Vol.2013-MUS-99 No.14
2013/5/11

IPSJ SIG Technical Report

Importance of Regularization inSuperresolution-Based Multichannel Signal Separation
with Nonnegative Matrix Factorization
Daichi Kitamura1,a)Hiroshi Saruwatari1 Kiyohiro Shikano1
Yu Takahashi2

Kazunobu Kondo2

Abstract: In this paper, we address a multichannel signal separation problem and propose a new hybrid method uti-

lizing both directional clustering and supervised nonnegative matrix factorization (NMF) with norm regularization
term. Conventional hybrid method concatenating supervised NMF after directional clustering has a problem that theextracted signal suﬀers from considerable spectral distortion because directional clustering yields spectral chasms. To
solve this problem, we propose a new supervised NMF algorithm that regards the spectral chasms as unseen observations and reconstructs the target source components via spectrogram extrapolation. In addition, this paper addresses anissue on importance of regularization introduced in the superresolution procedure. Our experimental results show thatthe proposed hybrid method with regularization greatly improves the separation performance for stereo signals.

1. Introduction
In recent years, music signal separation based on nonnegative
matrix factorization (NMF) [1], which is a type of sparse representation algorithm, has been a very active area of signal processing research [2], [3]. NMF for acoustical signals decomposes
an input spectrogram into the product of a spectral basis matrix
and its activation matrix. In particular, supervised NMF (SNMF)[4], [5], which includes a priori training with some sample soundsof a target instrument, can extract the target signal to some extent,
particularly in the case of a small number of instruments. However, for the case of a mixture consisting of many sources, such as
more realistic musical tunes, the source extraction performanceis markedly degraded when only single-channel observation isavailable.
Multichannel NMF, which is a natural extension of NMF fora stereo or multichannel music signal, has been proposed as anunsupervised method [6], [7]. However, such unsupervised separation is a diﬃcult problem, even if the signal has multichannel components, because the decomposition is underspecified.
Hence, these algorithms involve strong dependence on initial values and lack robustness.
As another means for addressing multichannel signal separation, directional clustering has also been proposed as an unsupervised method [8], [9]. This method quantizes directional information via time-frequency binary masking under the assumptionthat the sources are completely sparse in the time-frequency domain. However, there is an inherent problem that sources located12a)Nara Institute of Science and Technology, Ikoma, Nara 630–0192, Japan.
Yamaha Corporation, Iwata, Shizuoka 438–0192, Japan.
daichi-k@is.naist.jp

c 2013 Information Processing Society of Japan⃝in the same direction cannot be separated using the directional information. To cope with this problem, a hybrid method for multichannel signal separation, which concatenates SNMF after directional clustering, has been proposed [10]. However, this hybrid
method also has a problem that the extracted signal suﬀers fromconsiderable distortion because the signal obtained by directional
clustering has many spectral chasms. This results in the cascaded
SNMF being forced to incorrectly mimic such artificial spectral
chasms.
In this paper, we propose a new SNMF algorithm for the hybrid method. Using index information generated by binary masking, the proposed SNMF regards the spectral chasms as unseen
observations, and finally reconstructs the target source components via spectrogram extrapolation using supervised bases. In
other words, the proposed method can be categorized as superresolution because the degraded resolution in the time-frequency
domain resulting from the preceding binary masking can be recovered. In addition, this paper addresses an issue on importance of regularization introduced in the superresolution procedure. Our experimental results show that the proposed method
outperforms several conventional methods and that the distortionof the extracted signal can be mitigated by the eﬀectiveness of thesuperresolution-based method.

2. Conventional Method
2.1 Overview of Penalized SNMF
The unsupervised NMF approaches have diﬃculty in clustering decomposed spectral patterns into a specific target instrumental sound. Furthermore, each basis may be forced to include a multi-instrumental spectral pattern. To solve this prob-

1Vol.2013-MUS-99 No.14
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IPSJ SIG Technical Reportlem, SNMF has been proposed [4], [5]. In particular, a penalized
SNMF (PSNMF), which imposes a penalty term to force supervised bases and other bases to become uncorrelated with eachother, achieves good performance [5]. PSNMF consists of two
processes, a priori training and observed signal separation, as described below in detail.
2.1.1 Training process of supervision
In PSNMF, as the supervision training process, a priori spectral
patterns (bases) should be trained in advance to achieve source
separation. Hereafter, we assume that we can obtain specific soloplayed instrumental sounds, which is the target of the separation
task. The trained bases are constructed by NMF as
Ytarget ≃ F Q,

(1)swhere Ytarget (∈ RΩ×T
) is an amplitude spectrogram of the spe≥0cific signal for training, F (∈ RΩ×K
≥ 0 ) is a nonnegative matrix
that involves bases of the target signal as column vectors, ands
Q(∈ R≥K×T) is a nonnegative matrix that corresponds to the activa0
tion of each basis of F . In addition, Ω is the number of frequency
bins, T s is the number of frames of the training signal, and K is
the number of bases. Therefore, the basis matrix F constructedby Eq. (1) is used for the supervision of the target instrumental
signal.
2.1.2 Signal separation process
The following equation represents the decomposition of
PSNMF using the trained supervision matrix F :Y ≃ F G + HU ,

(2)

K×T
where Y (∈ RΩ×T
≥ 0 ) is an observed spectrogram, G(∈ R≥ 0 ) is anactivation matrix that corresponds to F , H(∈ RΩ×L
≥ 0 ) represents
the residual spectral patterns that cannot be expressed by F G,
and U (∈ R≥L×T
0 ) is an activation matrix that corresponds to H.
Moreover, T is the number of frames of the observed signal and L
is the number of bases of H. In PSNMF, the matrices G, H, andU are optimized under the condition that F is known in advance.
Hence, ideally, F G represents the target instrumental components and HU represents the components other than the target
sounds after the decomposition.
2.1.3 Cost function
In the decomposition of PSNMF, a cost function is defined using some measures of the distance between Y and F G + HU
as

JSNMF = D (Y |F G + HU ) + µ∥F T H∥2Fr ,(3)

where D (·|·) is an arbitrary distance function, e.g., ItakuraSaito divergence (IS-divergence), generalized Kullback-Leibler
divergence (KL-divergence), or the Euclidean distance (EUCdistance). In this study, we use EUC-distance in the cost function.
In addition, µ is the weighting parameter for the penalty term and∥·∥Fr represents the Frobenius norm. This penalty term indicatesthat F and H are forced to become uncorrelated with each otherto avoid sharing the same basis.
2.1.4 Multiplicative update rules of PSNMF
The update rules based on EUC-distance are given byc 2013 Information Processing Society of Japan⃝∑fω,k yω,tgk,t ← gk,t ∑ω,ω fω,k rω,t(4)∑t yω,t ul,t∑∑hω,l ← hω,l ∑,t ul,t rω,t + 2µ k fω,k ω′ fω′,k hω′,l∑hω,l yω,tul,t ← ul,t ∑ω,ω hω,l rω,t

(5)
(6)

where yω,t , fω,k , gk,t , hω,l , and ul,t are the nonnegative entries ofmatrices Y , F , G, H, and U , respectively, and∑∑rω,t =
fω,k′ gk′,t +hω,l′ ul′,t .
(7)
k′

l′

2.1.5 Problem of PSNMF
PSNMF can extract the target signal to some extent, particularly in the case of a small number of sources. However, for thecase of a mixture consisting of many sources, such as more realistic musical tunes, the source extraction performance is markedly
degraded because of the existence of instruments with similar
timbre.
2.2 Directional Clustering and Its Hybrid Method with
PSNMF
Decomposition methods employing directional information forthe multichannel signal have also been proposed as unsupervised
separation techniques [8], [9]. These methods quantize directional information via time-frequency binary masking under theassumption that the sources are completely sparse in the timefrequency domain. Such directional clustering works well, evenin an underdetermined situation where the number of sources isgreater than that of inputs. However, there is an inherent problem
that sources located in the same direction cannot be separated using the directional information. Furthermore, the extracted signal
is likely to be distorted because of the eﬀect of binary masking.
To solve this problem, a hybrid method that concatenates
PSNMF after the directional clustering has been proposed [10].
This hybrid method can eﬀectively extract the target instrument
because the directionally clustered signal contains only few instruments. Moreover, the residual interfering signal in the same
direction can be removed by PSNMF.

3. Proposed Method
3.1 Motivation and Strategy
The conventional hybrid method has a problem that the extracted signal suﬀers from the generation of considerable distortion. This is due to the binary masking in directional clustering.
The signal in the target direction, which is obtained by directional
clustering, has many spectral chasms because the assumption ofsparseness in the time-frequency domain does not always holdcompletely. In other words, the resolution of the spectrogram
clustered as the target-direction component is degraded by timefrequency binary masking. Figure 1 shows an example of thespectrum of a signal separated by directional clustering. The obtained spectrum has many chasms owing to the binary masking.
These spectral losses may deteriorate the performance of separation because PSNMF is forced to incorrectly fit these spectral
chasms using supervised bases.
To solve this problem, in this section, we propose a new2Vol.2013-MUS-99 No.14
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Frequency ofsource component

40
20
0
-20
-40
0

Fig. 1

200

400
600
800
Frequency [Hz]1000

Example of spectrum of signal separated by directional clustering.

Input stereo signal
L-ch
R-ch
STFT
Directional clustering
Center component

L-ch
Superresolutionbased SNMF

R-ch
Superresolutionbased SNMF

ISTFT

ISTFT

spectraMixingz

Fig. 2Signal flow of proposed hybrid method.

superresolution-based SNMF algorithm as an alternative to theconventional PSNMF for the hybrid method (see Fig. 2). This algorithm utilizes index information determined in directional clustering. For example, if the target instrument is localized in thecenter cluster along with the interference, superresolution-based
SNMF is only applied to the existing center components using index information. Therefore, the spectrogram of the target instrument is reconstructed using more matched bases because spectral
chasms are treated as unseen, and these chasms have no impact
on the cost function in SNMF. In addition, the components of thetarget instrument lost after directional clustering can be extrapolated using the supervised bases. In other words, the resolutionof the target spectrogram is recovered with the superresolution by
the supervised basis extrapolation.
To illustrate the separation mechanism step by step, Fig. 3 (a)shows the configuration of source components in the stereo signal, (b) shows the separated components that are clustered aroundthe center direction by directional clustering, and (c) shows theseparated target component obtained by superresolution-based
SNMF. In Fig. 3 (a), the source components are distributed inall directions with some overlapping. After directional clustering (Fig. 3 (b)), the center sources lose some of their components(i.e., the tails on both sides), and the other source components
leak in the center cluster. After SNMF, the proposed algorithm restores the lost components using the supervised bases (Fig. 3 (c)).
3.2 Regularization of Basis Extrapolation
The basis extrapolation in the proposed SNMF includes an underlying problem. If the time-frequency spectra are almost unseen in the spectrogram, which means that the indexes are almost
zero, a large extrapolation error may occur. Then, incorrect basesc 2013 Information Processing Society of Japan⃝

Fig. 3

CenterDirection(c) After
superresolutionbased SNMF

Left
Extracted signal

Right

CenterDirection(b) After
directional
clustering

Left
Frequency ofsource component

Index ofcenter cluster

Target source(a) Observed

Left
Frequency ofsource component

Amplitude [dB]

IPSJ SIG Technical Report

Right

Extrapolated
target source

CenterDirection

Right

Directional source distribution of (a) observed stereo signal, (b) separated components in center cluster, and (c) component separated andextrapolated by superresolution-based SNMF.

are chosen and fitted to a small number of spectral grids by incorrectly modifying the activation matrix G. In the worst case, theactivation matrix G contains very large values and the extracted
signal is overloaded.
Figure 4 shows an example of the spectrogram that is extracted
signal with basis extrapolation error at the end of signal. In thiscase, incorrect extrapolation occurred at the time frames that havespectral chasms as many as 99% of frequency bins. To avoid thiserror, we should add a new penalty term in the cost function, as
described below in detail.
3.3 Cost Function of Superresolution-Based SNMF
Here, the index matrix I(∈ RΩ×T
{0, 1} ) is obtained from the binary
masking preceding the directional clustering. This index matrix
has specific entries of unity or zero, which indicate whether or
not each grid of the spectrogram belongs to the target directional
cluster. The cost function in superresolution-based SNMF is defined using the index matrix I as
J = D(I ◦ Y |I ◦ (F G + HU ))+ µ∥F T H∥2Fr + λ∥I ◦ (F G)∥2Fr ,(8)

where µ and λ are the weighting parameters for each penalty
term, · represents the binary complement of each entry in the index matrix, and ◦ indicates the Hadamard product of matrices.
Since the divergence D is only defined in grids whose index isone, the chasms in the spectrogram are ignored in this SNMF decomposition. In addition, the first penalty term has the same property as the conventional method Eq. (3), and the second penalty
term forces the minimization of the norm ∥F G∥Fr in proportion3Vol.2013-MUS-99 No.14
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IPSJ SIG Technical Reportfω,k gk,t,k′ fω,k′ gk′,thω,l ul,tβl,ω,t = ∑.
l′ hω,l′ ul′,tFrequency [kHz]4αk,ω,t = ∑

3

(14)

Second, for the penalized terms (hereinafter, referred to as
J (penalty) ) in Eq. (9), the upper bound function Q(penalty) is defined using the auxiliary variables γω,k,l ≥ 0 and δk,ω,t ≥ 0 that∑∑satisfy ω γω,k,l = 1, and k δk,ω,t = 1. Similarly to Eq. (12), we
obtain
2

2

∑ ∑
∑  ∑



i


f
g
J (penalty) = µ
f
h
+
λ

ω,k k,t 
ω,k ω,l 

 ω,t

2

1

k,l

≤µ

0
0

1

2

3

4

to the number of zeros in the index matrix I. Hence, the supervised bases are chosen so as to minimize the norm ∥F G∥Fr toavoid the extrapolation error. In other words, this penalty term
regulates the extrapolation.
3.4 Auxiliary Function for Cost Function
In this section, we derive auxiliary functions for Eq. (8) basedon EUC-distance for superresolution-based SNMF. Here, we canrewrite Eq. (8) using EUC-distance as∑()
J=iω,t y2ω,t + vω,t + 2wω,t
ω,t


2
2

∑ ∑
∑  ∑ fω,k hω,l  + λiω,t+µfω,k gk,t ,
ω

ω,t

(9)

k

where iω,t is the entry of index matrix I, which maps the valuesof one and zero onto the time-frequency (ω-t) region. In addition,vω,t and wω,t are given by

2 
2
∑
 ∑
vω,t = fω,k gk,t  +  hω,l ul,t  ,
(10)
k

l




∑
 ∑wω,t = fω,k gk,t   hω,l ul,t k

− yω,t∑lfω,k gk,t − yω,t∑khω,l ul,t .

(11)

l

vω,t ≤k

αk,ω,t+∑ h2ω,l u2l,t≡ Q(vω,t ) ,lβl,ω,t(12)

where the equality in (12) holds if and only if the auxiliary variables are set as

c 2013 Information Processing Society of Japan
⃝

+λ

iω,t

k

2 2
gk,t
fω,k

ω,t,k

δk,ω,t(15)

where the equality in Eq. (15) holds if and only if the auxiliary
variables are set as follows:
γω,k,l = ∑δk,ω,t = ∑fω,k hω,l,(16)

fω,k gk,t.
k′ fω,k′ gk′,t(17)

ω′ fω′,k hω′,lFinally, using Eqs. (12) and (15), we can define the upperbound function J + for J as∑()
J ≤ J+ =iω,t y2ω,t + Q(vω,t ) + 2wω,t + Q(penalty) . (18)ω,t3.5 Multiplicative Update Rules
In this section, we drive the update rules based on EUCdistance. The update rules with respect to each variable are determined by setting the gradient to zero. From ∂J + /∂gk,t = 0, we
obtain
 2

2
∑
∑
∑
fω,k
gk,t
 fω,k gk,tiω,t 
+ fω,khω,l ul,t − yω,t fω,k  + λ
iω,t
= 0.
αk,ω,tδk,ω,tωωl(19)
By substituting Eqs. (13), (14), (16), and (17) into Eq. (19), we
can rewrite Eq. (19) as∑∑∑∑iω,t fω,k rω,t + λiω,t fω,kfω,k′ gk′,t =iω,t yω,t fω,k .
ω

ω

k′

ω

(20)

Since it is diﬃcult to analytically derive the optimal G, H, andU that minimize Eq. (9), we define an auxiliary function that represents the upper bound of J as described below.
First, for vω,t , the upper bound function Q(vω,t ) is defined using∑
auxiliary variables αk,ω,t ≥ 0 and βl,ω,t ≥ 0 that satisfy k αk,ω,t = 1∑
and l βl,ω,t = 1. Applying Jensen’s inequality to this, we have
2 2∑ fω,kgk,tγω,k,lω,t∑≡ Q(penalty) ,Spectrogram of extracted signal with basis extrapolation error. This
example incorrectly includes overloaded artifact around 4.5 sec.

k,l

ω

2 2
∑ fω,khω,lk,l,ωTime [s]Fig. 4(13)

Then we can obtain the update rule by multiplying both sides ofEq. (20) by gk,t as follows:∑ω iω,t yω,t fω,kgk,t ← gk,t ∑. (21)∑∑ifr+
λ ω iω,t fω,k k′ fω,k′ gk′,tω ω,t ω,k ω,tThe update rules of the other variables based on EUC-distanceare similarly obtained as follows:∑t iω,t yω,t ul,t∑∑hω,l ←hω,l ∑
,
(22)
i
u
r
+
µ k fω,k ω′ fω′,k hω′,l
ω,t
l,t
ω,t
∑t
iω,t yω,t hω,lul,t ←ul,t ∑ω.
(23)ω iω,t hω,l rω,t4

Vol.2013-MUS-99 No.14
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IPSJ SIG Technical Report
Center

Left

Right

44
2

3
1

Fig. 5

Scores of each instrument.

4. Evaluation Experiment
4.1 Experimental Conditions
To confirm the eﬀectiveness of the proposed algorithm, we
compared five methods, namely, simple directional clustering [8],simple PSNMF [5], multichannel NMF based on IS-divergence[7], the conventional hybrid method using PSNMF after directional clustering [10], the proposed hybrid method using
superresolution-based SNMF without regularization after directional clustering (hereafter, referred to as Proposed method 1),
and the proposed hybrid method using superresolution-based
SNMF with regularization after directional clustering (hereafter,
referred to as Proposed method 2), in terms of their ability to separate simulated music signals. In this experiment, we used stereo
signals containing four instruments, an oboe sound (Ob.), a flute
sound (Fl.), a trombone sound (Tb.), and a piano sound (Pf.), as
the signal source. The scores of each instrument are depicted inFig. 5. These signals were artificially generated by a MIDI synthesizer, and the observed signals Y were produced by mixing
four sources with an input SNR of 0 dB. The observed signal contained one source in the left and right directions and two sourcesin the center direction based on a sine law (see Fig. 6). The target instrument is always located in the center direction along withanother interfering instrument, and the left and right sources arelocated at 40◦. In addition, we used the same MIDI sounds of thetarget instruments as supervision for a priori training. The training sounds contained two octave notes that cover all notes of thetarget signal in the observed signal. The sampling frequency ofall signals was 44.1 kHz. The spectrograms were computed using a 92-ms-long rectangular window with a 46-ms overlap shift.
The number of iterations for the training was 500 and that for the
separation was 400. Moreover, the number of clusters used in directional clustering was 3, the number of a priori bases was 100,and the number of bases for matrix H was 30. In this experiment,
the weighting parameters µ and λ were empirically determined.
4.2 Experimental Results
We used the signal-to-distortion ratio (SDR), source-tointerference ratio (SIR), and sources-to-artifacts ratio (SAR) defined in [11] as the evaluation scores. Here, the estimated signalŝ (t) is defined asŝ (t) = starget (t) + sinterf (t) + sartif (t) ,(24)

where starget (t) is the allowable deformation of the target source,c 2013 Information Processing Society of Japan⃝

Fig. 6 Panning of four sources with sine law used in experiment. Numbered
black circles represent locations of instruments in stereo format. For
example, if target is Ob., No.1 is set to Ob. and Nos.2, 3, and 4 are
combinations of Fl., Tb., and Pf.

sinterf (t) is the allowable deformation of the sources that accountfor the interferences of the undesired sources, and sartif (t) is an
artifact term that may correspond to the artifacts of the separation algorithm, such as musical noise, or simply undesirable
deformation induced by the nonlinear property of the separation
algorithm. The formulae for SDR, SIR, and SAR are defined as∑
SDR = 10 log10 ∑t starget (t)2,2t {einterf (t) + eartif (t)}∑2t starget (t),SIR = 10 log10 ∑2
t einterf (t){}2∑t starget (t) + einterf (t)SAR = 10 log10.
∑2t eartif (t)

(25)
(26)

(27)

SDR indicates the quality of the separated target sound, SIR indicates the degree of separation between the target and othersounds, and SAR indicates the absence of artificial distortion.
Figure 7 show the average SDR, SIR, and SAR for eachmethod, where the four instruments are shuﬄed with 12 combinations. From the SDRs in Fig. 7, we can confirm that directional clustering and multichannel NMF do not have suﬃcientperformance because they cannot discriminate the sources in thesame direction. In contrast, the methods using SNMF can give
better results and Proposed method 2 outperforms all other methods. Furthermore, the evaluation scores of Proposed method 1
are markedly lower than Proposed method 2. This indicates thatsuperresolution-based SNMF in Proposed method 1 has a risk tocause the extrapolation error, whereas Proposed method 2 withregularization can mitigate such an error.

5. Conclusions
In this paper, we propose a new SNMF algorithm for thesuperresolution-based method to separate stereo signals and indicated an eﬀectiveness of the regularization. The proposed algorithm utilizes index information that indicates the direction ofeach component in the time-frequency domain, and restores thetarget signal via the extrapolation of supervision bases with regularization. From the experimental results, it can be confirmed thatthe proposed method with regularization increases the separation
performance for stereo signals compared with the conventional
methods.
5Vol.2013-MUS-99 No.14
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IPSJ SIG Technical Report(a)10

SIR [dB]

SDR [dB]

8
6
4
2
0

Fig. 7

24
22
20
18
16
14
12
10
8
6
4
2
0

10

(b)

(c)
8
SAR [dB]

12

6
4
2
0

Average scores when θ = 40◦: (a) shows SDR, (b) shows SIR, and (c) shows SAR for conventional
and proposed methods.

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c 2013 Information Processing Society of Japan⃝

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