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Bidirectional Quaternion Long-Short Term Memory Recurrent Neural Networks for Speech Recognition

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Abstract

Recurrent neural networks (RNN) are at the core of modern automatic speech recognition (ASR) systems. In particular, long-short term memory (LSTM) recurrent neu-ral networks have achieved state-of-the-art results in many speech recognition tasks, due to their efficient representation of long and short term dependencies in sequences of interdependent features. Nonetheless, internal dependencies within the element composing multidimensional features are weakly considered by traditional real-valued representations. We propose a novel quaternion long-short term memory (QL-STM) recurrent neural network that takes into account both the external relations between the features composing a sequence , and these internal latent structural dependencies with the quaternion algebra. QLSTMs are compared to LSTMs during a memory copy-task and a realistic application of speech recognition on the Wall Street Journal (WSJ) dataset. QLSTM reaches better performances during the two experiments with up to 2.8 times less learning parameters, leading to a more expressive representation of the information.
BIDIRECTIONAL QUATERNION LONG-SHORT TERM MEMORY
RECURRENT NEURAL NETWORKS FOR SPEECH RECOGNITION
Titouan Parcollet1,3, Mohamed Morchid1, Georges Linarès1, and Renato De Mori1,2
1Université d’Avignon, LIA, France
2McGill University, Montréal, Canada
3Orkis, Aix en provence, France
ABSTRACT
Recurrent neural networks (RNN) are at the core of
modern automatic speech recognition (ASR) systems. In
particular, long-short term memory (LSTM) recurrent neu-
ral networks have achieved state-of-the-art results in many
speech recognition tasks, due to their efficient representa-
tion of long and short term dependencies in sequences of
inter-dependent features. Nonetheless, internal dependencies
within the element composing multidimensional features are
weakly considered by traditional real-valued representations.
We propose a novel quaternion long-short term memory (QL-
STM) recurrent neural network that takes into account both
the external relations between the features composing a se-
quence, and these internal latent structural dependencies with
the quaternion algebra. QLSTMs are compared to LSTMs
during a memory copy-task and a realistic application of
speech recognition on the Wall Street Journal (WSJ) dataset.
QLSTM reaches better performances during the two experi-
ments with up to 2.8times less learning parameters, leading
to a more expressive representation of the information.
Index TermsQuaternion long-short term memory, re-
current neural networks, speech recognition
1. INTRODUCTION
During the last decade, deep neural networks (DNN) have
encountered a wide success in numerous domain applica-
tions. In particular, automatic speech recognition systems
(ASR) performances have been remarkably improved with
the emergence of DNNs. Among them, recurrent neural
networks [1] (RNN) have been shown to effectively encode
input sequences, increasing the accuracy of neural network
based ASR systems [2]. Nonetheless, vanilla RNNs suffer
from vanishing/exploding issues [3], or the lack of a mem-
ory mechanism to remember patterns in very-long or short
sequences. These problems have been alleviated by the intro-
duction of long-short term memory (LSTM) RNN [4] with
gates mechanism that allows the model to update or forget in-
formation in memory cells, and to select the content cell state
to expose in a network hidden state. LSTMs have reached
state-of-the art performances in many benchmarks [4, 5], and
are widely employed in recent ASR models, with the almost
unchanged acoustic input features used in previous systems.
Traditional ASR systems rely on multidimensional acous-
tic features such as the Mel filter bank energies alongside with
the first, and second order time derivatives to characterize
time-frames that compose the signal sequence. Considering
that these components describe three different views of the
same element, neural networks have to learn both the internal
relations that exist within these views, and external or global
dependencies that exist between the time-frames. Such con-
cerns are partially addressed by increasing the learning capac-
ity of neural network architectures. Nonetheless, even with a
huge set of free parameters, it is not certain that both local and
global dependencies are properly represented. To address this
problem, new quaternion-valued neural networks, based on a
high-dimensional algebra, are proposed in this paper.
Quaternions are hyper-complex numbers that contain a
real and three separate imaginary components, fitting per-
fectly to three and four dimensional feature vectors, such
as for image processing and robot kinematics [6, 7]. The
idea of bundling groups of numbers into separate entities
is also exploited by the recent capsule network [8]. With
quaternion numbers, LSTMs are conceived to encode latent
inter-dependencies between groups of input features during
the learning process with less parameters than real-valued
LSTMs, by taking advantage of the use of the quaternion
Hamilton product as the counterpart of the dot product.
Early applications of quaternion-valued backpropagation al-
gorithms [9, 10] have efficiently shown that quaternion neu-
ral networks can approximate quaternion-valued functions.
More recently, neural networks of hyper-complex numbers
have received an increasing attention, and some efforts have
shown promising results in different applications. In partic-
ular, a deep quaternion network [11, 12], a deep quaternion
convolutional network [13, 14], or a quaternion recurrent
neural network [15] have been successfully employed for
challenging tasks such as images, speech and language pro-
cessing. For speech recognition, in [14], quaternions with
only three internal features have been used to encode input
speech. An additional internal feature is proposed in this
paper to obtain a richer representation with the same number
of model parameters.
Based on all the above considerations, the contributions of
this paper can be summarized as follows: 1) The introduction
of a novel model, called bidirectional quaternion long-short
term memory neural network (QLSTM)1, that avoids known
RNN problems also present in quaternion RNNs, and shows
that QLSTMs achieve top of the line results on speech recog-
nition; 2) The introduction of a novel input quaternion that
integrates four views of speech time frames. The model is
first evaluated on a synthetic memory copy-task to ensure that
the introduction of quaternion into the LSTM model does not
alter the basic properties of RNNs. Then, QLSTMs are com-
pared to real-valued LSTMs on a realistic speech recognition
task with the Wall Street Journal (WSJ) dataset. The reported
results show that the QLSTM outperforms the LSTM in both
tasks with a higher long-memory capability on the memory
task, a better generalization performance with better word er-
ror rates (WER), and a maximum reduction of the number
of neural paramaters of 2.8times compared to real-valued
LSTM.
2. QUATERNION ALGEBRA
The quaternion algebra Hdefines operations between quater-
nion numbers. A quaternion Q is an extension of a complex
number defined in a four dimensional space as:
Q=r1 + xi+yj+zk,(1)
where r,x,y, and zare real numbers, and 1,i,j, and kare
the quaternion unit basis. In a quaternion, ris the real part,
while xi+yj+zkwith i2=j2=k2=ijk =1is the
imaginary part, or the vector part. Such a definition can be
used to describe spatial rotations. The Hamilton product
between two quaternions Q1and Q2is computed as follows:
Q1Q2=(r1r2x1x2y1y2z1z2)+
(r1x2+x1r2+y1z2z1y2)i+
(r1y2x1z2+y1r2+z1x2)j+
(r1z2+x1y2y1x2+z1r2)k.(2)
The Hamilton product is used in QLSTMs to perform trans-
formations of vectors representing quaternions, as well as
scaling and interpolation between two rotations following a
geodesic over a sphere in the R3space as shown in [16].
3. QUATERNION LONG-SHORT TERM MEMORY
NEURAL NETWORKS
Based on the quaternion algebra and with the previously
described motivations, we introduce the quaternion long-
1Code is available at https://github.com/Orkis-Research/Pytorch-
Quaternion-Neural-Networks
short term memory (QLSTM) recurrent neural network. In
a quaternion dense layer, all parameters are quaternions, in-
cluding inputs, outputs, weights and biases. The quaternion
algebra is ensured by manipulating matrices of real numbers
[14] to reconstruct the Hamilton product from quaternion al-
gebra. Consequently, for each input vector of size N, output
vector of size M, dimensions are split in four parts: the first
one equals to r, the second to xi, the third one is yj, and
the last one equals to zk. The inference process of a fully-
connected layer is defined in the real-valued space by the dot
product between an input vector and a real-valued M×N
weight matrix. In a QLSTM, this operation is replaced with
the Hamilton product ’ (Eq. 2) with quaternion-valued
matrices (i.e. each entry in the weight matrix is a quaternion).
Gates are core components of the memory of LSTMs.
Based on [17], we propose to extend this mechanism to
quaternion numbers. Therefore, the gate action is charac-
terized by an independent modification of each component
of the quaternion-valued signal following a component-wise
product (i.e. in a split fashion [18]) with the quaternion-
valued gate potential. Let ft,it,ot,ct, and htbe the forget,
input, output gates, cell states and the hidden state of a LSTM
cell at time-step t. QLSTM equations can be derived as:
ft=σ(Wfxt+Rfht1+bf),(3)
it=σ(Wixt+Riht1+bi),(4)
ct=ft×ct1+it×α(Wcxt+Rcht1+bc),(5)
ot=σ(Woxt+Roht1+bo),(6)
ht=ot×α(ct),(7)
with σand αthe sigmoid and tanh quaternion split activa-
tions [18, 11, 19, 10]. The quaternion weight and bias ma-
trices are initialized following the proposal of [15]. Quater-
nion bidirectional connections are equivalent to real-valued
ones [20]. Consequently, past and future contexts are added
together component-wise at each time-step. The full back-
propagtion of quaternion-valued recurrent neural network can
be found in [15].
4. EXPERIMENTS
This section provides the results for QLSTM and LSTM on
the synthetic memory copy-task (Section 4.1), and a descrip-
tion of the quaternion acoustic features (Section 4.2) that are
used as inputs during the realistic speech recognition experi-
ment with the Wall Street Journal (WSJ) corpus (Section 4.3).
4.1. Synthetic memory copy-task as a sanity check
The copy task originally introduced by [21] is a synthetic test
that highlights how RNN based models manage the long-term
memory. This characteristic makes the copy task a powerful
benchmark to demonstrate that a recurrent model can learn
long-term dependencies. It consists of an input sequence of a
length L, composed of Sdifferent symbols followed by a se-
quence of time-lags or blanks of size T, and ended by a delim-
iter that announces the beginning of the copy operation (after
which the initial input sequence should be progressively re-
constructed at the output). In this paper, the copy-task is used
as a sanity check to ensure that the introduction of quaternions
on LSTM models does not harm the basic memorization abili-
ties of the LSTM. The QLSTM is composed of 8K parameters
with one hidden layer of size 20, while the LSTM is made of
8.2K parameters with an hidden dimension of 40 neurons. It
is worth underlying that due to the nature of the task, the out-
put layer of the QLSTM is real-valued. Indeed, 9symbols
are one-hot encoded (S= 0, ..., 7for the sequence and 8for
the blank) and can not be split in four components. Different
values of T= 10,50,100 are investigated alongside with a
fixed sequence size of L= 10. Models are trained with the
Adam optimizer, with an initial learning rate λ= 5·103, and
without employing any regularization methods. The training
is performed on 2,000 epochs with the cross-entropy used as
the loss function. At each epoch, models are fed with a batch
of 10 randomly generated sequences.
Fig. 1. Evolution of the cross entropy loss, and of the ac-
curacy of both QLSTM (Blue curves) and LSTM (Orange
curves) during the synthetic memory copy-task for time lags
or blanks Tof 10,50 and 100.
The results reported in Fig.1 highlight a slightly faster
convergence of the QLSTM over the LSTM for all sizes (T).
It is also worth noticing that real-valued LSTM failed the
copy-task with T= 100 while QLSTM succeeded. It is
easily explained by the impact of quaternion numbers dur-
ing the learning process of inter-denpendencies of input fea-
tures. Indeed, the QLSTM is a smaller (less parameters), but
more efficient (dealing with higher dimensions) model than
real-valued LSTM, resulting in a higher generalization capa-
bility: 20 quaternion neurons are equivalent to 20 ×4 = 80
real-valued ones. Overall, the introduction of quaternions in
LSTMs do not alter their basics properties, but it provides a
higher long-term dependencies learning capability. We hy-
pothesis that such efficiency improvements alongside with a
dedicated input representation will help QLSTMs to outper-
form LSTMs in more realistic tasks, such as speech recogni-
tion.
4.2. Quaternion acoustic features
Unlike in [14], this paper proposes to use four internal fea-
tures in an input quaternion. The raw audio is first split ev-
ery 10ms with a window of 25ms. Then 40-dimensional log
Mel-filter-bank coefficients with first, second, and third order
derivatives are extracted using the pytorch-kaldi2toolkit and
the Kaldi s5 recipes [2]. An acoustic quaternion Q(f , t)asso-
ciated with a frequency band fand a time-frame tis formed
as follows:
Q(f, t) = e(f , t) + e(f, t)
∂t i+2e(f , t)
2tj+3e(f, t)
3tk.
(8)
Q(f, t)represents multiple views of a frequency band fat
time frame t, consisting of the energy e(f, t)in the filter
band at frequency f, its first time derivative describing a
slope view, its second time derivative describing a concavity
view, and the third derivative describing the rate of change
of the second derivative. Quaternions are used to construct
latent representations of the external relations between the
views characterizing the contents of frequency bands at given
time intervals. Thus, the quaternion input vector length is
160/4 = 40. Decoding is based on Kaldi [2] and weighted fi-
nite state transducers (WFST) that integrate acoustic, lexicon
and language model probabilities into a single HMM-based
search graph.
4.3. Speech recognition with the Wall Street Journal
QLSTMs and LSTMs are trained on both the 14 hour sub-
set ‘train-si84’, and the full 81 hour dataset ’train-si284’ of
the Wall Street Journal (WSJ) corpus. The ‘test-dev93’ de-
velopment set is employed for validation, while ’test-eval92’
composes the testing set. It is important to notice that eval-
uated LSTMs and QLSTMs are bidirectionals. Architecture
models vary in both number of layers and neurons. Indeed
the number of recurrent layers Lvaries from three to four,
while the number of neurons Nis included in a gap from 256
to 1,024. Then, one dense layer is stacked alongside with an
output dense layer. It is also worth noticing that the number of
quaternion units of a QLSTM layer is N/4. Indeed, QLSTM
neurons are four dimensional (i.e. a QLSTM layer that deals
with a dimension size of 1,024 has 1,024/4 = 256 effec-
tive quaternion neurons). Models are optimized with Adam,
2pytorch-kaldi is available at https://github.com/mravanelli/pytorch-kaldi
Table 1. Word error rates (WER %) obtained with both training set (WSJ14h and WSJ81h) of the Wall Street Journal corpus.
’test-dev93’ and ’test-eval92’ are used as validation and testing set respectively. Lexpresses the number of recurrent layers.
Models are bidirectional. Results are from an average of three runs.
Models WSJ14 Dev. WSJ14 Test WSJ81 Dev. WSJ81 Test Params
R-LSTM-3L-256 12.7 8.6 9.5 6.5 4.0M
H-QLSTM-3L-256 12.8 8.5 9.4 6.5 2.3M
R-LSTM-4L-256 12.1 8.3 9.3 6.4 4.8M
H-QLSTM-4L-256 11.9 8.0 9.1 6.2 2.5M
R-LSTM-3L-512 11.1 7.1 8.2 5.2 12.2M
H-QLSTM-3L-512 10.9 6.9 8.1 5.1 5.6M
R-LSTM-4L-512 11.3 7.0 8.1 5.0 15.5M
H-QLSTM-4L-512 11.1 6.8 8.0 4.9 6.5M
R-LSTM-3L-1024 11.4 7.3 7.6 4.8 41.2M
H-QLSTM-3L-1024 11.0 6.9 7.4 4.6 15.5M
R-LSTM-4L-1024 11.2 7.2 7.4 4.5 53.7M
H-QLSTM-4L-1024 10.9 6.9 7.2 4.3 18.7M
with vanilla hyper-parameters and an initial learning rate of
5·104. The learning rate is progressively annealed using
an halving factor of 0.5that is applied when no performance
improvement on the validation set is observed. The models
are trained during 15 epochs. All the models converged to a
minimum loss, due to the annealed learning rate. Results are
from a three folds average.
At first, it is important to notice that reported results on
Table 1 compare favorably with equivalent architectures [5]
(WER of 11.7% on ’test-dev93’), and are competitive with
state-of-the-art and much more complex models based on bet-
ter engineered features [22] (WER of 3.8% with the 81 hours
of training data, and on ’test-eval92’). Table 1 shows that
the proposed QLSTM always outperform real-valued LSTM
on the test dataset with less neural parameters. Based on the
smallest 14 hours subset, a best WER of 6.9% is reported in
real conditions (w.r.t to the best validation set results) with a
three layered QLSTM of size 512, compared to 7.1% for an
LSTM with the same size. It is worth mentioning that a best
WER of 6.8% is obtained with a four layered QLSTM of size
512, but without consideration for the validation results. Such
performances are obtained with a reduction of the number of
parameters of 2.2times, with 5.6M parameters for the QL-
STM compared to 12.2M for the real-valued equivalent. This
is easily explained by considering the content of the quater-
nion algebra. Indeed, for a fully-connected layer with 2,048
input values and 2,048 hidden units, a real-valued RNN has
2,04824.2M parameters, while, to maintain equal input
and output dimensions, the quaternion equivalent has 512
quaternions inputs and 512 quaternion hidden units. There-
fore, the number of parameters for the quaternion-valued
model is 5122×41M. Such a complexity reduction turns
out to produce better results and have other advantages such
as a smaller memory footprint while saving models on bud-
get memory systems. This reduction allows the QLSTM to
make the memory more “compact” and therefore, the re-
lations between quaternion components are more robust to
unseen documents from both validation and testing data-sets.
This characteristic makes our QLSTM model particularly
suitable for speech recognition conducted on low computa-
tional power devices like smartphones. Both QLSTMs and
LSTMs produce better results with the 81 hours of training
data. As for the smaller subset, QLSTMs always outperform
LSTMs during both validation and testing phases. Indeed, a
best WER of 4.3% is reported for a four layered QLSTM of
dimension 1,024, while the best LSTM performed at 4.5%
with 2.9times more parameters, and an equivalently sized
architecture.
5. CONCLUSION
This paper proposes to process sequences of traditional and
multidimensional acoustic features with a novel quaternion
long-short term memory neural network (QLSTM). The pa-
per introduce first a novel quaternion-valued representation
of the speech signal to better handle signal sequences depen-
dencies, and a LSTM composed with quaternions to repre-
sent in the hidden latent space inter-dependencies between
quaternion features. The proposed model has been evalu-
ated on a synthetic memory copy-task and a more realistic
speech recognition task with the large Wall Street Journal
(WSJ) dataset. The reported results support the initial intu-
itions by showing that QLSTM are more effective at learn-
ing both longer dependencies and a compact representation of
multidimensional acoustic speech features by outperforming
standard real-valued LSTMs on both experiments, with up to
2.8times less neural parameters. Therefore, and as for other
quaternion-valued architectures, the intuition that the quater-
nion algebra of the QLSTM offers a better and more compact
representation for multidimensional features, alongside with
a better learning capability of feature internal dependencies
through the Hamilton product, have been validated.
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