EFFICIENT MULTICHANNEL NONNEGATIVE MATRIX FACTORIZATION EXPLOITINGRANK-1 SPATIAL MODEL
Daichi Kitamura 1 , Nobutaka Ono 2, 1 , Hiroshi Sawada 3 , Hirokazu Kameoka 3, 4 , Hiroshi Saruwatari 4
1

The Graduate University for Advanced Studies (SOKENDAI), Kanagawa, Japan
2
National Institute of Informatics, Tokyo, Japan
3
Nippon Telegraph and Telephone Corporation, Tokyo, Japan
4
The University of Tokyo, Tokyo, Japan
ABSTRACTThis paper proposes a new eﬃcient multichannel nonnegative matrix factorization (NMF) method. Recently, multichannel NMF(MNMF) has been proposed as a means of solving the blind source
separation problem. This method estimates a mixing system ofsources and attempts to separate them in a blind fashion. However,this method is strongly dependent on its initial values because there
are no constraints in the spatial models. To solve this problem, we
introduce a rank-1 spatial model into MNMF. The proposed method
estimates a demixing matrix while representing sources using NMF
bases and can be optimized by the update rules of independent vector
analysis and conventional single-channel NMF. Experimental results
show the eﬃcacy of the proposed method in terms of robustness andconvergence speed.
Index Terms— blind source separation, nonnegative matrix factorization, independent vector analysis, rank-1 spatial model
1. INTRODUCTION
Blind source separation (BSS) is a technique for separating specificsources from a recorded sound without any information about therecording environment, mixing system, or source locations. In a
determined or overdetermined situation (number of microphones≥ number of sources), independent component analysis (ICA) [1]is the method most commonly used to solve the BSS problem,and many ICA-based techniques have been proposed [2]–[6]. On
the other hand, for an underdetermined situation (number of microphones < number of sources) including monaural recording,nonnegative matrix factorization (NMF) [7] has received much attention [8, 9]. BSS is generally used to solve speech separation
problems, but recently the use of BSS for music signals has also
become an active research area [10].
As a means of solving the permutation problem [11]–[13] intime-frequency domain ICA, independent vector analysis (IVA) [14]has been proposed. Such ICA-based methods assume independence
between the sources in order to estimate a demixing matrix. However, if the observed signal frequently contains co-occurring sources,
such as music signals, these methods cannot separate them with highaccuracy because the independence between the sources is weakened.
In NMF-based methods, the decomposed bases (spectral patterns) must be clustered so as to represent the specific sources. Oneeﬀective way of achieving this is to utilize a sample sound of the target signal [15, 16]. However, such supervision cannot be utilized in
BSS. To solve this problem, multichannel NMF (MNMF) has been

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proposed [17]–[22]. In particular, MNMF methods [20]–[22] estimate a mixing system of the sources via spatial covariance matrices,which is utilized for source clustering.
In [17]–[19], an instantaneous mixture in the time domain isassumed, and only the observed gain of each microphone is modeled by extending NMF to nonnegative tensor factorization. For
a convolutive mixture, we must treat complex values in the timefrequency domain. MNMF methods [20, 21] separately model the
mixing system (as spatial covariance matrices) and the sources (using the NMF representation). These methods can be considered
as a multichannel extension of NMF. Update rules based on the
expectation-maximization algorithm have been derived. In [22], aunified MNMF scheme was proposed. This method employs Hermitian positive definite covariance matrices to model both the spatialand source components, and multiplicative update rules have been
derived. However, it was reported that the algorithms [20]–[22] weresensitive to the initial values in source separation tasks.
In this paper, we propose a new eﬃcient MNMF method with arank-1 spatial model. Instead of estimating the mixing system, theproposed method estimates the demixing matrix while representing
the sources using NMF bases. In addition, the proposed method canbe optimized using the fast update rules of IVA and single-channel
NMF. This fact enabled us to reveal the relationship between IVA
and MNMF by analysis. The eﬃcacy of the proposed method is
confirmed experimentally.
2. CONVENTIONAL METHODS
2.1. Formulation
Let the numbers of sources and microphones (channels) be N andM, respectively. The multichannel source and the observed and separated signals in each time-frequency slot are described as
si j = (si j,1 · · · si j,N )t ,xi j = (xi j,1 · · · xi j,M )t ,yi j = (yi j,1 · · · yi j,N )t ,

(1)
(2)
(3)

where i = 1, . . . , I; j = 1, . . . , J; n = 1, . . . , N; and m = 1, . . . , M are
the integral indexes of the frequency bins, time frames, sources, andchannels, respectively, t denotes the vector transpose, and all the entries of these vectors are complex values. When the window sizein a short-time Fourier transform (STFT) is suﬃciently long compared with the impulse responses, we can approximately represent
the observed signal as
xi j = Ai si j ,(4)

ICASSP 2015

where Ai = (ai,1 · · · ai,N ) is an M × N mixing matrix and ai,n is thesteering vector for each source. In the determined situation (M = N),
we can define the demixing matrix Wi = (wi,1 · · · wi,M )h , and theseparated signal is represented as
y i j = Wi x i j ,Sourcesignals

MixingsystemObserved
signals

Demixing
systemSpatialcovariance Latent
Separated matrices variable
signals

Bases andactivations(5)hwhere denotes the Hermitian transpose.
2.2. IVA

Fig. 1. Conceptual model of MNMF (N = M = 2).

IVA is one of the techniques used to solve the permutation problem[13] and can be applied only in the overdetermined situation (M ≥N). For simplicity, let M be equal to N. In this method, we define
the source component as a vector that consists of all frequency bins,
given asy j,m = (y1 j,m · · · yI j,m )t .

3. PROPOSED METHOD(6)

IVA can be used to estimate the demixing matrix Wi by assuming
both independence between the sources (vectors) and a higher-ordercorrelation between the frequency bins in each source [14]. The costfunction of IVA is defined as follows:∑ 1∑∑
QIVA (W ) = m
G(y j,m ) − i log | det Wi |,(7)
J j
where J is the number of time frames and G(y j,m ) is a contrast function. When y j,m obeys a probabilistic density function p(y j,m ), thecontrast function G(y j,m ) is given as − log p(y j,m ). In IVA, G(y j,m ) =∥y j,m ∥2 is often used [14], which assumes a spherical Laplace distribution for the source prior, where ∥ · ∥2 denotes the L2 norm. For theminimization of (7), fast and stable update rules, which are derived
by an auxiliary function technique, have been proposed [23].
2.3. MNMF
MNMF is a natural extension of simple NMF for multichannel signals [22]. This method can be applied even in the underdeterminedsituation (M < N). The observed signal is represented as
Xi j = xi j xhi j ,where constant terms are omitted. Update rules to minimize (10)have been derived by the auxiliary function technique [22].

3.1. Motivation and strategy
The separation performance of IVA is degraded for music signals
because the independence between the sources is weakened by thefrequent co-occurrence. In addition, IVA assumes that all the frequency bins have the same time-varying gains. Therefore, IVA is notsuitable for signals that have harmonic structures. MNMF has problems in terms of the convergence speed and its strong dependence on
its initial values. This is because a huge number of variables shouldbe optimized using only one cost function, and because there is noconstraint for optimizing the spatial covariance matrices H.
To solve these problems, we propose a new eﬃcient MNMF
method that employs a rank-1 approximation for the spatial covariance matrices H. In this approach, similarly to in IVA, we assume
the determined situation and the linear time-invariant mixing system described by (4) to restrict the flexibility of the spatial model.
In addition, instead of estimating H, the proposed method estimates
the demixing matrix Wi while representing the sources using NMF
bases.
3.2. Derivation of cost function(8)

where Xi j is a Hermitian positive definite matrix of size M × M.
The diagonal elements of Xi j represent real-valued powers observedby the microphone, and the nondiagonal elements represent thecomplex-valued correlations between the microphones. The decomposition model of MNMF is expressed as
∑ (∑
)
(9)
Xi j ≈ X̂i j = k n Hi,n znk tik vk j ,where k = 1, . . . , K is the integral index of the spectral bases, andHi,n is an M×M Hermitian positive definite matrix, which comprises
the spatial covariance for each frequency i and source n. In addition,znk ∈ R[0, 1] is a latent variable that indicates whether the kth∑basis belongs to the nth source (znk = 1) or not (znk = 0) and satisfies n znk = 1;
tik ∈ R≥ 0 and vk j ∈ R≥ 0 are the elements of the basis matrix T ∈ R≥I ×K
0
and activation matrix V ∈ R≥K0× J . Figure 1 shows a conceptual modelof MNMF. In BSS, the mixing and demixing systems are unknown.
MNMF decomposes the observed signal into T and V and simultaneously optimizes the spatial covariance matrices H that correspondto each sources. Then, the sources are separated by associating thesevariables T and V with H by using a cluster-indicator latent variable Z ∈ R≥N0×K . The cost function based on Itakura-Saito divergenceis defined as [22]]∑ [−1
QMNMF = i, j tr(Xi j X̂i j ) + log det X̂i j ,
(10)

277

If we assume that the mixing system shown in Fig. 1 is represented
by the mixing matrix Ai = (ai,1 · · · ai,n ) appearing in (4), the spatial
covariance matrix Hi,n can be approximated by a rank-1 matrix thatis an outer product of the steering vector ai,n as follows:
Hi,n = ai,n ahi,n .

(11)

To introduce the rank-1 approximation into MNMF, we substitute(11) into (9) and reformulate X̂i j using the mixing matrix Ai as follows:)∑ (∑X̂i j = k n ai,n ahi,n znk tik vk j∑∑
= n ai,n ahi,n k znk tik vk j
= Ai Di j Ahi ,

(12)

where
(

)

Di j = diag di j,1 , . . . , di j,N ,(13)

di j,n =(14)∑

k znk tik vk j .

By substituting (12) into the MNMF cost function (10), we obtain
]
)
( )−1
∑ [ (
−1
+ log det Ai Di j Ahi . (15)D−1
Q = i, j tr xi j xhi j Ahi
i j AiTable 2. Experimental conditions

Table 1. Music sources
ID
Song
Source (1/2/3)
1bearlin-roads snip 85 99
acoustic guit main/bass/vocals
2 fort minor-remember the name snip 54 78
drums/violins synth/vocals
3
ultimate nz tour snip 43 61
guitar/synth/vocals
E2A impulse
Source 2
response from
Source 1
RWCP database
Reverberationtime: 300 ms

Number of bases K
Number of iterations

Source 3

Downsampled from 44.1 kHz to 16 kHz
512 ms
128 ms
Proposed method 1: L = 30 (K = 90)
MNMF+MWF and Proposed method 2: K = 90
200

Gaussian distributions in each time-frequency slot [24], similarly toconventional MNMF.

2m
20
50

Sampling frequency
FFT length
Window shift70

3.4. Update rules

2.83 cm 2.83 cm

Fig. 2. Recording conditions of room impulse response.
In the overdetermined situation (we let M equal N for simplicity), thedemixing matrix Wi exists and we can transform the variables, i.e.,the observed signal xi j and the mixing matrix Ai , to the separated
signal yi j = Wi xi j and the demixing matrix Wi = A−1
i respectively,
as follows:
)
(
)−1
∑ [ (
Q = i, j tr Wi−1 yi j yihj Wih Wih D−1
i j Wi()()]+ log (det Ai ) det Di j det Ahi
)
(
)−1
∑ [ (
= i, j tr Wi Wi−1 yi j yihj Wih Wih D−1ij]+ 2 log | det Ai | + log det Di j
]
)
∑ [ (
∑
= i, j tr yi j yihj D−1
m log di j,m
i j − 2 log | det Wi | +][|yi j,m |2∑∑∑ ∑− 2 log | det Wi | + m log k zmk tik vk j .
= i, j m ∑k zmk tik vk j
(16)
In the conventional MNMF, the separated signal is obtained by clustering H, T , and V into specific sources using a latent variable Z.
The proposed method estimates the demixing matrix Wi to obtainthe separated signal yi j , where we approximately decompose yi j into
zmk , tik , and vk j in each iteration.
3.3. Relationship between IVA and MNMF
The first and second terms in the cost function (16) are equivalent
to the IVA cost function (7), and the first and third terms in (16) areequivalent to a single-channel NMF cost function described as[]|yi j |2∑∑
(17)
QNMF = i, j ∑+ log l til vl j ,l til vl j
where l = 1, . . . , L indicates the basis index. This fact reveals therelationship between IVA and MNMF, namely, MNMF with a rank1 approximation, which assumes a linear time-invariant mixing
system in the time-frequency domain, is essentially equivalent to
IVA with a basis decomposition model. Therefore, the proposed
method can be considered as an intermediate model between IVA
and MNMF in terms of the degree of freedom. From the IVA
side, we introduced the basis decomposition model with NMF tocapture the actual spectral patterns, and from the MNMF side, anapproximation for the spatial model was introduced to make theoptimization more eﬃcient. However, the source priors of IVA
and the proposed method are diﬀerent. IVA generally assumes thespherical Laplace distribution, which has the same variance for all
frequency bins, as the source prior by setting G(y j,m ) = ∥y j,m ∥2 in(7). The proposed method with (16) assumes independent complex

278

If we eliminate zmk in (16), the diﬀerentials ∂Q/∂tik and ∂Q/∂vk j become identical to ∂QNMF /∂til and ∂QNMF /∂vl j , respectively. Therefore, when zmk ∈ {0, 1} and all the sound sources are modeled by thesame number of bases L (namely, L×M = K), the update rules of til,mand vl j,m are the same as those of single-channel NMF, i.e.,vuuut ∑ |y |2 v (∑ ′ t ′ v ′ )−2
l j,mj i j,ml il ,m l j,mtil,m ← til,m
(18)
(∑
)−1 ,
∑
j vl j,ml′ til′,m vl′ j,mvu∑uut |y |2 t (∑ ′ t ′ v ′ )−2
i i j,m il,ml il ,m l j,mvl j,m ← vl j,m
(19)
(∑
)−1 ,
∑
i til,ml′ til′,m vl′ j,m∑(20)
ri j,m = l til,m vl j,m ,where ri j,m is an estimated variance of each source [24]. Alternatively, if we employ a continuous-valued zmk to cluster the bases into
specific sources, we can derive the following update rules of zmk , tik ,and vk j by minimizing (16) by the auxiliary function technique;vuuut ∑ |y |2 t v (∑ ′ z ′ t ′ v ′ )−2
k mk ik k j
i, j i j,m ik k j
zmk ← zmk
(21)
(∑
)−1 ,
∑
i, j tik vk j
k′ zmk′ tik′ vk′ jvuuut ∑ |y |2 z v (∑ ′ z ′ t ′ v ′ )−2
mk k jj,m i j,mk mk ik k j(22)
tik ← tik
(∑
)−1 ,
∑
j,m zmk vk j
k′ zmk′ tik′ vk′ jvuuut ∑ |y |2 z t (∑ ′ z ′ t ′ v ′ )−2
mk ik
i,m i j,mk mk ik k j(23)
vk j ← vk j
)−1 ,
(∑
∑
k′ zmk′ tik′ vk′ ji,m zmk tik∑(24)
ri j,m = k zmk tik vk j ,∑∑where we calculate zmk ← zmk / m′ zm′k to ensure m zmk = 1 at each
iteration.
The demixing matrix Wi can be optimized by the update rule of
IVA because the diﬀerentials ∂Q/∂Wi and ∂QIVA /∂Wi are equivalent. Fast and stable update rules for IVA have been derived as [23]1∑ 1
xi j xhi j ,J j ri j,m()
wi,m ← Wi Vi,m −1 em ,()− 1hwi,m ← wi,m wi,mVi,m wi,m 2 ,Vi,m =

hxi j ,
yi j,m ← wi,m

(25)
(26)
(27)
(28)

where em denotes the unit vector with the mth element equal to unity.
We estimate all the variables that minimize (16) by iterating(18)–(20) or (21)–(24) and (25)–(28) alternately. Note that the scale

6
5
4
3
2
1
0

Source 1 Source 2 Source 3

22
20
18
16
14
12
10
8
6
4
2
0

(b)

Source 1 Source 2 Source 3

Proposed method 2
14
12
10
8
6
4
2
0
-2
-4

12
SDR improvement [dB](a)

SIR improvement [dB]SDR improvement [dB]7

Proposed method 1

MNMF+MWF

SAR [dB]

IVA
8

10

(c)

8
6
IVA
MNMF+MWF
Proposed method 1
Proposed method 2

4
2
0
-2

0

100

Source 1 Source 2 Source 3

Fig. 3. Average scores for ID1 data: (a) SDR improvement, (b) SIR
improvement, and (c) SAR.

200
Iteration steps

300

400

Fig. 6. Example of SDR convergence: result for Source 1 in ID3.
Table 3. Computational times for separation of ID1 (s)

12

28

(a)

SIR improvement [dB]SDR improvement [dB]

14

10
8
6
4
2
0

Proposed method 1

MNMF+MWF
24

(b)

12

(c)

16
12

8
6

8

4

4

2
0

Source 1 Source 2 Source 3

Source 1 Source 2 Source 3

Fig. 4. Average scores for ID2 data: (a) SDR improvement, (b) SIR
improvement, and (c) SAR.

(a)

10
8
6
4
2
0

Source 1 Source 2 Source 3

36
32
28
24
20
16
12
8
4
0

(b)
SAR [dB]SDR improvement [dB]12

Proposed method 1

MNMF+MWF
SIR improvement [dB]

IVA
14

IVA
91.6

MNMF+MWF
4498.4

Proposed method 1
121.0

Proposed method 2
173.4

10

20

0

Source 1 Source 2 Source 3

Proposed method 2
14

SAR [dB]

IVA

Source 1 Source 2 Source 3

Proposed method 2
18
16
14
12
10
8
6
4
2
0

(c)

Source 1 Source 2 Source 3

Fig. 5. Average scores for ID3 data: (a) SDR improvement, (b) SIR
improvement, and (c) SAR.
ambiguity exists between Wi and ri j,m because both of them can determine the scale. Therefore, the estimated variance ri j,m has a riskof diverging. To avoid this problem, we normalize Wi and ri j,m ateach iteration. The signal scale can be restored by applying a backprojection technique [12] after the optimization.
4. EXPERIMENT
4.1. Conditions

from RWCP database [26] with each source. The other experimental conditions are described in Table 2. As the evaluation scores, weused the signal-to-distortion ratio (SDR), source-to-interference ratio (SIR), and sources-to-artifacts ratio (SAR) defined in [27]. SDR
indicates the total separation performance, SIR indicates the degreeof separation, and SAR indicates the absence of artificial distortion.
4.2. Results
Figures 3–5 show the average scores and their deviations in 10 trials with various initializations. From these results, we confirm that
IVA cannot achieve satisfactory separation because this method is
not suitable for music signals. MNMF+MWF gives slightly better
performance than IVA, and its maximum scores are comparable tothe average scores of the proposed methods. However, owing to thelack of robustness, the error bars are relatively large. The proposed
methods achieve good and stable performance, which is particularly
evident in Figs. 4 and 5. This is because (a) the spectral patterns areeﬀectively modeled by the decomposition of bases compared with
IVA, and (b) the optimization of the demixing matrix using the IVA
update rules results in a stable separation performance.
Figure 6 shows an example of SDR convergence for eachmethod. Both IVA and the proposed methods show much fasterconvergence and better results than MNMF+MWF. The actual computational times in this experiment are shown in Table 3, where thecalculations were performed using MATLAB 8.3 (64-bit) with anIntel Core i7-4790 (3.60 GHz) CPU. The computational times of the
proposed methods are less than twice that of IVA. MNMF requires alonger computational time because of the eigenvalue decomposition
for each Hi,n .
5. CONCLUSION

To confirm the eﬃcacy of the proposed method, we conducted anevaluation experiment using professional music signals. In this experiment, we compared four methods, namely, IVA [23], MNMF
with a multichannel Wiener filter (referred to as MNMF+MWF)[22], Proposed method 1 (update using (18)–(20) and (25)–(28)),
and Proposed method 2 (update using (21)–(24) and (25)–(28)). Proposed method 1 models all the sources with the same fixed numberof bases, L. In Proposed method 2, we only set the total numberof bases, K, and the sources are flexibly modeled with the optimal
number of bases using the cluster-indicator Z. We used three musics
obtained from SiSEC [25] and selected three sources, as shown in
Table 1. In addition, the observed signal, which has three channels,
was created by convoluting the impulse response E2A (see Fig. 2)279This paper presents an eﬃcient MNMF method that includes a rank1 approximation of the covariance matrix. The proposed method canbe optimized using the fast update rules of IVA and single-channel
NMF. Also, we revealed that MNMF with the rank-1 approximation
is essentially equivalent to IVA with the basis decomposition model.
The experimental results show that the proposed method achieves
faster convergence and better results than the conventional methods.
6. ACKNOWLEDGMENT
This work was partially supported by Grant-in-Aid for JSPS Fellows
Grant Number 26·10796.

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