2017 IEEE INTERNATIONAL WORKSHOP ON MACHINE LEARNING FOR SIGNAL PROCESSING, SEPT. 25–28, 2017, TOKYO, JAPANMEL-GENERALIZED CEPSTRAL REGULARIZATION FOR DISCRIMINATIVENON-NEGATIVE MATRIX FACTORIZATION
Li Li1 , Hirokazu Kameoka2 and Shoji Makino1
1

2

University of Tsukuba, JapanNTT Communication Science Laboratories, NTT Corporation, Japan
ABSTRACTThe non-negative matrix factorization (NMF) approach has
shown to work reasonably well for monaural speech enhancement tasks. This paper proposes addressing two shortcomings of the original NMF approach: (1) the objectivefunctions for the basis training and separation (Wiener filtering) are inconsistent (the basis spectra are not trained sothat the separated signal becomes optimal); (2) minimizing spectral divergence measures does not necessarily leadto an enhancement in the feature domain (e.g., cepstral domain) or in terms of perceived quality. To address the firstshortcoming, we have previously proposed an algorithm for
Discriminative NMF (DNMF), which optimizes the same
objective for basis training and separation. To address thesecond shortcoming, we have previously introduced novel
frameworks called the cepstral distance regularized NMF(CDRNMF) and mel-generalized cepstral distance regularized NMF (MGCRNMF), which aim to enhance speech bothin the spectral domain and feature domain. This paper proposes combining the goals of DNMF and MGCRNMF byincorporating the MGC regularizer into the DNMF objectivefunction and proposes an algorithm for parameter estimation.
The experimental results revealed that the proposed method
outperformed the baseline approaches.
Index Terms— Discriminative non-negative matrix factorization, mel-generalized cepstral representation, speechenhancement, single-channel
1. INTRODUCTION
Speech enhancement is a technique for recovering a speech
signal underlying an observed noisy speech signal. Since
the presence of noise can significantly degrade the quality ofspeech transmission systems and the performance of such applications as speech recognition and speech conversion, manymethods have been proposed over the decades.
For monaural speech enhancement tasks, the non-negative
matrix factorization (NMF) approach [1, 2] has attracted a
lot of attention after being proposed as a powerful approach.
This work was supported by JSPS KAKENHI Grant Number 17H01763
and SECOM Science and Technology Foundation.

978-1-5090-6341-3/17/$31.00 c 2017 IEEE

In recent years, deep neural network-based approaches haveshown to be significantly effective particularly for supervised
speech enhancement tasks [3, 4]. However, the NMF approach still remains attractive under unsupervised or semisupervised settings or when only a limited amount of training data is available. The basic idea behind the NMF approach is that approximating the magnitude or power spectrum observed at each time frame as a linear sum of basis
spectra scaled by time-varying amplitudes amounts to factorizing the spectrogram of an observed signal, interpreted asa non-negative matrix, into the product of two non-negative
matrices. In a supervised/semi-supervised setting, NMF isfirst applied to the spectrograms of training instances to train
the basis spectra of speech (and noise). At test time, NMF
is applied to the spectrogram of a test mixture signal, where
each subset of the basis spectra is fixed at the pretrained spectra. The underlying speech components can then be separated
out using a Wiener filter constructed by the estimated power
spectrograms of speech and noise. Although this approach
has shown to work reasonably well, there are two shortcomings to be addressed: (1) the basis spectra obtained in theconventional way do not ensure that the separated signal will
be optimal at test time due to the inconsistency between theobjective functions for training and separation (Wiener filtering); (2) NMF does not necessarily lead to an enhancement in
the feature domain (e.g., cepstral domain) or in terms of perceived quality, which implies naively using NMF as a frontend processing for e.g., speech recognition and speech transmission systems does not always lead to satisfactory results.
To address the first shortcoming, we have previously
focused on a framework called the Discriminative NMF(DNMF) [5], which provides a way to train the basis spectraso that the output of the Wiener filter becomes as close to
the spectrogram of each training example as possible. Whilethe convergence of the basis training algorithm proposedin the original work of DNMF [5] is not guaranteed, we
have proposed a convergence-guaranteed algorithm based on
the majorization-minimization principle [6]. To overcome
the second shortcoming, we have previously proposed two
extensions of NMF, namely, cepstral distance regularized
NMF (CDRNMF) [7] and NMF with mel-generalized cepstral regularization (MGCRNMF) [8]. These methods have

allowed us to jointly enhance speech in both the spectral andfeature domains by optimizing a combined objective function of an NMF-based model-fitting criterion defined in the
spectral domain and a GMM-based probability distribution
defined in the feature domain. The MGCRNMF frameworkuses the mel-generalized cepstral (MGC) representation [9]of speech as the feature, which is widely used in parametric speech synthesis. Since the MGC representation is anauditory-motivated representation of speech spectra, using
this representation as the feature has shown to contribute toenhancing the perceived quality of enhanced speech.
This paper proposes a novel approach combining thegoals of DNMF and MGCRNMF by incorporating the MGC
regularizer into the DNMF objective function and proposes
an algorithm for the parameter estimation.
The remaining part of the paper proceeds as follows: webriefly review the conventional NMF approach for speech enhancement in subsec. 2.1 and the established methods DNMF
and MGCRNMF in the rest of sec. 2. In sec. 3, we introduce
the proposed method and derive the update rules for parameter estimation algorithm based on majorization-minimization
principle. In the experimental section (sec. 4), we define
the data set, investigate the hyperparameters of the model andcompare the proposed method with the established methods.
Conclusions are given in sec. 5.

=XYω,t log
ω,t


Yω,t
− Yω,t + [W H]ω,t ,[W H]ω,twhere [·]i,j denotes the {i, j}-th element of a matrix. Onces
the underlying speech components Xω,t= [W s H s ]ω,t andnnnnoise components Xω,t = [W H ]ω,t are estimated, the enhanced speech can be separated out using the Wiener filter
constructed with the estimated power spectrogram of speechand noiseX̂ s =

Given an observed power spectrogram of a noisy speech signal Y = (Yω,t )Ω×T ∈ R≥0,Ω×T , where ω and t are frequency and time indices, NMF factorizes it into the product of a non-negative basis matrix W = [W s , W n ] and a
non-negative coefficient (activition) matrix H = [H s ; H n ],llwhere W l = (Wω,k)Ω×K l ∈ R≥0,Ω×K with l = {s, n} ispretrained by applying NMF to the spectrograms of trainingl)Ω×T . A typical criterion for this issamples Rl = (Rω,t(1)

Wlwith a column-wise normalization of W l , where D is a costfunction that measures the difference between Rl and W l H l .
At test time, W is fixed at the trained basis spectra and H isthe variable to be estimated so that the objective function

2.2. Discriminative NMF
To eliminate the inconsistency between the objective functions for training (1) and separation (4), instead of applying
NMF to speech and noise training samples individually to
train the basis spectra, Weninger [5] proposed directly using
the reconstruction error of the separated signals as an objective function for the basis training


s
s
s W H
W = argmin DKL R
⊗M ,
(5)
WH
W
X
subject to ∀k,
Wω,k = 1,
where M denotes the spectrograms of the created mixture
signals by adding training speech data to noise data. This
framework is called discriminative NMF (DNMF) by analogy
with the discriminative models for classification or regression.
2.3. Mel-generalized cepstral regularization
We first introduce the mel-generalized cepstral (MGC) representation [9] which plays a key role in the method proposedin [8]. MGC representaion is a parametric model for spectral envelope of speech with frequency resolution similar tothe human auditory systems, which is described by M + 1
coefficients and two hyperparameters γ and α,µω = lγ−1

is minimized subject to non-negativity. NMF leads to different optimization problems according to the definition of thecost function D. Here we consider the generalized Kullback
Leibler (KL) divergence as a goodness-of-fit criterion.

X
M



jω
c(m)Ψmα (e )

(6)

m=0

=
(2)

H

DKL (Y |W H)(4)

ω

2.1. NMF for speech enhancement

H = argmin D(Y |W H),

W sH s⊗Y,
WH

where ·· and ⊗ here are element-wise operations.

2. CONVENTIONAL METHODS

W l = argmin D(Rl |W l H l )

(3)


1/γ
M
P

jω

 1+γ
c(m)Ψm(e)(0 < |γ| ≤ 1)
α
m=0

M
P


jω
exp
c(m)Ψmα (e ).

(γ = 0)

m=0

The function lγ−1 (· ) is the inverse of the generalized logarithmic function((ω γ − 1)/γ (0 < |γ| ≤ 1)lγ (ω) =
,
(7)
log ω
(γ = 0)parameterized by γ. Ψα (z) is an all-pass function given byΨα (z) =z −1 − α,
1 − αz −1(8)which can be seen as a frequency warping function parameterized by |α| < 1. The coefficients c = [c(0), . . . , c(M )]T
are called the MGC coefficients (MGCCs). Given a spectral sequence, the MGCCs are defined as the inverse Fouriertransform of the generalized logarithmic power spectrum calculated on a warped frequency scale. Note that MGC representation takes the all-pole spectral model and the cepstral
representation as special cases when (γ, α) = (−1, 0) and(γ, α) = (0, 0) respectively. When the sampling frequency is
16 kHz, the phase characteristic of the all-pass function becomes a good approximation to the mel scale with α = 0.42
and to the bark scale with α = 0.55 [11].
With the motivation to ensure that the features of the uns
in cepstral domain are also
derlying speech components Xω,tenhanced, by utilizing a codebook consisting of I prototype
spectra µi represented using MGC representation which ispretrained by K-means algorithm using clean speech samples, we have previously proposed mel-generalized cepstral
distance regularized NMF (MGCRNMF) combining a NMFbased model-fitting criterion (2) and a penalty term defineds=
as the Itakura-Saito (IS) divergence [10] between Xω,t[W s H s ]ω,t and a prototype spectrum Sω,t (θ) selected fromthe pretrained codebook
J (W s , H s , θ) =s
X  Xω,tω,tSω,t (θ)

− log

s

Xω,t
− 1 , (9)Sω,t (θ)

where Sω,t (θ) = βt,rt µω,rt . Here, θ = {rt , βt,rt } consists ofa set of cluster indicator variables rt ∈ {1, . . . , I} describing
to which of the I clusters the t-th speech spectrum is assigned
and the correspongding scaling parameters βt,rt introduced toeliminate the scaling indeterminacy. At test time, we find thess Tprototype spectrum µi closest to X st = [X1,t, . . . , XΩ,t]in terms of the IS divergence and the Sω,t (θ) can be easily
obtained by multiplying the selected µi to the optimal scaling
β̂t,i =s
1 X Xω,t

Ω

ω

µω,i

.

(10)

Note that the less (9) becomes, the more simliar speech envelope X s has to the clean speech one, which means the features in cepstral domain are enhanced.

Specifically, the proposed method uses DNMF for basis
training and considers an optimization problem combining a
NMF-based model-fitting criterion (2) and a novel MGC regssularizer obtained by replacing Xω,tby X̂ω,t,J˜(W , H, θ) =s
X  X̂ω,tω,tSω,t (θ)− log

In this section, we introduce a novel approach which combines the goals of DNMF and MGCRNMF by incorporating
the MGC regularizer into the DNMF objective function. Then
we derive an computationally efficient algorithm for parameter estimation.

(11)swhere X̂ω,t
are the enhanced speech spectra obtained by
Wiener filtering (4). With the new regularization term, theproblem we are interested in can be cast as

H = argmin F(W , H, θ),

(12)

H

F(W , H, θ) = DKL (Y |W H) + λJ˜(W, H, θ)with W is fixed at the pretrained spectral basis using (5). λ ≥0 here is a weight parameter to measure the importance of theregularization term relative to the NMF cost.
3.1. Majorization-minimization principle
Although minimizing the objective function including theregularization term (11) directly is analytically difficult, we
can derive a computationally efficient algorithm to find a
locally optimal solution based on majorization-minimization(MM) principle [12, 13].
Suppose F (Θ) is an objective function that we wish tominimize with respect to Θ. Majorization-minimization principle considers to construct a “majorizer” F + (Θ, α) defined
as a function satisfying F (Θ) = minα F + (Θ, α), where α iscalled an auxiliary parameter. An algorithm that consists ofiteratively minimizing F + (Θ, α) with respect to Θ and α isguaranteed to converge to a stationary point of the objectivefunction. It should be noted that this concept is adopted inmany existing algorithms [1, 14].
3.2. Update rules and algorithm
Here, we derive a majorizer for the objective function (12)with respect to H s and H n when target MGC representationSω,t (θ̂) with θ fixed to θ̂. First, DKL (Y |W H) involves a“log-of-sum” form of Wk,ω Hk,t . Since the negative logarithm function is a convex function, we can invoke Jensen’s
inequality to construct an upper bound of DKL (Y |W H)
having a “sum-of-logs” form in the same way as [1]+
DKL (Y |W H) ≤ DKL(Y |W H)

3. PROPOSED METHOD

s

X̂ω,t
−1 ,
Sω,t (θ)

(13)

+
DKL
(Y |W H)
c

=

X
ω,t

− Yω,tXk
Wk,ω Hk,t+ Xω,t ,ζk,ω,t log
ζk,ω,twhere =c denotes equality up to a constantterm and ζk,ω,t isP
a positive weight that sums to unity, k ζk,ω,t = 1. It can bewhere γk,ω,t is a positive weight that sums to unity. The
equality of (23) holds if and only ifshown that equality of (13) holds if and only if
ζk,ω,t = PK

Wk,ω Hk,tk0 =1 Wk ,ω Hk ,t
0

.

(14)

Then, we focus on the regularization termX  Yω,t Gsω,tc− log Gsω,t + log Gω,t ,J˜(W , H; θ̂) =Sω,t (θ̂)Gω,t
ω,t
(15)
PK
PKs
s
s
Wk,ω
Hk,t
and Gω,t = k=1 Wk,ω Hk,t .
where Gsω,t = k=1
To construct an upper bound for the first term of (15), we can
invoke the Lemma 1 introduced in [6]Gsω,tτω,t Gsω,t 2
1
≤
+
.
Gω,t
2
2τω,t G2ω,t

(16)

1
.
τω,t = sGω,t Gω,t(17)

Since a quadratic function is convex, we can apply Jensen’s
inequality to Gsω,t 2 , which yields
Kss 2 s 2
X
Wω,k
Hk,t
k=1

αk,ω,t

,

(18)

where
Pαk,ω,t > 0 is also a positive number that sums to unity,i.e., k αk,ω,t = 1. The equality of (18) holds if and only if(19)

We can use the fact that 1/x2 is convex in the first quadrant
and use Jensen’s inequality to obtain a majorizer:K3X
ξk,ω,t1≤(20)
2 H2 ,
G2ω,t
Wk,ω
k,t
k=1
P
where ξk,ω,t > 0 and k ξk,ω,t = 1. It can be proved thatthe equality of this inequality holds if and only if

ξk,ω,t = PK

Wk,ω Hk,tk0 =1 Wk ,ω Hk,t
0

.
0

Ks
K
s 2 s 2
3
X
X
τω,t Wk,ω
Hk,tξk,ω,tGsω,t≤+
2 H 2 . (22)Gω,t2αk,ω,t2τω,t Wk,ωk,tk=1

As regards the second term, Jensen’s inequality can be invoked again since − log Gsω,t is convex in Gsω,t ,− log Gsω,t ≤ −

Ks
X
k=1

γk,ω,t log

log Gω,t ≤

s
s
Wk,ω
Hk,t
,
γk,ω,tK
X
Wk,ω Hk,t
k=1

ηω,t

+ log ηω,t − 1,

(23)

(25)

where ηω,t is an arbitrary positive number. The equality ofthis inequality holds if and only if(26)

From (22), (23) and (25), we can construct a majorizer for the
regularization term as
J˜(W , H; θ̂) ≤ J˜+ (W , H, Γ; θ̂)3s 2 s 2X
X τω,t Yω,t Wk,ωξk,ω,t
Yω,t
Hk,t
+
=
2 H2
2αk,ω,t Sω,t (θ̂)
2τω,t Sω,t (θ̂)Wk,ω
k,t
k,ω,t
k,ω,t
−

X

γk,ω,t logss
X Wk,ω Hk,t
Wk,ω
Hk,t
+
+ d,
γk,ω,tηω,tk,ω,twhere Γ = {ζk,ω,t , τω,t , γk,ω,t , ηω,t , αk,ω,t , ξk,ω,t } denotes
a set of all the auxiliary variables and d denotes a constantterm. The upper bound for the objective function can be easily
obtained by combining the majorizers for each term as+
F + (W , H, Γ; θ̂) = DKL(Y |W H) + λJ˜+ (W , H, Γ; θ̂).

The update rules for Hk,t can be obtained by setting at zeros
the partial derivatives of the derived majorizer with respect tosnHk,tand Hk,t. Thus, the update rules can be obtained as the
positive solution of the following quartic and cubic equations:
X
ω

(21)

By substituting (18) and (20) into (16), the majorizer for the
first term can be written as

k=1

The third term log Gω,t is concave in Gω,t . Hence, we canuse the fact that a tangent line to the graph of a differentiable
concave function lies entirely above the graph:

k,ω,t

s
s
Wk,ω
Hk,t
αk,ω,t = PKs.
ssk0 =1 Wk0 ,ω Hk0 ,t(24)

ηω,t = Gω,t .

The equality of (16) holds if and only if

Gsω,t 2 ≤

s
s
Wk,ω
Hk,t
γk,ω,t = PKs.
ssk0 =1 Wk0 ,ω Hk0 ,t0

τω,t Yω,t2αk,ω,t Sω,t (θ̂)
X
X
s 2
−
γk,ω,t Hk,t−ωn
X Wk,ωωs 2 s 4Wk,ωHk,t +

ηω,t

ω
n 3
Hk,t
−

s
X Wk,ω
ω

ηω,t

s 3
Hk,t

3
Yω,t ξk,ω,ts 2
2τω,t Sω,t (θ̂)Wk,ω

X

3
Yω,t ξk,ω,tωn 22τω,t Sω,t (θ̂)Wk,ω

= 0.

= 0, (27)(28)

It is noteworthy that all the parameters can be updated in parallel using these update rules, which means this algorithm iswell suited to parallel implementations. Furthermore, sinceeach of the update rules consists of a negative 0th-order term
and a negative 2nd-order term, it turns out that there is only
one positive solution, implying that there is no need to solve
a solution selection problem.
Algorithm. 1 shows the whole procedure.

Algorithm 1 Algorithm presented in subsec. 3.2
Require: pretrained speech basis W and I MGC prototypesµ, parameters λ and M axItersn
1: random initialize H and H
2: for iter = 1 to M axIter do
3:
if iter ≤ 50 then4:
update H s and H n using SNMF
lambdalambda5:else
6:calculate the enhanced speech X̂ s using (4)Fig. 2. Average SDR improvement [dB] (left) and MFCC
7:
for Frame t = 1 to T do
distance improvement [dB] (right) obtained by MGCRNMF
s8:
rt = arg mini DIS (X t , µi )and the proposed method using 50 test samples with λ from
9:compute βt,i using (10)0.1 to 1.5 at 0.1 intervals. The points draw the maximum of
10:
Sω,t (θ̂) = βt,rt µω,rtthe curves.
11:
end for12:
update auxiliary variables Γ using (14), (17), (19),13:(21) , (24) and (26)from the ATR503 database [15] and 5 types of measured
14:
update H s , H n by solving the equations (27) and (28) noise, namely BusTerminal-5dB, Square-5dB, BowlingAlley15:
end if5dB, SubwayStation0dB and DepartmentStore0dB, excerpted
16: end forfrom the ATR ambient noise sound database. The test data
were created by adding noise signals to clean speech signals
with the signal-to-noise ratios (SNRs) of -5, 0 dB. All the
MGCRNMF2
audio signals were monaural and sampled at 16KHz. The
1.5
1
STFT was computed using the Hanning window with 32ms
0.5long and 16ms overlap.
0
0
20
40
60
80
100
120
140
160
180
200
At training phase, 200 utterances spoken by 2 male and
MGCRDNMF
1.5
2 female speakers were used to train 40 speech basis spectra.
1
For noise we used the same number of basis spectra. We run
200 iterations for SNMF basis training and 25 iterations for0.5
DNMF training. For DNMF, the basis obtained by running
0
0
20
40
60
80
100
120
140
160
180
200
Iteration
NMF for 100 iterations was used as an initialization. The
same training set was also used for K-means training. The
cluster number was set at 1000. We used 20 order MGCCs
Fig. 1. Percentage of the number of frames which changewith hyperparameter (γ, α) = (−1, 0.42) since it has been
the cluster to be assigned during the updating of MGCRNMF
shown in [8] that this hyperparameters setting can achieve rel(upper) and MGCRDNMF (bottom). The average number of
atively stable high performance under all conditions.
10 samples randomly selected under 5 noise types are shown
At test time, 50 utterances selected randomly fromin the figure.
ATR503 database were used as a test data set. All methods were run 200 iterations. Since each frame should finallyconverge to one cluster, we do not need to select the closest
Table 1. A comparison of runtime [sec] between updatprototype spectrum at every iteration. Fig. 4 shows that there
ing indicator variables at each iteration and updating themwas only about 1 % frames would change the cluster to which
only at the first iteration using MGCRNMF and the proposed
one it assigned after updating the indicator variables {rt }method. The length of the test data was 5 seconds.
once so that we can set the updating number of the indicator
w/ update w/o update
variables at 1 with considering the balance of the time cost
MGCRNMF
135.5534
1.9471and the performance. Tab. 4 shows a comparison of the runMGCRDNMF 259.9060
126.9996
time between updating indicator variable every iteration andonly updating it once using MGCRNMF and the proposed
method. The programs were run in the MATLAB 2015b with
Inter(R) Xeon E3-1505M V5 CPU @2.80GHz 64bit and 16.0
4. EXPERIMENTS
GB memory. The results show a significant improvement inTo evaluate the effect of the proposed method for speech enruntime between w/o update and w/ update. The performance
hancement task, we tested supervised NMF (SNMF) [2], Disdegradation caused by decreasing the updating number was
criminative NMF (DNMF) [6], NMF with mel-generalized
quite small which was only about 0.05 dB. It is noteworthy
cepstral regularization (MGCRNMF) [8] and the proposedthat without updating every iteration, MGCRNMF can realize
method (MGCRDNMF) using the speech data excerpted
a real-time running.
4

6.5

MGCRDNMF

6.3

MGCRDNMF

6.2
6.1
6

5.9
5.8

MGCRNMF

5.7

MFCC Improvement [dB]SDR Improvement [dB]

6.4

3.5

3

MGCRNMF

2.5

2

5.6

Percentage [%]Percentage [%]

5.5
0

0.2

0.4

0.6

0.8

1

1.2

1.4 1.5

0

0.2

0.4

0.6

0.8

1

1.2

1.4 1.5

Table 2. From top to bottom, there are respectively average SDR, SIR, MFCC distance Improvement [dB] evaluatedunder 5 noise conditions. The highest score of each term isshown in bold font type.
Noise TypeSNMF MGCRNMF DNMF Proposed
BusTerminal
10.71
11.22
11.57
11.94
Square
6.19
6.45
6.76
6.88
BowlingAlley
3.37
3.40
4.01
4.30
SubwayStation
3.90
3.78
4.22
4.46
DepartmentStore 4.73
4.95
4.76
4.97
Noise TypeSNMF MGCRNMF DNMF Proposed
BusTerminal
13.85
15.07
17.43
18.32
Square
8.61
9.29
10.32
11.14
BowlingAlley
5.27
5.77
6.72
7.40
SubwayStation
6.74
7.71
8.27
8.87
DepartmentStore 7.09
7.89
8.92
10.42
Noise TypeSNMF MGCRNMF DNMF Proposed
BusTerminal
1.79
2.27
3.24
3.66
Square
1.87
2.10
1.84
3.05
BowlingAlley
1.32
1.97
2.81
3.13
SubwayStation
1.81
2.52
2.85
3.51
DepartmentStore 1.79
2.27
2.36
3.22

We investigated the weight parameter λ during 0.1 to 1.5
at 0.1 intervals and the results are shown in Fig. 4. According
to the Fig. 4, we set λ = 0.4 for MGCRNMF and λ = 0.5
for the proposed method. We used Signal-to-distortion ratios(SDRs), signal-to-interference ratios (SIRs) [16] and MFCC
distance for the evaluation. Given two D-dimension MFCC
sequences x[d] and y[d] calculated from N frequency bins,
the MFCC distance is defined as follow:
v
u D
X
20D u
t2
(x[d] − y[d])2 .
(29)
Dist =
N ln 10method outperformed the existing methods SNMF, DNMF
and MGCRNMF in terms of SDR, SIR and MFCC distance
improvements, which showed the effectiveness of the proposed method.
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dTab. 4 shows the results of average SDR, SIR and MFCC
distance improvement [dB] obtained using SNMF, DNMF,
MGCRNMF and the proposed method under 5 noise conditions. The proposed method outperformed the other methods
under all the conditions in terms of all the evaluation criteria.
5. CONCLUSION
This paper proposed an unified approach combining a DNMF
basis training phase and mel-generalized cepstral regularization NMF which considers an optimization problem combining a NMF-based model-fitting criterion and a MGC regularization which measures IS divergence between the pretrained
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