Real to H-Space Encoder for Speech Recognition

Abstract

Deep neural networks (DNNs) and more precisely recurrent neural networks (RNNs) are at the core of modern automatic speech recognition systems, due to their efficiency to process input sequences. Recently, it has been shown that different input representations, based on multidimensional algebras, such as complex and quaternion numbers, are able to bring to neural networks a more natural, compressive and powerful representation of the input signal by outperforming common real-valued NNs. Indeed, quaternion-valued neural networks (QNNs) better learn both internal dependencies, such as the relation between the Mel-filter-bank value of a specific time frame and its time derivatives, and global dependencies, describing the relations that exist between time frames. Nonetheless, QNNs are limited to quaternion-valued input signals, and it is difficult to benefit from this powerful representation with real-valued input data. This paper proposes to tackle this weakness by introducing a real-to-quaternion encoder that allows QNNs to process any one dimensional input features, such as traditional Mel-filter-banks for automatic speech recognition.

Full text

Real to H-space Encoder for Speech Recognition
Titouan Parcollet1,2, Mohamed Morchid1, Georges Linarès1, Renato De Mori3
1Avignon Université, LIA, France
2ORKIS, Aix-en-provence, France
3McGill University, Montréal, QC, Canada
titouan.parcollet@alumni.univ-avignon.fr, {firstname.lastname}@univ-avignon.fr
Abstract
Deep neural networks (DNNs) and more precisely recurrent
neural networks (RNNs) are at the core of modern automatic
speech recognition systems, due to their efficiency to process
input sequences. Recently, it has been shown that different in-
put representations, based on multidimensional algebras, such
as complex and quaternion numbers, are able to bring to neural
networks a more natural, compressive and powerful representa-
tion of the input signal by outperforming common real-valued
NNs. Indeed, quaternion-valued neural networks (QNNs) better
learn both internal dependencies, such as the relation between
the Mel-filter-bank value of a specific time frame and its time
derivatives, and global dependencies, describing the relations
that exist between time frames. Nonetheless, QNNs are limited
to quaternion-valued input signals, and it is difficult to benefit
from this powerful representation with real-valued input data.
This paper proposes to tackle this weakness by introducing a
real-to-quaternion encoder that allows QNNs to process any one
dimensional input features, such as traditional Mel-filter-banks
for automatic speech recognition.
Index Terms: quaternion neural networks, recurrent neural net-
works, speech recognition
1. Introduction
Automatic speech recognition (ASR) systems have been widely
impacted by machine learning, and more precisely by the resur-
gence of deep neural networks (DNNs). In particular, recurrent
neural networks (RNNs) have been designed to learn parame-
ters of sequence to sequence mapping, and various models were
successfully applied to ASR with a remarkable increasing in the
ASR system performance. In order to avoid parameter estima-
tion problems, RNNs with long short-term memory [1, 2], and
gated recurrent unit (GRU) [3] have been proposed to mitigate
vanishing and exploding gradients when learning long input se-
quences. Nevertheless, less attention has been paid to model
input features with multiple views of speech spectral tokens.
A noticeable exception is the use of complex-valued num-
bers in neural networks (CVNNs) to jointly represent am-
plitude and phase of spectral samples [4]. More recently,
quaternion-valued neural networks (QNNs) have been inves-
tigated to process the traditional Mel-frequency cepstral co-
efficients (MFCCs), or Mel-filter-banks plus time derivatives
[5, 6, 7] as composed entities. Superior accuracy, with up
to four times less model parameters, has been observed with
quaternion-valued models compared to results obtained with
real-valued equivalent models. In fact, common real-valued
neural networks process energies and time derivatives inde-
pendently, learning both global dependencies between multiple
time frames, and local values in a specific time frame, without
considering the relations between a value and its derivatives. In-
stead, the quaternion algebra allows QNNs [5, 6, 8, 9, 10, 11]
to process time frames as composed entities, with internal re-
lations learned within the specific algebra, and global depen-
dencies learned with the neural network architecture, while re-
ducing by an important factor the number of neural parameters
needed. As a side effect, with quaternion algebra, the number
of neural parameters to be estimated is reduced by an impor-
tant factor. Nonetheless, QNNs input features must be encoded
as quaternion numbers, requiring a preliminary definition of in-
put views that cannot be modified by the learning process. In
many cases, it looks advantageous to have multiple input fea-
ture views, but there may be different choices of them and it is
not clear how to make a selection. Examples could be views
based on temporal or spectral relations. In this paper, a real-
to-quaternion encoder (R2H) is proposed to let a quaternion-
valued neural architecture learn hidden representations of input
feature views. The R2H layer acts as an encoder to train QNNs
with any real-valued input vector. Indeed, this encoder allows
the model to learn in an end-to-end architecture, a latent and
quaternion-valued representation of the input data. This repre-
sentation is then used as an input to a quaternion-valued clas-
sifier, to exploit the capabilities of quaternion neural networks.
For achieving this objective, the contributions of this paper are:
• Investigate different real-to-quaternion (R2H) encoders
to learn an internal representation of any real-valued in-
put data (Section 4).
• Merge the R2H with the previously introduced quater-
nion long short-term memory neural network (QLSTM,
Section 3)[6]1.
• Evaluate this approach on the TIMIT, and Librispeech
speech recognition tasks (Section 5).
Results improvements on both TIMIT and Librispeech speech
recognition tasks are reported with the introduction of a R2H
encoder in a QLSTM architecture, with input made of 40 Mel-
filter-bank coefficients, and with more than three times fewer
neural parameters than with real-valued LSTMs.
2. Quaternion algebra
The quaternion algebra Hdefines operations between quater-
nion numbers. A quaternion Q is an extension of a complex
number to the hyper-complex plane defined in a four dimen-
sional 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
1Code is available at: https://github.com/Orkis-Research/Pytorch-
Quaternion-Neural-Networks

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. A quaternion Qcan also be summarized into
the following matrix of real numbers, that turns out to be more
suitable for computations:
Qmat =


r−x−y−z
x r −z y
y z r −x
z−y x r


.(2)
The conjugate Q∗of Qis defined as:
Q∗=r1−xi−yj−zk.(3)
Then, a normalized or unit quaternion Q/is expressed as:
Q/=Q
|Q|,(4)
with |Q|the norm of Q defined as:
|Q|=pr2+x2+y2+z2.(5)
Finally, the Hamilton product ⊗between two quaternions Q1
and Q2is computed as follows:
Q1⊗Q2=(r1r2−x1x2−y1y2−z1z2)+
(r1x2+x1r2+y1z2−z1y2)i+
(r1y2−x1z2+y1r2+z1x2)j+
(r1z2+x1y2−y1x2+z1r2)k.(6)
The Hamilton product is used in QNNs to perform transforma-
tions 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 [12].
3. Quaternion long short-term memory
neural networks
Long short-term memory neural networks (LSTM) are a well-
known and investigated extension of recurrent neural networks
[1, 13]. LSTMs offer an elegant solution to the vanishing and
exploding gradient problems, alongside with a stronger learn-
ing capability of long and short-term dependencies within se-
quence. Following these strengths, a quaternion long short-term
memory neural network (QLSTM) has been proposed [5].
In a quaternion-valued layer, all parameters are quaternions,
including inputs, outputs, weights and biases. The quaternion
algebra is ensured by manipulating matrices of real numbers
[7, 11] 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 be-
tween an input vector and a real-valued M×Nweight matrix.
In a QLSTM, this operation is replaced with the Hamilton prod-
uct ’⊗’ (Eq. 6) with quaternion-valued matrices (i.e. each entry
in the weight matrix is a quaternion).
Both LSTM and QLSTM networks rely on a gate action
[14], that allows the cell-state to retain or discard information
from the past, and the future in the case of a bidirectional
(Q)LSTM. Gates are defined in the quaternion space follow-
ing [5]. Indeed, the gate mechanism implies a component-wise
product of the components of the quaternion-valued signal with
the gate potential in a split manner [15]. Let ft,it,ot,ct, and ht
be the forget, input, output gates, cell states and the hidden state
of a QLSTM cell at time-step t. QLSTM equations are defined
as:
ft=σ(Wf⊗xt+Rf⊗ht−1+bf),(7)
it=σ(Wi⊗xt+Ri⊗ht−1+bi),(8)
ct=ft×ct−1+it×α(Wc⊗xt+Rc⊗ht−1+bc),(9)
ot=σ(Wo⊗xt+Ro⊗ht−1+bo),(10)
ht=ot×α(ct),(11)
with σand αthe Sigmoid and Tanh quaternion split activations
[15, 9].
Bidirectional connections allow (Q)LSTM networks to con-
sider the past and future information at a specific time step, en-
abling the model to capture a more global context [2]. Quater-
nion bidirectional connections are identical to real-valued ones.
Indeed, past and future contexts are added together component-
wise at each time step.
An adapted scheme initialization for quaternion neural net-
works parameters has been proposed in [6]. In practice, bi-
ases are set to zero while weights are sampled following a Chi-
distribution with four degrees of freedom. Finally, QRNNs re-
quire a specific backpropagation algorithm detailed in [6].
4. R2H encoder
As mentioned in the introduction, having input features repre-
sented by quaternions requires to have predefined a number of
views for the same input token. This prevents the use of quater-
nion networks when prior knowledge suggests to use multiple
views whose number and type cannot be exactly defined. For
example, it is known that time relations between a speech spec-
trum and its neighbor spectra may improve the classification of
the phoneme whose utterance produced the spectrum. Never-
theless, these relations may not be limited to time derivatives of
all spectra samples in a spoken sentence. To overcome this lim-
itation, a new method is proposed. It consists in introducing a
real-valued encoder directly connected to the real-valued input
signal. The real-to-quaternion (R2H) encoder is trained jointly
to the rest of the model in an end-to-end manner, such as any
other layer. After training, the encoder is expected to allow a
mapping from the real space of the input features, to a latent in-
ternal representation meaningful for the following quaternion
layers. The trained model is thus able to directly deal with
real-valued input features, while internally processing quater-
nion numbers. The R2H encoder is a traditional dense layer
followed by a quaternion activation function and normalization.
The number of neurons contained in the layer must be a multi-
ple of four for the quaternion representation. Let W,Xand B
be the weight matrix, the real-valued input and the bias vectors
respectively. Q/
out is the unit quaternion vector obtained at the
output of the projection layer and is expressed as:
Q/
out =Qout
|Qout|,(12)
with,
Qout =α(W.X +B),(13)
and αany quaternion split activation function. In practice, Qout
and Q/
out follow the quaternion internal representation defined

Figure 1: Illustration of the R2H encoder, used as an input layer to a QLSTM. Inputs are real, before being turned into quaternions,
and finally unitary quaternions within the R2H encoder.
in Section 3. Consequently, the input is split in four features
from a latent sub-space, that are interpreted as quaternion com-
ponents: the first one equals to r, the second to xi, the third
one is yj, and the last one equals to zk, making it possible to
apply the quaternion normalization, and the activation function.
At the end of training Q/
out capture an internal latent mapping
of the real-valued input signal Xtrough a vector of unit quater-
nions. Adding the R2H encoder as an input layer to QLSTMs
or to any other QNNs allow the model to deal with real-valued
inputs, while taking the strengths of QNNs (Figure 1).
5. Experiments
Model architectures used for the experiments are presented in
Section 5.1. Then, the R2H encoder is compared to the tradi-
tional and naive quaternion representation on the TIMIT and
Librispeech speech recognition tasks (Section 5.2)
5.1. Model architectures
QLSTMs have already been investigated for speech recognition
in [5] and [6]. Consequently, and based on these previous re-
searches, QLSTMs are composed with four bidirectional QL-
STM layers with an internal real-valued size of 1,024, equiva-
lent to 256 quaternion neurons. Indeed, 256 ×4=1,024 real
numbers. The R2H encoder size varies from 256 to 1,024 to ex-
plore the best latent quaternion representation. Tanh, HardTanh
and ReLU activation functions are investigated to compare the
impact of bounded (Tanh, Hardtanh) and unbounded (ReLU)
R2H encoders. In fact, the quaternion normalization allows a
numerical reduction of the internal representation, but the ReLU
counteracts the latter effect by integrating high real and positive
values to the encoding. The final layer is real-valued and corre-
sponds to the HMM states obtained with the Kaldi [16] toolkit.
A dropout of 0.2is applied across all the layers except the out-
put. The Adam optimizer [17] is used to train the models with
vanilla hyperparameters. The learning rate is halved every-time
the loss on the validation set is below a certain threshold fixed
to 0.001 to avoid overfitting. Finally, models are implemented
with the Pytorch-Kaldi toolkit [18]. While the effectiveness of
QLSTM over LSTM has been demonstrated, an LSTM network
trained in the same conditions and based on [5] is considered as
a baseline. All the models are trained during 30 epochs, and
the results on both the validation and test sets are saved at this
point.
5.2. Phoneme recognition with the TIMIT corpus
The training process is based on the standard 3,696 sentences
uttered by 462 speakers, while testing is conducted on 192 sen-
tences uttered by 24 speakers of the TIMIT [19] dataset. A val-
idation set composed of 400 sentences uttered by 50 speakers
is used for hyper-parameter tuning. The raw audio is processed
with a window of 25ms and an overlap of 10ms. Then, 40-
dimensional log Mel-filter-bank coefficients are extracted with
the Kaldi toolkit. In previous work with QLSTMs [6, 5], first,
second and third time order derivatives were composed with
spectral energies to build a multidimensional input representa-
tion with quaternions. In this paper, the time derivatives are no
longer used. Instead, latent representations are directly learned
from the R2H encoder, fed with the 40 log Mel-filter bank co-
efficients. For the sake of comparison, an input quaternion is
naively composed with input features from four consecutive
Mel-filter-bank coefficients, before being fed to a standard QL-
STM.
Figure 2 reports the results obtained for the investigation
of the R2H encoder size and the impact of the activation layer.
Results are from an average of three runs and are not obtained
w.r.t to the validation set. Indeed, performances on the test set
are evaluated only once at the end of the training phase. It is
first interesting to note that a layer of 1,024 neurons always
gives better results than a layer of size 256 or 512, without
even considering activation functions. In the same manner, the
Tanh activation outperforms both ReLU and Hardtanh activa-
tion function with all the layer size, with an average phoneme
error rate (PER) on the TIMIT test set of 15.6% compared to
16.7and 16% for the ReLU and HardTanh activations. It is im-
portant to note that the ReLU activation gives the worst results.
An explanation of such phenomenon is the definition interval of
the ReLU function. When dealing with ReLU, outputs of the
R2H layer are not bounded in the positive domain before being
normalised. Therefore, the dense layer can output large values
that are then squashed by the quaternion normalization, and it
can be hard for the neural network to learn such mapping. Con-
versely, both Hardtanh and Tanh functions are bounded by −1
and 1, making it easier to learn, since values of the R2H layer
before and after normalization vary on the same range. The
HardTanh function also hardly saturates at −1and 1in the same
manner as the ReLU activation for negative numbers, while the
Tanh smoothly tends to these bounds. Consequently, the Hard-
Tanh gives slightly worst results than the Tanh. Finally, a best

PER of 15.4% is obtained with a normalised R2H encoder of
size 1,024 based on the Tanh activation function, compared to
16.5% and 15.9% with ReLU and Hardtanh functions.
ReLU HardTanh Tanh
15.5
16
16.5
17 16.9
16.2
15.9
16.7
15.9
15.5
16.5
15.9
15.4
PER %
R2H Size 256 R2H Size 512 R2H Size 1024
Figure 2: Phoneme Error Rate (PER %) obtained on the test
set of the TIMIT corpus with different activation functions, and
different R2H encoder size for a QLSTM. Results are from an
average of three runs.
Table 1 presents a summary of the results observed on the
TIMIT phoneme recognition task with a QLSTM and basic
quaternion features, compared to the proposed QLSTM coupled
with the best R2H encoder from Fig. 2. For fair comparison, a
real-valued LSTM is also tested. As highlighted in [6], QL-
STMs models require less neural parameters than LSTMs due
to their internal quaternion algebra. Therefore, an LSTM with
1,024 neurons per layer is composed of 46.0million parame-
ters, while corresponding QLSTMs only need 15.5Mparam-
eters. It is first interesting to note that the R2H encoder helps
the QLSTM to obtains the same PER as the real-valued LSTM,
while dealing with a real-valued input signal. Indeed, both mod-
els performed at 15.4% on the test set, while the QLSTM still
requires more than three times fewer neural parameters.
Table 1: Phoneme error rate (PER%) of the models on the de-
velopment and test sets of the TIMIT dataset. “Params" stands
for the total number of trainable parameters. "R2H-Norm" and
"R2H" correspond to R2H encoders with and without normal-
ization. Results are from an average of 3runs.
Models Dev. Test Params
LSTM 14.5 15.4 46.0M
QLSTM 14.9 15.9 15.5M
R2H-QLSTM 14.7 15.7 15.5M
R2H-Norm-QLSTM 14.4 15.4 15.5M
Then, it is worth underlining that the basic QLSTM without
R2H layer obtains the worst PER of all models with 15.9% on
the test set, due to the inappropriate input representation. Then,
the impact of the quaternion normalization process is investi-
gated by comparing a R2H encoder without normalization, to
a normalised one. As expected, the quaternion normalization
helps the input to fit the quaternion representation, and thus
gives better results with 15.4% of PER compared to 15.7%
for the non-normalized R2H encoder. It is important to men-
tion that such results are obtained without batch-normalization,
speaker adaptation or rescoring methods.
5.3. Speech recognition with the Librispeech corpus
The experiments are extended to the larger Librispeech dataset
[20]. Librispeech is composed of three distinct training subsets
of 100,360 and 500 hours of speech respectively, represent-
ing a total training set of 960 hours of read English speech. In
our experiments, the models are trained following the setups de-
scribed in Section 5.1, and based on the train_clean_100 subset
containing 100 hours. Results are reported on the test_clean set.
Input features are the same as for the TIMIT experiments, and
the best activation function reported in Figure 2 is used (Tanh).
No regularization techniques such as batch-normalization are
used, and no rescoring methods are applied at testing time.
Table 2: Word error rate (WER%) of the models on
test_clean set of the Librispeech dataset with a training on the
train_clean_100 subset. “Params" stands for the total number
of trainable parameters. "R2H-Norm" and "R2H" correspond
to R2H encoders with and without normalization. No rescoring
technique is applied.
Models Test Params
LSTM 8.1 49.0M
QLSTM 8.5 17.7M
R2H-QLSTM 8.3 17.7M
R2H-Norm-QLSTM 8.0 17.7M
The total number of neural parameters is slightly differ-
ent when compared with the TIMIT experiments due to the in-
creased number of HMM states, and therefore neurons, of the
output layer for the Librispeech task. Nonetheless, the number
of parameters is still lowered by a factor of 3when using QL-
STM networks, compared to the real-valued LSTM. Similarly
to the TIMIT experiments, the QLSTM with a normalized R2H
layer reaches slightly better performances in term of word error
rate (WER), with 8.0% compared to 8.1% for the LSTM. More-
over, the R2H encoder allows the QLSTM WER to decrease
from 8.5% to 8.0%, representing an absolute gain of 0.5%. The
reported results on the larger Librispeech dataset demonstrate
that the R2H encoder solution scales well with more realistic
speech recognition tasks.
6. Conclusions
Summary. This paper addresses one of the major weakness
of quaternion-valued neural networks known as the inability
to process non quaternion-valued input signal. A new real-
to-quaternion (R2H) encoder is introduced, making it possi-
ble to learn in a end-to-end manner a latent quaternion rep-
resentation from any real-valued input data. Such representa-
tion is then processed with QNNs such as a quaternion LSTM.
The experiments conduced on the TIMIT phoneme recognition
task demonstrate that this new approach outperforms a naive
quaternion representation of the input signal, enabling the use
of QNNs with any type of inputs.
Future work. Split activation functions and current quaternion
gate mechanisms do not fully respect the quaternion algebra
by considering each elements as uncorrelated components. A
future work will consist on the investigation of purely quater-
nion recurrent neural networks, involving well-adapted activa-
tion functions, and proper quaternion gates.

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