[1,2,3]

[1]The proposed method in this study won the 1st place on the leaderboard of
the AICovidVN 115M Challenge (https://www.covid.aihub.vn/).

[2]The study is conducted by the FruitLab team.

[3]A full version is in preparation and will be updated in an early date.

[] [1] Propose ideas, develop software, and join the challenge

[orcid=0000-0002-8086-6722] [1] Propose ideas, draft the paper, and join the
challenge

[] Develop software and deploy the service

[] Develop software and deploy the service

[] Revise the paper and support the development

[] Revise the paper and support the development

[] Revise the paper and support the development

[2] Supervise the team, provide methodology and appropriate guidance, and
revise the paper

[cor1]Co-first authors: these authors have contributed equally to this work.
[cor2]Corresponding author.

# Fruit-CoV: An Efficient Vision-based Framework for Speedy Detection and
Diagnosis of SARS-CoV-2 Infections Through Recorded Cough Sounds (a Short
Report)

Long H. Nguyen<EMAIL_ADDRESS>Faculty of Information Technology,
Ton Duc Thang University, Ho Chi Minh City, Vietnam  Nhat Truong Pham
<EMAIL_ADDRESS>Division of Computational Mechatronics,
Institute for Computational Science, Ton Duc Thang University, Ho Chi Minh
City, Vietnam  Van Huong Do<EMAIL_ADDRESS>ASICLAND, Suwon, South
Korea  Liu Tai Nguyen<EMAIL_ADDRESS>Faculty of Information
Technology, Ton Duc Thang University, Ho Chi Minh City, Vietnam  Thanh Tin
Nguyen<EMAIL_ADDRESS>Human Computer Interaction Lab, Sejong University,
Seoul, South Korea  Van Dung Do<EMAIL_ADDRESS>Hanoi University of
Industry, Hanoi, Vietnam  Hai Nguyen<EMAIL_ADDRESS>Khoury College of
Computer Science, Northeastern University, Boston, USA  Ngoc Duy Nguyen
<EMAIL_ADDRESS>Institute for Intelligent Systems Research and
Innovation, Deakin University, Victoria, Australia

###### Abstract

SARS-CoV-2 is colloquially known as COVID-19 that had an initial outbreak in
December 2019. The deadly virus has spread across the world, taking part in
the global pandemic disease since March 2020. In addition, a recent variant of
SARS-CoV-2 named Delta is intractably contagious and responsible for more than
four million deaths over the world. Therefore, it is vital to possess a self-
testing service of SARS-CoV-2 at home. In this study, we introduce Fruit-CoV,
a two-stage vision framework, which is capable of detecting SARS-CoV-2
infections through recorded cough sounds. Specifically, we convert sounds into
Log-Mel Spectrograms and use the EfficientNet-V2 network to extract its visual
features in the first stage. In the second stage, we use 14 convolutional
layers extracted from the large-scale Pretrained Audio Neural Networks for
audio pattern recognition (PANNs) and the Wavegram-Log-Mel-CNN to aggregate
feature representations of the Log-Mel Spectrograms. Finally, we use the
combined features to train a binary classifier. In this study, we use a
dataset provided by the AICovidVN 115M Challenge, which includes a total of
7371 recorded cough sounds collected throughout Vietnam, India, and
Switzerland. Experimental results show that our proposed model achieves an AUC
score of 92.8% and ranks the 1st place on the leaderboard of the AICovidVN
Challenge. More importantly, our proposed framework can be integrated into a
call center or a VoIP system to speed up detecting SARS-CoV-2 infections
through online/recorded cough sounds.

###### keywords:

Sound classification COVID-19 Recorded cough sounds Delta variant EfficientNet
SARS-CoV-2 infections Deep learning Neural network Machine vision Remote
detection Speedy detection PANNs Log-Mel Spectrogram Self-testing service
Wavegram

##  1 Introduction

Delta is an instantly contagious variant of SARS-CoV-2 that has caused a
multi-continent disaster. Specifically, a study from the World Health
Organization (WHO) reports that the COVID-19 pandemic is responsible for
4,400,284 deaths and 209,876,613 confirmed cases over the world until August
2021 111https://covid19.who.int/. In addition to the traditional COVID-19
testing methods that require a physical presence of patients, it is crucial to
complement a free-of-charge self-testing service from home with an instant
outcome. In this study, we investigate a method of detecting and diagnosing
SARS-CoV-2 infections through cellphone recorded cough sounds.

In particular, we propose a vision-based method that transforms a sound into a
Log-Mel Spectrogram. We then introduce a two-stage deep neural architecture
and employ a supervised training on the network to classify between infected
or non-infected people. The first stage includes the baseline EfficientNet-V2
(EffNetV2) network that extracts visual features of the Log-Mel Spectrograms.
In the second stage, we use the Wavegram-Log-Mel CNN, followed by 14
convolutional layers extracted from the PANNs to further process feature
representations of the Log-Mel Spectrograms. The aggregated features are then
used to train the network to detect infected cough sounds.

In this study, we use a dataset provided by the AICovid-VN 115M Challenge to
evaluate the efficiency of the proposed framework. The dataset combines two
public datasets from India and Switzerland and a collection of recorded cough
sounds in Vietnam. As a result, the dataset includes 4,508 training sounds,
1,230 public testing sounds, and 1,627 private testing sounds. Our proposed
framework achieves an AUC score of 92.8% in a private test.

The rest of this paper is organized as follows. Related studies are summarized
in section 2. Fruit-CoV is presented in section 3. In section 4, we present
the evaluation results. Finally, section 5 concludes the paper.

##  2 Related Work

Researchers have a great interest in employing low-cost applications toward
detecting SARS-CoV-2 through cough and breath sounds. For instance, the
authors in (Laguarta et al., 2020) used transfer learning to develop an AI-
based diagnostic system using recorded cough sounds. In particular, the
authors extracted Mel Frequency Cepstral Coefficients (MFCCs) from sounds to
train a classifier based on three pre-trained ResNet50s and a Poisson
biomarker layer. The model achieves an accuracy of 95% and 99.9% on groups of
25 people with five and three positive cases, respectively.

The authors in (Brown et al., 2020) adopted a machine learning binary
classifier to identify the presence of COVID-19 using cough and breath sounds
in a crowdsourced respiratory dataset. In (Pahar et al., 2021), the authors
used a wide range of machine learning and deep learning techniques to detect
infected cough sounds. For instance, logistic regression (LR), support vector
machine (SVM), multilayer perceptrons (MLP), convolutional neural networks
(CNN), long short-term memory (LSTM), and ResNet50 were used. SMOTE
oversampling was adopted to the minor class, which accounts for 7.86% of cough
sounds. The authors in (Valdés et al., 2021) used both unsupervised and
supervised learning approaches to analyze different types of cough sounds,
e.g., dry, wet, whooping, and COVID-19 coughs. However, the method only uses
raw audio signals from a limited dataset. An ensemble learning technique as in
(Mohammed et al., 2021) consists of a shallow MLP, a CNN, and a pre-trained
CNN. However, the accuracy of this technique is not greater than 90%.

Many datasets of cough sounds have been created to encourage the research
community into the combat with SARS-CoV-2. For example, the authors in (Sharma
et al., 2020) created a COVID-19 dataset, namely Coswara, which comprises
breathing, coughing, and voice sounds. The dataset includes 6,507 clean
recorded sounds and 1,117 noisy ones collected from 941 people. Additionally,
the authors in (Cohen-McFarlane et al., 2020) created a dataset of 73 samples
to complement a lack number of COVID-19 datasets. Another dataset named
COUGHVID was provided to the research community by (Orlandic et al., 2021).
This dataset contains approximately 20,000 recorded cough sounds accompanying
the bioinformation such as age, gender, geographic location, and COVID-19
positive/negative testing outcome.

##  3 Proposed Method

###  3.1 Dataset

The AICovidVN 115M Challenge is a well-known community project in Vietnam
toward finding an efficient method of testing COVID-19 based on cough sounds.
The winning team is selected to deploy the testing service on a broader scale
in Vietnam. The dataset contains two classes, i.e., the positive label and the
negative label. It comprises 4,508 training and 2,863 testing samples. A wide
array of different devices are utilized to record the sounds.

###  3.2 Fruit-CoV

In this paper, we used the baseline EfficientNet-V2 in the first stage to
extract high-level visual representations from the Log-Mel Spectrograms. In
the second stage, a pre-trained CNN with 14 layers from the PANNs is used to
extract additional embedding features of the Log-Mel Spectrograms. Because the
PANNs was trained on a large-scale AudioSet dataset that consists of many
respiratory audio classes, the extracted features can be used as bio-embedding
features. Embedding features of two branches are concatenated to eventually
train a binary classifier. Figure 1 describes an overview of our proposed
scheme.

Figure 1: The proposed scheme of two-stage framework.

The dataset includes three major sampling rates: 4 kHz, 8 kHz, and 48 kHz. We
apply the following data processing:

  * •

In the case of 4 kHz: we convert cough sounds into Log-Mel Spectrograms with a
sampling rate of 4 kHz. The Log-Mel Spectrograms are used as inputs of both
stages. In the first stage, besides EfficientNet-V2, a GeM Pooling is added to
seek for important feature representations. Such representations are
propagated into a fully connected layer to classify infected or non-infected
cases. In the second stage, sounds are converted into Log-Mel Spectrograms
using a sampling rate of 8 kHz before feeding into the CNN14. Moreover, to
adapt with the embedding features (Embedding 1) extracted from the trained
EffNetV2, we select the features from the ‘conv_block6’ of the CNN14, followed
by a GeM Pooling layer to generate the embedding features, i.e., Embedding 2.

  * •

In the case of 8 kHz: we apply a similar procedure as of 4 kHz except the Log-
Mel Spectrograms are processed by a sampling rate of 8 kHz in both two stages.
Additionally, the Embedding 2 is chosen from the ‘embedding’ layer of the
CNN14 without adding the GeM Pooling layer.

  * •

In the case of 48 kHz: The procedure is different from the first two cases.
Specifically, we convert cough sounds into Log-Mel Spectrograms with a
sampling rate of 48 kHz in the first stage. However, in the second stage, to
avoid losing the quality of sounds and unsynchronizing with the pre-trained
model’s weights, we feed the Wavegram-Log-Mel-CNN sounds with a sampling rate
of 32 kHz before applying the CNN14 to generate the Embedding 2.

  * •

For other sampling rates, we convert sounds into Log-Mel Spectrograms using
the closest sampling rate as listed in case 1, case 2, or case 3.

##  4 Performance Evaluation

###  4.1 Hyperparameter Settings

Hyperparameters are shown in Table 1, where the SpecAugment augmentation
method is applied to train the baseline, i.e. EffNetV2, in the first stage of
our proposed framework. Additionally, we use the k-fold cross-validation
method to generalize the model with $k=5$.

Table 1: The hyper-parameter settings of our proposed framework. Params | Configure/Setup
---|---
Learning rate | $e^{-3}$
Learning rate scheduler | Cosine ($e^{-5}$)
Batch size | 16
Epochs | 30
Optimization | CosAngularGrad
Loss | BCE with logits
Augmentation | SpecAugment

To convert cough sounds into Log-Mel Spectrograms, we apply a Mel bin of 256
in the case of 4 kHz, while other cases use a Mel bin of 128. Librosa is used
to convert cough sounds into Log-Mel Spectrograms in this study.

###  4.2 Evaluation Results

Figure 2 shows AUC scores of the first stage with different sampling rates
while AUC scores of the second stage are shown in Figure 3. In Figure 2, the
scores reach 93%, 96%, and 94% in the first stage with sampling rates of 4
kHz, 8 kHz, and 48 kHz, respectively. Similarly, in the second stage, the
scores reach 95%, 97%, and 99%.

Figure 2: AUC Scores on the first stage with 4, 8, and 48 kHz, respectively
(from left to right).

Figure 3: AUC Scores on the second stage with 4, 8, and 48 kHz, respectively
(from left to right).

Comparing to existing studies, our proposed method is more flexible with
different sampling rates. The AUC score reaches 92.8% on the private test of
the AICovidVN 115M Challenge.

##  5 Conclusion

This study proposed a vision-based framework, namely Fruit-CoV, which provides
high performance in detecting and diagnosing SARS-CoV-2 infections through
recorded cough sounds. We introduced a hybrid deep neural network architecture
that utilizes state-of-the-art architectures, i.e., EfficientNet-V2 and PANNs.
We also meticulously analyze the dataset and apply appropriate data
preprocessing. Experimental results show that our proposed Fruit-CoV framework
attains an AUC score of 92.8% and won the 1st place on the leaderboard of the
AICovidVN 115M Challenge. In addition, Fruit-CoV can be integrated into a call
center, a VoIP system, or a mobile application to deploy a self-testing
service of COVID-19 from home rapidly. We continuously improve the Fruit-CoV
framework by aggregating recorded cough sounds, automating the data
processing, exploring other state-of-the-art techniques, and generating an
evaluation benchmark of existing methods.

## Acknowledgment

The authors would like to thank the organizer of the AI-Covid-VN 115M
Challenge for providing the recorded cough sound dataset.

## References

  * Brown et al. (2020) Brown, C., Chauhan, J., Grammenos, A., Han, J., Hasthanasombat, A., Spathis, D., Xia, T., Cicuta, P., Mascolo, C., 2020.  Exploring automatic diagnosis of covid-19 from crowdsourced respiratory sound data, in: Proceedings of the 26th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, pp. 3474–3484.
  * Cohen-McFarlane et al. (2020) Cohen-McFarlane, M., Goubran, R., Knoefel, F., 2020.  Novel coronavirus cough database: Nococoda.  IEEE Access 8, 154087–154094.
  * Laguarta et al. (2020) Laguarta, J., Hueto, F., Subirana, B., 2020.  Covid-19 artificial intelligence diagnosis using only cough recordings.  IEEE Open Journal of Engineering in Medicine and Biology 1, 275–281.
  * Mohammed et al. (2021) Mohammed, E.A., Keyhani, M., Sanati-Nezhad, A., Hejazi, S.H., Far, B.H., 2021.  An ensemble learning approach to digital corona virus preliminary screening from cough sounds.  Scientific Reports 11, 1–11.
  * Orlandic et al. (2021) Orlandic, L., Teijeiro, T., Atienza, D., 2021.  The coughvid crowdsourcing dataset, a corpus for the study of large-scale cough analysis algorithms.  Scientific Data 8, 1–10.
  * Pahar et al. (2021) Pahar, M., Klopper, M., Warren, R., Niesler, T., 2021\.  Covid-19 cough classification using machine learning and global smartphone recordings.  Computers in Biology and Medicine , 104572\.
  * Sharma et al. (2020) Sharma, N., Krishnan, P., Kumar, R., Ramoji, S., Chetupalli, S.R., Ghosh, P.K., Ganapathy, S., et al., 2020.  Coswara–a database of breathing, cough, and voice sounds for covid-19 diagnosis.  arXiv preprint arXiv:2005.10548 .
  * Valdés et al. (2021) Valdés, J.J., Xi, P., Cohen-McFarlane, M., Wallace, B., Goubran, R., Knoefel, F., 2021\.  Analysis of cough sound measurements including covid-19 positive cases: A machine learning characterization, in: 2021 IEEE International Symposium on Medical Measurements and Applications (MeMeA), IEEE. pp. 1–6.


DESY-21-102

Infrared photons and asymptotic symmetries

Zhengwen Liu

Deutsches Elektronen-Synchrotron DESY, Notkestrasse 85, 22607 Hamburg, Germany

Pujian Mao

Center for Joint Quantum Studies and Department of Physics,
School of Science, Tianjin University, 135 Yaguan Road, Tianjin 300350, China

> Abstract  S-matrix elements exhibit universal factorization when multiple
> infrared photons are emitted in scattering processes. We explicitly show
> that the leading soft factorization of tree-level amplitudes with the
> emission of any number of soft photons can be interpreted as the Ward
> identity of the asymptotic symmetry of gauge theory.

##  1 Introduction

Understanding the factorization property of scattering amplitudes in various
special kinematics, such as soft, collinear and Regge limits, plays an
important role in both making precision predictions for physical observables
and revealing the hidden structure of quantum field theory. In particular, on-
shell scattering amplitudes may display universal factorization when one or
more soft particles are emitted in scatterings, in e.g. gauge theories and
gravity [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]. The universal soft factorization of
scattering amplitudes is often referred to as soft theorems in the literature.

In recent years there have been renewed interests on soft theorems from a more
theoretical side. The new enthusiasm comes in part from the discovery of a
remarkable connection between soft theorems and symmetries at null infinity
[11, 12]. The soft theorems can be rewritten as Ward identities of certain
asymptotic symmetries. In particular, the large gauge transformations of
electromagnetism and non-Abelian gauge theory are behind the soft photon
theorem [13] and the soft gluon theorem [14, 15]. The constraints on emission
of soft gravitons come from the supertranslation [16], the Abelian ideal part
of the Bondi-Metzner-Sachs (BMS) transformations [17, 18, 19]. More details on
this connection can be found in the review [20].

While soft theorems in a variety of theories have been intensively
investigated from various perspectives, the study of the related asymptotic
symmetries has been limited to single and double soft limits. The purpose of
this work is to extend the relationship between asymptotic symmetries and soft
theorems to any multiplicity for soft particles. For simplicity, we choose
tree-level amplitudes with multiple soft-photon emission as our testing ground
in this work. First, using Feynman diagrams and the eikonal approximation, we
derive the leading soft theorem for any number of soft photons. We then extend
the mysterious relation between the single soft photon theorem and the large
gauge transformations of electromagnetism to any number of soft photons.

##  2 The multi-soft photon current

In this section we present the soft-current for the emission of any number of
soft photons from a tree-level amplitude with an arbitrary number of hard
particles, in order to reveal its relation with asymptotic symmetries in
subsequent sections.

We consider an on-shell scattering amplitude with any $n$ hard external legs
and $m$ soft photons ${\cal M}_{m+n}(p_{1},\ldots,p_{n};q_{1},\ldots,q_{m})$,
where $\\{p_{1},\ldots,p_{n}\\}$ and $\\{q_{1},\ldots,q_{m}\\}$ denote hard
and soft momenta respectively. Introducing an infinitesimal parameter
$\lambda$, $q_{i}=\mathcal{O}(\lambda)$ and $p_{i}=\mathcal{O}(1)$ in the soft
limit $\lambda\to 0$. In the multi-soft limit, any on-shell amplitude is
expected to factorizes into a product of a universal operator acting on the
on-shell amplitude with only hard particles [7]

$\displaystyle{\cal
M}_{m+n}(p_{1},\ldots,p_{n};q_{1},\ldots,q_{m})\,=\,\mathscr{J}_{m}(q_{1},\ldots,q_{m})\,{\cal
M}_{n}(p_{1},\ldots,p_{n})+\cdots,$ (1)

where the ellipsis denotes terms that are power-suppressed in the soft limit.
It is convenient to define the soft current with the polarization tensor
structure stripped off

$\displaystyle\mathscr{J}_{m}(q_{1},\ldots,q_{m})\,\equiv\,\varepsilon^{\mu_{1}}(q_{1})\cdots\varepsilon^{\mu_{m}}(q_{m})\mathcal{J}_{\mu_{1}\cdots\mu_{m}}(q_{1},\ldots,q_{m}),$
(2)

where $\varepsilon^{\mu_{i}}(q_{i})$ is the polarization vector of the soft
photon $i$, we assume which satisfies $q_{i}\cdot\varepsilon(q_{i})=0$.

It is well-known that the current for the emission of a single soft photon
takes the form in the all-outgoing convention

$\displaystyle\mathcal{J}^{\mu}(q)\,=\,\sum_{i=1}^{n}e_{i}\,{\cal
E}_{i}^{\mu}(q),\qquad{\cal E}_{i}^{\mu}(q)\,\equiv\,{p_{i}^{\mu}\over
p_{i}\cdot q},$ (3)

where $e_{i}$ denotes the charge of the hard particle $i$, and ${\cal
E}_{i}^{\mu}(q)$ is the famous eikonal vertex. It is straightforward to verify
that the eikonal current is conserved, i.e. $q^{\mu}\mathcal{J}_{\mu}(q)=0$,
due to the conservation of charge.

It is clear that the leading soft current receives the contributions from only
diagrams where soft particles are emitted from external hard (charged) lines.
To derive the multi-soft photon current, let us introduce a generalization of
the eikonal vertex. We define a $m$-point soft effective vertex as a sum of
all possible contributions for the emission of $m$ soft photons from a single
hard external line. Schematically,

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}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
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\pgfsys@invoke{\lxSVG@closescope
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Evidently, the single soft effective vertex is just the eikonal vertex (3)

$\displaystyle{\cal J}_{i}^{\mu}(q)\,=\,e_{i}\,{\cal E}_{i}^{\mu}(q).$ (5)

There are two different configurations for the emission of two soft partons
from a single line. To be precise, the double soft effective vertex is given
by

$\displaystyle{\cal J}_{i}^{\mu_{1}\mu_{2}}(q_{1},q_{2})\,$
$\displaystyle=\,e_{i}^{2}\,\Big{(}{\cal E}_{i}^{\mu_{2}}(q_{12})\,{\cal
E}_{i}^{\mu_{1}}(q_{1})+{\cal E}_{i}^{\mu_{1}}(q_{12})\,{\cal
E}_{i}^{\mu_{2}}(q_{2})\Big{)}$ $\displaystyle=\,e_{i}^{2}\,{\cal
E}_{i}^{\mu_{1}}(q_{1})\,{\cal E}_{i}^{\mu_{2}}(q_{2})$ (6)
$\displaystyle=\,{\cal J}_{i}^{\mu_{1}}(q_{1})\,{\cal
J}_{i}^{\mu_{2}}(q_{2}),$

with $q_{a\cdots b}=q_{a}+\cdots+q_{b}$. Similarly, there are $3!=6$ diagrams
that contribute to the soft effective vertex when three soft photons are
emitted from a hard line. Therefore the triple soft effective vertex is given
by

$\displaystyle{\cal J}_{i}^{\mu_{1}\mu_{2}\mu_{3}}(q_{1},q_{2},q_{3})\,$
$\displaystyle=\,e_{i}^{3}\,\Big{(}{\cal E}_{i}^{\mu_{3}}(q_{123})\,{\cal
E}_{i}^{\mu_{2}}(q_{12})\,{\cal
E}_{i}^{\mu_{1}}(q_{1})+\text{permutations}\big{)}$
$\displaystyle=\,e_{i}^{3}\,{\cal E}_{i}^{\mu_{1}}(q_{1})\,{\cal
E}_{i}^{\mu_{2}}(q_{2})\,{\cal E}_{i}^{\mu_{3}}(q_{3})$ (7)
$\displaystyle=\,{\cal J}_{i}^{\mu_{1}}(q_{1})\,{\cal
J}_{i}^{\mu_{2}}(q_{2})\,{\cal J}_{i}^{\mu_{3}}(q_{3}).$

More generally, by induction it is easy to obtain the soft effective vertex
for arbitrary $m$ soft photons

$\displaystyle{\cal J}_{i}^{\mu_{1}\cdots\mu_{m}}(q_{1},\ldots,q_{m})\,$
$\displaystyle=\,{\cal J}_{i}^{\mu_{1}}(q_{1})\,\cdots\,{\cal
J}_{i}^{\mu_{m}}(q_{m}).$ (8)

Very nicely, the multi-soft photon effective vertex in (8) has a fully
factorized form, i.e. a $m$-point soft effective vertex factorizes into a
product of $m$ eikonal vertices. As we will see below, soft effective vertices
defined above not only are useful building blocks to construct full soft
currents, but also capture the key structure of the full soft current [21].

In the following, we show how to use soft effective vertices to derive the
full soft current. Evidently, the single-soft current is simply the summation
of eikonal vertices over all hard legs,

$\displaystyle\mathcal{J}^{\mu}(q)\,=\,\sum_{i=1}^{n}{\cal J}_{i}^{\mu}(q).$
(9)

To compute the double-soft photon current, we need to consider all tree-level
Feynman diagrams involving two photons and $n$ external hard lines that carry
momenta $p_{i}$ and charges $e_{i}$. There are two types of diagrams that
contributes to the result: two soft photons emit from one and two hard lines
respectively

$\displaystyle\mathcal{J}^{\mu_{1}\mu_{2}}(q_{1},q_{2})\,$
$\displaystyle=\,\Bigg{(}\sum^{n}_{i,j=1\atop i\neq
j}\mathcal{J}_{i}^{\mu_{1}}(q_{1})\mathcal{J}_{j}^{\mu_{2}}(q_{2})\Bigg{)}+\Bigg{(}\sum^{n}_{i=1}\mathcal{J}_{i}^{\mu_{1}\mu_{2}}(q_{1},q_{2})\Bigg{)}$
$\displaystyle=\,\Bigg{(}\sum^{n}_{i=1}\mathcal{J}_{i}^{\mu_{1}}(q_{1})\Bigg{)}\Bigg{(}\sum^{n}_{j=1}\mathcal{J}_{j}^{\mu_{2}}(q_{2})\Bigg{)}$
(10)
$\displaystyle=\,\mathcal{J}^{\mu_{1}}(q_{1})\,\mathcal{J}^{\mu_{2}}(q_{2}).$

More generally, for the multi-soft current, the $m$ soft photons can emit from
at most $m$ external hard lines. Using the property that any $m$-point soft
photon effective vertex factorizes into a product of $m$ eikonal vertices,
i.e. (8), a sum over all possible diagrams gives

$\displaystyle\mathcal{J}^{\mu_{1}\cdots\mu_{m}}(q_{1},\ldots,q_{m})\,=\,\mathcal{J}^{\mu_{1}}(q_{1})\,\cdots\,\mathcal{J}^{\mu_{m}}(q_{m}).$
(11)

As expected, the soft-current for the emission of any number of soft photons
takes a nicely factorized form.

##  3 Soft theorem as Ward identity of asymptotic symmetries

In this section, we show how to recover the soft photon theorems with any
number of soft photons presented in the previous section as Ward identities of
large gauge transformation. We begin by reviewing the derivation of the single
soft photon theorem from large gauge transformation in [13, 22].

###  3.1 Single-soft photon theorem

The Minkowski spacetime has two null boundaries, past null infinity ${\cal
I}^{-}$, and future null infinity ${\cal I}^{+}$. Here we will concentrate on
${\cal I}^{+}$, while everything can be repeated on ${\cal I}^{-}$ in a
similar way. The retarded spherical coordinates will be applied with the
following change of coordinates:

$u=t-r\,,\quad x^{1}+ix^{2}=\frac{2rz}{1+z\bar{z}}\,,\quad
x^{3}=r\,\frac{1-z\bar{z}}{1+z\bar{z}}\,.$ (12)

where $r=\sqrt{x^{i}x_{i}}$. The line element of Minkowski spacetime now
becomes

$\text{d}s^{2}=-\text{d}u^{2}-2\text{d}u\text{d}r+2r^{2}\gamma_{z\bar{z}}\text{d}z\text{d}\bar{z}\,,\quad\gamma_{z\bar{z}}=\frac{2}{(1+z\bar{z})^{2}}\,.$
(13)

The piece $2\gamma_{z\bar{z}}\text{d}z\text{d}\bar{z}$ is just the metric of
the round sphere $S^{2}$. ${\cal I}^{+}$ is the submanifold
$r\rightarrow\infty$, with topology $S^{2}\times\mathbb{R}$. We will denote
the sphere at $u=\pm\infty$ by ${\cal I}^{+}_{\pm}$.

The soft momentum $q$ can be characterized by an energy $\omega_{q}$ and a
direction on the sphere $(w,\bar{w})$, as

$q_{\mu}=\omega_{q}\left(1,\frac{w+\bar{w}}{1+w\bar{w}},i\frac{\bar{w}-w}{1+w\bar{w}},\frac{1-w\bar{w}}{1+w\bar{w}}\right)\
.$ (14)

Then the $n$ hard momenta $p_{k}$ can be parameterized as (14) by energies
$\omega_{k}$ and directions $(w_{k},\bar{w}_{k})$ in a similar way. The single
soft photon theorem (3) can be then rewritten in the position space as

$\displaystyle\lim_{\omega_{q}\to
0}\langle\textrm{out}|\omega_{q}\,\mathfrak{a}_{+(-)}(q){\cal
S}|\textrm{in}\rangle=\frac{1+|w|^{2}}{\sqrt{2}}\sum_{k=1}^{n}\frac{e_{k}}{w-w_{k}}\,\langle\textrm{out}|{\cal
S}|\textrm{in}\rangle\ ,$ (15)

where $e_{k}$ is the electric charge of the $k$-th particle, and
$\mathfrak{a}_{+(-)}(q)$ is the annihilation operator that creates outgoing
negative(positive)-helicity soft photon with momentum $q$.

To connect with asymptotic symmetries, we recall that if a symmetry is
generated by a charge $Q$, the associated Ward identity reads as follows:

$\langle\rm{out}|Q^{\textrm{out}}{\cal S}-{\cal
S}Q^{\textrm{in}}|\rm{in}\rangle=0\ ,$ (16)

which is a consequence of $[Q,{\cal S}]=0$. The charge for a spontaneously
broken symmetry must act non-linearly on the states, otherwise it would
annihilate the vacuum. So we can decompose the charge into linear and non-
linear pieces $Q=Q_{\rm{L}}+Q_{\rm{NL}}$. The Ward identity for the charge of
a broken symmetry becomes

$\langle\rm{out}|Q_{\rm{NL}}^{\textrm{out}}{\cal S}-{\cal
S}Q^{\textrm{in}}_{\rm{NL}}|\rm{in}\rangle=-\langle\rm{out}|Q^{\textrm{out}}_{\rm{L}}{\cal
S}-{\cal S}Q^{\textrm{in}}_{\rm{L}}|\rm{in}\rangle\,.$ (17)

Since $Q_{\textrm{NL}}$ creates zero-momentum Goldstone boson, equation (17)
is very similar to the single soft photon theorem (15).

The charge responsible for the single soft photon theorem is given by [23, 24,
13]

$Q_{\varepsilon_{\textrm{out}}}=-\int_{{\cal
I}^{+}_{-}}\text{d}z\text{d}\bar{z}\,\gamma_{z\bar{z}}\,\varepsilon(z,\bar{z})\,A_{u}^{0}\
,$ (18)

where $\varepsilon(z,\bar{z})$ is an arbitrary function on the sphere that
generates the large (residual) gauge transformations of the form $\delta
A_{\mu}=\partial_{\mu}\varepsilon(z,\bar{z})$ in the radial gauge condition
$A_{r}=0$. It is a symmetry of the S-matrix following [11, 13]. The gauge
field and the conserved current that coupled to the gauge field are assumed to
follow a $\frac{1}{r}$-expansion as

$A_{u}=\frac{A^{0}_{u}(u,z,\bar{z})}{r}+{\cal O}\Big{(}\frac{1}{r^{2}}\Big{)}\
,\quad
A_{z(\bar{z})}=A^{0}_{z(\bar{z})}(u,z,\bar{z})+\sum\limits_{m=1}^{\infty}\frac{A^{m}_{z(\bar{z})}(u,z,\bar{z})}{r^{m}}\
,$ (19) $J_{u}=\frac{J^{0}_{u}(u,z,\bar{z})}{r^{2}}+{\cal
O}\left(\frac{1}{r^{3}}\right)\ ,\quad
J_{z(\bar{z})}=\frac{J^{0}_{z(\bar{z})}(u,z,\bar{z})}{r^{2}}+\sum\limits_{m=1}^{\infty}\frac{J^{m}_{z(\bar{z})}(u,z,\bar{z})}{r^{m+2}}\
.$ (20)

We have used the ambiguities of a conserved current to set the radial
component of the current to zero. This is consistent with working in the
radial gauge. Maxwell’s equations yield

$e\,\partial_{u}A^{0}_{u}=\gamma_{z\bar{z}}^{-1}\partial_{u}(\partial_{z}A^{0}_{\bar{z}}+\partial_{\bar{z}}A^{0}_{z})+J^{0}_{u}\
.$ (21)

The charge $Q$ splits into

$\displaystyle Q_{\textrm{NL}}=\int_{{\cal
I}^{+}}\text{d}z\text{d}\bar{z}\text{d}u\,\varepsilon\,\partial_{u}\partial_{\bar{z}}A^{0}_{z}\
,$ (22) $\displaystyle Q_{\textrm{L}}=\frac{1}{2}\int_{{\cal
I}^{+}}\text{d}z\text{d}\bar{z}\text{d}u\,\gamma_{z\bar{z}}\,\varepsilon\,J_{u}^{0}\,.$
(23)

For notational brevity, we will suppress the out label and keep only the anti-
holomorphic term.

For the gauge field, we perform a stationary-phase approximation of the mode
expansion:

$A^{0}_{z(\bar{z})}(x)=-\frac{i}{8\pi^{2}}\frac{\sqrt{2}}{1+z\bar{z}}\int_{0}^{\infty}\text{d}\omega_{q}\left[\mathfrak{a}_{+(-)}(\omega_{q}\hat{x})\,e^{-i\omega_{q}u}-\mathfrak{a}^{\dagger}_{-(+)}(\omega_{q}\hat{x})\,e^{i\omega_{q}u}\right]\
,$ (24)

where the creation and annihilation operators satisfy the standard commutation
relations. Then, using the Fourier relation (defining
$F(u)=\int_{-\infty}^{\infty}\text{d}\omega\,e^{i\omega u}\tilde{F}(\omega)$):

$\int_{-\infty}^{\infty}\text{d}u\,\partial_{u}F(u)=2\pi i\lim_{\omega\to
0}\left[\omega\tilde{F}(\omega)\right]\ ,$ (25)

and the special choice

$\varepsilon(z,\bar{z})=\frac{1}{w-z}\ ,$ (26)

we obtain for the non-linear piece of the charge111 We only keep the anti-
holomorphic part of the charge (22), namely the charge only contains
$\partial_{\bar{z}}\varepsilon$. In particular one needs to split (23) via
$J^{0}_{u}\to\frac{1}{2}J^{0}_{u}+\frac{1}{2}J^{0}_{u}$. Otherwise an extra
factor of 2 that arise from a proper treatment of the radiative phase space
[13, 25] is needed. :

$\displaystyle\langle\textrm{out}|Q_{\textrm{NL}}{\cal S}|\textrm{in}\rangle$
$\displaystyle=\frac{1}{4}\frac{\sqrt{2}}{1+|w|^{2}}\lim_{\omega_{q}\to
0}\langle\textrm{out}|\omega_{q}\,\mathfrak{a}_{+}(q){\cal
S}|\textrm{in}\rangle\ ,$ (27)

For the linear piece, considering the complex scalar charged (with charge
$Q_{e}$) matter as example where the current at leading order is
$J^{0}_{\mu}=iQ_{e}(\bar{\Phi}^{0}\partial_{\mu}\Phi^{0}-\Phi^{0}\partial_{\mu}\bar{\Phi}^{0})$,
we just need to use the boundary canonical commutation relation [26]:

$[\bar{\Phi}^{0}(u,z,\bar{z}),\Phi^{0}(u^{\prime},w,\bar{w})]=\frac{i}{4}\gamma_{w\bar{w}}^{-1}\,\Theta(u^{\prime}-u)\delta^{2}(z-w)\
,$ (28)

to obtain that

$\displaystyle\langle\textrm{out}|Q_{\textrm{L}}{\cal S}|\textrm{in}\rangle$
$\displaystyle=\sum_{k=1}^{n}-\frac{e_{k}}{4(w-w_{k})}\langle\textrm{out}|{\cal
S}|\textrm{in}\rangle\ .$ (29)

Assembling the two expressions, the soft theorem (15) can be recovered from
(17).

We have only paid attention to the out part of (17). The analysis of the in
part can be carried out in the same way. The generators on ${\cal I}^{+}$ and
${\cal I}^{-}$ are connected by an anti-podal identification, see e.g.[13] for
details.

###  3.2 Double-soft photon theorem

After having the connection between the single soft photon theorem and the
asymptotic symmetries, it is natural to ask about the case involving more soft
photons. Let us start with the double soft case, for which we need to consider
a family of Ward identity

$\langle\textrm{out}|[Q_{1},[Q_{2},{\cal S}]]|\textrm{in}\rangle=0\ .$ (30)

Since $[Q_{2},{\cal S}]=0$, the crucial part for deriving the connection
between the Ward identity and double soft photon theorem is

$[Q_{1NL},[Q_{2},{\cal S}]]=0\ ,$ (31)

which is equivalent to

$[Q_{1NL},[Q_{2NL},{\cal S}]]=-[Q_{1NL},[Q_{2L},{\cal S}]]\ .$ (32)

Similar to the gravity case [27], this expression can be rewritten using the
Jacobi identity among $Q_{1NL}$, $Q_{2L}$ and ${\cal S}$ as

$[Q_{1NL},[Q_{2NL},{\cal S}]]=[Q_{2L},[{\cal S},Q_{1NL}]]+[{\cal
S},[Q_{1NL},Q_{2L}]]\ .$ (33)

The bracket structure should be considered as the large gauge transformation
by the first generator on the second one. Recalling the expression of the
linear piece (22) and non-linear piece (23) of the charge, we can obtain that

$[Q_{1},Q_{2L}]=0\ ,\quad[Q_{1L},Q_{2L}]=0\ ,$ (34)

which lead to $[Q_{1NL},Q_{2L}]=0$. Substituting this back to (33) and using
the fact that $[{\cal S},Q_{1}]=0$, one can derive

$[Q_{1NL},[Q_{2NL},{\cal S}]]=(-1)^{2}[Q_{2L},[Q_{1L},{\cal S}]]\ .$ (35)

Using the last expression and repeating the derivation in the previous
subsection by inserting twice the charges, one can recover the double soft
photon theorem in a straightforward way.

###  3.3 Triple-soft photon theorem

The derivation in the previous subsection can be easily extended to triple
soft photon case where the crucial ingredient is

$[Q_{1NL},[Q_{2NL},[Q_{3},{\cal S}]]]=0\ .$ (36)

After splitting the third charge into linear and non-linear pieces, one can
obtain

$[Q_{1NL},[Q_{2NL},[Q_{3NL},{\cal S}]]]=-[Q_{1NL},[Q_{2NL},[Q_{3L},{\cal
S}]]]\ .$ (37)

First, repeating the derivation in the double soft case for $Q_{2NL}$,
$Q_{3L}$, and ${\cal S}$, the Ward identity (37) can be rewritten as

$[Q_{1NL},[Q_{2NL},[Q_{3NL},{\cal S}]]]=[Q_{1NL},[Q_{3L},[Q_{2L},{\cal S}]]]\
.$ (38)

Then, using the Jacobi identity among $Q_{1NL}$, $Q_{3L}$ and $[Q_{2L},{\cal
S}]$ and the commutation relation between $Q_{1NL}$ and $Q_{3L}$, we obtain

$[Q_{1NL},[Q_{2NL},[Q_{3NL},{\cal S}]]]=[Q_{3L},[Q_{1NL},[Q_{2L},{\cal S}]]]\
.$ (39)

Finally, repeating the derivation in the double soft case for $Q_{1NL}$,
$Q_{2L}$, and ${\cal S}$, we get

$[Q_{1NL},[Q_{2NL},[Q_{3NL},{\cal S}]]]=(-1)^{3}[Q_{3L},[Q_{2L},[Q_{1L},{\cal
S}]]]\ .$ (40)

This is the key ingredient for recovering the triple soft photon theorem from
the Ward identity of asymptotic symmetry.

###  3.4 Multi-soft photon theorem

In this subsection, we will show that the multi-soft photon theorem can be
derived from the Ward identity

$[Q_{1NL},[Q_{2NL},......,[Q_{(m-1)NL},[Q_{n},{\cal S}]......]=0\ .$ (41)

We will prove it by induction. Suppose that the $(m{-}1)$-soft photon theorem
can be obtained from the Ward identity

$[Q_{1NL},[Q_{2NL},......,[Q_{(m-2)NL},[Q_{m-1},{\cal S}]......]=0\ ,$ (42)

which means that the Ward identity can be rewritten as

$[Q_{1NL},[Q_{2NL},......,[Q_{(m-1)NL},{\cal
S}]......]=(-1)^{m-1}[Q_{(m-1)L},[Q_{(m-2)L},......,[Q_{1L},{\cal S}]......]\
.$ (43)

Hence

$(-1)^{m}[Q_{(m-1)L},......,[Q_{1L},{\cal
S}]......]=[Q_{1NL},......,[Q_{(m-2)NL},[Q_{(m-1)L},{\cal S}]......]\ .$ (44)

For the case of $m$ soft photons, after splitting the $m$th charge into the
linear and non-linear pieces, the Ward identity becomes

$[Q_{1NL},......,[Q_{mNL},{\cal
S}]......]=-[Q_{1NL},......,[Q_{(m-1)NL},[Q_{mL},{\cal S}]......]\ .$ (45)

First, applying the relation in (44) for $Q_{2NL}$,……,$Q_{(m-1)NL}$, $Q_{mL}$
and ${\cal S}$, the Ward identity can be rewritten as

$[Q_{1NL},[Q_{2NL},......,[Q_{mNL},{\cal
S}]......]=(-)^{m-1}[Q_{1NL},[Q_{mL},[Q_{(m-1)L},......,[Q_{2L},{\cal
S}]......]\ .$ (46)

Then, using the Jacobi identity among $Q_{1NL}$, $Q_{mL}$ and
$[Q_{(m-1)L},......,[Q_{2L},{\cal S}]......]$ and the commutation relation
between $Q_{1NL}$ and $Q_{mL}$, we obtain

$[Q_{1NL},[Q_{2NL},......,[Q_{mNL},{\cal
S}]......]=(-)^{m-1}[Q_{mL},[Q_{1NL},[Q_{(m-1)L},......,[Q_{2L},{\cal
S}]......]\ .$ (47)

Repeating such procedure for $m-2$ times to put the $Q_{1NL}$ to the right, we
get

$[Q_{1NL},[Q_{2NL},......,[Q_{mNL},{\cal
S}]......]=(-)^{m-1}[Q_{mL},[Q_{(m-1)L},......,[Q_{2L},[Q_{1NL},{\cal
S}]......]\ .$ (48)

Finally, using $[Q_{1},{\cal S}]=0$, we obtain

$[Q_{1NL},......,[Q_{mNL},{\cal
S}]......]=(-1)^{m}[Q_{mL},......,[Q_{1L},{\cal S}]......]\ .$ (49)

This is precisely what we need to recover the multiple soft photon theorem
from the Ward identity (41) of asymptotic symmetry.

##  4 Conclusions and Discussions

To conclude, we have generalized the equivalence between the single-soft
photon theorem and the Ward identities of asymptotic symmetries to any number
of soft photons.

Let us comment on several directions for future research. First, our analysis
in this work was restricted to leading soft theorems. It would be interesting
to continue our exploration beyond leading order in the multiple soft limit.
Second, it would be interesting to investigate the relations between multi-
soft theorems and asymptotic symmetries in non-Abelian gauge theory and
gravity. Third, it has been observed that the single soft photon theorem is
equivalent to the electromagnetic memory effect that is a residual velocity to
the test charges in the detector after the passage of electromagnetic
radiation [28]. It is natural to ask for the memory effect connected to the
multi-soft limit of amplitude.

Finally, while the study of asymptotic symmetries until now has been
restricted to theoretical interests, their use to study scattering amplitudes
relevant to real-life collider processes has somewhat been limited. It would
be very tempting to see if these novel ideas provide insight into the
structure of the infrared singularities of amplitudes in gauge theory. This
would help us improve theoretical predictions for multi-parton processes at
the Large Hadron Collider.

## Acknowledgments

ZL would like to thank Claude Duhr for collaboration on related topics. We are
grateful to Volker Schomerus and Jun-Bao Wu for comments and a careful reading
of the manuscript. The work of ZL was supported by the Deutsche
Forschungsgemeinschaft (DFG) under Germany’s Excellence Strategy (EXC 2121)
‘Quantum Universe’ (No. 390833306), and by the European Research Council (ERC)
under the European Union’s Horizon 2020 research and innovation programme
through the ERC Consolidator Grant “Precision Gravity: From the LHC to LISA”
(No. 817791). The work of PM is supported in part by the National Natural
Science Foundation of China under Grants No. 11905156 and No. 11935009.

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# Advancing Full-Text Search Lemmatization Techniques with Paradigm Retrieval
from OpenCorpora

Dmitriy Kalugin-Balashov

(May 2023)

##  1 Introduction

In full-text search applications, the primary goal is to effectively retrieve
and match relevant documents based on user queries. By focusing on finding the
first form, or the lemma, of a word, the search process can be streamlined and
optimized. The lemma serves as a normalized representation of a word’s
different inflected forms, allowing for a more accurate comparison between
user queries and document content. This approach reduces the complexity and
computational overhead associated with full morphological analysis, which
includes extracting all possible forms of a word along with their grammatical
properties. By prioritizing lemma retrieval, full-text search engines can
achieve faster response times and more precise results, while minimizing the
resources required for processing large volumes of text data.

Consequently, building upon the foundation of pymorphy[1], the golemma library
was developed to address the challenge of efficiently identifying the first
form, or lemma, of words in the Russian language.

##  2 Challenges with Russian Language

Lemmatization and stemming both reduce words to their base forms but operate
differently. Stemming, a simple rule-based process, removes suffixes without
considering context, often yielding invalid words. Lemmatization, conversely,
uses a vocabulary and morphological analysis to derive the base form, or
"lemma," considering context and generating valid words.

Stemming is designed for English and other Western languages with simpler
inflectional structures, whereas Russian’s complex inflectional structure
poses challenges for stemming algorithms. Russian words can have multiple
inflections, stems, and can form compound words, making rule-based systems
less effective.

Lemmatization is generally more effective for languages with complex
inflectional structures, such as Russian. It employs dictionaries and
morphological rules to determine a word’s base form, considering grammatical
context and accounting for multiple stems and compound words. Lemmatization
returns valid dictionary words, making it useful for natural language
processing tasks like text classification, information retrieval, and machine
translation. Its flexibility in handling various grammatical forms and tenses
makes it suitable for tasks like text generation and summarization.

In summary, lemmatization is a sophisticated, accurate approach suitable for
handling Russian’s complexity, yielding valid words for a range of NLP tasks.

##  3 Definition of Paradigm

In the context of the pymorphy2 library, a paradigm refers to a collection of
inflected forms of a word that possess the same grammatical properties and
share a common lemma. A paradigm is characterized by a range of grammatical
categories, such as number, tense, gender, and others, and may encompass
multiple forms of a word for each category.

For instance, the paradigm for the Russian word "бежать" (to run) would
comprise forms like "бегу" (I run), "бежишь" (you run), "бежит" (he/she/it
runs), "бежим" (we run), "бежите" (you run), and "бежат" (they run), as well
as all potential forms of the word in various tenses and aspects.

The pymorphy2 library utilizes the OpenCorpora project’s morphological
dictionary as its data source, which supplies comprehensive information on the
grammatical properties and inflectional forms of Russian words.

When employing pymorphy2, the library loads the morphological dictionary and
establishes a set of paradigms for each word. These paradigms are then
utilized to generate the inflected forms of a word, in addition to determining
the lemma and grammatical properties of a given word form. Furthermore, the
paradigms enable the generation of all possible forms of a word, which proves
beneficial for tasks such as text generation, summarization, and question
answering.

In golemma, the concept of a paradigm found in pymorphy has been reevaluated
and simplified. We consider the following structure as a paradigm:

⬇ type Paradigm struct {  CutPrefix int CutSuffix int AddPrefix string
AddSuffix string }
---

The first form of a word is obtained by applying the paradigm to it. Applying
a paradigm entails removing $CutPrefix$ characters from the left, $CutSuffix$
characters from the right, and then concatenating $AddPrefix$ on the left and
$AddSuffix$ on the right.

For example,

$\textup{зарубил}\xrightarrow{(2,1,\textup{<<>>},\textup{<<ть>>})}\textup{рубить}.$

The reason for this reevaluation and simplification of the concept of a
paradigm in golemma, compared to pymorphy2, is to increase the efficiency and
speed of retrieving the first form of a word. This is a critical aspect in
full-text search, where the main goal is not necessarily to generate all
possible forms of a word or to determine its grammatical properties, but
rather to quickly and accurately identify the base form, or lemma, of a given
word.

The simplified paradigm structure in golemma reduces the computational
overhead associated with handling morphological data and focuses on the
essential elements needed for lemmatization. The use of ’CutPrefix’,
’CutSuffix’, ’AddPrefix’, and ’AddSuffix’ provides a straightforward way to
transform a word form back to its lemma, which can be done with minimal
processing. This leads to a more responsive and efficient full-text search
process, especially when dealing with large volumes of text.

##  4 Retrieving Paradigms

OpenCorpora is an invaluable resource for paradigm retrieval due to its
comprehensive and accurate morphological dictionary for the Russian language.
The project contains a vast collection of annotated linguistic data, including
grammatical properties and inflectional forms for Russian words. By utilizing
OpenCorpora, developers can access a reliable source of information to create
efficient and precise algorithms for paradigm extraction. This, in turn,
enhances the effectiveness of natural language processing tasks, such as
lemmatization, which greatly benefits from accurate paradigm retrieval.

The SAX (Simple API for XML)111https://docs.python.org/3/library/xml.sax.html
parser provides numerous advantages. As an event-driven parser, it does not
store the entire XML document in memory. Instead, it reads and processes the
document sequentially, making it highly memory-efficient, particularly when
handling large XML files like those found in OpenCorpora. Owing to its event-
driven nature, the SAX parser is generally faster than other parsing methods,
such as DOM (Document Object Model), which loads the entire XML document into
memory prior to processing. This speed is especially beneficial when parsing
large datasets like OpenCorpora, as it can save significant time and
resources.

The <lemmata> section in OpenCorpora’s dictionary
XML222http://opencorpora.org/files/export/dict/dict.opcorpora.xml.bz2 contains
information about the lemmas, or base forms, of words in the Russian language.
Each entry in the <lemmata> section represents a lemma and is accompanied by
its grammatical properties, such as part of speech, gender, case, and number.
Additionally, the section provides details about the inflected forms of each
lemma, which are essential for understanding the different ways a word can
appear in a text.

The dictionary format we get after parsing is a Python dictionary where each
key is a unique integer identifier $u$, and the corresponding value is a tuple
containing two elements:

  * •

The first element of the tuple is the normal form $n$ of the word.

  * •

The second element of the tuple is a set of inflected forms $i_{k}$ of the
word.

$u\rightarrow(n,\\{i_{0},i_{1},\ldots\\})$

This dictionary is a source for paradigm retrieval algorithm.

In simple words, the paradigm function takes two inputs, a normal form and an
inflected form of a word, and calculates their Longest Common Substring
(LCSS). Then, it extracts the prefixes and suffixes of both forms by removing
the LCSS. Finally, the function returns a tuple containing the lengths of the
inflected form’s prefix and suffix, as well as the normal form’s prefix and
suffix. This tuple essentially represents the paradigm that connects the
normal form and the inflected form.

The goal is to build two Python dictionaries ($p_{k}$ refers to the identifier
of a particular paradigm):

  * •

$(\textup{cut\\_prefix},\textup{cut\\_suffix},\textup{add\\_prefix},\textup{add\\_suffix})\rightarrow
p_{k}$

  * •

$\textbf{{murmur3}}(n)\rightarrow\\{(i_{0},p_{0}),(i_{1},p_{1}),\ldots\\}$

MurmurHash3333https://pypi.org/project/mmh3/ is a non-cryptographic hash
function that is popular for its speed and uniform distribution of hash values
which make it a good choice for full-text search applications.

As of May 2023, we’ve successfully extracted 2488 unique paradigms from a
grand total of 391,842 lemmas in the OpenCorpora dataset

##  5 Building Dictionary

Building dictionary is done in two main steps:

  * •

Saving the Paradigms: The function first retrieves the paradigms from the
retriever object, rearranges them as pairs of (value, key), sorts them based
on the value and then writes them into the file. The paradigms are saved as a
map with pairs using the MessagePack (msgpack) packer.

  * •

Preparing and Writing the Dictionary: The dictionary is structured as a map
where the key is the hash of the word’s normal form (calculated using
MurmurHash3), and the value is a list of tuples, each containing a word form
and its paradigm ID. This structure is reversed to a form where the key is the
word form, and the value is a tuple containing two lists: one with the hashes
and one with the paradigm IDs. Then, the function sorts the items based on the
word form, and groups together entries with the same word form into a single
entry. Finally, these items are written to the file as a map with pairs using
the MessagePack packer.

The file format, therefore, consists of two main sections: The first is a map
of paradigms, and the second is a map of words, each associated with a list of
hashes and a list of paradigm IDs.

$\mbox{ip}\rightarrow(\mbox{cut\\_prefix},\mbox{cut\\_suffix},\mbox{add\\_prefix},\mbox{add\\_suffix})$
---
$\mbox{if}\rightarrow([\textbf{\mbox{murmur3}}(\mbox{nf}_{1}),\textbf{\mbox{murmur3}}(\mbox{nf}_{2}),\ldots],[\mbox{ip}_{1},\mbox{ip}_{2},\ldots])$

The MessagePack binary
format444https://github.com/msgpack/msgpack/blob/master/spec.md is used for
efficient storage and quick retrieval. The generated file serves as a vital
tool for efficient lemmatization in full-text search, mapping different forms
of words to their first form. Its binary structure, combined with the use of
MurmurHash3, allows for rapid retrieval and accurate results, thus enhancing
the performance of full-text searches.

We’ve developed a compact and efficient way to store the entirety of a
language dictionary, optimizing it specifically for the retrieval of word’s
first forms. This has been achieved by combining advanced data structures,
hashing techniques, and compression methods. The resulting system offers a
significantly reduced storage footprint, along with improved retrieval speeds.
This makes it an ideal solution for enhancing the performance of full-text
searches, where rapid and accurate lemmatization is key.

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