphones, and/or sources. Experimental results
showed that (1) the proposed method achieved better separation performance than the conventional MNMF, (2) its localization performance was comparable to that of conventional methods, and (3) itworked weil in an unknown environment whose acoustic characteristics were significantly different from those of an anechoic room.
With further extensions, it would be possible to estimate thenumber of the sources in a nonparametric Bayesian manner and todevelop an online source separation algorithm. Other future works
include examining the effect of prior learning of basis matrices andcomparing the proposed method with various methods such as otherextensions of MNMF [11, 19].

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