#2019 The Acoustical Society of Japan

Acoust. Sci. & Tech. 40, 3 (2019)

INVITED REVIEW

Nonnegative matrix factorization based on complex generative model
Daichi Kitamura
Department of Electrical and Computer Engineering, National Institute of Technology,
Kagawa College, Chokushi 355, Takamatsu, 761–8058 Japan
Abstract: Nonnegative matrix factorization (NMF) is a powerful technique of extracting meaningful
patterns from an observed matrix and has been used for many applications in the audio signal
processing ﬁeld. In this article, the principle of NMF and some extensions based on a complex
generative model are reviewed. Also, their application to audio source separation is presented.
Keywords: Nonnegative matrix factorization, Complex generative model, Audio source separation
PACS number: 43.60.Cg, 43.60.Uv

1.

[doi:10.1250/ast.40.155]

INTRODUCTION

Nonnegative matrix factorization (NMF) [1,2] is analgorithm for extracting a limited number of meaningful
nonnegative patterns from an observed nonnegative matrix.
Since essential features in observed data are useful formany applications, NMF has been utilized in many ﬁelds,
especially audio signal processing.
Let a nonnegative matrix be X 2 RIJ
0 , where I and J
are the numbers of rows and columns in X, respectively. In
NMF, we approximately decompose X as
X  X^ ¼ WH
ð1ÞX
wk hTk ;ð2Þ
¼kwhere W ¼ ðw1    wK Þ 2 RIK
0 is called the basis matrix
that includes K meaningful nonnegative patterns (basis
vectors) wk 2 RI1
as column vectors and H ¼
0KJTðh1    hK Þ 2 R0 is called the activation matrix that
includes the coeﬃcient vectors hk 2 RJ1
0 for wk as row
vectors. Equation (1) can be rewritten in an element-wise
equation asX
xij  x^ij ¼
wik hkj ;ð3Þkwhere xij , x^ij , wik , and hkj are the nonnegative elements in^ W, and H, respectively, i and j are the indexes ofX, X,
rows and columns in X, and k is the index of bases in W.
When the number of bases, K, is set to be small as
K  minðI; JÞ, the NMF decomposition (1) becomes a
low-rank approximation, resulting in the extraction of
meaningful patterns.
e-mail: kitamura-d@t.kagawa-nct.ac.jp

The variables W and H in NMF can be estimated by
the following optimization:
min DðXjWHÞ s.t. wik ; hkj  0 8i; j; k;

ð4Þ

W;H

where DðjÞ is a similarity function between two input
matrices for which the squared Euclidean distance [2],generalized Kullback–Leibler (KL) divergence [2], andItakura–Saito (IS) divergence [3] are often used. Thus, (4)estimates the low-rank model WH that approximately
represents the observed data X. Since the closed-form
solution of (4) cannot be obtained, W and H are estimatedusing an iterative optimization algorithm [4,5] with anarbitrary initialization scheme (e.g., [6]).
In audio signal processing, to obtain an observed
nonnegative matrix X, a complex-valued spectrogram
C 2 CIJ , which is obtained by applying short-time Fouriertransform (STFT) to a time-domain signal, is transformed
into the nonnegative matrix as X ¼ jCj: p , where theoperator j  j: p for matrices denotes the element-wiseabsolute and pth-power operations. In this case, I and J
correspond to the numbers of frequency bins and timeframes, respectively. In particular, an amplitude spectrogram X ¼ jCj:1 or a power spectrogram X ¼ jCj:2 is often
used. Figure 1 shows an example of NMF decompositionof a power spectrogram. The observed spectrogram
includes two tones with diﬀerent pitches, and the model
WH represents their spectral patterns (w1 and w2 ) andtime-varying gains (h1 and h2 ) when K ¼ 2. Therefore,
NMF can be interpreted as unsupervised learning where
frequently appearing spectral patterns and their activations
are extracted as wk and hk , respectively.
NMF has been applied to many applications including
audio source separation [7–11], automatic music transcription [12,13], acoustic event detection [14], and super-155

Acoust. Sci. & Tech. 40, 3 (2019)

Fig. 1 NMF decomposition for audio signals.

Mixture in waveform domain (time domain)

STFT andtaking th powerNot equivalent
Mixture in nonnegative spectrogram domain
Rank-1matrices

NMF
modeling

Fig. 2 Inappropriate mixing model assumed in NMF foraudio signals.

in some tones, phase spectra signiﬁcantly aﬀect perception,e.g., white noise and impulsive sound. Moreover, a phase
spectrogram is required when we apply inverse STFT withthe estimated model spectrogram WH to recover a timedomain signal. As the most commonly used method, thephase spectrogram of C is added to WH to recover thesignal. Wiener ﬁltering or phase recovery [16,17] is anotherapproach often used.
Many algorithms have been proposed to solve theabove-mentioned problem. In this article, a complexvalued extension of NMF (complex NMF: CNMF)[18–20] and NMF based on complex generative models[3,21–23] are reviewed.

3.

CNMF EMPLOYING PHASE SPECTRA

The conventional NMF estimates nonnegative rank-1
spectrogram components wk hTk while ignoring the phase
spectrogram of X. In CNMF [18], the components areextended from nonnegative matrices to complex-valued
matrices, namely, CNMF approximates the observed
complex-valued spectrogram C asX
c^ij;kð5Þ
cij  c^ij ¼
kresolution of audio signals [15]. In this article, we review
some extended NMF algorithms that are based on acomplex generative model, and that enable us to deal withnot a nonnegative spectrogram X but a complex-valued
spectrogram C. Also, their application to audio source
separation is presented.

2.

PROBLEM IN NMF-BASED MODELINGIn NMF, X is approximated by the sum of rank-1
nonnegative matrices wk hTk as (2). This decomposition isvalid for inherently nonnegative observed data (e.g., grayscale pictures and counting data of customers’ buying).
However, for audio signals, there exists the problem
described below.
As shown in Fig. 2, audio mixing is the sum of timedomain signals and is identical to the sum of the ‘‘complexvalued’’ spectrograms in the time-frequency domain.
However, NMF decomposes the nonnegative spectrogram
jCj: p into K rank-1 nonnegative spectrograms wk hTk , andthis decomposition assumes the additivity of nonnegative
spectra. In general, for two complex values c1 and c2 , theadditivity of their nonnegative values jc1 j p and jc2 j p(jc1 þ c2 j p ¼ jc1 j p þ jc2 j p ) does not hold when p 6¼ 0. For
this reason, NMF decomposition of nonnegative spectrograms is an inappropriate model. More precisely, wave
cancellation by phase shifting in audio mixing is ignored in
the estimation of W and H. This is a speciﬁc problem inaudio modeling based on NMF. Although the modelingerror of phase spectra is not so critical for human audition,

156

¼

X

wik hkj ejij;k ;ð6Þkwhere cij is the element of C, c^ij;k 2 C is the complexvalued spectral component (model) thatpsatisﬁes
jc^ij;k j ¼
ﬃﬃﬃﬃﬃﬃﬃ
wik hkj and argðc^ij;k Þ ¼ ij;k , and j ¼ 1. Therefore,
CNMF directly decomposes the complex-valued spectrogram C into complex-valued spectrogram components C^
whose amplitude and phase are wk hTk and k 2 RIJ
½0;2Þ ,respectively, as
X
C  C^ ¼
ð7Þ
C^ k :k

The variables in CNMF are wik , hkj , and ij;k , and they
can be estimated by the maximum likelihood (ML) sensewith the following generative model:
cij ¼ c^ij þ "ij ;
"ij

ð8Þ

N C ð0;  2 Þ;

ð9Þ



1
jc  j2
2
N C ðc; ;  Þ ¼
exp 
;
 2
2

ð10Þ

where N C ðc; ;  2 Þ is an isotropic complex Gaussian
distribution with the mean  and the variance  2 > 0, as
depicted in Fig. 3. CNMF approximates the observed
spectrogram C as the sum of a limited number of components C^ k , and its approximation error "ij is assumed toobey Fig. 3 independently deﬁned in each time-frequency
slot. By ML estimation, we obtain the following optimization problem for estimating the variables in CNMF:

D. KITAMURA: NMF BASED ON COMPLEX GENERATIVE MODEL(ISNMF) is obtained, and its optimization problem can be
rewritten as
!
X
X
xij
P
min
þ log
wik hkj
W;Hk wik hkji; jks.t. wik ; hkj  0 8i; j; k;where the constant terms are omitted. The statisticalinterpretation of (12) when X ¼ jCj:2 is described below.
Let us assume that the observed complex-valued
spectrum cij can be decomposed into K complex-valued
P
components as cij ¼ k cij;k , where each spectral component cij;k obeys the zero-mean isotropic complex Gaussian
distribution (Fig. 3) as

Fig. 3 Isotropic complex Gaussian distribution.


2

X
X

c^ij;k mincij 

wik ;hkj ;ij;ki; jkXs.t. wik ; hkj  0 8i; j; k;wik ¼ 1 8k:

cij;k

i

This minimization can be interpreted as a complex version
of NMF using the squared Euclidean distance. Similar to
NMF, the iterative update rules for CNMF can be derived
by an auxiliary function technique [18]. In recent years, thesimilarity function in CNMF is generalized to -divergence, which includes the generalized KL divergence and
IS divergence [19,20].
As described above, CNMF assumes the additivity ofcomplex-valued spectrogram components and the low rankof the amplitude spectrogram, resulting in an appropriate
decomposition model without ignoring phase information.
However, since we must estimate not only the bases andactivations but also their phase spectrograms, its optimization is unstable and strongly depends on the initializationof variables [24].

4.

ð12Þ

2
N C ð0; ij;k
Þ:

ð13Þ

2
The variance ij;k> 0 is a nonnegative parameter that
ﬂuctuates depending on both frequency i and time j. Since
the complex Gaussian distribution has a stable (orreproductive) property, the observed spectrum cij , whichis the sum of cij;k , also obeys the same generative model
P 2
with the variance ij2 ¼ k ij;kascij

N C ð0; ij2 Þ P 2 :
¼ N C 0; k ij;kð14Þ

The generative model (14) (hereafter referred to as pðcij Þ) is
depicted in Fig. 4. A time-frequency slot with a large
spectral power has a wide distribution and easily generates
complex values with a large amplitude. On the other hand,a slot with a small spectral power has a narrow distribution
and generates almost zero values. Since pðcij Þ is a zeromean and isotropic distribution, the generative model ofphase argðcij Þ always shows a uniform distribution. The
variance ij2 corresponds to the expectation value of cij asij2 ¼ E½jcij j2

NMF BASED ON COMPLEX
GENERATIVE MODELSIt has been revealed that NMF based on a particularsimilarity function can be interpreted as the ML estimation
assuming complex generative models for the observed
data. On the basis of these generative models, the additivityof nonnegative spectrograms is justiﬁed in a statisticalsense. In this section, NMF based on complex generativemodels is reviewed.
4.1. NMF Based on IS Divergence and Its Statistical
Interpretation
IS divergence between c and  is deﬁned asDðcjÞ ¼

jcj2
jcj2

log
 1:
2
2

ð11Þ

When we set the similarity function in (4) to (11), jcj2 ¼
P
xij , and  2 ¼ k wik hkj , NMF based on IS divergenceFig. 4 Local Gaussian model assumed in ISNMF.

157

Acoust. Sci. & Tech. 40, 3 (2019)X¼

E½jcij;k j2 ;

ð15Þ

k

where E½ denotes the expectation of the observed data. By
assuming that pðcij Þ is mutually independent w.r.t. i andj, we can deﬁne the generative model of the observed
complex-valued spectrogram C asCpðc11 ; c12 ;    ; cIJ Þ
Ypðcij Þ
¼
i; j

¼

Y

 P
2
:
N C 0; k ij;k

ð16Þ

i; j

The generative model (16) is often called the local
Gaussian model (LGM) [25] and is extended to multichannel signals as a multivariate generative model[26–28].
2
in (16) can be estimated by ML
The variance ij;kestimation. The likelihood function of the observed data C
can be obtained as
 P
Y
2
L¼
N C 0; k ij;k
i; j

¼

Y
i; j 

1
P

2
k ij;k2

ð17Þ

and its negative log-likelihood function is log L ¼

X
i; j

!
X
jcij j22
P 2 þ log
ij;k þ log  :k ij;kk

ð18Þ

Thus, the ML estimator of the variance is obtained by2minimizing (18) w.r.t. ij;k. Note that the minimizations of(18) and (12) become equivalent when xij ¼ jcij j2 and2¼ wik hkj up to the constant terms.
ij;kFor the reasons mentioned above, it is revealed that
applying ISNMF to the power spectrogram X ¼ jCj:2 is2equivalent to the ML estimation of the variance ij;kbased
on LGM (16). Also, in LGM, the sum of nonnegative
components wik hkj corresponds to the sum of variances2
ij;k¼ E½jcij;k j2 . This result shows that the mixture ofP
complex-valued spectral components (cij ¼ k cij;k ) can be
represented by the sum of nonnegative parameters
P 2(ij2 ¼ k ij;k) by assuming LGM, and the ad hoc process
in NMF-based audio modeling, which is taking a power ofcomplex-valued data to make them nonnegative, can bejustiﬁed in the expectation sense. Therefore, the validityof ISNMF-based modeling, which approximates a power
spectrogram jCj:2 by the sum of nonnegative spectrogram
components wk hTk as shown in Fig. 2, is justiﬁed eventhough the mixing of audio signals is the sum of complexvalued spectrograms. The formulation based on LGM isapplied to a multichannel audio source separation task
[27–30].

158

c1

N C ð0; 12 Þ;

c1 þ c2

c2

N C ð0; 22 Þ

N C ð0; 12 þ 22 Þ:

ð19Þ

Similarly, if we employ the zero-mode isotropic complex
Cauchy distribution [31], which is deﬁned as
CC ðc; 0; Þ ¼!

jcij jexp  P 2 ;k ij;k4.2. NMF Based on Generalized LGM
To justify the additivity of nonnegative spectrogram
components in the expectation sense, the generative modelneeds to belong to a stable distribution family [31], which
has the stable property. The stable property satisﬁes thefollowing requirement: for two random variables (r.v.s)
v1 and v2 independently generated from the same distribution, their linear combination av1 þ bv2 and another r.v.
dv þ e also obey the same distribution, where a > 0,
b > 0, d > 0, and e are constants. If we assume such stable
distribution as a generative model, the sum of r.v.s canbe modeled by the sum of the parameters of theirdistributions. Indeed, the complex Gaussian distribution
is a special case of stable distribution, and the sum of r.v.s
can be modeled by the sum of the second-order expectations (variances) as21=2 3

2½jcj2 þ ð21=2 Þ2 2

;

ð20Þ

the sum of r.v.s can be modeled by the sum of the ﬁrstorder expectations (scale parameters)  > 0. The complexvalued spectral components cij;k are assumed to obey (20),and the scale parameters deﬁned in each time-frequency
slot correspond to the expectation of amplitude values asij;k ¼ E½jcij;k j . Since the complex Cauchy distribution has
the stable property, the generative model of the observed
P
spectrum cij ¼ k cij;k also becomes P
cij CC 0; k ij;k ;ð21Þ
thereby justifying the additivity of amplitude spectral
components ij;k ¼ wik hkj in the expectation sense.
NMF based on (21) is called Cauchy NMF [21].
Moreover, the complex Student’s t distribution is also
employed in NMF (tNMF) [22], where the complex
Student’s t distribution has a degree-of-freedom parameter > 0. tNMF is a generalization of Cauchy NMF and
ISNMF because it coincides with them when  ¼ 1 and ! 1, respectively. Thus, tNMF can represent theintermediate model between Cauchy NMF and ISNMF,
although the stable property in the complex Student’s t
distribution holds only when  ¼ 1 and  ! 1. tNMF
is also applied to multichannel audio source separation[32,33].

5. NMF BASED ON COMPLEX
GENERALIZED GAUSSIAN DISTRIBUTION
In this section, a new generalization of LGM, NMF
based on the complex generalized Gaussian distribution

D. KITAMURA: NMF BASED ON COMPLEX GENERATIVE MODEL

Fig. 5 Isotropic complex GGD with ij ¼ 1:5.

(GGDNMF) [23], is reviewed. Also, its application tosparse noise reduction is presented.
5.1. Generative Model in GGDNMF
In GGDNMF, LGM assumed in ISNMF is generalized
to the complex generalized Gaussian distribution (GGD).
The zero-mean and isotropic complex GGD GC ðcij ; 0; ; ij Þ
is assumed as a generative model of a complex-valued
spectrogram C:YGC ðcij ; 0; ; ij Þ
C
i; j

¼

i; j

ijp ¼

1 2

Y
X

1

2

2

exp 
ij2 ð2= Þ

wik hkj ;
2 jcij j
ij

;

ð22Þ
ð23Þ

k

where > 0 is the shape parameter, ij > 0 is the scaleparameter deﬁned in each i and j, ðÞ is the gammafunction, and p is the parameter that deﬁnes the domain of
NMF decomposition as jCj: p . As depicted in Fig. 5, thecomplex GGD GC ðcij ; 0; ; ij Þ coincides with the complex
Gaussian and complex Laplace distributions when ¼ 2
and ¼ 1, respectively. Also, it becomes sub-Gaussian(platykurtic) and super-Gaussian (leptokurtic) distributions
when > 2 and < 2, respectively.
5.2. Divergence Derived from Complex GGD
To clarify the relationship between the generativemodel (22) and the similarity function in NMF, we derive
the divergence based on the complex GGD by calculating
deviance. The log-likelihood function of (22) becomeslog L ¼ log GC ðc; 0; ; Þ
 
1 2
2 jcj
:
¼ log
 2 log  2
21 ð2= Þ
From @ log L=@ ¼ 0, the ML estimator of  is given asML ¼ jcj. The deviance D ¼ log LðML Þ  log LðÞ  0
is obtained asDðckÞ ¼ 2 log jcj 
¼

2



jcj




2

þ 2 log  þ

 
2 jcj


 
jcj
 log
1 :


ð24Þ

Since the deviance (24) is nonnegative and becomes zero if
and only if  ¼ jcj, it satisﬁes the axiom of divergence. By
comparing (24) and IS divergence (11), we can conﬁrm
that (24) is a generalization of IS divergence w.r.t. . Also,
it is revealed [23] that (24) is a special case in a moregeneralized divergence called - divergence [34].
5.3. Optimization Algorithm
The cost function in GGDNMF is given as
2
XX
jcij j4Dðcij jij Þ ¼
P

p
i; j
i; j
k wik hkj
3Xþ log
wik hkj 5;
p
k

ð25Þ

where the constant terms are omitted. The iterative update
rules for estimating the minimizers wik and hkj can beobtained as follows [23]:
2
3 p
þp
P
zij
2 hkj 7
6 j P
6
7
k0 wik0 hk0 j
7 ;
wik
wik 6
ð26Þ
6 P
7
1
4
5
P
hkj
j
k0 wik0 hk0 j
2

P

zij

3 p

2 wik 7
6 i P
6
7
0 wik 0 hk 0 j
k
7
hkj 6
hkj
6 P
7
1
4
wik 5
iP
k0 wik0 hk0 j

p
1 p
p
:
zij ¼ jcij j ij

þp

;

ð27Þ

ð28Þ

159

Acoust. Sci. & Tech. 40, 3 (2019)

Fig. 6 Power spectrograms of (a) original and (b)observed noisy signals.

Fig. 8 Average SDR improvements of (a) tNMF and(b) GGDNMF for various , , and p.

Fig. 7 Examples of power spectrogram estimated by(a) ISNMF (SDR: 13:52 dB), (b) Cauchy NMF(SDR: 3.77 dB), (c) tNMF (SDR: 7.26 dB), and (d)GGDNMF (SDR: 7.38 dB).

5.4. Application to Sparse Noise Reduction
As an application of NMF based on a heavy-taildistribution such as < 2 in GGDNMF, some experimental results of sparse noise reduction are presented. The
signal used in this experiment is shown in Fig. 6. Sparsenoise is the components sparsely distributed in the timefrequency domain, and such noise is called musical noiseand often arises after applying nonlinear signal processing,e.g., spectral subtraction. ISNMF with p ¼ 2, Cauchy
NMF with p ¼ 1, tNMF, or GGDNMF was applied to theobserved signal (Fig. 6(b)). The obtained model spectrogram jWHj:ð2= pÞ is shown in Fig. 7, where K was set to 30
for all NMFs. Also,  and p were respectively set to 2 and
0.5 in tNMF, and and p were set to 0.1 in GGDNMF. As
the evaluation score, the source-to-distortion ratio (SDR)[35] was used. From Fig. 7, we can conﬁrm that the sparsenoise is reduced in all NMFs except for ISNMF. This isbecause the ML estimation based on the heavy-tailed
distribution can ignore the sparse noise as outliers and ﬁnds
the parameters WH that have a low-rank structure in thecontaminated observed data.
Figure 8 shows the averaged scores of the same
experiment as in the case of Fig. 7 with various observed

160

signals. From this result, we can conﬁrm the transition of
the optimal generative model (the value of  orthat
achieves the highest SDR) depending on the domain
parameter p. This is because the eﬀect of the sparse noise
components varies depending on the spectrogram domain.
Regarding SDR, tNMF and GGDNMF achieve almost thesame performance for sparse noise reduction.

6.

CONCLUSION

In this article, the inappropriate assumption in NMFbased audio modeling was explained. As the solution tothis problem, CNMF and ISNMF were reviewed. Also,
the extensions of ISNMF, Cauchy NMF, tNMF, andGGDNMF, were explained, and their application to sparsenoise reduction was presented.

ACKNOWLEDGMENTS
This work was partly supported by SECOM Scienceand Technology Foundation, Yamaha corporation, and
JSPS KAKENHI Grant Numbers JP17H06572 and
19K20306.
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Daichi Kitamura received the Ph.D. degreefrom SOKENDAI, Japan. He joined the University of Tokyo in 2017 as a Research
Associate, and he moved to National Instituteof Technology, Kagawa Collage as an AssistantProfessor in 2018. His research interests include
audio source separation and statistical signal
processing. He received Awaya Prize Young
Researcher Award from The Acoustical Societyof Japan (ASJ) in 2015, Ikushi Prize from Japan Society for the
Promotion of Science in 2017, Best Paper Award from IEEE Signal
Processing Society Japan in 2017, and Itakura Prize Innovative
Young Researcher Award from ASJ in 2018.

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