2019 27th European Signal Processing Conference (EUSIPCO)

Generalized Multichannel Variational Autoencoder
for Underdetermined Source Separation
Shogo Seki1 , Hirokazu Kameoka2 , Li Li3 , Tomoki Toda1 , Kazuya Takeda1
1

Nagoya University, Nagoya, JapanNTT Communication Science Laboratories, Atsugi, Japan
3
University of Tsukuba, Tsukuba, Japanseki.shogo@g.sp.m.is.nagoya-u.ac.jp, kameoka.hirokazu@lab.ntt.co.jp
2

Abstract—This paper deals with a multichannel audio source
separation problem under underdetermined conditions. Multichannel Non-negative Matrix Factorization (MNMF) is one ofthe powerful approaches, which adopts the NMF concept forsource power spectrogram modeling. It works reasonably wellfor particular types of sound sources, however, one limitation isthat it can fail to work for sources with spectrograms that do
not comply with the NMF model. To address this limitation, anovel technique called the Multichannel Variational Autoencoder(MVAE) method was recently proposed, where a Conditional VAE(CVAE) is used instead of the NMF model for source power
spectrogram modeling. This approach has shown to perform
impressively in determined source separation tasks thanks to therepresentation power of DNNs. This paper generalizes MVAE
originally formulated under determined mixing conditions sothat it can also deal with underdetermined cases. The proposed
method was evaluated on an underdetermined source separation
task of separating out three sources from two microphone inputs.
Experimental results revealed that the generalized MVAE method
achieved better performance than the conventional MNMF
method.
Index Terms—Underdetermined source separation, Variational
audoencoder, Non-negative matrix factorization

I. I NTRODUCTION
Blind source separation (BSS) refers to a problem of
separating out individual source signals from microphonearray inputs where the transfer functions between the sources
and microphones are unknown. The frequency-domain BSS
approach allows the utilization of various models for the timefrequency representations of source signals and/or array responses. For example, Independent Vector Analysis (IVA) [1],[2] offers a way of jointly solving frequency-wise source
separation and permutation alignment under the assumptionthat the magnitudes of the frequency components originatingfrom the same source are likely to vary coherently over time.
Other approaches involve multichannel extensions of Nonnegative Matrix Factorization (NMF) [3]–[6]. NMF was originally applied to music transcription and monaural source
separation tasks [7], [8] where the idea is to interpret thepower spectrogram of a mixture signal and approximate it as
the product of two non-negative matrices. This can be viewed
as approximating the power spectrum of a mixture signal
observed at each time frame by the sum of basis spectra scaledby time-varying magnitudes. Multichannel NMF (MNMF) is978-9-0827-9703-9/19/$31.00 ©2019 IEEE

TABLE I: Comparison with the conventional methods
MethodILRMA [4], [6]MNMF [3], [5]MVAE [10]Proposed

Separation
Determined
Underdetermined
Determined
Underdetermined

Source model
NMF
NMF
VAE
VAE

an extension of this approach to a multichannel case thatallows for the use of spatial information. It can also beseen as an approach to frequency-domain BSS using spectral
templates as a clue for jointly solving frequency-wise source
separation and permutation alignment.
The original MNMF [3] was formulated under a general
problem setting where sources can outnumber microphones
and a determined version of MNMF was subsequently proposed in [4]. While the determined version is applicable
only to determined cases, it allows an implementation of asignificantly faster algorithm than the general version. The
determined MNMF framework was later called Independent
Low-Rank Matrix Analysis (ILRMA) [9]. The MNMF framework including ILRMA is notable in that the optimization
algorithm is guaranteed to converge, however, one limitationis that it can fail to work for sources with spectrograms that
do not comply with the NMF model.
To address this limitation, a technique called the Multichannel Variational Autoencoder (MVAE) method was recently
proposed in [10]. It is an extension of ILRMA in which a
Variational Autoencoder (VAE) [11] is used instead of the
NMF model to estimate the power spectrograms of the sourcesin a mixture. Several studies [12], [13] have demonstrated
that the VAE-based power spectrogram modeling is effective
in a speech enhancement task. MVAE, which focuses on asource separation task, allows the estimation of the separation
matrices by employing a single Conditional VAE (CVAE) [14],trained using the spectrograms of speech samples with speaker
ID labels, as a generative model of the speech spectrogramsof multiple speakers. This approach is noteworthy in thatit can exploit the benefits of the representation power ofneural networks for source power spectrogram modeling andhas shown to outperform ILRMA on a determined source
separation task.
While the original MVAE method was formulated under
determined mixing conditions, this paper generalizes it sothat it can also deal with underdetermined cases (TABLE I).

2019 27th European Signal Processing Conference (EUSIPCO)

From the formulation assuming a mixing system, we derivea separation algorithm. Experimental results demonstrate thatthe proposed method outperforms the conventional MNMF
method.
II. P ROBLEM F ORMULATIONs(f, n) = [s1 (f, n), · · · , sJ (f, n)]T ∈ CJ ,x(f, n) = [x1 (f, n), · · · , xI (f, n)] ∈ C ,
I

(1)
(2)

where (·)T denotes the transpose. Now, we use a mixing
system of the form
x(f, n) = A(f )s(f, n),A(f ) = [a1 (f ), · · · , aJ (f )] ∈ CI×J ,

(3)
(4)

to describe the relationship between s(f, n) and x(f, n) whereA(f ) is called the mixing matrix.
Here, we assume that sj (f, n) independently follows a zeromean complex Gaussian distribution with variance vj (f, n)sj (f, n) ∼ NC (sj (f, n)|0, vj (f, n)).

(5)(5) is called the Local Gaussian Model (LGM) [15]. Whensj (f, n) and sj ′ (f, n) are independent for j ̸= j ′ , s(f, n)followss(f, n) ∼ NC (s(f, n)|0, V(f, n)),(6)where V(f, n) is a diagonal matrix with diagonal entries
v1 (f, n), · · · , vJ (f, n). From (3) and (6), x(f, n) is shown
to follow
x(f, n) ∼ NC (x(f, n)|0, A(f )V(f, n)AH (f )),(7)

where (·)H denotes the conjugate transpose. Thus, given theobserved mixture signals X = {x(f, n)}f,n , using the mixing
matrices A = {A(f )}f and the variances of source signals
V = {vj (f, n)}j,f,n , the log-likelihood is given byclog p(X |A, V) =
(
)−1
∑ [
−
tr(xH (f, n) A(f )V(f, n)AH (f )x(f, n))f,n()]+logdet A(f )V(f, n)AH (f ) ,(8)c

A. MNMF
The spatial covariance of the observed mixture signal canbe rewritten as the linear sum of the outer products of aj (f )multiplied by vj (f, n):A(f )V(f, n)AH (f ) =We consider a situation where J source signals are observedby I microphones. Let sj (f, n) and xi (f, n) be the ShortTime Fourier Transform (STFT) coefficient of the j-th source
signal and the i-th observed signal, where f and n are thefrequency and time indices, respectively. We denote the vectors
containing s1 (f, n), · · · , sJ (f, n) and x1 (f, n), · · · , xI (f, n)
byTIII. R ELATED W ORK

where = denotes the equality up to constant terms. If there
is no constraint imposed on vj (f, n), (8) will be split intofrequency-wise source separation problems. This indicates that
there is a permutation ambiguity in the separated components
for each frequency since permutation of j does not affect thevalue of the log-likelihood. Thus, we usually need to performpermutation alignment after A is obtained.

∑aj (f )vj (f, n)aHj (f )()∑=
vj (f, n)Rj (f ) = X̂(f, n) ,
j
j

(9)

where Rj (f ) represents the spatial covariance of source j.
As with IVA, MNMF makes it possible to jointly solve
frequency-wise source separation and permutation alignment
by imposing a constraint on vj (f, n). Specifically, vj (f, n)
is modeled as the linear sum of Kj spectral templates
hj,1 (f ), · · · , hj,Kj (f ) ≥ 0 scaled by time-varying activations
uj,1 (n), · · · , uj,Kj (n) ≥ 0:
vj (f, n) =

∑Kj
k=1

hj,k (f )uj,k (n).

(10)

It is also possible to allow all the spectral templates to beshared by every source and let the contribution of the k-th
spectral template to source j be determined in a data-driven
manner. Thus, vj (f, n) can also be expressed as
vj (f, n) =∑K
k=1

bj,k hk (f )uk (n),(11)

where bj,k∑ ∈ [0, 1] is a continuous indicator variable thatsatisfiesk bj,k = 1. Here, bj,k can be interpreted as the
expectation of a binary indicator variable that describes theindex of the source to which the k-th template is assigned.
The optimization algorithm of MNMF consists of iteratively
updating the spatial covariance matrices R = {Rj (f )}j,f , andthe source models H1 = {hj,k (f )}j,k,f , U1 = {uj,k (n)}j,k,nor B = {bj,k }j,k , H2 = {hk (f )}k,f , U2 = {uk (n)}k,n .
We can derive update equations using the principle of the
Majorization-Minimization (MM) algorithm [16].
B. ILRMAILRMA is a special class of MNMF designed to solve
determined source separation problems. Unlike MNMF, whichuses the mixing system (3), ILRMA uses a separation systemof the forms(f, n) = WH (f )x(f, n),W(f ) = [w1 (f ), · · · , wJ (f )] ∈ C

(12)
I×J

,

(13)

assuming the mixing matrix is invertible. The inverse matrix
WH (f ) is called the separation matrix. From (6) and (12),x(f, n) is shown to follow
x(f, n) ∼ NC (x(f, n)|0, (WH (f ))−1 V(f, n)(W(f ))−1 ).

(14)

Given the observed signals X , using the separation matrices
W = {W(f )}f and V, the log-likelihood is given byclog p(X |W, V) = 2N
−

∑

∑

f

log |detWH (f )|
[

f,n,j

]
|wjH (f )x(f, n)|2,
log vj (f, n) +vj (f, n)(15)where vj (f, n) is modeled as (10) or (11) as with MNMF.

2019 27th European Signal Processing Conference (EUSIPCO)

As with MNMF, we can derive the MM-based update
equations for H1 , U1 or B, H2 , U2 . Since ILRMA is anatural extension of IVA, we can use a fast update rule called
the Iterative Projection (IP) [17] for the separation matrices,originally developed for IVA.
C. MVAEOne limitation of the MNMF framework including ILRMA
is that since vj (f, n) is restricted to (10) or (11), it can fail towork for sources with spectrograms that do not actually follow
this form. The MVAE method is an extension of ILRMA that
replaces (10) with a pretrained CVAE. Let S̃ = {s(f, n)}f,n be
the complex spectrogram of a particular sound source. MVAE
models the generative model of S̃ using a CVAE with an
auxiliary input c. Here, we assume that c is represented asa one-hot vector, indicating the class of a source. Thus, theelements of c must sum to unity. For example, if we consider
speaker identities as the source class, each element of c will
be associated with a different speaker.
The CVAE consists of an encoder network and a decoder
network, which are assumed to be trained using labeled training examples {S̃m , cm }M
m=1 prior to separation. The encoder
distribution qϕ (z|s̃, c) is expressed as a Gaussian distribution:qϕ (z|S̃, c) =∏kN (z(k)|µϕ (k; S̃, c), σϕ2 (k; S̃, c)),(16)

where z denotes a latent space variable and z(k), µϕ (k; S̃, c),
σϕ2 (k; S̃, c) represent the k–th elements of z, µϕ (S̃, c),
σϕ2 (S̃, c), respectively. The decoder distribution pθ (S̃|z, c, g)
is expressed as a zero-mean complex Gaussian distribution:pθ (S̃|z, c, g) =∏f,n

NC (s(f, n)|0, v(f, n)),v(f, n) = g · σθ2 (f, n; z, c),

(17)
(18)

where σθ2 (f, n; z, c) represents the (f, n)–th element of thedecoder output σθ2 (z, c) and g is the global scale of thegenerated spectrogram. Both the encoder and decoder network
parameters ϕ, θ are trained using the following objectivefunction[J (ϕ, θ) = E(S̃,c)∼pD (S̃,c) Ez∼q(z|S̃,c) [log p(S̃|z, c)]]−KL[q(z|S̃, c)||p(z)] ,Fig. 1: Illustration of generalized MVAE
IV. G ENERALIZED MVAE
Fig. 1 shows an illustration of generalized MVAE andMNMF with the source model given by (10). As with theoriginal MVAE method, we use the decoder network of the
pretrained CVAE as the generative model of source power
spectrograms.
Since the decoder distribution is given in the same form
as the LGM, we can use pθ (S̃j |zj , cj , gj ) to develop the loglikelihood of the form (8). Hence, we can derive an iterativealgorithm for estimating R, G = {gj }j , and Ψ = {zj , cj }jin the same way as the derivation of the MM-based algorithm
for MNMF. From [16], we can show the following inequality:
L = − log p(X |A, V)[c ∑ ∑tr(X(f, n)Pj (f, n)R−1
j (f )Pj (f, n))≤jf,nvj (f, n)]+ vj (f, n)tr(K−1 (f, n)Rj (f )) ,(20)

where the equality holds when
Pj (f, n) = vj (f, n)Rj (f )(∑jHvj (f, n)Rj (f ))−1

K(f, n) = X(f, n) = x(f, n)x (f, n).
(19)

where E(S̃,c)∼pD (S̃,c) [·] denotes the sample mean over the training examples and KL[·||·] is the Kullback-Leibler divergence.
The trained decoder distribution pθ (S̃|z, c, g) is considered
as a universal generative model that is capable of generating spectrograms of all the sources involved in the training
examples. MVAE employs the decoder part of the CVAE as
the source model vj (f, n) in (15) and treats the input z andc to the decoder as the model parameters to be estimated.
The optimization algorithm of MVAE consists of updating
the separation matrices using IP, the global scale using the
MM algorithm, and the input to the pretrained decoder using
backpropagation. The advantage of the MVAE is that it canleverage the strong representation power of VAE for source
power spectrogram modeling.

,

(21)
(22)

Thus, we can use the right-hand side of (20) as a majorizerof L where P = {Pj (f, n)}j,f,n and K = {K(f, n)}f,n
are auxiliary variables. An iterative algorithm consists ofminimizing this majorizer with respect to R, G, and Ψ andupdating P and K at (21) and (22). The optimal update of R
is analytically obtained as
Rj (f ) ← Λ−1
j (f )#(Rj (f )Ωj (f )Rj (f )),(23)

where # denotes the geometric mean of two positive semidefinite matrices [18]:
1

1

1

1

1

G#H = G 2 (G− 2 HG− 2 ) 2 G 2 .

(24)

Λj (f ), Ωj (f ) are given as follows:Λj (f ) =Ωj (f ) =

∑
∑

n
n

vj (f, n)X̂
vj (f, n)X̂−1
−1(f, n),(25)−1(f, n)X(f, n)X̂(f, n).

(26)

2019 27th European Signal Processing Conference (EUSIPCO)

Algorithm 1 MVAE algorithm
Train ϕ and θ with (19)
Initialize R, Ψ, and G
repeatfor each j do
Update Rj = {Rj (f )}f using (24)
Update ψj = {zj , cj } with (20) using backpropagation
Update gj using (28)
end foruntil converge

m
0
.0
5

0.80 m; 60 deg.
0.10 m0.80 m; -35 deg.

m
0
0
.2
2.00 mSince the majorizer is split into source-wise terms, Ψ can be
updated parallelly using backpropagation. Note that we musttake account of the sum-to-one constraints when updating cj .
This can be easily implemented by inserting an appropriately
designed softmax layer that outputs cj
cj = softmax(dj ),(27)

and treating dj as the parameter to be estimated instead. The
optimal update of G is obtained as
gj ← gj
v
u∑
−1
−1
u
2
u f,n σθ (f, n; zj , cj )tr(X̂ (f, n)X(f, n)X̂ (f, n)Rj (f ))
,
×t
∑
−1
2
(f, n)Rj (f ))f,n σθ (f, n; zj , cj )tr(X̂(28)

The source separation algorithm of the generalized MVAE issummarized as Algorithm 1.
V. E XPERIMENTAL E VALUATIONS
A. Settings
The proposed method was evaluated on an underdeterminedsource separation task of separating out three sources fromtwo microphone inputs. As the experimental data, we used
speech samples of the Voice Conversion Challenge (VCC)
2018 dataset [19], which contains recordings of 6 female
and 6 male US English speakers. From the dataset, we usedutterances of 2 female and 2 male speakers, ’SF1’, ’SF2’,’SM1’, and ’SM2’. 81 utterances and 35 utterances of eachspeaker were used for training and evaluation, respectively.
Fig. 2 shows the room configuration of the evaluation, where
d and × shows the locations of microphones and sources,
respectively. Using the evaluation data, all the 3-speaker
patterns were prepared: ’SF1+SF2+SM2’, SF1+SM1+SF2’,’SF1+SM1+SM2’, and ’SM1+SF2+SM2’. For each speaker
pattern, 10 speech mixtures were generated by randomly
choosing utterances of individual speakers and randomly
placing them at × in Fig. 2. All the speech mixtures weregenerated at two different reverberant conditions: T60 = 78 ms
and T60 = 351 ms.
All the speech signals were resampled at 16 kHz and STFT
analysis was conducted with 256 ms frame length and 128 ms
hop length. We designed the encoder and decoder networks of
the CVAE as in Fig. 3. In this experiment, speaker identities
are considered as the source class category: latent code cj
shown in Fig. 1 is represented as a four-dimensional one-hot

0.75 m; 30 deg.

6.00 mFig. 2: Room configuration
vector. The Adam [20] algorithm with learning rate 0.0002 wasused to train the CVAE and the SGD algorithm with learning
rate 0.0005 was used to update the VAE source model Ψ.
Following methods including the proposed method were
evaluated for comparison.
MNMF1: MNMF with source model given by (10)MVAE1: MVAE initialized by MNMF1• MNMF2: MNMF with source model given by (11)
• MVAE2: MVAE initialized by MNMF2• SB-MNMF: MNMF2 with spectral dictionaries• SB-MVAE: MVAE initialized by SB-MNMF••

The source separation algorithms were run for 300 iterations
for the conventional methods and 100 iterations for the proposed methods. The parameters of the proposed methods were
initialized using the MNMF methods run with 200 iterations
through the encoder network. The numbers of basis spectrain the all MNMF algorithms were set to 10 per speaker. We
also evaluated the MNMF in semi-blind condition, where thespectral dictionaries are trained with the same dataset as the
CVAE. The spectral dictionary of each speaker were obtainedby Itakura-Saito NMF (IS-NMF) [8] with 1000 iterations.
As the evaluation metrics, the Signal-to-Distortion Ratio(SDR), the source Image-to-Spatial distortion Ratio (ISR), theSignal-to-Inference Ratio (SIR), and the Signal-to-Artifact Ratio (SAR) [21] between the reference signals and the separated
signals were calculated for each mixture and averaged.
B. Results
The separation performance under each reverberant condition is shown in Fig. 4. We can see that the proposed methods
consistently achieves better performances than the baseline
methods (MNMF1, MNMF2). Since the difference between
the baseline methods and the proposed methods is simply typesof source model, these results imply that the use of VAE source
model successfully contributed to improving the separation
performance. In the comparison with the MNMF in semiblind condition, the proposed method achieves better SDR
performances. Furthermore, the proposed method significantly
improves the SIR performances. This results show us that the
proposed method using VAE as source model can accurately
estimate the spatial covariance and improves suppression performance for the interference.

2019 27th European Signal Processing Conference (EUSIPCO)

8.0

6.0

8.0

MNMF1
MVAE1
MNMF2
MVAE2
SB-MNMF
SB-MVAE

6.0

4.0

4.0

2.0

2.0

MNMF1
MVAE1
MNMF2
MVAE2
SB-MNMF
SB-MVAE(a) Encoder network
0.0

SDR

ISR

SIR

SAR

0.0

SDR

ISR

SIR

SAR

(a) T60 = 78 ms(b) T60 = 351 ms
Fig. 4: Averaged separation performances [dB] at reverberant
conditions of (a) T60 = 78 ms and (b) T60 = 351 ms(b) Decoder network
Fig. 3: Network configurations of (a) encoder and (b) decoder,where [c, l] denotes the input channel and frame length. Bothconvolution and deconvolution represents 1-dimensional operation. (k, s, p) represents the kernel size, the stride size along
frame, and the zero padding size at both ends, respectively.
VI. C ONCLUSION
This paper proposed generalization of the MVAE method
originally formulated under determined conditions so that it
can also be applied to underdetermined source separation
tasks. The separation algorithm of the proposed method wasderived from the formulation assuming a mixing system. Experimental results revealed that the generalized MVAE method
achieved better performance than the conventional methods
and demonstrated that VAE source model successfully contributed to improving the separation performance.
We plan to compare the proposed method with other DNNbased method [22] and investigate the performance in musicsource separation.
ACKNOWLEDGEMENT
This work was partly supported by JSPS KAKENHI Grant
Number 17H01763.
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