BAYESIAN DRUM TRANSCRIPTION BASED ON NONNEGATIVE MATRIX FACTORDECOMPOSITION WITH A DEEP SCORE PRIOR
Shun Ueda Kentaro Shibata Yusuke Wada Ryo Nishikimi Eita Nakamura Kazuyoshi Yoshii
Graduate School of Informatics, Kyoto University, Japan
ABSTRACTThis paper describes a statistical method of automatic drum transcription that estimates a musical score of bass and snare drums andhi-hats from a drum signal separated from a popular music signal.
One of the most effective approaches for this problem is to apply
nonnegative matrix factor deconvolution (NMFD) for estimating the
temporal activations of drums and then perform thresholding for estimating a drum score. Such a pure audio-based approach, however,cannot avoid musically unnatural scores. To solve this, we propose
a unified Bayesian model that integrates an NMFD-based acoustic
model evaluating the likelihood of a drum score for a drum spectrogram, with a deep language model serving as a prior (constraint) of
the score. The language model can be trained with existing drumscores in the framework of autoencoding variational Bayes and has
more expressive power than the conventional statistical models. We
derive an inference algorithm using Gibbs sampling, which is a marriage of the solid formalism of Bayesian learning with the expressive power of deep learning. It is shown that the proposed method
not only slightly improved the F-measure score but also increased
musical naturalness of the transcribed drum scores than NMFD.
Index Terms— Drum transcription, musical language model,
NMF, VAE, deep Bayesian learning
1. INTRODUCTION
Automatic drum transcription (ADT) has actively been investigatedfor describing the rhythmic characteristics of popular music in thefield of music information retrieval (MIR) [1]. Although many different types of drum instruments such as floor, low, and high toms
and ride and crash cymbals are included in a drum kit, three kinds ofdrum instruments, i.e., bass and snare drums and hi-hats, have commonly been focused on because they form the rhythmic backboneof popular music. Most studies on ADT aim to estimate drum rolls
describing the onset times of those drums in a similar way that moststudies on automatic music transcription (AMT) aim to estimate piano rolls describing the onset and offset times of pitched musicalinstruments. To complete ADT, it is thus necessary to convert drumrolls to drum scores by quantizing the onset times of the drums. Sucha process is called rhythm transcription in AMT [2, 3], but this has
scarcely been investigated in ADT.
A typical approach to ADT is to use nonnegative matrix factorization (NMF) for decomposing a drum spectrogram into the basis
spectra and temporal activations of the three drums [4–6]. NMF has
often been used for AMT and is especially suitable for ADT because drum sounds appear repeatedly with different combinationsand volumes and the magnitude spectrogram of a drum part can thus
This work was supported in part by JST ACCEL No. JPMJAC1602,
JSPS KAKENHI No. 16H01744 and No. 16J05486, and the Kyoto University Foundation.

      

               

Fig. 1. A hierarchical generative model of a drum-part spectrogram
integrating a pretrained DNN-based drum score model (score prior)with an NMFD-based acoustic model (score likelihood). Given adrum-part spectrogram as observed data, a drum score and all variables are estimated by using both models in a Bayesian manner.
be approximated as a low-rank matrix. Since the acoustic characteristics of each drum cannot be fully represented by a basis spectrum, Smaragdis [7] proposed a convolutional extension of NMF
called nonnegative matrix factor decomposition (NMFD) that approximates a drum-part spectrogram as a patchwork consisting ofoverlapping basis spectrograms of the drums. To detect the onset
times of the drums, simple peak-picking or thresholding is typicallyapplied to the estimated activations. To avoid such a separate postprocessing, Liang et al. [8] proposed beta-process NMF (BP-NMF)
that introduces binary variables (masks) describing the presence orabsence of basis components at each time.
Although NMF and its variants have been used successfullyfor ADT, musically unnatural drum rolls are often obtained. If adictionary of drum patterns is available, one can categorize eachsegment of the estimated drum rolls into one of the registered patterns [9]. This approach, however, cannot deal with unregistereddrum patterns. Recurrent neural networks (RNNs) have recently
been used for learning direct conversion of a drum-part spectrogram
to a drum roll in a supervised manner and significantly improved theperformance [10, 11]. However, musically unnatural scores cannot
be avoided because RNNs are used for learning the temporal dynamics of drum sound mixtures at the frame level and those of drumscores at the tatum level are not considered.
The limitation of these pure acoustic models calls for a music
language model defined on symbolic musical scores. Such language
models have recently been used successfully for AMT [12–14]. Abasic approach to representing the sequential dependency of musical notes is to use first- or lower-order Markov models or hidden
Markov models (HMMs) [13]. The expressive power of these models, however, is severely limited and higher-order models are computationally prohibitive. RNN-based language models have recently
been proposed to learn long-term dependency of musical notes andused for estimating musical scores from piano rolls estimated by an
NMF-like low-rank acoustic model [14]. Principled integration of alanguage model defined on discrete symbols and an acoustic model
defined on continuous values is still an open problem.
In this paper, we propose a new approach to ADT based on a unified Bayesian model integrating a DNN-based language model withan NMFD-based acoustic model (Fig. 1) under an assumption thattatum times (16th-note-level beat times) and bar lines are given inadvance (e.g. by a beat tracking method [15]). The acoustic model
evaluates the likelihood of a drum score (tatum-level binary variables) for a drum spectrogram and the language model evaluates theprior probability (musical appropriateness) of the score. While thephysical additivity of drum sounds can be represented well by a linear model based on NMFD, the complicated syntactic structures ofdrum scores are hard to be explicitly represented. We thus use a variational autoencoder (VAE) [16] for learning an implicit generativemodel of one-measure drum patterns with their latent feature representations from existing drum patterns in an unsupervised manner.
Given a drum spectrogram, a drum score (a sequence of one-measure
drum patterns) and all variables of the language and acoustic models
can be estimated in a principled manner via Gibbs sampling.
At the heart of this study is a marriage of the solid formalismof Bayesian learning with the expressive power of deep learning.
This is the first attempt that utilizes a powerful deep prior model for
ADT and that can be applied to more general types of AMT. A key
advantage of our deep Bayesian approach is that a huge amount ofdrum patterns available on the Web [17] can be used for learning thelanguage model, while a standard approach to end-to-end learningneeds time-aligned pair data for supervised learning.

2.2.1. NMFD-Based Acoustic Model (Score Likelihood)
The magnitude spectrogram X is approximated by using basis spec(K+1)×F ×M(K+1)×T
trograms W ∈ R+
, activation vectors H ∈ R+,
K×R
and binary masks S ∈ {0, 1}
as follows:def

Xf t ≈ Yf t =

M X
K
X

Yf tkm .

(1)

m=1 k=0

Here, Yf tkm is given by
Yf tkm = Wkf m Hk,t−m Sk,r(t−m)
Yf t0m = W0f m H0,t−m ,(k ≥ 1),(2)where M is the number of frames forming each basis spectrogram,{Wkf m }F
f =1 (k ≥ 1) is the basis spectrum of drum k at frame mand r(t) denotes the tatum to which frame t belongs. We have introduced an additional basis spectrogram W0f m and an activation vector H0t to represent possible noise added to the target drum sounds.
To evaluate the approximation error of Eq. (1), we use the KullbackLeibler (KL) divergence as in KL-NMF [19]. In terms of probabilistic modeling, the minimization of the KL divergence is equivalent to
the maximization of the Poisson likelihood given by
Xf t ∼ Poisson(Yf t ).

(3)

To complete Bayesian formulation, we put conjugate gamma
priors on W as follows:
Wkf m ∼ Gamma(akf m , bkf m ) (k ≥ 1),(4)
W0f m ∼ Gamma(a0 , b0 ),where Gamma(a∗ , b∗ ) denotes a gamma distribution with shapeand rate hyperparameters a∗ and b∗ . Similarly, we put conjugate
gamma priors H as follows.

Hkt ∼ Gamma(ck , dk ) (k ≥ 1),(5)
H0t ∼ Gamma(c0 , d0 ),where ck , dk c0 , and d0 are hyperparameters.

2. PROPOSED METHODThis section describes the proposed method that estimates a drumscore from a drum-part signal separated from a popular music signal
using harmonic/percussive source separation (HPSS) [18].
2.1. Problem Specification
The problem of ADT is formalized as follows:
F ×T
The magnitude spectrogram of a target signal X ∈ R+

Input:
with 16th-note-level tatum times and bar lines
Output: Drum score S ∈ {0, 1}K×R .

2.2.2. DNN-Based Language Model (Score Prior)
The binary masks S are assumed to independently follow Bernoulli
distributions as follows:Skr ∼ Bernoulli(πkr ),where πkr indicates the prior probability of the presence of the onsetof drum k at tatum r. For mathematical convenience, we rewrite thedrum- and tatum-wise representation given by Eq. (6) as a measurewise representation as follows:si ∼ Bernoulli(πi ),Here, F is the number of frequency bins, T the number of timeframes, K = 3 the number of drum instruments (snare and bass
drums and hi-hats), and R the number of tatums in the observed
signal. The target signal is assumed to include only the percussive
components obtained from the HPSS method [18]. The binary mask
Skr indicates whether drum k has an onset at tatum r. Note that S
can be divided into measures (drum patterns).
2.2. Model Formulation
We formulate a hierarchical generative model of a magnitude spectrogram X by integrating a DNN-based language model of binarymasks S with an NMFD-based acoustic model of X (Fig. 1).

(6)

(7)

where si and πi are 16K-dimensional binary and real-valued vectors consisting of Skr ’s and πkr ’s in measure i (0 ≤ i ≤ I − 1),
respectively. The core part of the proposed method is that πi is represented by an implicit deep generative model as follows:zi ∼ N (0, 1),
πi = DNNθ (zi ),

(8)
(9)

where DNNθ is a non-linear function with parameters θ that maps
zi to πi and zi is a V -dimensional latent representation of the drum
pattern of measure i. The deep score prior pθ (S) is obtained bymarginalizing out the latent variables Z from the implicit generativemodel given by pθ (S|Z)p(Z).

(QFRGHU
2XWSXW ࣌ଶ

ࣆ

)&

)&

)& WDQK)& WDQK,QSXW'HFRGHU

where the first and second terms of Eq. (14) or Eq. (15) indicate
the prior probability and the acoustic likelihood, respectively, andS¬(kr) denotes the subset of S excluding Skr . Note that π depends
on Z. The likelihood terms of Eq. (14) and Eq. (15) are given by࣊)& VLJPRLG)& WDQK

p(X|W, H, S¬(kr) , Skr = 0)
! Xf t)& WDQK)& WDQK‫ א ܁‬Թସ଼‫ א ܈‬Թସ

=

Y

Y

Yf¬k
t +

X

Wkf m Hk,t−mmt∈{r(t)=r} f()
X· exp −

Fig. 2. The VAE of one-measure drum patterns.

Wkf m Hk,t−m

,

(16)

m

2.3. Score Prior Learning
To estimate the deep score prior pθ (S), we train a variational autoencoder (VAE) for existing drum patterns S in an unsupervisedmanner. Our goal is to estimate the DNN parameters θ that maximize the likelihood given by pθ (S). Since the direct maximizationof pθ (S) is intractable, we derive the lower bound of log pθ (S) thatcan be maximized easily. More specifically, introducing an arbitrary
variational distribution q(Z) and using Jensen’s inequality, the lower
bound of log pθ (S) can be derived as follows:
log pθ (S) ≥ −KL[q(Z)|p(Z)] + Eq [log pθ (S|Z)].

(10)

As an instance of q(Z), we formulate a recognition model qφ (Z|S)
with parameters φ defined as follows:qφ (Z|S) =I−1
YN (zi |µφ (si ), σφ2 (si )),

(11)

i=0

where µφ and σφ2 are nonlinear functions defined with DNNs whose
input and output are 16K- and V -dimensional vectors, respectively.
The lower bound of log pθ (S) can be further written as follows:
log pθ (S)

1X
2
2
≥
1 + log σφ,v(si ) − µ2φ,v (si ) − σφ,v
(si )
2 i,v
X
+
Eq [Skr log πkr + (1 − Skr ) log(1 − πkr )],where Yf¬kt is given by
Yf¬k
t =

XX

Yf tkm(k ≥ 1).

(18)

l6=k m

2.4.2. Updating NMFD-Based Acoustic Model
To sample W, H, and S involved in Bayesian NMFD with binarymasks S, we extend a Gibbs sampling method proposed for Bayesian
NMF with binary masks called BP-NMF [8]. More specifically, conditioned by H and S, W is sampled as follows:
Wkf m ∼ Gamma(âkf m , b̂kf m ),Pâkf m = P t Xf t λf tkm + akf m (k ≥ 1),â0f m = t Xf t λf t0m + a0 ,Pb̂kf m = t Hk,t−m Sk,t−m + bkf m (k ≥ 1),P
b̂0f m = t H0,t−m + b0 ,λf tkm =

(12)

(19)
(20)
(21)

Yf tkm
.
Yf t

(22)

Similarly, conditioned by W and S, H is sampled as follows:k,rand φ because π is determined by Eq. (9). Both θ and φ are jointly
optimized such that the lower bound given by Eq. (12) is maximized
by a stochastic gradient descent method such as Adam [20].
2.4. Score Posterior Computation
Given X as observed data, we aim to compute the posterior distribution p(W, H, S, Z|X). Since this cannot be calculated analytically,
we use Gibbs sampling for iteratively and alternately updating W,
H, S, and Z in a stochastic manner.
2.4.1. Updating Drum Score
Using the acoustic model with W and H and the language modelwith Z, binary masks S are sampled as follows:
P1kr
Skr ∼ Bernoulli,(13)
P0kr + P1kr
P1kr ∝ πkr p(X|W, H, S¬(kr) , Skr = 1),(17)t∈{r(t)=r} fwhere λf tkm is an auxiliary variable given by

where µ2φ,v (si ) is the vth dimension of the V -dimensional output of2(si )is defined similarly. Eq. (12) is a function of θ
µ2φ (si ) and σφ,vP0kr ∝ (1 − πkr ) p(X|W, H, S¬(kr) , Skr = 0),p(X|W, H, S¬(kr) , Skr = 1)YY  ¬k Xf t
,
=
Yf t

(14)
(15)

Hkt ∼ Gamma(ĉkt , dˆkt ),Pĉkt = P f,m Xf t λf,t+m,km + ck (k ≥ 1),ĉ0t = f,m Xf t λf,t+m,0m + c0 ,(Pdˆkt = f,m Wkf m Skt + dk (k ≥ 1),P
dˆ0t = f,m W0f m + d0 .

(23)
(24)
(25)

2.4.3. Updating DNN-Based Language Model
Since it is difficult to analytically calculate the posterior distributionof Z, we use a Metropolis-Hastings method to update Z. A proposalof z∗i at each bar i is sampled in a way of random walk by using a
Gaussian distribution as follows:z∗i ∼ q(z∗i |zi ) = N (zi , 0.1).

(26)

The proposal z∗i is accepted as the next zi with the following acceptance rate az∗i |zi :p(z∗i )
az∗i |zi = min 1,p(zi )Yk,r∈{bar(r)=i}p(Skr |z∗i ) .
p(Skr |zi )(27)

Method

Part

P(%)

R(%)

F(%)

NMFD

HH
SD
BD

79.4
63.2
82.3

60.9
63.6
80.2

69.0
63.4
81.2

HH
SD
BD

80.9
67.6
83.0

61.4
65.4
79.4

69.8
66.5
81.2

VAE-NMFD

Table 1. Performances of ADT for RWC popular music database.
The “HH”, “SD”, and “BD” represent the hi-hats and snare and bass
drums, respectively.

rHere, bar(r) = b 16
c denotes the measure to which tatum r belongs.
To estimate Z effectively, we initialize Z with samples drawn fromthe recognition model qφ (Z|S) with initial estimates of S.

Input

Ground truth

HH
SD
BD

NMFD

HH
SD
BD
HH
SD
BD

VAE-NMFD

HH
SD
BD
HH
SD
BD

Blue: Detected notes (correct), Red: Delete error, Yellow: Insertion error

Fig. 3. Examples of drum scores estimated by NMFD (baseline)
and VAE-NMFD (proposed). For VAE-NMFD, activations obtained
after applying the masks are shown.

3. EVALUATION
conditions for forming a peak:
3.1. Experimental Setup

Hkt∗ Skt∗ ≥ 0.3 · max{Hkt Skt },t

For evaluation, we used audio signals in the RWC popular music
database [21]. Those signals were converted into monaural signals
and divided into segments of 30-second length. The second segmentof each piece was used for evaluation. We selected 64 pieces inwhich bass and snare drums and hi-hats are played at least once.
We split the selected audio signals into segments of 1 measure using
tatum times obtained from the annotations [22]. The tatum times weused were shifted 0.03 seconds earlier from the original annotations
to align them with the onset times of the drum sounds.
All songs were sampled at 44.1 kHz, and we obtained magnitude spectrograms using an STFT with a Hann window of 2048
points and a shifting interval of 441 points (10 ms). Moreover, weapplied HPSS [18] for the spectrograms to separate the drum part
spectrograms. Each magnitude spectrogram was normalized so thatthe average magnitude becomes unity.
To determine the hyperparameters akf m and bkf m (k ≥ 1), we
used the isolated sounds of bass and snare drums and hi-hats fromthe RWC musical instrument sound database [23]. The magnitude
spectrograms of those sounds were obtained similarly as above andthe template spectrogram of each drum was estimated by applying
NMFD with a single basis to the spectrogram obtained by concatenating all the spectrograms of the target drum. The hyperparameters
were set so that the means of the Gamma priors were the same as thebasis spectrograms and the variance of the Gamma priors was 0.01.
The other hyperparameters of the priors on basis spectrograms W
and activation vectors H were set as a0f m = 0.05, b0f m = 50.0,c0 = 50.0, d0 = 50.0, ck = 1.0, and dk = 50.0. We trained the
VAE network using 41474 bars obtained from drum scores of The
Beatles and Japanese popular music, which had no overlaps with thetest data. The number of frames forming each basis spectrogram was
M = 20. The dimension of the latent variable zi was V = 4.
Performance of ADT was measured by the precision and recall
rates and F-measure defined as follows:
Nc
Nc
2RP
P=
, R=
, F=
,
(28)
Ne
Ng
R+P
where Ne , Ng , and Nc are the numbers of estimated, ground-truth,and correct notes, respectively. For each k (≥ 1), note onsets t∗
are detected using the estimated from H and S by the following

Hkt∗ Skt∗ = ∗ max ∗{Hkt Skt }.

t −5≤t≤t +5

(29)
(30)

When the time difference between an estimated note and a groundtruth note was within 50 ms, we judged the estimated note as correct.
3.2. Experimental Results
The experimental results of ADT are shown in Table 1. For snare
drum and hi-hats, the proposed method significantly outperformed
NMFD in all the metrics. For bass drum, the recall rate for the
proposed method was slightly worse than that of NMFD and theF-measure was even. In the example in Fig. 3, the snare drum part
obtained by NMFD (acoustic model) had unnatural rhythmic patterns (for example in the last half measure) whereas that obtainedby the proposed method was musically natural. These results indicate that the proposed method integrating the DNN-based language
model and the NMFD-based acoustic model not only improved theobjective evaluation metrics but also increased the musical naturalness of the transcribed scores. These results clearly demonstrate theeffectiveness of the proposed method.

4. CONCLUSION
This paper has presented a statistical method of ADT that integrates
an NMFD-based acoustic model with a VAE-based deep language
model in a unified Bayesian manner. A key advantage of our deep
Bayesian approach is that the language model can be learned frommusical scores, while a standard approach to end-to-end learningneeds time-aligned pair data for supervised learning. This approach
can be applied to more general types of music transcription. The
experimental results showed that the proposed method can estimate
musically natural scores by leveraging the powerful deep score prior.
A future direction is to integrate the present method with a statistical method of beat and downbeat detection for joint estimationof drum scores and beat times, similarly as in [24]. We also plan torepresent the temporal dependency and repetitive structures of drum
patterns by using a time-series or recurrent extension of the VAE.

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