INTERSPEECH 2016
September 8–12, 2016, San Francisco, USA

Semi-Supervised Joint Enhancement ofSpectral and Cepstral Sequences of Noisy Speech
Li Li1 , Hirokazu Kameoka1,2 , Takuya Higuchi2 and Hiroshi Saruwatari1
1

Graduate School of Information Science and Technology, The University of Tokyo, Japan
2
NTT Communication Science Laboratories, NTT Corporation, Japan{li li, hiroshi saruwatari}@ipc.i.u-tokyo.ac.jp{kameoka.hirokazu, higuchi.takuya}@lab.ntt.co.jp

Abstractfrom those of the training condition for these techniques to work
successfully. Other feature domain methods have more or lessthe same limitations [4–7].
In contrast to the feature domain methods, spectral domain
methods are particularly noteworthy in that they can work successfully even without any prior knowledge about the noisecharacteristics. This is because we can make use of a reasonable spectrum model to estimate the underlying speech components in an observed spectrum thanks to the additive natureof speech and noise components in the spectral domain. The
semi-supervised non-negative matrix factorization (NMF) approach [8] is an example of such methods, and it has attracted
a lot of attention in recent years. Factorizing the magnitude (orpower) spectrogram of a mixture signal, interpreted as a nonnegative matrix, into the product of two non-negative matrices can be interpreted as approximating the observed spectra ateach time frame as a linear sum of basis spectra scaled by timevarying amplitudes. This amounts to decomposing the observed
spectrogram into the sum of low rank spectrograms. In a semisupervised setting, the basis spectra of speech are ﬁrstly trainedusing clean speech samples. NMF is then applied to an observed
noisy speech spectrogram, where a subset of the basis spectra is
ﬁxed at the pretrained spectra. In this way, we can separate out
the underlying speech components using the Wiener ﬁlter obtained with the estimated speech and noise spectrograms. Although this approach is powerful in terms of a signal-to-noise
ratio measure or some subjective criteria, one drawback is thatthey do not necessarily lead to an improvement in the qualityof the enhanced speech in the feature domain. This implies thatnaively using these algorithms as front-end processing for applications such as speech recognition and voice conversion does
not always lead to satisfactory results.
As stated above, feature domain methods (e.g., SPLICE)
and spectral domain methods (e.g., NMF) have their own advantages and disadvantages. To address the drawbacks andcombine the advantages of these methods, this paper proposesa novel approach that aims to jointly enhance the spectral andcepstral sequences of noisy speech, by optimizing a combined
objective function consisting of an NMF-based model-ﬁtting
criterion deﬁned in the spectral domain and a GMM-based
probability distribution deﬁned in the cepstral domain. We derive a novel majorizer for this objective function, which allows
us to derive a convergence-guaranteed iterative algorithm basedon a majorization-minimization scheme for optimization.

While spectral domain speech enhancement algorithms using non-negative matrix factorization (NMF) are powerful interms of signal recovery accuracy (e.g., signal-to-noise ratio),
they do not necessarily lead to an improvement in the quality of the enhanced speech in the feature domain. This implies that naively using these algorithms as front-end processing for e.g., speech recognition and speech conversion does
not always lead to satisfactory results. To address this problem, this paper proposes a novel method that aims to jointly
enhance the spectral and cepstral sequences of noisy speech,by optimizing a combined objective function consisting of an
NMF-based model-ﬁtting criterion deﬁned in the spectral domain and a Gaussian mixture model (GMM)-based probability
distribution deﬁned in the cepstral domain. We derive a novel
majorizer for this objective function, which allows us to derive a convergence-guaranteed iterative algorithm based on amajorization-minimization scheme for the optimization. Experimental results revealed that the proposed method outperformed the conventional NMF approach in terms of both signalto-distortion ratio and cepstral distance.
Index Terms: speech enhancement, Gaussian mixture model,
non-negative matrix factorization, mel-frequency cepstral coefﬁcients, majorization-minimization

1. Introduction
The presence of noise in speech can signiﬁcantly degrade thequality of speech transmission systems and the performanceof speech processing systems including speech recognition andvoice conversion. Many speech enhancement algorithms havebeen proposed with the goal of overcoming this problem. Conventional speech enhancement methods can be roughly divided
into two types, according to the domain in which the enhancement is performed, namely feature domain methods and spectral domain methods. The former aims to recover clean speechfeatures (typically cepstral features) whereas the latter aims torecover clean speech spectra (or signals).
Since feature domain methods directly enhance the featuresof speech, they are particularly useful as a front-end for speechprocessing systems that use speech features as inputs. Stereo
piecewise linear compensation for environment (SPLICE) [1,2]is a typical example of such methods. With this approach, aGaussian mixture model (GMM) is used to model the joint
distribution of clean and noisy speech features. The GMM istrained using stereo synchronous data of noisy and clean speechsamples. By using the trained GMM, a mapping function froma noisy speech feature to its clean version is deﬁned as the conditional expectation of a clean speech feature given a noisy observation. Since each of the Gaussians represents a linear transform, the mapping is piecewise linear. Although it has beenshown that it yields a steady improvement in speech recognition performance, one drawback is that the performance tends
to be poor when the test condition does not match the training
condition (when we face unseen noise). Adaptation techniques
can be used to compensate for this mismatch [3], but the noisecharacteristics of the test condition must not differ signiﬁcantly

Copyright © 2016 ISCA

2. Formulation
2.1. Problem setting
We start by reviewing the formulation of the NMF approach.
Let us denote an observed magnitude (or power) spectrogramas Y = (Yω,t )Ω×T ∈ R≥0,Ω×T , where ω and t are frequency and time indices. Given an observed spectrogram Y,
we consider approximating it with the sum of the speech and
(s)
(n)
(s)
noise components, Xω,t = Xω,t + Xω,t , where Xω,t and(n)Xω,t are represented by non-negative linear combinations of(s)(s)
Ks speech basis spectra H1,ω , . . . , HKs ,ω and Kn noise basis3753

http://dx.doi.org/10.21437/Interspeech.2016-1286

(n)

(n)

spectra H1,ω , . . . , HKn ,ω :(s)Xω,t =Ks(s)

2.2. Proposed method seen as regularized NMF
(s)

(n)

Hk,ω Uk,t , Xω,t =

k=1

Kn


(n)

(n)

Hk,ω Uk,t .

The present problem setting can be seen as a regularized variant of NMF if we interpret −K(X (s) ) as a regularization term.
When some of the speech and noise basis spectra become similar, the decomposition of an observed spectrum into speech andnoise components will not be unique. In such a case, components originating from speech can be misinterpreted as noise
components and vice versa, leading to an inaccurate separation.
The regularization term is expected to play the role of eliminating this kind of indeterminacy by keeping the estimated speech
spectra within a proper range in the MFCC domain. In thissense, the regularization term can contribute more to improving the separation performance than the regular (unregularized)
NMF. This effect will be conﬁrmed in Sec. 3.

(1)

k=1

In a semi-supervised setting, the speech basis spectra(s)(s)
H1,ω , . . . , HKs ,ω are pretrained using clean speech samples.
(s)Thus, U , H (n) and U (n) are the unknown variables to be
estimated in the separation process. NMF leads to different optimization problems according to the deﬁnition of the measureof the difference between Y and X = (Xω,t )Ω×T . Here weuse the I divergence


Yω,t
Yω,t log− Yω,t + Xω,t , (2)I(Y |X) =Xω,tω,t2.3. Majorization-minimization algorithm
Although it is difﬁcult to solve the above optimization problem
analytically, we can develop a computationally efﬁcient algorithm to ﬁnd a locally optimal solution based on a majorizationminimization method. Note that the majorization-minimization
method itself is not an algorithm, but a description of how
to construct an optimization algorithm. When applying the
majorization-minimization method to the problem of minimizing a certain objective function, the ﬁrst step is to design an auxiliary function called a “majorizer” that never lies below the objective function. Suppose F (Θ) is an objective function that wewish to minimize with respect to Θ. A majorizer F + (Θ, α) isthen deﬁned as a function satisfying F (Θ) = minα F + (Θ, α),where α is called an auxiliary parameter. An algorithm that consists of iteratively minimizing F + (Θ, α) with respect to Θ andα is guaranteed to converge to a stationary point of the objective function. It should be noted that this concept is adopted
in many existing algorithms. For example, the expectationmaximization (EM) algorithm [11] builds a surrogate for a likelihood function of latent variable models by using Jensen’s inequality. It is also well known for its use in devising an algorithm for NMF [9, 10].
In this section,
we derive a majorizer ofJ (U (s) , H (n) , U (n) ), according to which we obtain the
update equations for U (s) , H (n) and U (n) . First, I(Y |X)
involves a “log-of-sum” form of Hk,ω Uk,t . Since the negative
logarithm function is a convex function, we can invoke Jensen’s
inequality to construct an upper bound of I(Y |X) with a“sum-of-logs” form in the same way as [9]as the goodness-of-ﬁt criterion. Once U (s) , H (n) and U (n) are
obtained with an NMF algorithm, we can separate out the underlying speech components using the Wiener ﬁlter constructed(s)(n)
with X (s) = (Xω,t )Ω×T and X (n) = (Xω,t )Ω×T . As statedabove, this method does not necessarily lead to an improvementin the quality of the enhanced speech in the feature domain, implying that naively using this method as a front-end for suchapplications as speech recognition and voice conversion does
not always produce satisfactory results. When performing separation, we would want to ensure that the cepstral feature of
the separated component is also enhanced. With this as motivation, we introduce a probability density function over the cepstral representation of X (s) to deﬁne an additional cost function. Here, we deﬁne it with the logarithm of a GMM in the
MFCC domain


(s)
2
wm
N (Xn,t ; μn,m , σn,m), (3)K(X (s) ) = log(s)Xn,t =

tmcn,l logn(s)fl,ω Xω,t ,

(4)

ω

l
(s)



(s)

(s)

= (X0,t , . . . , XN −1,t )T is an N -dimensional
where X t
mel-frequency cepstral coefﬁcient (MFCC) vector of(s)(s)
X0,t , . . . , XΩ−1,t . Here, fl,ω is the lth coefﬁcient of the
mel-ﬁlter banks and {cn,l }0≤n≤N −1,0≤l≤L−1 are the coefﬁcients of the discrete cosine transform. Note that this expression
reduces to the log mel-spectrum when cm,n = δm,n (where δdenotes Kronecker’s delta), and the log-power spectrum whenfn,ω = δn,ω and cm,n = δm,n , implying that the following
derivation does not restrict the feature vector deﬁnition to
MFCC alone. θ = {μm , Σm , wm }1≤m≤M is a set consistingof the GMM parameters where μm = (μ1,m , . . . , μN,m )T ,
Σm = diag(σ1,m , . . . , σN,m ) and wm are the mean, covariance and weight of the mth Gaussian components. As with the
speech basis spectra, θ is pretrained from clean speech samples.
Thus, the greater Eq. (4) becomes, the more likely X (s) is to
be enhanced in the feature domain.
The proposed method considers an optimization problem
that consists of minimizing a combined objective function of
Eqs. (2) and (4)J (U (s) , H (n) , U (n) ) = I(Y |X) − λK(X (s) ),I(Y |X) ≤ I + (Y |X),
(6)




Hk,ω Uk,tc− Yω,tI + (Y |X) =ζk,ω,t log+ Xω,t ,ζk,ω,tω,tkwhere =c denotes equality up to a constantterm and ζk,ω,t is
a positive weight that sums to unity, k ζk,ω,t = 1. It can beshown that the equality of Eq. (6) holds if and only if
ζk,ω,t = 

Hk,ω Uk,t.
k Hk ,ω Uk ,t(7)

Note that for convenience of notation here we have deﬁned
(5)

(s)

Hk,ω = Hk,ω (k = 1, . . . , Ks ),where λ ≥ 0 weighs the importance of the MFCC-GMM term
relative to the NMF cost. This optimization problem is mathematically challenging in the sense that the objective function
simultaneously involves a spectral distance term I(Y |X) anda cepstral distance term K(X (s) ). Although K(X (s) ) is simply
a log-GMM when viewed as a function of cepstral parameters{X (s) }, it becomes more complicated when viewed as a function of spectral parameters {X (s) }. This paper proposes a novel
general framework for solving this class of optimization problems.

(n)

Hk+Ks ,ω = Hk,ω (k = 1, . . . , Kn ),(s)

Uk,t = Uk,t (k = 1, . . . , Ks ),(n)

Uk+Ks ,t = Uk,t (k = 1, . . . , Kn ).
Next, we derive a majorizer of the MFCC-GMM term
−K(X (s) ). In the same way as Eq. (6), we can construct
an upper bound of the negative logarithm function by invoking3754

where ρl,k,ω,t is a positive weight that sums to unity,ω,k ρl,k,ω,t = 1. It can be shown that the equality of thisinequality holds if and only if

Jensen’s inequality−K(X (s) ) ≤ −

αm,t logwm2n N (Xn,t ; μn,m , σn,m )

αm,t

t,m
c

=



αm,t

 (Xn,t − μn,m )2

t,m

n

2
2σn,m

ρl,k,ω,t = ,(8)Since the quadratic function is a convex function, we can employ the inequality used in [12, 13] to construct an upper boundof Eq. (8) (cn,l log Gl,t − ϕl,n,m,t )2βl,n,m,tl,where φl,t is an arbitrary positive number. This is simply givenby the fact that a tangent line to the graph of a differentiable
concave function lies entirely above the graph and that a logarithm function is a concave function. We can easily conﬁrm thatthe equality of this inequality holds if and only if(10)

φl,t = Gl,t .

where βl,n,m,t is an arbitrary positive number that sums to
βl,n,m,t = 1, and ϕl,n,m,t is a real number that sums
unity, l 
to μn,m , l ϕl,n,m,t = μn,m . For convenience of notation(s)
we use Gl,t to denote ω fl,ω Xω,t . It can be shown that theequality of this inequality holds if and only if
ϕl,n,m,t = cn,l log Gl,t + βl,n,m,t (μn,m − Xn,t ).

Bl,t log Gl,t ≤ Bl,t(11)

υk,l,ω,t = 

where d is a constant term that does not depend on Hk,ω and
Uk,t . Here, for convenience of notation, we have deﬁned
αm,t c2n,l,22σn,m βl,n,m,t
n,mBl,t = −(14)

1(log Gl,t ) ≤+ p(ξl,t )Gl,t + q(ξl,t ),Gl,t2 log ξl,t
1
+ 2 ,
ξl,t
ξl,t

q(ξl,t ) = (log ξl,t )2 − 2 log ξl,t −(15)− K(X (s) )


Al,t
≤
t,l

(17)







.

(23)

(24)
ρ2l,k,ω,t

ω,k

t,l

to construct an upper bound of the ﬁrst term of Eq. (12). We
can conﬁrm that the equality of this inequality holds if and onlyifξl,t = Gl,t .

+ p(ξl,t )Gl,t + q(ξl,t )fl,ω Hk,ω Uk,t



Gl,t
δBl,t ≥0 |Bl,t |+ log φl,t − 1
+
φl,t

(16)
2
,
ξl,t

fl,ω Hk,ω Uk,t
ω  ,k fl,ω Hk ,ω Uk ,twhere δ is an indicator function that takes the value 1 if its argument is true and 0 otherwise.
To sum up, a majorizer of −K(X (s) ) can be written as

wherep(ξl,t ) =

(22)

ω,k

Since Al,t is non-negative, we can use the inequality [14]:2fl,ω Hk,ω Uk,t,υk,l,ω,t

By combining Eqs. (20) and (22), we can write an upper boundof Bl,t log Gl,t asGl,tBl,t log Gl,t ≤δBl,t ≥0 |Bl,t |+ log φl,t − 1φl,tfl,ω Hk,ω Uk,tυk,l,ω,t log,−δBl,t <0 |Bl,t |υk,l,ω,t(13)

 αm,t cn,l ϕl,m,n,t.
2σn,mβl,n,m,tn,mυk,l,ω,t log

where υk,l,ω,t > 0 is a positive weight that sums to unity,k,ω υk,l,ω,t = 1. It can be shown that the equality of thisinequality holds if and only if

(12)




ω,k

t,l

Al,t =(21)

Second, when Bl,t is negative, Bl,t log Gl,t becomes convex in
Gl,t , and so we can invoke Jensen’s inequailty to construct anupper boundFrom Eqs. (10) and (8), we obtain

−K(X (s) ) ≤Al,t (log Gl,t )2 +Bl,t log Gl,t + d,
t,l

(19)

Care must be taken of the fact that Bl,t can be either nonnegative or negative. Thus, we consider applying different inequalities to the second term of Eq. (12) according to the signof Bl,t . First, when Bl,t is non-negative, we can show thatGl,tBl,t log Gl,t ≤ Bl,t
+ log φl,t − 1 ,
(20)
φl,t

where αm,t is also a positive weight that sums to unity. The
equality of this inequality holds if and only if
2
wm n N (Xn,t ; μn,m , σn,m)
αm,t = 
.
(9)
2

, σ
w
N
(X
;
μ
n,t

m
n,m
m
n
n,m )(Xn,t − μn,m )2 ≤fl,ω Hk,ω Uk,t.
fl,ω Hk ,ω Uk ,tω  ,k

−


t,l

(18)

δBl,t <0 |Bl,t |ω,kυk,l,ω,t log

fl,ω Hk,ω Uk,t
+ d.
υk,l,ω,t

By combining this with the majorizer of I(Y |X), we can construct a majorizer of the objective function of interest. This majorizer is particularly noteworthy in that it is given as the sum ofa reciprocal function, logarithm functions and a ﬁrst order function of Hk,ω Uk,t , which can be minimized analytically withrespect to Hk,ω and Uk,t . We have already seen that the update
equations for the auxiliary parameters are given by Eqs. (9),(11), (18), (19), (21) and (23). The next step is to derive theupdate equations for Hk,ω and Uk,t .

By focusing on the fact that a reciprocal function is convex in
the positive domain and that fl,ω Hk,ω Uk,t is positive, we canapply Jensen’s inequality to 1/Gl,tρ2l,k,ω,t
1
1
= 
≤
,
Gl,t
fl,ω Hk,ω Uk,tω,k fl,ω Hk,ω Uk,t
ω,k

3755

2.4. Update rules
We can obtain the update rules for Hk,ω and Uk,t by setting
the partial derivatives of the proposed majorizer with respect to
H (s) , U (s) , H (n) and U (n) at zero. Note that the update rulesof the ﬁrst two are given as the solutions to quadratic equations.
Since H (s) and U (s) must be non-negative, a positive solution
must be selected. Thus, we ﬁnally arrive at the following update
equations2ak,ω
ζk+Ks ,ω,t Yω,tt,t Uk+Ks ,t

Uk,t + λtbk,ω = −,ζk,ω,t Yω,t − λtck,ω = −λ

(27)

4.5

4

3.5

NMF

(28)

φl,t

ζk,ω,t Yω,t − λproposedfl,ω Uk,t ,δBl,t <0 |Bl,t |υk,l,ω,t

8
6
4

0

Babble noise

12

NMF
proposed

10

10

8
6
4
2

closed female closed male open female0

open maleclosed female closed male open female

open male

Figure 3: SDR improvement with proposed method and semi10



φl,t

Museum noise

SDR improvement [dB]fl,ω Hk,ω ,7
NMF
proposed

9
8

6

7
6
5
4
3

Background music noise
NMF
proposed

5
4
3
2

2
1

1
0

δBl,t <0 |Bl,t |υk,l,ω,t ,closed female closed male open female

open male0

closed female closed male open female

open male

Figure 4: SDR improvement with proposed method and semi-supervised NMF. With measured museum noise (left) and backgroundmusic noise (right).

ω,lfl,ω Hk,ω

White noise2 δBl,t ≥0 |Bl,t |
ω,l

ω,l

NMF

supervised NMF. With white noise (left) and babble noise (right).

ωfk,t = −λ4.5

proposed

NMF
proposedω,l Al,t ρ2l,k,ω,t5

semi-supervised NMF. With measured museum noise (left) and background music noise (right).

,+λek,t = −6

5.5

Figure 2: MFCC distance improvement with proposed method andfl,ω Uk,t
Hk,ω + λAl,t p(ξl,t )fl,ω Hk,ωdk,t =

proposed

Background music noise

5

l,t

ω

MFCC distance improvement [dB]

MFCC distance improvement [dB]

Museum noisel,t Al,t ρ2l,k,ω,t

NMF

MFCC distance improvement [dB], δBl,t ≥0 |Bl,t |


3
2.8

proposed

5.5

12

l,t

3.2

2.6

(25)

Al,t p(ξl,t )fl,ω Uk,t

+λ

3.4

semi-supervised NMF. With white noise (left) and babble noise (right).

l,t



4.6

3.6

SDR improvement [dB]ak,ω =

4.8

NMF

(26)

2dk,t

ζ
k+K
(n)
s ,ω,t Yω,tω
Uk,t = ,H
k+Ks ,ω
ω
where

5

3.8

6.5

e2k,t − 4dk,t fk,t−ek,t +

5.2

SDR improvement [dB](s)

Uk,t =



5.4

4

Figure 1: MFCC distance improvement with proposed method and

MFCC distance improvement [dB](n)Hk,ω =

b2k,ω − 4ak,ω ck,ω−bk,ω +

4.2

5.6

SDR improvement [dB](s)
Hk,ω =

Babble noise

White noise
5.8.

NMF that was run for 500 iterations. The enhanced speech signals were obtained using the Wiener ﬁlter constructed by theestimated speech and noise spectrograms.
Fig. 1–4 show the results of the proposed method tested onfour types of noise. As the results show, the proposed method
yielded a 0.6 dB higher cepstral distance improvement in average and a slightly higher SDR improvement over conventionalsemi-supervised NMF.

We can conﬁrm that when λ = 0 the update rules reduce tothose of the regular NMF with the I divergence.

3. Experiments
To conﬁrm the effect of the proposed method, we evaluated the
cepstral distance (the mean square distance in the MFCC domain) between the estimated and clean speech signals and thesignal-to-distortion ratios (SDRs) of the estimated speech signals using the ATR503 database [15]. We chose the conventional semi-supervised NMF method [8] for comparison.
We used four types of noise: white noise, babble noise,measured museum noise and background music noise. The
test data were created by adding noise sources to clean speechsources set with different signal-to-noise ratios (SNRs), ranging from -15dB to 5dB. All the audio signals were monauraland sampled at 16kHz. The STFT was computed using a Hanning window that was 32ms long with a 16ms overlap. In the
training process, the MFCC vectors were extracted from clean
speech samples uttered by 6 female and 4 male speakers (450sentences for each speaker) and a GMM with 30 components
was trained using the EM algorithm. We set the dimension of
the MFCC vector at 13. In the separation process, we set λ at 1.
The parameters were initialized using a regular semi-supervised

4. Conclusions
This paper proposed a novel approach for jointly enhancing
the spectral and cepstral sequences of noisy speech. The
method optimizes the combined objective function consistingof an NMF-based model-ﬁtting criterion deﬁned in the spectral domain and a Gaussian mixture model (GMM)-based probability distribution deﬁned in the cepstral domain based on amajorization-minimization scheme. Experimental results revealed that the proposed method outperformed the conventional
NMF approach in terms of both SDR and cepstral distance.

5. Acknowledgements
This work was supported by JSPS KAKENHI Grant Numbers
JP26730100 and JP26280060, and SECOM Science andTechnology Foundation.

3756

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