2016 24th European Signal Processing Conference (EUSIPCO)

A Unified Bayesian Model of Time-frequency
Clustering and Low-rank Approximation
for Multi-channel Source Separation
Kousuke Itakura, Yoshiaki Bando, Eita Nakamura, Katsutoshi Itoyama, Kazuyoshi Yoshii
Graduate School of Informatics, Kyoto University, Japan
Abstract—This paper presents a statistical method of multichannel source separation, called NMF-LDA, that unifies nonnegative matrix factorization (NMF) and latent Dirichlet allocation (LDA) in a hierarchical Bayesian manner. If the frequency
components of sources are sparsely distributed, the source spectrograms can be considered to be disjoint with each other inmost time-frequency bins. Under this assumption, LDA has beenused for clustering time-frequency bins into individual sources
using spatial information. A way to improve LDA-based source
separation is to consider the empirical fact that source spectrograms tend to have low-rank structure. To leverage both thesparseness and low-rankness of source spectrograms, our method
iterates an LDA-step (hard clustering of time-frequency bins) thatgives deficient source spectrograms and an NMF-step (low-rankmatrix approximation) that completes the deficient bins of those
spectrograms. Experimental results showed the proposed method
outperformed conventional methods.

I. I NTRODUCTION
Microphone array processing forms the basis of computational auditory scene analysis that aims to understand individual auditory events in a sound mixture. One promising
approach to multi-channel source separation is time-frequency(TF) clustering [1]–[4]. If the frequency components of eachsource are sparsely distributed, as is often the case with harmonic sounds, the source spectrograms can be considered tobe disjoint with each other in most TF bins, i.e., one of thesources is dominant at each bin. This assumption, called Wdisjoint orthogonality [5], is reasonable because the additivityof source spectrograms does not hold exactly and a loud soundmasks softer sounds at each TF bin. Under this assumption,
Otsuka et al. [4] proposed a Bayesian mixture model inspiredby latent Dirichlet allocation (LDA) [6] for clustering TF bins
into sources at the same time as clustering those sources into
different directions. Such unified source separation and localization can circumvent the permutation problem of conventional frequency-domain separation methods such as independent component analysis (ICA) [7].
Mainly in the context of single-channel source separation,
nonnegative matrix factorization (NMF) has gained a lot of
attention [8]. It approximates the power spectrogram of anobserved mixture signal as the product of a basis matrix (a setof basis spectra) and an activation matrix (a set of temporal activations). For multi-channel source separation, multi-channel
extensions of NMF (MNMF) were proposed [9], [10]. MNMF
decomposes the complex spectrograms of mixture signals into
basis spectra, temporal activations, and spatial information. To
reduce the sensitivity of MNMF to parameter initialization,

978-0-9928-6265-7/16/$31.00 ©2016 IEEE

Source 1
Spatialcorrelation
matrixBinary mask

Source 2
Observed spectrograms
Spatialcorrelation
matrixBinary mask

Bases

Activations

Estimated with NMF

Estimated with LDA

Fig. 1. The generative story of NMF-LDA. Source spectrograms are generated
from bases and activations. Masks denote which source is dominant in each
TF bin. The masked source spectrograms are given phases according to spatial
correlation matrices and then gathered to yield observed spectrograms.

Kitamura et al. [11] proposed a method that restricts the spatial
correlation matrices to rank-1 matrices. This rank-1 MNMF
can be considered a model unifying NMF and independentvector analysis (IVA) [12], [13]. Although MNMF can leverage the low-rankness of source spectrograms, their sparsenessis not taken into account because the sources in every TF bin
are allowed to be active simultaneously.
In this paper we propose a hierarchical Bayesian model,
called NMF-LDA, that improves multi-channel source separation by leveraging both the sparseness and low-ranknessof source spectrograms. As illustrated in Fig. 1, the complex
spectrograms of observed multi-channel mixture signals are
generated as follows: the power spectrogram of each sourceis stochastically determined as the product of a basis matrix
and an activation matrix and then the phases of the mixture
signals at each TF bin are stochastically determined according
to the spatial correlation matrix of the dominant source at thatbin. Using Gibbs sampling, the basis and activation matrices
are estimated in the framework of NMF at the same time asa dominant source is identified at each TF bin and the spatial
correlation matrix is estimated in the framework of LDA.
II. R ELATED W ORK
A standard approach to multi-channel source separation is toestimate a linear “unmixing” filter that separates the complex
spectra of mixture signals into those of source signals in thefrequency domain [7], [12]–[14]. Mixture signals are usuallymodeled as the sum of source signals convolved with the impulse responses of the corresponding source directions. This isequivalent to an instantaneous mixing process in the frequency
domain, i.e., the complex spectra of mixture signals are thesum of source spectra multiplied with impulse-response spectra. Using such linearity between mixture and source spectra,

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2016 24th European Signal Processing Conference (EUSIPCO)

ytf = [ytf 1 , · · · , ytf K ] ∈ C .

(1)
(2)

Assuming an instantaneous mixing process in the frequency
domain, the observation xtf is represented as!K
xtf = k=1 af k ytf k ,(3)
where af k is a steering vector of source k at frequency f .

A. Mixture Modeling Approach: Latent Dirichlet Allocation
The LDA-based method [4] simultaneously performs source
separation and localization. Let D be the number of possible
directions (angles), e.g., D = 72 if 360 degrees are discretizedwith an interval of 5 degrees. Using the sparseness of source
spectrograms, Eq. (3) is replaced with
!K,D
xtf = k,d=1 ztf k skd af d ytf k ,(4)where af d ∈ CM is a steering vector of direction d at frequency f . ztf k takes 1 when source k is dominant at frame t
and frequency f and otherwise takes 0. Similarly, skd takes 1
when source k exists in direction d and otherwise takes 0.
A standard way to represent the complex spectrum ytf k ofsource k is to use a complex Gaussian distribution as follows:
ytf k | λtf k ∼ NC (ytf k | 0, λtf k ),(5)

where λtf k is the power spectrum density of source k. Using
Eq. (4) and Eq. (5), the complex spectrum xtf is found tofollow a mixture of complex Gaussian distributions as follows:$#%ztf k skd"K,D$
xtf | λ, G, Z, S ∼ k,d=1 NC xtf $ 0, λtf k G−1
, (6)
fd

Time

Source 2Time

Frequency

K

Observation

Frequency

T

Frequency

xtf = [xtf 1 , · · · , xtf M ]T ∈ CM ,Time

Source 2

Frequency

Let us consider that K sources are observed with M microphones. Each TF bin in the complex spectrograms of observed
and source signals is defined as follows:

Factorial model
Source 1

Observation

Frequency

III. C ONVENTIONAL M ETHODS

Mixture model
Source 1
Frequency

frequency-domain ICA can estimate a linear unmixing filter at
each frequency bin [7]. The permutation of separated source
spectra, however, is not aligned between different frequency
bins. One possibility to solve the permutation ambiguity is
to focus on the directions and inter-frequency correlations ofsources [14]. IVA [12], [13] is an extension of ICA that canjointly deal with all frequency components in a vectorial manner. These methods can be used only under a determinedor overdetermined condition that the number of sources isequal to or less than that of microphones. A promising way
to cope with the underdetermined condition is to focus on thesparseness of source spectra [15].
Another popular approach to source separation is to perform
nonlinear time-frequency masking based on the sparseness(disjointness) of source spectra [1]–[4]. If each TF bin is
independently classified into one of sound sources [3], thepermutation ambiguity arises as in ICA. To solve this problem,
Otsuka et al. [4] proposed a statistical method that jointly
clusters each TF bin into different sources and directions. Thismethod can work well under an underdetermined condition.

Time

Time

Time

Fig. 2. Comparison of a mixture model and a factorial model. In the mixture
model, one of sources is stochastically selected at each TF bin. In the factorialmodel, all sources are accumulated in a weighed manner at each TF bin.
H
where G−1
f d = af d af d is a spatial correlation matrix for direction d at frequency f .

B. Factorial Modeling Approach: Multichannel NMF
Multichannel NMF (MNMF) [9] can perform blind source
separation. Unlike the LDA-based method, all sources are assumed to be activated in a weighted manner at every TF bin.
Assuming Eq. (5) as in the LDA-based method and using
Eq. (3), the complex spectrum xtf is found to follow a complex Gaussian distribution as follows:
$ !
#
%
$
K
xtf | λ, G ∼ NC xtf $ 0, k=1 λtf k G−1
(7)
fk ,
H
where G−1
f k = af k af k is a spatial correlation matrix forsource k at frequency f .
A key idea of MNMF is to decompose the power spectrum
density λtf k of source k by using a low-rank approximation technique (e.g., NMF and canonical polyadic decomposition [16]) as follows:!L
λtf k = l=1 ulk wlf hlt ,(8)where wlf is the power of the l-th basis spectrum at frequency
f , hlt is the volume of the l-th basis at frame t, !
and ulk is aK
contribution of the l-th basis to source k such that k=1 ulk =1. Comparing the likelihood of LDA given by Eq. (6) with thatof MNMF Eq. (7), we can see the clear difference between
mixture and factorial modeling approaches (Fig.2).
IV. P ROPOSED M ETHODThis section explains a hybrid of factorial and mixture models called NMF-LDA that integrates TF clustering based on
LDA with low-rank approximation of source spectrograms
based on NMF (Fig. 1). Our model was inspired by the LDAbased method [4] and MNMF [9]. The sparseness of source
spectrograms justifies an assumption that only one of the sources
is activated at each TF bin. This calls for mixture modeling thatstochastically chooses one of sources at each bin. Using the
low-rankness of source spectrograms, the power spectrogramof each source is approximated by the product of basis spectra
and temporal activations in a way of factorial modeling.
A. Model Formulation
The proposed model consists of LDA and NMF parts thatare integrated in a hierarchical Bayesian manner.

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2016 24th European Signal Processing Conference (EUSIPCO)

1) LDA Part: According to the LDA-based model describedin Section III-A, the likelihood function for the observation
xtf is given by Eq. (6). The total likelihood function over all
frames and all frequencies is thus given by"T,F,K,D
ztf k skd
X|λ, G, Z, S ∼ t,f,k,d=1 NC (xtf |0, λtf k G−1
. (9)
fd )

Low-rank approximation

NMF
Multi-channel inputs

LDA Time-frequency clustering

As in the LDA-based method [4], the latent variables Z and
S are drawn from the following categorical distributions:ztf | πt ∼ Categorical(ztf | πt ),(10)

sk | φ ∼ Categorical(sk | φ).

(11)πt ∼ Dirichlet(πt | aπ0 1K ),(12)For mathematical convenience, conjugate prior distributions
are put on the model parameters π, φ, and G as follows:φ ∼ Dirichlet(φ | aφ0 1D ),Gf d ∼ WC (Gf d | ν, G0f d ),(13)

hklt ∼ Gamma(hklt | ah0 , bh0 ),NMF
Completed source
spectrograms

(16)
(17)

Deficient source
spectrograms

Fig. 3. Overview of the proposed iterative optimization algorithm.

1) Updating LDA Part: G, Z, and S are alternately sampled from conditional posterior distributions given by
Gf d | X, Θ¬Gf d ∼ WC (Gf d | νf′ d , G′f d ),(14)

where 1N is a N -dimensional vector with all entries one andWC is a complex Wishart distribution(see Appendix). Notethat estimation of source indicators Z over all TF bins andestimation of direction indicators S over K sources correspondto source separation and localization, respectively.
2) NMF Part: The power spectrum density λtf k in Eq. (9)is factorized in a similar way to MNMF described in Section
III-B except for two ways. Unlike Eq. (8), λtf k is considered
as a random variable drawn from a gamma distribution witha factorized scale parameter as follows:
$
&
'
$
α
$
λtf k | Wk , Hk ∼ Gamma λtf k $ α, !L, (15)$
l=1 wklf hklt
!L
where E[λtf k |Wk , Hk ] = l=1 wklf hklt and α is a hyperparameter controlling how likely λtf k is to be close to an exact!L
low-rank structure (λtf k → l=1 wklf hklt when α → ∞).
Another modification is that the power spectrum density ofeach source k is represented with a unique set of basis spectra unlike Eq. (8). These modifications enable our model toflexibly represent a wide variety of sound spectrograms.
For mathematical convenience, conjugate prior distributions
are put on the model parameters W and H as follows:w
wklf ∼ Gamma(wklf | aw
0 , b0 ),Observed mixture
spectrograms′
ztf | X, Θ¬ztf ∼ Categorical(ztf | πtf),′
sk | X, Θ¬sk ∼ Categorical(sk | φk ),

(19)
(20)

where Θ is a set of all parameters and Θ¬∗ indicates a set of
all parameters excluding ∗. The conditional posterior parame′ters νf′ d , G′f d , πtf, and φ′k are given by!T,K
νf′ d = ν + t,k=1 ztf k skd ,(21)
H!
T,K x x0 −1
G′−1
+ t,k=1 tfλtftfk ztf k skd ,(22)
f d = (Gf d )
($
)
*+skd
$
xH
$ Gf d $¬tf "D′πtf Gf d xtf
πtf,
k = (a0 + ntk )d=1 $ λtf k $ exp −
λtf k
(23)
($
)
*+ztf k$"T,FxH
$ Gf d $tf Gf d xtf
φ′kd = (aφ0 + c¬k,d )t,f =1 $ λtf k $ exp −
λtf k
(24)

where n¬tf
tk indicates the number of TF bins assigned to source
k without a sample at frame t and frequency f , and c¬kd is thenumber of sources assigned to direction d without source k2) Updating NMF Part: λ, W , and H are alternately sampled by using a non-collapsed Gibbs sampler and a MetropolisHastings (MH) algorithm [17]. Since the gamma distribution(Eq. (15)) is a special case of the generalized inverse Gaussian(GIG) distribution [18] (see Appendix) and the GIG distribution is a conjugate prior for the Gaussian distribution (Eq. (9)),λ can be analytically sampled as follows:

where a∗0 and b∗0 are hyperparameters.

λtf k | X, Θ¬λtf k ∼ GIG(λtf k | γtf k , ρtf k , τtf k ),B. Posterior Inference

(18)

(25)

where γtf k , ρtf k , and τtf k are given by

Our goal is to find optimal model parameters such that theposterior probability p(G, Z, S, π, φ, λ, W , H|X) is maximized. Since the true posterior is analytically intractable, we
use partially-collapsed Gibbs sampling after marginalizing out
π and φ. As shown in Fig. 3, LDA and NMF are iterated until
the likelihood converges. G, Z, and S are updated with LDA
and λ, W , and H are updated with NMF. Given the optimal
parameters, frequency-domain source signals can be recovered
using multichannel wiener filtering [9].

γtf k = α − M ztf k , ρtf k = !Lτtf k =

!Dα

l=1 wkf l hklt

d=1 xtf Gf d xtf ztf k skd .

,

(26)
(27)

Since the true conditional posterior distributions of W and
H are analytically intractable, the MH algorithm [17] is used
for sampling W and H instead of Gibbs sampling. The key ofthe MH algorithm is the design of a proposal distribution thatstochastically generates a candidate of a next sample based on2282

2016 24th European Signal Processing Conference (EUSIPCO)a previous sample. If the proposal is close to the true posterior,the candidate is set to the next sample with a high acceptanceratio. Otherwise, the previous value is set to the next sample.
In this study, approximated posterior distributions of W and
H are calculated by variational Bayesian (VB) inference andthen used as proposal distributions for MH sampling.
Although the GIG distribution can be a conjugate prior on
the scale parameter of the gamma distribution, neither Eq. (16)nor Eq. (17) has no direct conjugacy with Eq. (15) because thesum operation is involved in the scale parameter. The complete
log-likelihood given by Eq. (15), Eq. (16), and Eq. (17) isthus lower bounded by an auxiliary function L, using Jensen’s
inequality and a first-order Taylor approximation in the same
way as Bayesian NMF [19] as follows:!T,F,K!Lψ
L = − t,f,k=1 λtf k l=1 wkflktf− log(ωktf )l hklt!L wkf l hklt
q(wkf l )q(hklt )− l=1 ωktf − log p(wkf l )p(hklt ) ,(28)
where φlktf and ωktf are auxiliary variables. The original loglikelihood can be recovered by maximizing the lower bound
L with respect to φlktf and ωktf as follows:
wkf l hklt
ψlktf = !L,
l=1 wkf l hklt

ωktf =!L

l=1 wkf l hklt .

(29)

Since the GIG-gamma conjugacy is recovered in L, the variational posterior distributions of W and H used as proposal
distributions for MH sampling are obtained as follows:ww
q(wkf l ) = GIG(wkf l | aw
0 , ρkf l , τkf l ),(30)hq(hklt ) = GIG(hklt | ah0 , ρhklt , τklt
),

(31)

w
h
h
where ρw
kf l , τkf l , ρklt and τklt are given byw
ρw
kf l = b0 +

ρhklt = bh0 +

!T

hklt
t=1 ωktf ,

w
τkf
l =

wkf l
f =1 ωktf ,

h
τklt
=

!F

!T

t=1

!F

2
ψlktf
λtf k

hklt
2
ψlktf
λtf k

f =1

wkf l

,

(32)

.

(33)

V. E VALUATIONThis section presents source separation results obtained withsimulated convolutive mixture signals. The experiments wereconducted in underdetermined conditions (the number of microphones M < the number of sources K) and overdetermined
conditions (M > K). The proposed NMF-LDA was comparedwith IVA [12], MNMF [9] and the LDA-based method (simply
written as LDA in this paper) [4] in overdetermined conditions
and with MNMF and LDA in underdetermined conditions
because IVA can not be used in underdetermined conditions.
Although the original model of LDA [4] can estimate the number of sources, for fair evaluation we conducted experiments
under the condition that the number of sources is fixed. Morespecifically, Eq. (15) is replaced with:p(λtf k ) = Gamma(λtf k |1, 1).

(34)

A. Experimental Conditions
Figure 4 shows the locations of microphones and sources.
Three sources were convoluted using impulse responses measured in a room where the reverberation time RT60 was 400

0Mic. 11008080Mic. 428

Mic. 2

Mic. 3Fig. 4. Locations of microphones and sources.

ms. The number of microphones M was 2 or 4; mic. 2 andmic. 4 (shown in black) were used when M = 2 and all
microphones were used when M = 4. 30 mixtures were used
for evaluation; 10 were mixtures of music signals (including
guitar, bass, vocal, hi-hat, piano sounds), 10 were mixtures ofspeech signals, and 10 were mixtures of music and speech
signals. The music and speech signals were selected fromthe SiSEC data set [20] and the JNAS phonetically balanced
Japanese utterances [21], respectively. The audio signals were
sampled at 16000 Hz and a short-time Fourier transformation
was carried out with a 512 pt Hamming window and a 256
pt shift size. The steering vectors âf d were measured in an
anechoic room such that D = 72 with 5◦ resolution. Hyperparameters were set as follows: ν = M , G0f d = (âf d âH
fd +wh0.01 × I)−1 , L = 20, aπ0 = aφ0 = 10, aw
=
b
=
a
0
0
0 = 1,
bh0 = L, α = 10. Signal-to-distortion ratio (SDR) [22] wasused to evaluate separation performances. The larger the SDR,
the better the separation performance.
B. Experimental Results
Figures 5 and 6 show the SDR improvements. In all conditions, NMF-LDA achieved best separation performance of allcompared methods. For speech data, NMF-LDA slightly outperformed LDA. For music data, although LDA was inferior to
MNMF, NMF-LDA achieved better performance than MNMF.
This indicates that the low-rankness of music spectrograms ismuch stronger than that of speech spectrograms.
Tables I and II show the average SDR improvements. The
average SDR improvement obtained by NMF-LDA was at
most 3.3 dB greater than that obtained by LDA. While inthe case of two microphones NMF-LDA archived much better
performance than LDA, in the case of four microphones theperformance difference was small. This indicates that the lowrankness of source spectrograms effectively helps estimate thepower spectrogram density when a fewer number of microphones are available.
Although steering vectors are required for setting appropriate prior distributions on spatial correlation matrices in NMFLDA (Eq. (14)) as in LDA, this is not a problem in practice.
The experimental results showed that NMF-LDA works wellin an unknown environment whose acoustic characteristics are
significantly different from those of an anechoic room.
VI. C ONCLUSION
This paper presented a hierarchical Bayesian model of multichannel source separation that combines LDA and NMF con-2283

2016 24th European Signal Processing Conference (EUSIPCO)SDR improvement (dB)

MNMF

LDA

GIG(y|γ, ρ, τ ) =NMF-LDAexp{(γ − 1) log y − τ y − τ /y},√2τ γ/2 Kγ (2 ρτ )(36)

where Kγ is a modified Bessel function of the second kind.
ACKNOWLEDGMENT

Music data

Music and speech data

Speech data

R EFERENCES

Fig. 5. SDR improvements in the case of two microphones.
SDR improvement (dB)

IVAMusic data

MNMF

LDA

NMF-LDA

Speech data

Music and speech data

Fig. 6. SDR improvements in the case of four microphones.
TABLE IAVERAGE SDR IMPROVEMENTS IN THE CASE OF TWO MICROPHONES .
MNMFLDANMF-LDAMusic
5.7 dB
3.2 dB
6.5 dB

Speech
5.1 dB
6.0 dB
6.7 dB

Music and speech
6.6 dB
5.7 dB
7.5 dBTABLE IIAVERAGE SDR IMPROVEMENTS IN THE CASE OF FOUR MICROPHONES .
IVA
MNMF
LDA
NMF-LDAMusic
5.7 dB
6.4 dB
6.3 dB
7.5 dB

Speech
7.8 dB
9.2 dB
10.0 dB
10.1 dB

Music and speech
7.1 dB
8.8 dB
9.5 dB
9.5 dBsidering both the sparseness and low-rankness of source spectrograms. A dominant source at each TF bin is identified andthe spatial correlation matrix for each source is estimated in theframework of LDA. The power spectrogram of each source isdecomposed into basis spectra and temporal activations, which
are estimated in the framework of NMF. Experimental results
showed that the proposed method achieved better source separation performance than conventional methods.
To estimate the number of sound sources as the same time as
optimizing the number of basis spectra, we plan to formulate
a nonparametric Bayesian extension of NMF-LDA based on
the Dirichlet, gamma, and/or beta processes. Another possible
extension is to allow spatial correlation matrices to smoothly
vary over time for dealing with moving sound sources. In thiscase, the NMF part is expected to help because basis power
spectra are scarcely affected by the locations of sound sources.
To track moving sources in real time, we also plan to develop
an efficient algorithm of online Bayesian inference.
A PPENDIX
The probability density function of the complex Wishart
distribution and that of the GIG distribution are given by:
WC (G|ν, G0 ) =This study was partially supported by Grant-in-Aid for Scientific Research Nos. 24220006 and 15K12063.

|G|ν−M exp(−tr(G(G0 )−1 )), (35)"M −1|G0 |ν π M (M −1)/2 m=0 Γ(ν − m)

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