2018 26th European Signal Processing Conference (EUSIPCO)

Generative adversarial network-based approach tosignal reconstruction from magnitude spectrogram
Keisuke Oyamada∗ , Hirokazu Kameoka† , Takuhiro Kaneko † ,Kou Tanaka † , Nobukatsu Hojo † , Hiroyasu Ando ∗∗ University of Tsukuba, Japan,

Email: s1720584@s.tsukuba.ac.jp† NTT Communication Science Laboratories, NTT Corporation, Japan,

Abstract—In this paper, we address the problem of reconstructing a time-domain signal (or a phase spectrogram) solely
from a magnitude spectrogram. Since magnitude spectrograms
do not contain phase information, we must restore or infer phase
information to reconstruct a time-domain signal. One widely usedapproach for dealing with the signal reconstruction problem was
proposed by Griffin and Lim. This method usually requires many
iterations for the signal reconstruction process and dependingon the inputs, it does not always produce high-quality audio
signals. To overcome these shortcomings, we apply a learningbased approach to the signal reconstruction problem by modelingthe signal reconstruction process using a deep neural network andtraining it using the idea of a generative adversarial network.
Experimental evaluations revealed that our method was able toreconstruct signals faster with higher quality than the GriffinLim method.
Index Terms—phase reconstruction, deep neural networks,
generative adversarial networks

I. I NTRODUCTION
This paper addresses the problem of reconstructing a timedomain signal solely from a magnitude spectrogram.
The magnitude spectrograms of real-world audio signals
tend to be highly structured in terms of both spectral andtemporal regularities. For example, pitch contours and formant
trajectories are clearly visible from a magnitude spectrogram
representation of speech compared with a time-domain signal.
Therefore, there are many cases where processing magnitude
spectrograms can deal with problems more easily than directly
processing time-domain signals. In fact, many methods formonaural audio source separation are applied to magnitude
spectrograms [1]–[3]. Furthermore, a magnitude spectrogram
representation was recently found to be reasonable and effective for use with speech synthesis systems [4], [5].
Since a magnitude spectrogram does not contain phase
information, we must restore or infer phase information toreconstruct a time-domain signal. This problem is called thesignal (or phase) reconstruction problem. One widely usedmethod for solving the signal reconstruction problem was
proposed by Griffin and Lim [6] (hereafter referred to as the
Griffin-Lim method). One of the drawbacks of the Griffin-Lim
method is that it usually requires many iterations to obtain
This work was conducted as a part of research undertaken by the Center
for Artificial Intelligence Science, University of Tsukuba and supported by
JSPS KAKENHI Grant Number 17H01763.

ISBN 978-90-827970-1-5 © EURASIP 2018high-quality audio signals. This makes it particularly difficult
to apply it to real-time systems. Furthermore, there are somecases where high-quality audio signals can never be obtainedeven though the algorithm is run for many iterations. To
overcome these shortcomings of the Griffin-Lim method, weapply a learning-based approach to the signal reconstruction
problem. Specifically, we propose modeling the reconstruction
process of a time-domain signal from a magnitude spectrogramusing a deep neural network (DNN) and propose introducing
the idea of the generative adversarial network (GAN) [7] fortraining the signal generator network.
The remainder of the paper is organized as follows. We
provide an overview of the phase reconstruction problem in
Section II, introduce the Griffin-Lim method in Section III,and present our GAN-based approach in Section IV. Experimental evaluations, and supplements for training our model
are provided in Section V. Finally, we offer our conclusions
in Section VI.
II. S IGNAL R ECONSTRUCTION P ROBLEM
In this section, we provide an overview of the signal
reconstruction problem.
We use x = [x(0), . . . , x(T − 1)]T ∈ RT to denotea time domain signal and cf,n ∈ C to denote the time
frequency component of x where f and n indicate frequency and time indices, respectively. By defining wf,n =[wf,n (0), . . . , wf,n (T − 1)]T ∈ CT as a complex sinusoidof frequency ωf modulated by a window function centeredat time tn , cf,n is defined by the inner product between x
and wf,n , namely cf,n = wH
f,n x. With a short-time Fouriertransform (STFT), tn corresponds to the center time of framen and wf,n is the modulated complex sinusoid padded withzeros over the range outside the frame. By using c ∈ CF N
to denote a vector obtained by stacking all the time-frequency
components cf,n , the relationship between c and x can be
written as
c = Wx,(1)

where W is a F N × T matrix where each row is wHf,n .
Hereafter, we call c a complex spectrogram. Since the total
number F N of time frequency points is usually set at morethan the number T of sample points of the time domain signal,c is a redundant representation of x. Namely, c belongs to2528

2018 26th European Signal Processing Conference (EUSIPCO)a T -dimensional linear subspace C spanned by each column
vector of W. With an STFT, all the elements of a complex
spectrogram must satisfy certain conditions to ensure thatthe waveforms within the overlapping segment of consecutive
frames are consistent. By using a to denote the magnitude
spectrogram of c where each element of a is given by theabsolute value of the element of c, the signal reconstruction
problem can be cast as an optimization problem of estimating
x solely from a using the redundancy constraint as a clue.

difficult to apply to real-time systems. Furthermore, there aresome cases where high-quality audio signals can never beobtained even though the algorithm is run for many iterations,for example when a is an artificially created magnitude
spectrogram. In the next section, we propose a learning-based
approach to the phase reconstruction problem to overcome
these shortcomings of the Griffin-Lim method.
IV. GAN- BASED SIGNAL RECONSTRUCTION
A. Modeling phase Reconstruction Process

III. G RIFFIN -L IM M ETHOD
One widely used way of solving the phase reconstruction
problem involves the Griffin-Lim method [6]. In this section,
we derive the iterative algorithm of the Griffin-Lim method
following the derivation given in [8].
Whether or not a given c satisfies the redundancy constraint
so that c is a complex spectrogram associated with a time
domain signal can be evaluated by examining whether or
not the orthogonal projection WW+ c of c to the subspace
C matches c. Here, W+ is a pseudo inverse matrix of W
satisfying
W+ c = argmin ∥c − Wx∥22x= (WH W)−1 WH c.

(2)With an STFT, (2) corresponds to an inverse STFT. Thus,
WW+ c is the STFT of the inverse STFT of c. Now, byusing ϕ to denote a vector where each element is the phase
ϕf,n ≡ eȷθf,n , the phase reconstruction problem for a given a
is formulated as an optimization problem of estimating ϕ thatminimizes
J (ϕ) = ∥a ⊙ ϕ − WW+ (a ⊙ ϕ)∥22 ,(3)

where ⊙ denotes an element-wise product. From (2),
WW+ (a ⊙ ϕ) is the point closest to a ⊙ ϕ in the subspace
C. Thus, we can rewrite (3) as
J (ϕ) = min ∥a ⊙ ϕ − c̃∥22 .
c̃∈C

(4)

According to the principle of the majorization-minimization
algorithm [9], it can be shown that J + (ϕ, c̃) ≡ ∥a ⊙ ϕ − c̃∥22
is a majorizer of J (ϕ) where c̃ ∈ C is an auxiliary variable
and a stationary point of J (ϕ) can be found by iteratively
performing the following updates:c̃ ← argmin ∥a ⊙ ϕ − c̃∥22 = WW+ (a ⊙ ϕ),(5)

ϕ ← argmin ∥a ⊙ ϕ − c̃∥22 = ∠c̃.

(6)

c̃∈C
ϕ

Here ∠· denotes an operation that divides each element ofa vector by its absolute value. With an STFT, Eq. (5) canbe interpreted as the inverse STFT of a ⊙ ϕ followed by
the STFT whereas Eq. (6) is a procedure for replacing thephase ϕ with the phase of c̃ updated via (5). This algorithm
is procedurally equivalent to the Griffin-Lim method [6].
The Griffin-Lim method usually requires many iterations to
obtain a high-quality audio signal. This makes it particularlyISBN 978-90-827970-1-5 © EURASIP 2018

By using ϕ(0) to denote the initial value of ϕ, and defining
h(a, ϕ) ≡ WW+ a⊙ϕ and g(c) ≡ ∠c, the iterative algorithmof the Griffin-Lim method can be expressed as a multilayercomposite functionĉ = h(a, g(· · · g(h(a, g(h(a, ϕ(0) )))) · · · )).

(7)

Here, h is a linear projection whereas g is a nonlinear operation
applied to the output of h. Hence, (7) can be viewed as a deepneural network (DNN) where the weight parameters and theactivation functions are fixed. From this point of view, findingan algorithm that converges more quickly to a better solution
than the Griffin-Lim algorithm can be regarded as learning theweight parameters (and the activation functions) of the DNN.
This idea is inspired by the deep unfolding framework [10],which uses a learning strategy to obtain an improved version ofa deterministic iterative inference algorithm by unfolding theiterations and treating them as layers in a DNN. Fortunately,an unlimited number of pair data of c and {a, ϕ} can becollected very easily by computing the complex, magnitude
and phase spectrograms of time domain signals. This is very
advantageous for efficiently training our DNN.
In the following, we consider a DNN that uses a and ϕ as
inputs and generates c (or x) as an output. We call this DNN
a generator G and express the relationship between the input
and output as ĉ = G(a, ϕ).
B. Learning Criterion
For the generator training, one natural choice for thelearning criterion would be a similarity metric (e.g., the ℓ1norm) between the generator output and a target complex
spectrogram (or signal). Manually defining a similarity metric amounts to assuming a specific form of the probability
distribution of the target data (e.g., a Laplacian distribution
for the ℓ1 norm). However, the data distribution is unknown.
If we use a similarity metric defined in the data space as
the learning criterion, the generator will be trained in sucha way that the outputs that averagely fit the target data areconsidered optimal. As a result, the generator will learn togenerate only oversmoothed signals. This is undesirable as theoversmoothing of reconstructed signals causes audio quality
degradation. To avoid this, we propose using a similarity metric implicitly learned using a generative adversarial network(GAN) [7]. In addition to the generator network, we introduce
a discriminator network D that learns to correctly discriminate
the complex spectrograms ĉ generated by the generator andthe complex spectrograms of real audio signals. Given a target2529

2018 26th European Signal Processing Conference (EUSIPCO)

complex spectrogram c, the discriminator D is expected tofind a feature space where ĉ and c are as separate as possible.
Thus, we expect that minimizing the distance between ĉ andc measured in a hidden layer of the discriminator would
make ĉ indistinguishable from c in the data space. By using
D(·, a) ∈ R to denote the discriminator network D, we firstconsider the following criteria for the discriminator[]
1
V (D) = E(c,a)∼pc,a (c,a) (D(c, a) − 1)22[]
1+ Ea∼pa (a),ϕ∼pϕ (ϕ) D(G(a, ϕ), a)2 .
(8)2Here, the target label corresponding to real data is assumed
to be 1 and that corresponding to the data generated by thegenerator G is 0. Thus, (8) means that V (D) becomes 0
only if the discriminator D correctly distinguishes the “fake”
complex spectrograms generated by the generator G and the“real” complex spectrograms of real audio signals. Therefore,
the goal of D is to minimize V (D). As for the generator G,
one of the goals is to deceive the discriminator D so as tomake the “fake” complex spectrograms as indistinguishable
as possible from the “real” complex spectrograms. This canbe accomplished by minimizing the following criterion[]
1
U (G) = Ea∼pa (a),ϕ∼pϕ (ϕ) (D(G(a, ϕ), a) − 1)2 . (9)2Another goal for G is to make ĉ = G(a, ϕ) as close as
possible to the target complex spectrogram c. By using Dl (·)
to denote the output of the l-th layer of the discriminator D,
we would also like G to minimize
L∑
wl ∥Dl (c) − Dl (G(a, ϕ))∥22 ,(10)
I(G) =
l=0

where wl is a fixed weight, which weighs the importance ofthe l-th layer feature space. Here, the 0-th layer corresponds
to the input layer, namely D0 (c) = c.
The learning objectives for D and G can thus be summarized as follows:
D : V (D) → minimize,
G : U (G) + λI(G) → minimize,

(11)
(12)

where λ is a fixed weight.
A general framework for training a generator network insuch a way that it can deceive a real/fake discriminator
network is called a generative adversarial network (GAN)[7]. The novelty of our proposed approach is that we havesuccessfully adapted the GAN framework to the signal reconstruction problem by incorporating an additional term (10).
The GAN framework using (8), (9) as the learning criteria iscalled the least squares GAN (LSGAN) [11]. Note that GAN
frameworks using other learning criteria such as [12] have also
been proposed. Thus, we can also use the learning criteria
employed in [7], [12] or others instead of (8), (9).
V. E XPERIMENTAL E VALUATION
We tested our method and the Griffin-Lim method using
real speech samples.

ISBN 978-90-827970-1-5 © EURASIP 2018

A. Experimental Settings
1) Dataset: We used clean speech signals excerpted from[13] as the experimental data. The speech data consisted ofutterances of 30 speakers. The utterances of 28 speakers wereused as the training set and the remaining utterances were usedas the evaluation set. For the mini-batch training, we divided
each training utterance into 1-second-long segments with anoverlap of 0.5 seconds. All the speech data were downsampled
to 16 kHz. Magnitude spectrograms were obtained with an
STFT using a Blackman window that was 64 ms long with a
32 ms overlap.
2) Network Architecture: Fig. 1 shows the network architectures we constructed for this experiment. The left halfshows the architecture of the generator G and the right halfshows that of the discriminator D. The light blue blocks indicate convolutional layers, and k, s, and c on each convolutional
layer represent hyper-parameters. The yellow blocks indicate
activation functions. PReLU [14] was used for the generator G
and Leaky ReLU [15] was used for the discriminator D. The
violet blocks indicate element-wise sums, and the green blockindicates the concatenation of features along the channel axis.
The red blocks indicate fully-connected layers. Blocks without
symbols have the same hyper-parameters as the previous
blocks. Note that we referred to [16] when constructing thesearchitectures. The generator G is fully convolutional [17], thus
allowing an input to have an arbitrary length. The weight
constant wl was set to 0 for l = 0 and 1 for l ̸= 0. λ was set
to 1. RMSprop [18] was used as the optimization algorithm
and the learning rate was 5 × 10−5 Cα = 0.5. The mini-batch
size was 10 and the number of epochs was 73.
Instead of directly feeding an input magnitude spectrogramand a randomly-generated phase spectrogram into the generator G, we used a complex spectrogram reconstructed using
the Griffin-Lim method after 5 iterations as the G input. Boththe input and output of the generator G have 2 channels, one
corresponding to the real part and the other corresponding
to the imaginary part of the complex spectrogram. For preprocessing, we normalized the complex spectrograms of the
training data to obtain zero-mean and unit-variance at eachfrequency. At test time, the scale of the generator output at
each frequency was restored.
We added a block that applies an inverse STFT to thegenerator output before feeding it into the discriminator D.
We found this particularly important as the training did notwork well without this block.
B. Data Augmentation
It is a well-known fact that the difference between signals
is hardly perceptible to human ears when the magnitude spectrograms and the inter-frame phase differences are the same.
This implies that there is an arbitrariness in the initial phasesof spectrograms that are perceived similarly. By utilizing thisproperty, we can augment the training data for G and D by
preparing many different waveforms that are the same except
for the initial phases. We expect that this data augmentation
would allow the generator to concentrate on learning a way

2530

2018 26th European Signal Processing Conference (EUSIPCO)!"#"$%&'$(!!

./0F7!
17!
3F7!

3"%045%6"!

/!

E,;;@<
3#$(3A(C!

/G75!

)(:.@2'()*!

"#$%!

)(:.@2'()*!

"#$3:A!

./0F7!
17!
345!

E,;;@<
3#$(3A(C!

)(:.@2'()*!

E,;;@<
3#$(3A(C!

"#$%!

)(:.@2'()*!

"#$%!

B:9$8A,C(<
1D(3A?#9?:-!

)(:.@2'()*!
+89$:;<=:%(>#?-!

H$%(?1(<+IEI!

.'-/0"1(+/",&$'2$%-!

"#$%!

.'-/0"1(+/",&$'2$%-!

&'()*!

+,-!

"#$%!

"#$%!

+,-!

.6062!
17!
3F7!

.//0//!
1/!
37!

.606!
1/!
345!

"#$%!

"#$%!

.6062!
1/!
345!

&'()*!

"#$%!

&'()*!

.'-/0"1(+/",&$'2$%-!

.//0//!
1/2!
345!

)*+,$*-*#%&'$()!

Fig. 1. Network architectures of generator and discriminator. Light blue blocks indicate convolutional layers. In each convolutional layer, k, s, and c represent
kernel size, stride size, and number of channels, respectively. Here, k1 × ∗ indicates a one-dimensional convolutional layer whose kernel size is ∗. Red blocks
indicate fully connected layer. In each fully connected layer, the numbers represents size of output unit.
&'())(*+,(-.-"/012.3455.(/"'6/(1*%7"!!"#$"(")**)+,-)./.&01#'/2344/)0&"50)#+%6!

8'191%"2.-"/012"!"#$#%&'!

789/:#+*)'&+:&/)+0&";5<!

Fig. 2. Result of the AB test. The orange area indicates the rate of the A
and B pairs for which the listeners preferred A (proposed). The black bar
indicates the 95% confidence interval.

!%"#$":6';"/.26/6"

of inferring appropriate inter-frame phase differences given amagnitude spectrogram, thus facilitating efficient learning.
!%"#$"

C. Dimensionality Reduction
Note that the real and imaginary parts of the Fourier transform of a real-valued signal become even and odd functions,
respectively. Owing to this symmetric structure, it is sufficient
to restore/infer spectral components within the frequency range
from 0 up to the Nyquist frequency. We can therefore restrict
the sizes of the input and output of the generator to thisfrequency range.
D. Subjective Evaluation
We compared our proposed method with the Griffin-Lim
method in terms of the perceptual quality of reconstructedsignals by conducting an AB test, where “A” and “B” werereconstructed signals obtained respectively with the proposedand baseline methods. With this listening test, “A” and “B”
were presented in random orders to eliminate bias as regards
the order of stimuli. Five listeners participated in our listening
test. Each listener was presented with {“A”,“B”} ×10 signals
and asked to select “A”or “B” for each pair. The Griffin-Lim
method was run for 400 iterations. The signals were 2 to 5
seconds long.
The preference scores are shown in Fig. 2. As the resultshows, the reconstructed signals obtained with the proposed
method were preferred by the listeners for 76% of the 50
pairs.
E. Generalization ability
To confirm the generalization ability of the proposed
method, we tested it on musical audio signals excerpted from[19]. Examples of the reconstructed signals are shown in
Fig. 3. With these examples, we can observe a discontinuous
point in the reconstructed signal obtained with the Griffin-Lim
method. On the other hand, the proposed method appears toISBN 978-90-827970-1-5 © EURASIP 2018

Fig. 3. Waveforms of reconstructed music data [19]. The first row shows theacoustic signal reconstructed with the Griffin-Lim method, the second shows
the proposed method, and the third is the target acoustic signal (real-world
acoustic signal).

have worked successfully, even though the model was trainedusing speech data.
F. Comparison of Processing Times
We further compared the proposed method with the GriffinLim method in terms of the processing times needed toreconstruct time domain signals. For comparison, we measured
the processing times for various speech lengths. We used
speech data shorter than 6 seconds for the evaluation. Here, thenetwork architecture of our proposed method was the same as
Fig. 1, and the Griffin-Lim method was run for 400 iterations.
The CPU used in this experiment was “Intel Core i7-6850K
CPU @ 3.60GHz”. The GPU was “NVIDIA GeForce GTX
1080”. We implemented the Griffin-Lim method using the fast
Fourier transform function in NumPy [20]. We implemented
our model with Chainer [21]. Fig. 4 shows the result. As
the speech data become longer, the processing time increases
linearly. When executing the proposed method using the GPU,the time needed to reconstruct a signal was only about onetenth the length of that signal. On the other hand, the GriffinLim method executed using the CPU took about the same timeas the length of the signal. Therefore, if we can use a GPU,
the proposed method can be run in real time. However, whenusing the CPU, the proposed method took about three times
longer than the length of the signal. If we want to execute theproposed method in real-time using a CPU, we would needto construct a more compact architecture than that shown in
Fig. 1. One simple way would be to replace the convolutional
layers with downsampling and upsampling layers.

2531

2018 26th European Signal Processing Conference (EUSIPCO)

Fig. 4. The change in processing time with respect to the change in speech
length. Blue points show the processing time with the proposed method aGPU. Green points show the time with the Griffin-Lim method with a CPU.
Orange points show the time with the proposed method with a CPU.

VI. C ONCLUSION
This paper proposed a GAN-based approach to signal
reconstruction from magnitude spectrograms. The idea was tomodel the signal reconstruction process using a DNN and train
it using a similarity metric implicitly learned using a GAN
discriminator. Through subjective evaluations, we showed thatthe proposed method was able to reconstruct higher quality
time domain signals than the Griffin-Lim method, which
was run for 400 iterations. Furthermore, we showed that the
proposed method can be executed in real-time when using aGPU. Future work will include the investigation of a networkarchitecture appropriate for CPU implementations.

[11] X. Mao, Q. Li, H. Xie, R.Y. Lau, Z. Wang, et al., “Least squares
generative adversarial networks,” in Proc. ICCV, pp. 2813–2821, 2017.
[12] M. Arjovsky, S. Chintala, and L. Bottou, “Wasserstein GAN,” arXiv
preprint arXiv:1701.07875, 2017.
[13] C. Valentini-Botinhao, “Noisy speech database for training
speech enhancement algorithms and TTS models, [dataset],”University of Edinburgh. School of Informatics. CSTR, 2016.
http://dx.doi.org/10.7488/ds/1356.
[14] K. He, X. Zhang, S. Ren, and J. Sun, “Delving deep into rectifiers:
Surpassing human-level performance on imagenet classification,” inProc. ICCV, pp. 1026–1034, 2015.
[15] A.L. Maas, A.Y. Hannun, and A.Y. Ng, “Rectifier nonlinearities improveneural network acoustic models,” in Proc. ICML, vol. 30, no. 1, pp. 3,
2013.
[16] C. Ledig, L. Thesis, F. Huszár, J. Caballero, A. Cunningham, etal., “Photo-realistic single image super-resolution using a generative
adversarial network,” arXiv preprint arXiv:1609.04802, 2016.
[17] J. Long, E. Shelhamer, and T. Darrell, “Fully convolutional networks
for semantic segmentation,” in Proc. CVPR, pp. 3431–3440, 2015.
[18] T. Tieleman and G. Hinton, “Lecture 6.5-rmsprop: Divide the gradientby a running average of its recent magnitude,” COURSERA: Neuralnetworks for machine learning, vol. 4, no. 2, pp. 26–31, 2012.
[19] CAFÉ DEL CHILLIA, “In The Story That We Say,”
https://www.jamendo.com/track/1455877/in-the-story-that-we-say,
2017.
[20] S. Walt, S.C. Colbert, and G. Varoquaux, “The NumPy array: a structure
for efficient numerical computation,” Computing in Science & Engineering, vol. 13, no. 2, pp. 22-30, 2011.
[21] S. Tokui, K. Oono, S. Hido, and J. Clayton, “Chainer: a next-generation
open source framework for deep learning,” in Proc. LearningSys in thetwenty-ninth annual conference on NIPS, vol. 5, 2015.

R EFERENCES[1] P. Smaragdis, C. Févotte, G. J. Mysore, N. Mohammadiha, and M.
Hoffman, “Static and dynamic source separation using nonnegative
factorizations: A unified view,” IEEE Signal Processing Magazine, vol.
31, no. 3, pp. 66–75, 2014.
[2] T. Virtanen, J. Florent Gemmeke, B. Raj, and P. Smaragdis, “Compositional models for audio processing: Uncovering the structure of soundmixtures,” IEEE Signal Processing Magazine, vol. 32, no. 2, pp. 125–
144, 2015[3] H. Kameoka, “Non-negative matrix factorization and its variants foraudio signal processing,” in Applied Matrix and Tensor Variate Data
Analysis, T. Sakata (Ed.), Springer Japan, 2016.
[4] S. Takaki, H. Kameoka, and J. Yamagishi, “Direct modeling of frequency spectra and waveform generation based on phase recovery for
DNN-based speech synthesis,” in Proc. Interspeech, pp. 1128–1132,
2017.
[5] Y. Wang, RJ. Skerry-Ryan, D. Stanton, Y. Wu, R.J. Weiss, et al.,“Tacotron: A fully end-to-end text-to-speech synthesis model,” arXiv
preprint arXiv:1703.10135, 2017.
[6] D. W. Griffin and J. S. Lim, “Signal estimation from modified shorttime Fourier transform,” IEEE Trans. ASSP, vol. 32, no. 2, pp. 236–243,
1984.
[7] I. Goodfellow, J. Pouget-Abadie, M. Mirza, B. Xu, D. Warde-Farley, etal., “Generative adversarial nets,” in Adv. NIPS, pp. 2672–2680, 2014.
[8] J. Le Roux, H. Kameoka, N. Ono, S. Sagayama, “Fast signal reconstruction from magnitude STFT spectrogram based on spectrogram
consistency,” in Proc. DAFx, pp. 397–403, 2010.
[9] D. R. Hunter and K. Lange, “Quantile regression via an MM algorithm,”
Journal of Computational and Graphical Statistics, vol. 9, pp. 60–77,
2000.
[10] J. R. Hershey, J. Le Roux, and F. Weninger, “Deep unfolding:Model-based inspiration of novel deep architectures,” arXiv preprint
arXiv:1409.2574.

ISBN 978-90-827970-1-5 © EURASIP 20182532