LP -NORM NON-NEGATIVE MATRIX FACTORIZATION
AND ITS APPLICATION TO SINGING VOICE ENHANCEMENT
Tomohiko Nakamura† and Hirokazu Kameoka†,‡†

Graduate School of Information Science and Technology, The University of Tokyo.
7-3-1, Hongo, Bunkyo-ku, Tokyo, 113-8656, Japan.
‡
NTT Communication Science Laboratories, Nippon Telegraph and Telephone Corporation.
3-1, Morinosato Wakamiya, Atsugi, Kanagawa, 243-0198, Japan
Tomohiko_Nakamura@ipc.i.u-tokyo.ac.jp,kameoka.hirokazu@lab.ntt.co.jp
ABSTRACT
Measures of sparsity are useful in many aspects of audio signal processing including speech enhancement, audio coding and singing
voice enhancement, and the well-known method for these applications is non-negative matrix factorization (NMF), which decomposesa non-negative data matrix into two non-negative matrices. Althoughprevious studies on NMF have focused on the sparsity of the two
matrices, the sparsity of reconstruction errors between a data matrix
and the two matrices is also important, since designing the sparsity
is equivalent to assuming the nature of the errors. We propose a
new NMF technique, which we called L p -norm NMF, that minimizes
the L p norm of the reconstruction errors, and derive a computationally eﬃcient algorithm for L p -norm NMF according to an auxiliary
function principle. This algorithm can be generalized for the factorization of a real-valued matrix into the product of two real-valued
matrices. We apply the algorithm to singing voice enhancement andshow that adequately selecting p improves the enhancement.
Index Terms— Non-negative matrix factorization, L p norm, auxiliary function principle
1. INTRODUCTION
Non-negative matrix factorization (NMF) [1] is a powerful techniquethat approximates a data matrix Y by using the product of two nonnegative matrices W and H. NMF has been actively studied in many
scientific and engineering fields in recent years (see [2]). In particular, in the field of music signal processing, successful results wereobtained by regarding a magnitude spectrogram as a non-negative
matrix [3].
NMF is formulated as the problem of minimizing a measure between a data matrix and a model. One standard measure is the Frobenius norm, and it corresponds to assuming that the reconstruction
errors are additive Gaussian noise. While NMF with the Frobenius
norm works well for small errors, NMF performance deteriorates
for large errors such as spike noise, even if there are a few entries
with errors. We thus need to explore appropriate measures for large
errors.
Measures of sparsity (e.g. L1 norm) have been widely used inmany audio signal processing techniques such as NMF and robust
principle component analysis [4, 5]. Many previous studies of NMF
have used sparseness measures [6, 7], regularizers [8] and priors [9]This work was partly supported by JSPS Grant-in-Aid for Young Scientists B Grant Number 26730100.

978-1-4673-6997-8/15/$31.00 ©2015 IEEE

2115

on W and H, which induce sparse solutions. However, it is diﬃcult
to use the measures between a data matrix and a model directly since
the measures are often non-linear and not diﬀerentiable, and it isintractable to solve the NMF problem.
In this paper, we propose a new NMF technique, which we called
L p -norm NMF, that minimizes the L p norm of reconstruction errors
between a data matrix and a model. We formulate L p -norm NMF
with 0 < p ≤ 2 and derive a convergence-guaranteed algorithm thatconsists of multiplicative update equations for W and H based on
the auxiliary function principle [10, 11]. The constant p controls thesparsity of the reconstruction errors. To examine the eﬀect of varying
p, we apply the proposed algorithm to singing voice enhancement
for monaural audio signals.
We henceforth denote sets of real values and non-negative realvalues as R and R≥0 , respectively.
2. LP -NORM NON-NEGATIVE MATRIX FACTORIZATION
2.1. Problem setting
Let us define frequency, time, and basis indexes, respectively, as ω ∈[0, Ω − 1], t ∈ [0, T − 1] and k ∈ [0, K − 1] such that K < Ω andK < T . Given a non-negative data matrix Y := {yω,t }0≤ω≤Ω−1,0≤t≤T −1 ,L p -norm NMF is the problem of minimizing the L p norm∑∑p
L(W, H) := ∥Y − WH∥ pp =wω,k hk,t(1)
yw,t −
ω,tsubject to∀k,∑Wω,k = 1.

k

(2)

ω

Here W = {wω,k }ω,k is a Ω × K non-negative matrix and H = {hk,t }k,t
is a K × T non-negative matrix. Eq. (2) is introduced to avoid anindeterminacy in the scaling. When Y is a magnitude spectrogram,the columns of W represent spectral templates and the rows of H
represent the temporal activities of the spectral templates.
The constant p controls the sparsity of the reconstruction error.
A smaller p induces the sparsity, and most of the entries of the estimated data matrix WH are the same as those of Y, while the otherentries diﬀer greatly from those of Y. On the other hand, a larger p
does not induce the sparsity, and all the entries of the estimated data
matrix may be non-zero. We thus assume that the L p norm satisfies
0 < p < 2, which promotes sparsity. Note that when p = 2, theobjective function equals the NMF with Frobenius norm.

ICASSP 2015

2.2. Eﬃcient algorithm based on auxiliary function principle
Since L(W, H) involves a summation over k in the L p norm, it isintractable to solve the current minimization problem analytically.
However, we can develop a computationally eﬃcient algorithm tofind a locally optimal solution based on the auxiliary function principle [10, 11].
When applying an auxiliary function principle to the minimization problem, the first step is to define an upper bound function forthe objective function. Introducing auxiliary variables ξ = {ξω,t ∈R≥0 }ω,t , we derive
∑
∑
p
2
p−2
p
yw,t −wω,k hk,t ≤ pξω,tyw,t −wω,k hk,t + (2 − p)ξω,t
. (3)
k

k

The proof of the inequality is described in Lemma 2 of [12]. The
equality of Eq. (3) holds if and only if∑
ξω,t = yω,t −(4)wω,t hk,t .
k

The term in the square function of Eq. (3) includes a summation
over k, and we further derive the upper bound of the right-hand sideof Eq. (3). Since the square function is a convex function, we can
invoke Jensen’s inequality:)2 ∑ 1(∑wω,k hk,t ≤(wω,k hk,t )2(5)
λω,t,kkkwhere
λ = {λω,t,k ∈ R≥0 }ω,t,k are auxiliary variables that sum to unity:∑k λω,t,k = 1. The equality of Eq. (5) holds if and only ifwω,k hk,t
λω,t,k = ∑.
k′ wω,k′ hk′ ,t(6)

The upper bound of L(W, H) can thus be described as
L+ (W, H, λ, ξ)
∑
∑ 1
∑
{
}
p−2 2
=
pξω,t
yw,t − yω,twω,k hk,t +(wω,k hk,t )2
λ
ω,t,k
ω,t
k
k
∑
p
+
(2 − p)ξω,t.

J(W, H) = ∥Y − WH∥ pp ,∑subject to ∀k,
Wω,k = 1
min

W∈RΩ×K ,H∈RK×Tω

where 0 < p < 2 and J(W, H) is the objective function.
3.2. Iterative algorithm based on auxiliary function principle
As in Sec. 2, J(W, H) involves summation over k in the L p norm,
and we consider the upper bound of J(W, H) to apply the auxiliary function principle to the intractable minimization problem. The
inequality used in Eq. (3) is applicable to J(W, H), and the upperbound of J(W, H) is the same as the right-hand side of Eq. (3). To
derive the upper bound of the square function of Eq. (3), we can use
the generalization of Jensen’s inequality for yω,t , wω,k , hk,t ∈ R, which
was employed in [12]:∑∑ |αω,t,k − wω,k hk,t |22yω,t −wω,k hk,t ≤(11)
βω,t,kkkwhere α = {αω,t,k ∈∑R}ω,t,k , β = {β∑ω,t,k ∈ [0, 1]}ω,t,k are auxiliary
variables subject to k αω,t,k = yω,t , k βω,t,k = 1. The equality ofEq. (11) holds if and only if
)
(∑
(12)
wω,k′ hk′ ,t − yω,t .
αω,t,k = wω,k hk,t − βω,t,kk′

The upper bound of J + (W, H) can thus be described as∑∑ |αω,t,k − wω,k hk,t |2p−2pJ + (W, H, ξ, α, β) =pξω,t+ (2 − p)ξω,t
.
β
ω,t,k
ω,t
k
(13)
+

(7)

ω,t

We can derive update equations for the parameters, using the above
upper bound. By setting the partial derivative of L+ (W, H, ξ, λ) withrespect to W and H at zero and using Eqs. (4) and (6), we obtain
W ←W ⊙ [{(Y ⊘ C)H ⊤ } ⊘ {(WH ⊘ C)H ⊤ }]H ←H ⊙ [{W ⊤ (Y ⊘ C)} ⊘ {W ⊤ (WH ⊘ C)}]

factorization (L p -norm MF).
L p -norm MF is the problem of finding real-valued matrices W, H
for a given real-valued data matrix Y such that

(8)
(9)

where ⊙, ⊘ denote element-wise product and division, and C is anΩ × T matrix, whose (ω, t)-th element is∑2−pcω,t = yω,t −wω,k hk,t .
(10)k

It is worth noting that each update equation consists of multiplication
by a non-negative factor. Hence, it is guaranteed that the entries of
W and H are always non-negative when their initial values are set at
non-negative values.
3. LP -NORM MATRIX FACTORIZATION
3.1. Problem setting
Although the algorithm in the previous section is for non-negative
matrices, it can be generalized for real-valued matrices, according to
the auxiliary function principle. We call the problem L p -norm matrix

2116

By diﬀerentiating J (W, H, ξ, α, β) partially with respect to W and
H and setting them at zero, we can obtain the update equations:)∑∑ −1 ( −1k′ wω,k′ hk′ ,tt cω,t hk,t βω,t,k wω,k hk,t + yω,t −(14)
wω,k ←∑ −1 −1 2t cω,t βω,t,k hk,t
)
(
∑ −1
∑
−1
ω cω,t wω,k βω,t,k wω,k hk,t + yω,t −k′ wω,k′ hk′ ,thk,t ←(15)∑ −1 −1 2ω cω,t βω,t,k wω,kwhere cω,t is defined as Eq. (10).
3.3. Relation to L p -norm NMF
The ∑parameters β can be chosen arbitrarily subject to βω,t,k ∈ R≥0
and k βω,k,t = 1. Choosing β as in [12], we obtain the multiplicative
update equations as in Eqs. (8) and (9). Therefore, we can say thatthe algorithm of L p NMF is a special case of that of L p MF.
4. APPLICATION TO SINGING VOICE ENHANCEMENT
4.1. Singing voice enhancement
In this section, we apply L p -norm NMF to singing voice enhancement whose aim is to extract a singing voice from a monaural mixed
audio signal consisting of vocal and accompaniment parts. Singing
voice enhancement is often used in music information retrieval(MIR) applications such as automatic lyrics recognition [13, 14],
automatic singer identification [15], and automatic karaoke generators [16].

We utilize the fact that there is a diﬀerence in spectrogram between
accompaniments and singing voices. Accompaniment sounds are
generated by musical instruments, which reproduce approximately
the same sounds every time they are played. We can see the spectrograms of accompaniment signals as a low-rank matrix. In constrast, singers fluctuate their singing voices to obtain musical eﬀects
such as vibrato. The spectrograms of singing voices are relatively
high rank and sparse, and corresponds to reconstructions errors of
L p -norm NMF.
As mentioned in Sec. 2.1, the model spectrogram WH of L p -norm
NMF is a non-negative matrix with the rank of K, and small p induces sparsity of the reconstruction errors. Setting K and p at suﬃciently small values, we expect the low-rank matrix WH to containaccompaniment signals and the reconstruction errors Y −WH to contain singing voice signals.
Let the estimated magnitude spectrogram of a singing voice be
Ŝ = { ŝω,t }0≤ω≤Ω−1,0≤t≤T −1 . First, an input signal containing a singing
voice and accompaniment is converted into a complex spectrogramwith the short-time Fourier transform (STFT). We regard the magnitude spectrogram of the input signal as a non-negative matrix, andapply L p -norm NMF to the magnitude spectrogram. The obtainedmodel spectrogram WH should correspond to the accompaniment,
and the reconstruction errors between the observed magnitude spectrogram and the model spectrogram corresponds to the singing voice.
The time-frequency components of the model spectrogram may belarger than those of the observed spectrogram, and we derive Ŝ as

∑
∑


yω,t − k wω,k hk,t (yω,t ≥ k wω,k hk,t ).
(16)ŝω,t = ∑0(yω,t < k wω,k hk,t )The estimated magnitude spectrogram with the phase of the originalcomplex spectrogram is converted into an audio signal by the inverse
STFT.
5. EXPERIMENTAL EVALUATION
5.1. Experimental conditions
To evaluate the performance of the proposed method, we conducted
two experiments on singing voice enhancement: an evaluation of
the eﬀect of p, which controls sparsity, and frame length F, and acomparison of our results with the state-of-the-art [5, 17, 18].
The criteria for evaluating the singing voice enhancement were
the normalized signal-to-distortion ratio (NSDR) and the global
NSDR (GNSDR), given as
NSDR({ fˆt }t ; { ft }t , {xt }t ) = SDR({ fˆt }t , { ft }t ) − SDR({xt }t , { ft }t ), (17)∑wi NSDR({ fˆi,t }t ; { fi,t }t , {xi,t }t )∑,(18)
GNSDR = i
i wi
∑ ˆ
ft ft
SDR({ fˆt }t , { ft }t ) = 10 log10 ( ∑ )( ∑t ) ∑
,
(19)
ˆ
ˆ
f
f
t
t
t t −
t ft ft
where fˆi,t , fi,t and xi,t denote the estimated signal, the target signal
and the input signal of the i-th piece. NSDR represents the improvement in SDR, and GNSDR denotes the weighted averages of2117

Frequency [Hz]4.2. Enhancement algorithm

3000
2500
2000
1500
1000
500
0 0

2

4

6

8

10

12

Time [s](a) Spectrogram of input audio signal.
3000

Frequency [Hz]When input audio signals are multichannel, we can use spatial
cues, based on the fact that the vocal parts of music audio signals
are frequently mixed in the center of the sound field. However, formonaural inputs, we need other cues instead of the spatial cues.

2500
2000
1500
1000
500
0 0

2

4

6

8

10

12

Time [s](b) Spectrogram of audio signal after enhancement.
Fig. 1. Spectrogram examples of (a) a mixed audio signal and (b)the singing-voice-enhanced audio signal obtained with the proposed
method.
the NSDR of all the music pieces by the length of the i-th piece,{wi }i . These criteria have also been employed in many previous studies [5, 17–21]. To calculate the SDR, we used the BSS Eval Toolbox [22, 23].
As an evaluation dataset, we used the MIR-1K dataset [24], following the evaluation framework in [5, 17, 18]. The dataset consistsof 1000 Chinese song clips performed by amateur singers. The durations of the clips range from 4 to 13 s, and the audio signals aremonaural with a sampling rate of 16 kHz. The accompaniment andvocal parts were recorded separately, and we could mix them withany signal-to-noise ratio (SNR), where the SNR corresponds to thevoice to accompaniment ratio. The accompaniment and vocal partsfor each clip were mixed at −5 dB (accompaniment is louder), 0 dB
(same level) and 5 dB (vocal is louder) SNRs.
5.2. Eﬀect of sparsity and frame lengths
We first compared the proposed method in p and F. We used
p = 0.1, 0.2, · · · , 2.0 and F = 512, 1024, 2048, 4096 sample points.
For STFT, the window function was the sine window, and the frame
shifts were half the length of the frames. The number of bases was
set at K = 10. The entries of W and H were initialized randomly.
There were 200 iterations, which is supposed to be suﬃcient empirically.
Fig. 1 shows one of the enhanced results obtained with the proposed method. The figures show the spectrograms of the input signal and the enhanced result. We can see that most of the accompanying sounds (vertically and horizontally smooth components) aresuppressed, and the singing voice component of the spectrogram isclearer than that of the input spectrogram.

0

0.5

1
p

1.5

Global normalized
signal-to-distortion ratio [dB]Global normalized
signal-to-distortion ratio [dB]

4
3.5
3
2.5
2
1.5
1
0.5
0

frame: 512
frame: 10242(a) GNSDR at −5 dB SNR.

2
1
0
-1
-2
-3
-4
-5
-6

0

0.5

frame: 512
frame: 1024

frame: 2048
frame: 4096

1
p

1.5

(b) GNSDR at 0 dB SNR.

2

Global normalized
signal-to-distortion ratio [dB]frame: 2048
frame: 4096

frame: 512
frame: 1024

2
0
-2
-4
-6
-8
-10
-12
-14
-16

0

0.5

frame: 2048
frame: 4096

1
p

1.5

2

(c) GNSDR at 5 dB SNR.

Fig. 2. Global normalized signal-to-distortion ratio (GNSDR) of the proposed method with respect to p of the L p norm and frame length F.
Red, blue, green and purple points correspond to F = 512, 1024, 2048, 4096 sample points. The results are for (a) −5 dB, (b) 0 dB, and (c) 5dB SNRs.

5

10

Fig. 3 shows the distributions of NSDRs for each SNR. The results were for (p, F) = (1.7, 2048), (1.0, 2048), (0.8, 1024) for SNRsof −5, 0, 5 dB. Since most of the NSDRs exceeded 0 dB, and we canconfirm that the proposed method worked well for most of the input
signals.
5.3. Comparison with previous studies

0

Finally, we compared the proposed method with the state-of-theart [5, 17, 18]. The results are summarized in Tab. 1. The proposed
method with F = 1024 outperformed two previous methods for all
input SNRs. While the GNSDRs of the proposed method were lower
than those of [5] at SNRs of 0 and 5 dB, the GNSDR with the proposed method was 0.4 to 2.4 dB larger than those of three previous
methods at a SNR of −5 dB. This result indicates that the proposed
method works well particularly in a low SNR environment.

−5

Normalized signal−to−distortion ration [dB]

Table 1. Global normalized signal-to-distortion ratio (GNSDR) comparison with previous studies. F denotes the length of a frame in sample
point. Hsu, Rafii and Huang represent the corresponding methods [5, 17, 18].
Input
Proposed method
Proposed method
Hsu [17] Rafii [18] Huang [5]
SNR [dB]F = 1024
F = 2048−52.84 (p = 1.6)
3.70 (p = 1.7)
−0.51
0.52
1.51
0
1.93 (p = 1.0)
1.95 (p = 1.0)
0.91
1.11
2.37
1.43 (p = 0.8)
1.04 (p = 0.8)
0.17
1.10
2.57
5

−5

0

5

Input signal−to−noise ration [dB]Fig. 3. Box plot of the normalized signal-to-distortion ratio (NSDR)of the proposed method for the MIR-1K dataset. The results for
SNRs of −5, 0 and 5 dB are for (p, F) = (1.7, 2048), (1.0, 2048) and(0.8, 1024), respectively.
As illustrated in Fig. 2 for SNRs of −5, 0, 5 dB, the results showthat the GNSDRs depended strongly on p for all frame lengths. The
highest GNSDRs for all SNRs were 3.7 at (p, F) = (1.7, 2048) for−5 dB SNR, 1.95 at (p, F) = (1.0, 2048) for 0 dB SNR, and 1.43
at (p, F) = (0.8, 1024) for 5 dB SNR. We can see that p at which
GNSDR was the highest for each input SNR decreased as the input
SNR became higher. With a high input SNR, the non-zero timefrequency components of the singing voice spectrogram are large,
and increasing the sparsity is preferred. On the other hand, with a
low input SNR, the non-zero time-frequency components are small,and decreasing the sparsity is preferred. The obtained results are
consistent with this idea.

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6. CONCLUSION
We proposed a new NMF that minimizes the L p norm of the reconstruction errors between a data matrix and the model. A computationally eﬃcient algorithm was derived according to the auxiliary
function principle, and it has multiplicative update equations, whichguarantee the non-negativity of W and H. The algorithm of L p -norm
NMF can be generalized for that of L p -norm MF for real-valued
matrices. We applied L p -norm NMF to singing voice enhancement
and showed by experiments that adequately selecting p improves theenhancement quality, and the proposed method outperformed threeprevious works under a low SNR situation.
There are several ways to extend L p -norm NMF to other applications. One promising application is speech enhancement, since thespectrogram of background noise is sometimes approximated as low
rank and the speech spectrogram is relatively sparse.

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