# “Subverting the Jewtocracy”: Online Antisemitism Detection Using Multimodal
Deep Learning

Mohit Chandra  International Institute of Information Technology,
HyderabadHyderabadIndia<EMAIL_ADDRESS>,  Dheeraj Pailla
International Institute of Information Technology, HyderabadHyderabadIndia
<EMAIL_ADDRESS>,  Himanshu Bhatia  International Institute
of Information Technology, HyderabadHyderabadIndia
<EMAIL_ADDRESS>,  Aadilmehdi Sanchawala  International
Institute of Information Technology, HyderabadHyderabadIndia
<EMAIL_ADDRESS>,  Manish Gupta  International Institute of
Information Technology, HyderabadHyderabadIndia<EMAIL_ADDRESS>,
Manish Shrivastava  International Institute of Information Technology,
HyderabadHyderabadIndia<EMAIL_ADDRESS>and  Ponnurangam Kumaraguru
IIIT, DelhiNew DelhiIndia<EMAIL_ADDRESS>
###### Abstract.

The exponential rise of online social media has enabled the creation,
distribution, and consumption of information at an unprecedented rate.
However, it has also led to the burgeoning of various forms of online abuse.
Increasing cases of online antisemitism have become one of the major concerns
because of its socio-political consequences. Unlike other major forms of
online abuse like racism, sexism, etc., online antisemitism has not been
studied much from a machine learning perspective. To the best of our
knowledge, we present the first work in the direction of automated multimodal
detection of online antisemitism. The task poses multiple challenges that
include extracting signals across multiple modalities, contextual references,
and handling multiple aspects of antisemitism. Unfortunately, there does not
exist any publicly available benchmark corpus for this critical task. Hence,
we collect and label two datasets with 3,102 and 3,509 social media posts from
Twitter and Gab respectively. Further, we present a multimodal deep learning
system that detects the presence of antisemitic content and its specific
antisemitism category using text and images from posts. We perform an
extensive set of experiments on the two datasets to evaluate the efficacy of
the proposed system. Finally, we also present a qualitative analysis of our
study.

hate speech, antisemitism, multimodal classification, deep learning

††ccs: Computing methodologies Neural networks††ccs: Information systems
Social networking sites††ccs: Computing methodologies Supervised learning by
classification††ccs: Social and professional topics Hate speech††ccs:
Information systems Web mining

##  1\. Introduction

Online social media (OSM) platforms have gained immense popularity in recent
times due to their democratized nature, enabling users to express their views,
beliefs, and opinions, and easily share those with millions of people. While
these web communities have empowered people to express themselves, there have
been growing concerns over the presence of abusive and objectionable content
on these platforms. One of the major forms of online abuse which has seen a
considerable rise on various online platforms is that of
antisemitism.111http://www.crif.org/sites/default/fichiers/images/documents/antisemitismreport.pdf

Text: Even grandma can see what’s going on.

Text: I see the blews are at it again.

Figure 1. In both these examples, the post text appears to be non-antisemitic,
due to absence of an explicit reference to Jews. But when looked along with
the image, it can be classified as antisemitic.

According to International Holocaust Remembrance Alliance
(IHRA)222https://www.holocaustremembrance.com/, “Antisemitism is a certain
perception of Jews, which may be expressed as hatred toward Jews. Rhetorical
and physical manifestations of antisemitism are directed toward Jewish or non-
Jewish individuals and/or their property, toward Jewish community institutions
and religious facilities.”. Unlike some forms of hate-speech like sexism,
cyberbullying, xenophobia etc., antisemitism originates from multiple aspects.
In addition to the discrimination on the basis of race, antisemitism also
includes discrimination on the basis of religion (e.g.difference from
Christianity), economic activities (e.g.money lending) and political
associations (e.g.Israel-Palestine issue, holding power at influential
positions).

In recent times, online antisemitism has become one of the most widespread
forms of hate-speech on major social media
platforms333https://ec.europa.eu/information_society/newsroom/image/document/2016-50/factsheet-
code-conduct-8_40573.pdf. This recent trend of increased online hatred against
Jews can also be correlated to the increasing real-world crime. According to
the Anti-Defamation League’s 2019 report, there has been a 12% jump in the
total cases of antisemitism amounting to a total of $2,107$ cases, and a
disturbing rise of 56% in antisemitic assaults as compared to 2018 across the
U.S.444https://www.adl.org/news/press-releases/antisemitic-incidents-hit-all-
time-high-in-2019 Unlike studies on some major forms of online abuse like
racism (Badjatiya et al., 2017), cyber-bullying (Chatzakou et al., 2017) and
sexism (Parikh et al., 2019), online antisemitism present on the web
communities has not been studied in much detail from a machine learning
perspective. This calls for studying online antisemitism in greater depth so
as to protect the users from online/real world hate crimes.

In previous studies, it has been shown that real-world crimes can be related
to incidents in online spaces (Yang et al., 2018). Hence, it is essential to
build robust systems to reduce the manifestation of heated online debates into
real-world hate crimes against Jews. Due to the myriad of content on OSMs, it
is nearly impossible to manually segregate the instances of antisemitism,
thereby calling for the automation of this moderation process using machine
learning techniques. Although antisemitic content detection is such a critical
problem, there is no publicly available benchmark labeled dataset for this
task. Hence, we gather multimodal data (text and images) from two popular
social media platforms, Twitter and Gab.

Oftentimes abusive content flagging policies across various social media
platforms are very minimalistic and vague, especially for a specific abuse
sub-category, antisemitism. Although various social media platforms have taken
measures to curb different forms of hate-speech, more efforts are required to
tackle the problem of online antisemitism.555https://www.adl.org/holocaust-
denial-report-card#the-online-holocaust-denial-report-card-explained- As a
result, we provide a detailed categorization methodology and label our
datasets conforming to the same. Besides the dataset challenge, building a
robust system is arduous because of the multimodal nature of the social media
posts. A post with benign text may as well be antisemitic due to a hateful
image (as shown in Fig. 1). Thus, it becomes essential to take a more holistic
approach rather than just inferring based on text. As a result, we take a
multimodal approach which extracts information from text as well as images in
this paper.

Fig. 2 shows a detailed architecture of our proposed multimodal antisemitism
detection system which leverages the recent progress in deep learning
architectures for text and vision. Given a (text, image) pair for a social
media post, we use Transformer (Vaswani et al., 2017)-based models to encode
post text, and Convolutional Neural Networks (CNNs) to encode the image. Next,
we experiment with multiple fusion mechanisms like concatenation, Gated MCB
(Fukui et al., 2016), MFAS (Multimodal Fusion Architecture Search) (Perez-Rua
et al., 2019) to combine text and image representations. The fused
representation is further transformed using a joint encoder. The joint encoded
representation is decoded to reconstruct the fused representation and also
used to predict presence of antisemitism or an antisemitism category.

We believe that the proposed work can benefit multiple stakeholders which
includes – 1) users of various web communities, 2) moderators/owners of social
media platforms. Overall, in this paper, we make the following contributions:
(1) We collect and label two datasets on online antisemitism gathered from
Twitter and Gab with 3,102 and 3,509 posts respectively. Each post in both the
datasets is labeled for presence/absence as well as antisemitism category. (2)
We propose a novel multimodal system which learns a joint text+image
representation and uses it for antisemitic content detection and
categorization. The presented multimodal system achieves an accuracy of
$\sim$91% and $\sim$71% for the binary antisemitic content detection task on
Gab and Twitter respectively. Further, for 4-class antisemitism category
classification, our approach scores an accuracy of $\sim$67% and $\sim$68% for
the two datasets respectively, demonstrating its practical usability. (3) We
provide a detailed qualitative study to analyse the limitations and challenges
associated with this task and hate speech detection in general.

##  2\. Related Work

In this section, we present the past work, we broadly discuss– (1) studies on
antisemitism popular in sociology domain (2) popular hate speech datasets for
Twitter and Gab, (3) deep learning studies for broad detection of hateful text
content, and (4) overview of applications of multimodal deep learning.

###  2.1. Previous Studies on Antisemitism

Antisemitism as a social phenomenon has been extensively studied as part of
social science literature (Salwen, 2010; Schwarz-Friesel and Reinharz, 2017).
These studies have helped to explore the history behind antisemitism in-depth
but lack quantitative analyses. Apart from the studies based on Sociology,
there have been a few empirical studies. In one of the primary works,
researchers collected around 7 million images and comments from ‘4chan’ and
‘Gab’ to study the escalation and spread of antisemitic memes in a
longitudinal study (Zannettou et al., 2020). Ozalp et al. (2020) trained
statistical machine learning models on data collected from twitter to detect
antisemitic content. In contrast to this, our work focuses on detection of
antisemitism through a multimodal deep learning framework. Recently, in an
another work, researchers focused on detailed annotation analysis based on
IHRA’s guidelines for antisemitic content (Jikeli et al., 2019). Our work on
the other hand, encompasses the IHRA’s guidelines and incorporates fine-
grained classes of antisemitism. Although antisemitism detection is such a
critical problem, unfortunately, there hasn’t been any rigorous work on this
problem from a deep learning perspective. We fill this gap in this paper.

###  2.2. Hate Speech Datasets for Twitter and Gab

Hatespeech detection has become a popular area for research and there have
been quite a few works on dataset creation. Waseem and Hovy (2016) annotated
16,914 tweets, including 3,383 as ‘sexist’, 1,972 as ‘racist’ and 11,559 as
‘neither’. Davidson et al. (2017) annotated $\sim$24K tweets for ‘hate
speech’, ‘offensive language but not hate’ or ‘neither’. Another recent work
presented a dataset comprising of $44,671$ posts from various social media
platforms and annotated them as offensive or not. Gab was launched in 2016 and
hence there are only a few dataset based studies. Qian et al. (2019) proposed
a hate-speech dataset on Gab with 33,776 posts annotated for hate versus non-
hate. Chandra et al. (2020) annotated 7,601 Gab posts for ‘Biased Attitude’,
‘Act of Bias and Discrimination’ or ‘Violence and Genocide’. Unlike the
previous studies which have focused on general hate-speech or some popular
sub-categories (like racism, sexism), we focus specifically on data related to
antisemitism.

###  2.3. Deep Learning Methods for Detection of Various forms of Online Abuse

Deep learning has emerged as one of the most popular methods for hate-speech
detection especially in Text-only NLP problems. Founta et al. (2019) proposed
a Recurrent Neural Networks (RNN) based framework for classification of racism
& sexism, offensive speech, and cyberbullying using text and metadata. In
contrast to this, we propose a general framework which doesn’t require any
metadata. Serrà et al. (2017) showed that character level based LSTMs (Long
Short-Term Memory networks) can also be effective for abuse classification. In
a more recent work, Parikh et al. (2021) proposed a neural framework for
classifying sexism and misogyny.

Apart from LSTMs, Convolutional Neural Networks (CNNs) have been shown to be
fairly successful for this task as they retain the spatial information to
extract position invariant features. Gambäck and Sikdar (2017) used CNNs to
classify the tweets into racist, sexist, both or none. Park and Fung (2017)
proposed a two-step hybrid approach for classification on hateful text into
sexist or racist. They presented a hybrid CNN-based architecture which used
sentence and word embeddings. Badjatiya et al. (2017) compared multiple deep
learning architectures to classify tweets into racist, sexist, or neither.
Unfortunately, there has been no previous work on exploring deep learning for
antisemitism detection. Also, recently Transformer (Vaswani et al., 2017)
based methods have shown to outperform traditional deep learning methods like
RNNs and LSTMs. Hence, we resort to methods like Bidirectional Encoder
Representations from Transformers (BERT) and Robust BERT Approach (RoBERTa).
Besides text, we also leverage semantics extracted from the accompanying image
for improved antisemitism detection.

###  2.4. Applications of Multimodal Deep Learning

With the huge availability of multimodal data, multimodal deep learning has
been harnessed to improve the accuracy for various tasks like Visual Question
Answering (VQA) (Singh et al., 2019), fake news/rumour detection (Khattar et
al., 2019), etc. Inspired by the success of Optical Character Recognition
(OCR) on images for textVQA (Singh et al., 2019), we also run OCR on the post
images and use them for the classification task. Recently, (Sabat et al.,
2019; Yang et al., 2019; Gomez et al., 2020) have explored use of multimodal
deep learning for general abuse detection from datasets like Reddit + Google
images, and Twitter. Our proposed system differs from these previous methods
in two important aspects: (1) they use traditional deep learning recurrent
text methods like RNNs and LSTMs, while we investigate the application of more
promising Transformer-based methods, (2) while previous methods were proposed
for general hate, we focus on antisemitism.

##  3\. Antisemitism Categorization

Besides annotating every post as antisemitic or not, we also annotate them for
finer categories of online antisemitism. While there exists a good amount of
literature exploring the ways in which antisemitism manifests itself, we
primarily followed the categorization proposed by Brustein (Brustein, 2003) as
it covers the major aspects of antisemitism. In his work, he explored the
history behind the hatred against Jews and has categorized antisemitism into
four categories, namely: (1) Political (2) Economic (3) Religious (4) Racial.
We augmented this categorization with additional inputs from the detailed
IHRA’s666https://www.holocaustremembrance.com/working-definition-antisemitism
definition. We describe each category of antisemitism in detail in the
following.

Table 1. Basic statistics for the two datasets. | #Total Posts | #Antisemitic posts | #Political posts | #Economic posts | #Religious posts | #Racial posts
---|---|---|---|---|---|---
Twitter | 3,102 | 1,428 | 639 | 183 | 124 | 482
Gab | 3,509 | 1,877 | 736 | 118 | 144 | 879
Table 2. Frequent Unigrams and Bigrams for each of the Antisemitism Categories. N-Grams |  Political Antisemitism |  Economic Antisemitism |  Religious Antisemitism |  Racial Antisemitism
---|---|---|---|---
Unigrams |  jews, zionist, zog, israel, media, control. world, government, politics, conspiracy |  jewish, money, cash, finance, wealth, business, bankers, kosher |  jews, christ, jesus, killer, rabbi, expel, satan, christians, messiah |  jews, jewish, fake, holocaust, hitler, white, hebrew, ridiculous, pinocchio
Bigrams |  world domination, zionist jews, zionist occupied, terrorist zionist, jews state |  jewish money, money politics, money everything, money launderers, zionist bankers |  christ killer, read torah, jesus killer, ultra orthodox, rabbi israel, jewish ritual |  jewish man, jews attacks, anti semitism, jewish people, race mixing

###  3.1. Political Antisemitism

Political Antisemitism can be defined as the hatred toward Jews based on the
belief that Jews seek national and/or world power. In many of the cases lying
in this category, Jews are portrayed to be controlling major political
parties, governments, and decision making bodies. We also include the cases
where they are accused of controlling media for promoting their interests
(printing, Hollywood, etc). Furthermore, in some other cases, Jews are accused
of being more loyal to Israel and blamed for the various socio-political
crises. For example, The jews run congress through threats and intimidation.

###  3.2. Economic Antisemitism

Economic Antisemitism is based on the implicit belief that Jews perform and
control the economic activities which are harmful for others or the society.
This notion further exhibits multiple facets like Jews are undeservedly
wealthy, greedy, dishonest, materialistic or cheaters. For example, the
driving force behind globalism is jewish finance and greed.

###  3.3. Religious Antisemitism

Religious Antisemitism deals with bias and discrimination against Jews due to
their religious belief in Judaism. Cases belonging to this category portray
Jews as anti-Christ, Christ-killers, or against the teachings of the Bible.
Oftentimes, the posts also target Jewish religious institutions as well as
their spiritual leader (Rabbi). For example, may god strike down each and
every one of these filthy jewish antichrists.

###  3.4. Racial Antisemitism

Unlike religious antisemitism, racial antisemitism is based on the prejudice
against Jews as a race/ethnic group. Posts belonging to this category display
a sense of inferiority for the Jewish race by portraying them as degenerate or
attaching certain negative character as naturally inherited by them. Many
posts in this category refer to false Jewish conspiracy for racial intermixing
and blame them for LGBTQ+ related issues. Along with considering everything
which discriminates Jews based on ethnic grounds in this category, we have
also included instances talking about Holocaust and its denial, since racial
prejudice against Jews was one of the main cause for the aforementioned
event.777https://www.britannica.com/topic/anti-Semitism/Nazi-anti-Semitism-
and-the-Holocaust For example, white is right which makes the jews always
wrong.

Figure 2. Proposed multimodal system architecture.

##  4\. Online Antisemitism Datasets

We collect datasets from two popular social media platforms Twitter and Gab.
In this section, we discuss details related to data collection, annotation and
basic statistics.

###  4.1. Data Collection

We choose Twitter and Gab as the OSM platforms to gather data for our study.
While Twitter has strict anti-abuse policies and an active content moderation
team, Gab is an alt-right social media website with relaxed moderation
policies. Recently, Gab has gained massive popularity especially among those
who have been banned from mainstream web communities for violating their hate
speech policy (Zannettou et al., 2018). After gathering a massive collection
of posts from Twitter as well as Gab, we retained only those posts which
contained text as well as images. Further, we ensured that each post included
at least one term from a high precision lexicon. This lexicon contains common
racial slurs used against Jews along with other words like ‘Jewish’,
‘Hasidic’, ‘Hebrew’, ‘Semitic’, ‘Judaistic’, ‘israeli’, ‘yahudi’, ‘yehudi’ to
gather non-antisemitic posts as well, thereby helping maintain a balanced
class distribution. Presence of these terms does not necessarily indicate
presence/ absence of antisemitism and hence we manually annotate the posts.

###  4.2. Data Annotation

For the annotation task, we selected four undergraduate students who are
fluent in English. The annotators were given a detailed guideline along with
examples to identify instances of antisemitism. Moreover, to ensure that the
annotators had enough understanding of the task, we conducted multiple rounds
of test annotations followed by discussions on disagreements. In the
annotation procedure, each example was annotated by three annotators and the
disagreements were resolved through discussion between all annotators. Each
example was annotated on two levels after looking at the text as well as the
image – (1) binary label (whether the example in antisemitic or not), and (2)
if the example is antisemitic then assign the respective antisemitism
category. We used Fleiss’ Kappa score (Fleiss et al., 1971) to compute the
inter-annotator agreement. The Fleiss’ kappa score came out to be 0.707 which
translates to a substantial agreement between the annotators. We removed all
user sensitive information and followed other ethical practices to ensure user
privacy.

###  4.3. Data Statistics

Table 1 shows the post distribution across various classes for the two
datasets. As observed, majority of posts in both datasets lie in either
political antisemitism or the racial antisemitism category. We believe that
this trend is inline with the phenomenon of ‘New
antisemitism’.888https://en.wikipedia.org/wiki/New_antisemitism Table 2 shows
the frequent unigrams and bigrams for each of the antisemitism categories.
Overall, 84% of the total images had some form of text in them. This motivated
us to use an OCR module in the proposed system. On average, post text has
$\sim$45 and $\sim$27 words, while the OCR output is $\sim$50 and $\sim$51
words long, after pre-processing for Gab and Twitter respectively. We make the
data publicly available.999https://github.com/mohit3011/Online-Antisemitism-
Detection-Using-MultimodalDeep-Learning

##  5\. Multimodal Antisemitism Categorization System

Fig. 2 illustrates the architecture of our proposed multimodal system for
online antisemitism detection with RoBERTa text+OCR encoder, ResNet-152 image
encoder and the MFAS fusion module. Given a (text, image) pair for a social
media post, we use Transformer (Vaswani et al., 2017)-based models to encode
(1) post text and (2) OCR text extracted from the image. We use CNNs to encode
the image. Next, the fusion module combines the text and image
representations. The fused representation is further transformed using a joint
encoder. The joint encoded representation is decoded to reconstruct the fused
representation and also used to predict presence of antisemitism or an
antisemitism category. The entire network is trained end-to-end using back-
propagation. Since our datasets are relatively small, we fine-tune the pre-
trained networks on the presented datasets for multimodal classification. The
code for the proposed system can be found here. We now describe each module in
detail.

Table 3. Comparison of (5-fold cross validation) performance of popular text-only and image-only classifiers. The best performing method is highlighted in bold separately for both the text and image blocks. | Twitter | Gab
---|---|---
| Binary | Multiclass | Binary | Multiclass
| Accuracy | F-1 | Accuracy | F-1 | Accuracy | F-1 | Accuracy | F-1
Text only | GloVe+Dense |  .630$\pm$.009 |  .621$\pm$.013 |  .490$\pm$.013 |  .268$\pm$.025 |  .651$\pm$.027 |  .612$\pm$.040 |  .540$\pm$.031 |  .276$\pm$.018
FastText+Dense |  .540$\pm$.000 |  .351$\pm$.000 |  .467$\pm$.031 |  .223$\pm$.099 |  .566$\pm$.017 |  .429$\pm$.045 |  .532$\pm$.030 |  .269$\pm$.019
GloVe+att-RNN |  .583$\pm$.048 |  .552$\pm$.081 |  .416$\pm$.019 |  .239$\pm$.033 |  .630$\pm$.039 |  .624$\pm$.045 |  .460$\pm$.039 |  .240$\pm$.028
BERT+Dense |  .701$\pm$.015 |  .700$\pm$.016 | .669$\pm$.047 | .676$\pm$.036 | .889$\pm$.008 | .889$\pm$.009 |  .623$\pm$.025 |  .575$\pm$.038
RoBERTa+Dense | .733$\pm$.007 | .733$\pm$.008 |  .663$\pm$.039 |  .662$\pm$.050 |  .874$\pm$.010 |  .874$\pm$.010 | .632$\pm$.032 | .583$\pm$.039
Img only | ResNet-152 | .579$\pm$.014 | .578$\pm$.015 | .416$\pm$.028 | .317$\pm$.040 |  .587$\pm$.008 |  .583$\pm$.008 | .456$\pm$.020 | .275$\pm$.010
Densenet-161 |  .567$\pm$.033 |  .566$\pm$.033 |  .405$\pm$.033 |  .281$\pm$.011 | .610$\pm$.017 | .607$\pm$.015 |  .446$\pm$.031 |  .274$\pm$.027

###  5.1. Text + OCR Encoder

We remove URLs and non alpha-numeric characters. The cleaned text is tokenized
and then encoded using the BERT/RoBERTa tokenizer and encoder respectively
(Wolf et al., 2019). For getting the OCR output from the images we
experimented with three different services (Google’s Vision API, Microsoft’s
Computer Vision API and Open source tesseract). We found the Google’s Vision
API to perform the best for the broad range of images we had in the dataset
(from newspaper articles to memes). The extracted OCR text goes through the
same pre-processing process as the post text. We experiment with BERT and
RoBERTa for text encoding since they have been shown to lead to high accuracy
across multiple NLP tasks.

BERT (Devlin et al., 2018) is a transformer encoder with 12 layers, 12
attention heads and 768 dimensions. We used the pre-trained model which has
been trained on Books Corpus and Wikipedia using the MLM (masked language
model) and the next sentence prediction (NSP) loss functions. The input to the
BERT model is obtained as a concatenation as follows: (CLS, post text, SEP,
OCR text) where CLS and SEP are the standard classification and separator
tokens respectively. The 768-dimensional representation $T$ for the “CLS”
token from the last encoder layer is used as input by the fusion module.

RoBERTa (Liu et al., 2019) is a robustly optimized method for pretraining
natural language processing (NLP) systems that improves on BERT. RoBERTa was
trained with much more data – 160GB of text instead of the 16GB dataset
originally used to train BERT. It is also trained for larger number of
iterations up to 500K. Further, it uses larger byte-pair encoding (BPE)
vocabulary with 50K subword units instead of character-level BPE vocabulary of
size 30K used for BERT. Finally, compared to BERT, it removes the next
sequence prediction objective from the training procedure, and a dynamically
changing masking pattern is applied to the training data.

###  5.2. Image Encoder

We perform Gaussian normalization for each image, and resize images to
224x224x3 size and feed them to a CNN. We augment our training data using
image transformations such as random cropping and random horizontal flipping.
We connect the output of second last layer from these networks to a 768 sized
dense layer. Output from this layer $I$ is used as the image representation.

Few CNN architectures are popular like: AlexNet (Krizhevsky et al., 2012),
InceptionV3 (Szegedy et al., 2016), VGGNet-19 (Simonyan and Zisserman, 2014),
Resnet-152 (He et al., 2016) and Densenet-161 (Huang et al., 2017). We chose
Resnet-152 and Densenet-161 since they have been shown to outperform the other
CNN models across multiple vision tasks.

###  5.3. Fusion

To combine the features obtained from the Text + OCR and the image encoder
modules, $T$ and $I$, we experiment with three different techniques of fusion
– (1) Concatenation (2) Gated MCB (Fukui et al., 2016) and (3) MFAS (Perez-Rua
et al., 2019). Gated MCB (Multimodal Compact Bilinear) pooling combines
multimodal features using an outer product followed by a sigmoid non-linearity
. We also experimented with Hadamard inner product but found it to be worse
compared to gated MCB, in line with previous literature on multimodal deep
learning. As shown in Fig. 3 in (Perez-Rua et al., 2019), MFAS (Multimodal
Fusion Architecture Search) first concatenates text and image representations
from an intermediate hidden layer, applies a sigmoid non-linearity, and then
concatenates this with final layer text and image representations along with a
sigmoid non-linearity.

###  5.4. Joint Encoder/Decoder and Classifier

The fused representation $F$ is then passed through a series of Dense layers
(768 and 384) to obtain a joint encoded vector $J$. $J$ is fed to two modules:
joint decoder and classifier. The joint decoder again consists of dense layers
of sizes 768 and $|F|$. The classifier feeds the output $J$ to a dense layer
of size 128 and then finally to the output log-softmax layer. The joint
decoder aims to reconstruct $F$ and uses MSE (mean squared error) loss, while
the classifier aims to predict presence/absence of antisemitism or
antisemitism category. We use the sum of these two losses to train the model.

##  6\. Experiments

In this section, we discuss hyper-parameter settings; results using text-only,
image-only and multimodal classifiers; and qualitative analysis using
attention visualization, error analysis and case studies.

###  6.1. Hyper-Parameter Settings for Reproducibility

We use the following experimental settings. We perform 5 fold cross validation
where we split our labeled data in 64:16:20 as our train, validation, test
split for each fold. All hyper-parameters were tuned using validation set. For
the MFAS fusion module, we use the block 2 output as the intermediate layer
output since it gave us the best results compared to output from other blocks
(on validation set). For Gated-MCB all experimental settings were used as
suggested by the reference paper. For MFAS, $|F|=|F^{\prime}|$=2,816 (which is
3*768+512); for other fusion methods $|F|=|F^{\prime}|$=1,536 (which is
2*768).

For all experiments, we use Adam optimizer (Kingma and Ba, 2014). We
experimented with a range of learning rates and found $lr=2e^{-6}$ as the best
one. To improve the stability of the system across the samples we used a batch
normalization layer before the Dense layers. For the Dense layers, we use
dropouts with a drop probability of $0.2$. We used RELU non-linearity after
all our dense layers except the final output layer. We train our system for a
max of 50 epochs with early stopping, with a batch size of 4. For all the
results, we report 5-fold cross-validation accuracy and macro-F1. For further
details of hyper-parameters, we refer the reader to look at our
codebase101010https://github.com/mohit3011/Online-Antisemitism-Detection-
Using-MultimodalDeep-Learning.

Table 4. Comparison of (5-fold cross validation) performance of multimodal classifiers with RoBERTa as text encoder and ResNet-152 as image encoder. We also compare the performance of the proposed architecture with a baseline from Gomez et al.(Gomez et al., 2020) (FCM). | Twitter | Gab
---|---|---
Method | Binary | Multiclass | Binary | Multiclass
| Accuracy | F-1 | Accuracy | F-1 | Accuracy | F-1 | Accuracy | F-1
FCM (Gomez et al., 2020) |  .564$\pm$.015 |  .545$\pm$.038 |  .445$\pm$.006 |  .164$\pm$.022 |  0.607$\pm$0.014 |  .595$\pm$.028 |  .468$\pm$.005 |  .182$\pm$.027
Concatenation |  .710$\pm$.012 |  .708$\pm$.013 |  .662$\pm$.027 |  .664$\pm$.027 |  .905$\pm$.005 |  .905$\pm$.005 |  .653$\pm$.052 |  .616$\pm$.046
Gated MCB |  .690$\pm$.026 |  .683$\pm$.036 |  .679$\pm$.030 | .677$\pm$.043 |  .904$\pm$.014 |  .903$\pm$.014 |  .654$\pm$.039 |  .618$\pm$.043
MFAS | .715$\pm$.013 | .714$\pm$.014 | .680$\pm$.035 |  .675$\pm$.023 | .906$\pm$.007 | .906$\pm$.007 | .665$\pm$.029 | .625$\pm$.032

Figure 3. Text + OCR encoder module attention visualization
Figure 4. Image encoder module attention visualization (Best viewed in color)

###  6.2. Results using Text-only and Image-only Classifiers

We experiment with five popular pre-trained text embedding/network based
classifiers and two pre-trained image network classifiers. For the text-only
classifiers we use GloVe (Pennington et al., 2014), FastText (Joulin et al.,
2016), BERT (Devlin et al., 2018) and RoBERTa (Liu et al., 2019). We also
experiment with Founta et al. (Founta et al., 2019)’s method which is an
attentional RNN model with GloVe embeddings. For the image only classifiers we
experiment with ResNet-152 (He et al., 2016) and DenseNet-161 (Huang et al.,
2017). We also experimented with VGG-19 but did not see any better results.

Table 3 provides the comparative results. We make the following observations:
(1) Compared to the text-only methods, the image-only models provide much
lower accuracy. We believe this is because images related to antisemitic posts
are usually memes, screenshots, or news articles that usually don’t carry any
spatial-visual features. (2) Among the text-only classifiers, Transformer
based methods performed better than the rather shallow approaches like
FastText and GloVe. BERT and RoBERTa lead to very similar results. (3) Among
the image-only models, ResNet-152 performs the best except for binary
classification on Gab.

###  6.3. Results using Multimodal Classifiers

In this experiment we tested different fusion mechanism for our proposed
multimodal classifier. From Table 3, we observe that ResNet-152 is the best
image encoder and RoBERTa is the best text (post text + OCR) encoder. Hence,
we perform multimodal experiments with these encoders only. We show the
results obtained for this multimodal experiment in Table 4. In addition to
this, we also compared the performance of our proposed architecture with the
baseline model from (Gomez et al., 2020) (FCM). FCM uses GloVe for text
encoder and InceptionV3 for image encoder.

We make the following observations: (1) Each of the three variants of the
proposed architecture beat the baseline by a huge margin. (2) Results in Table
4 are much better compared to those in Table 3 except for the binary
classification for Twitter. For Gab, we see a massive increase of $\sim$2 and
$\sim$4 percentage points in accuracy and F1 for both binary and multi-class
tasks respectively. The improvement in results when using both text and images
(i.e., across Tables 4 and 3) is better for Gab overall compared to Twitter.
This is because the images on Gab are much more rich and informative compared
to those on Twitter. (3) MFAS based multimodal fusion approach outperforms the
Gated MCB and concatenation based fusion approaches.

###  6.4. Qualitative analysis: Attention Visualization

To gain better insights into the the proposed system, we visualize attention
weights for both the Text + OCR (using bertviz (Vig, 2019)) and the Image
encoder (using GradCAM (Selvaraju et al., 2017)). Figure 3 shows the
visualization for a self-attention head from the last encoder layer in the
Text + OCR module. We took an antisemitic example having the text content as
“some people have jew parasites embedded in their brains” and the OCR text
being “liberals”. We observe high attention between the word ‘parasites’ with
‘jew’ apart from the standard RoBERTa [SEP] tokens showcasing that the system
identified that this text refers Jews as parasites. Similarly, in Fig. 3
(right), the word ‘liberals’ present in the OCR text output shares higher
attention weights with the word ‘jew’ from the post text content showing the
cross-attention learnt by the system.

Fig. 4 shows the GradCAM visualization of the image in the post, we observe
higher attention on the region of the Happy Merchant Meme face which is
usually used as a symbol of antisemitism. Another interesting observation is
that the text in the image doesn’t get much attention, which makes our choice
of adding OCR suitable.

Table 5. Top: Correctly predicted examples. Bottom: Examples with erroneous predictions. Post text |  OCR Text/Image Description |  Actual Class |  Predicted Class |  Explanation
---|---|---|---|---
shabbat shalom to all my jewish friends may the lord bless you |  shabbat shalom everyone |  Non-Antisemitic |  Non-Antisemitic |  The terms “friends”, “Shabbat”, “Shalom” are good clues.
no more jewish wars for israel |  I see dead people wherever jews have the power |  Antisemitic |  Antisemitic |  The terms “dead”, “jewish” and “wars” are good clues.
Zog (2020): The heartwarming story of a magical dragon who eventually takes control of the entertainment industry. |  ZOG (with a picture of a dragon) |  Antisemitic |  Non-Antisemitic |  This post presents a case of sarcasm where ZOG (the dragon cartoon) is used to refer zionist occupied government (ZOG)
Beautiful woman. Not this are zionist woman. They have weapons everywhere. |  (No Text) |  Racial Antisemitism |  Political Antisemitism |  The presence of word ‘zionist’ causes confusion
Banksters jews and the blood from white people |  (image with people carrying money bags and dead people) |  Economic Antisemitism |  Racial Antisemitism |  Reference to ‘white people’ causes confusion.

###  6.5. Qualitative analysis: Error Analysis and Case Studies

Table 6. Confusion matrix for the binary classification task (Gab). The entries represent the sum on test set examples over 5-fold cross validation. Actual |  | Predicted
---|---|---
| Non-antisemitic | Antisemitic
Non-antisemitic | 1470 | 162
Antisemitic | 167 | 1710
Table 7. Confusion matrix for the multiclass classification task (Gab). The entries represent the sum on test set examples over 5-fold cross validation. Actual |  | Predicted
---|---|---
| Political | Economic | Religious | Racial
Political | 441 | 49 | 33 | 213
Economic | 14 | 82 | 3 | 19
Religious | 9 | 1 | 102 | 32
Racial | 141 | 40 | 76 | 622

Tables 6 and 7 show the confusion matrices for the proposed MFAS-based
multimodal system for binary and multi-class cases respectively for Gab.
Similarly, Tables 8 and 9 show confusion matrices for Twitter. Each entry in
the confusion matrices represents the sum of examples in the test sets over
5-fold cross validation. As observed in Table 6 and Table 8, the classifier
has higher percentage of False Positives than the False Negatives (where the
positive class is ‘Antisemitic’). We believe that this was due to many
borderline cases which confused the classifier on topics like ‘Anti-Israel
hate’, ‘Issue of Israel-Palestine conflict’ etc. Additionally, from Table 7
and Table 9 we observe that across both the datasets, the classifier is most
confused between the ‘Political’ and ‘Racial’ classes. This could be because
many politically oriented posts against Jews also used racial prejudices.

Table 8. Confusion matrix for the binary classification task (Twitter). The entries represent the sum on test set examples over 5-fold cross validation. Actual |  | Predicted
---|---|---
| Non-antisemitic | Antisemitic
Non-antisemitic | 1106 | 568
Antisemitic | 317 | 1111
Table 9. Confusion matrix for the multiclass classification task (Twitter). The entries represent the sum on test set examples over 5-fold cross validation. Actual |  | Predicted
---|---|---
| Political | Economic | Religious | Racial
Political | 470 | 35 | 11 | 123
Economic | 16 | 149 | 9 | 9
Religious | 15 | 4 | 79 | 26
Racial | 160 | 12 | 37 | 273

Finally, in Table 5, we present a few examples where our system produced
correct/incorrect (top/bottom part) predictions. The last “Explanation” column
details the plausible reason for the erroneous cases.

Figures 5 and 6 present two interesting instances where our multimodal system
misclassifies. The post referred in Figure 5 had the post text as “calling a
Jewish man a penny pincher is anti semitic”. The post is condemning
antisemitic behaviour but the complex structure of text present in the image
makes it hard for the model to extract information correctly. Also, the model
does not understand that the original tweet was posted by some other user and
not this user. Similarly, the post referred in Figure 6 had the post text as
“@usermention teams up with another antisemite this time a guy who tweeted an
image depicting Jews as controlling the world and adding that Jewish money
crushes the little people”. This post reports an antisemitic behaviour by
someone else through the screenshot of the tweet. But, due to the absence of
any additional context about the image being a screenshot of the tweet from
someone else, makes the system to commit the mistake. These two cases help us
surface a broader problem with the current systems capturing information from
multiple modalities.

Figure 5. In this example, the image contains screenshot of multiple
tweets/posts/articles stitched together posted by someone else. Figure 6. In
this example, the image is a screenshot of a hateful tweet posted by someone
else.

##  7\. Limitation

The presented task of antisemitism detection share similar set of limitations
as with other hate speech detection tasks like racism/sexism detection. We
list a few of the limitations here:

  * •

Keyword Bias: As with the other hate speech detection systems, keywords play
an important role in the classification of content. We observed that posts
containing certain keywords like zionists, holocaust, Hitler, Christians,
Torah were prone to be classified as antisemitic since majority of training
posts containing these keywords were labelled as antisemitic. This observation
is in line with the past research which claims that deep learning models learn
this kind of keyword biases.

  * •

Subtlety in the expression of hate: Another set of examples which were
misclassified belonged to the category of sarcasm/trolling/subtle hate. It
becomes extremely difficult for the system to extract the real intent behind
posts expressing views in a subtle manner. This in turn creates a dilemma of
freedom of speech/curbing hate speech which is a common problem across various
other forms of hate speech.

  * •

Noise from multiple modalities: Though we showed that adding information from
multiple modalities overall helps in antisemitism detection, in a few cases
noise present in one of the modalities caused misclassification (as shown in
Figures 5 and 6).

##  8\. Discussion

In this work we presented the first systematic study on the problem of
detection and categorization of antisemitism. We collected and labeled two
datasets for antisemitism detection and categorization. We hope that these
will accelerate further research in this direction. We proposed a multimodal
system which uses text, images and OCR for this task and demonstrated its
efficacy on the two datasets. We experimented with single-modal as well as
multimodal classifiers and found that combining data from multiple modalities
improves the performance and robustness of the system to a small extent for
Twitter but massively for Gab. Finally, we also performed a qualitative
analysis of our multimodal system through attention visualisation and error
analysis. We observed that the complexities in images and subtlety of hate in
text can lead to errors. Images with multiple screenshots, multi-column text
and texts expressing irony, sarcasm or indirect references posed problems for
the classifier.

Similar to images, videos have become increasingly common. It will be
interesting to develop multimodal systems involving text and videos for
detecting antisemitism in the future. Another interesting direction involves
usage of contextual information like user profiles for the classification
task.

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RE-PARAMETERISATION OF FOUR LIMB DARKENING LAWS AND THEIR IMPLEMENTATION INTO
THE JKTEBOP CODE

By John Southworth

Astrophysics Group, Keele University, Staffordshire, ST5 5BG, UK

Limb darkening (LD) is typically parameterised using a range of functional
‘laws’ in models of the light curves of eclipsing binary and transiting
planetary systems. The two-coefficient LD laws all suffer from a strong
correlation between their coefficients, preventing a reliable determination of
both coefficients from high-quality light curves. We use numerical simulations
to propose re-parameterisations of the quadratic, logarithmic, square-root and
cubic LD laws that show much weaker correlations, and implement them into the
jktebop code. We recommend that these re-parameterisations are used whenever
both LD coefficients are fitted. Conversely, when fitting for only one
coefficient, the standard laws should be used to avoid problems with fixing
coefficients at poor values. We find that these choices have little effect on
the other fitted parameters of a light curve model. We also recommend that the
power-2 LD law should be used as default because it provides a good fit to
theoretical predictions, and that the quadratic and linear laws should be
avoided because they do not.

## Introduction

Limb darkening (LD) is a universal phenomenon which modifies the brightness of
stars across their disc. LD results in a wavelength-dependent decrease in
brightness from the centre of the observed disc to the limb, and in a steeper
drop-off closer to the limb compared to near the centre. It arises because
sightlines which enter the surface of the star at an angle (“slant viewing
geometry”) penetrate less deep into the atmosphere, see cooler plasma than a
perpendicular sightline, and so perceive a lower flux.

LD was first noticed in our Sun by Luca Valerio in 1612 1, and was first
measured by Pierre Bouguer in 1729 2. It must be accounted for in any
observing project which involves spatially resolving a star, specifically
interferometry, eclipsing binaries (EBs) and transiting planetary systems
(TEPs). All analysis methods that the author is aware of for EBs and TEPs
include a treatment of LD in order to properly represent the characteristics
of the object(s) being considered.

In this work we describe the implementation of multiple LD laws into the
jktebop***http://www.astro.keele.ac.uk/jkt/codes/jktebop.html code 3; 4 for
modelling the light and radial velocity curves of EBs and TEPs. The novelty of
this work lies primarily in the re-parameterisation of the two-coefficient LD
laws to mitigate the strong correlations between the two coefficients. We
begin with a reminder of the different LD laws in use, present the re-
parameterisations we adopt, and conclude with advice on using the LD
functionality now included in jktebop.

## Limb darkening laws

For the analysis of the light curves of EBs, LD was implemented in the
pioneering Russell-Merrill method 5; 6; 7; 8; 9 using the linear law 5; 10:

$\frac{F(\mu)}{F(1)}=1-u_{\rm lin}(1-\mu)\,\,,$ (1)

where $F(\mu)$ is the flux at position $\mu=\cos\gamma$ on the stellar disc,
$\gamma$ is the angle between the observer’s line of sight and the surface
normal, $F(1)$ is the flux at the centre of the disc, and $u_{\rm lin}$ is the
linear LD coefficient. The strength of the LD is specified by $u_{\rm lin}$,
which is normally between unity (no limb darkening) and zero (surface flux
decreases to zero at the limb).

The linear LD law has been known for over a century to be an inadequate
representation of the solar LD 11; 12; 13; 14 prompting more sophisticated
laws to be proposed: the quadratic LD law (Kopal 15):

$\frac{F(\mu)}{F(1)}=1-u_{\rm quad}(1-\mu)-v_{\rm quad}(1-\mu)^{2}~{}~{},$ (2)

the logarithmic law (Klinglesmith & Sobieski 16):

$\frac{F(\mu)}{F(1)}=1-u_{\rm log}(1-\mu)-v_{\rm log}\mu\ln\mu~{}~{},$ (3)

the square-root law (Díaz-Cordovés & Giménez 17):

$\frac{F(\mu)}{F(1)}=1-u_{\rm sqrt}(1-\mu)-v_{\rm sqrt}(1-\sqrt{\mu})~{}~{},$
(4)

the cubic law (van’t Veer 18):

$\frac{F(\mu)}{F(1)}=1-u_{\rm cub}(1-\mu)-v_{\rm cub}(1-\mu)^{3}~{}~{},$ (5)

the power-2 law (Hestroffer 19):

$\frac{F(\mu)}{F(1)}=1-c(1-\mu^{\alpha})~{}~{},$ (6)

and the four-parameter law proposed by Claret 20:

$\frac{F(\mu)}{F(1)}=1-\sum_{n=1}^{4}u_{n}(1-\mu^{n/2})~{}~{}.$ (7)

The ebop code 21; 22, on which jktebop is based, used the linear LD law. Díaz-
Cordovés & Giménez 17 and Giménez & Díaz-Cordovés 23 modified ebop to include
the quadratic and square-root LD laws. The current author subsequently added
these and the logarithmic, cubic and four-parameter laws into jktebop
(versions 12, 15 and 31). We have now added the power-2 law (jktebop version
43) which means that all the laws given above are now implemented in jktebop.
The cubic law was included specifically because it was expected that the
greater functional difference between the two terms (compared to the quadratic
law) would make the two coefficients less correlated; it is shown below that
this is indeed the case. It is possible within jktebop to use different LD
laws for the two stars, with the exception of the four-parameter law.

## Review of published re-parameterisations of the LD laws

Our experience of using the LD laws in jktebop for a wide range of EBs and
TEPs is that: the linear law is adequate for most ground-based data but not
for light curves from space missions such as Kepler, CoRoT and TESS; results
from the two-parameter laws are typically in excellent agreement; one should
fit for one of the two LD coefficients when possible because theoretical
predictions are imperfect; fitting for both LD coefficients in the two-
coefficient laws is not recommended because they can be severely correlated.
Strong correlations are a particular issue for Markov chain Monte Carlo (MCMC)
codes as they cause a long autocorrelation length and thus decrease the number
of independent samples in the Markov chains. Support for these statements can
be found in correlation plots 24; 25 and supplementary material for the
Homgeneous Studies publications 25; 26; 27; 28. The strong correlations have
also been noticed by other researchers, e.g. refs. 29 and 30.

The correlations could be decreased by changing the parameterisation of the LD
laws, and a range of re-parameterisations have been proposed for the quadratic
law. Brown et al. 31 fitted for the sum and difference of the LD coefficients:

$u^{\prime}=u_{\rm quad}+v_{\rm quad}$ (8) $v^{\prime}=u_{\rm quad}-v_{\rm
quad}$ (9)

Holman et al. 32 used another:

$u^{\prime}=2\,u_{\rm quad}+v_{\rm quad}$ (10) $v^{\prime}=u_{\rm
quad}-2\,v_{\rm quad}$ (11)

and Pál 30 generalised these to

$u^{\prime}=u_{\rm quad}\sin\theta-v_{\rm quad}\cos\theta$ (12)
$v^{\prime}=u_{\rm quad}\sin\theta+v_{\rm quad}\cos\theta$ (13)

where $\theta$ depends on the properties of the system being studied but is
usually between 35∘ and 40∘. Kipping 33 has explored these in detail, and
Howarth 34 has discussed the comparison between observed and theoretical LD
coefficients.

Maxted 35 proposed a re-parameterisation of the power-2 LD law to depend on
the coefficients $h_{1}$ and $h_{2}$ where

$h_{1}=\frac{F(0.5)}{F(1)}=1-c\,(1-2^{-\alpha})$ (14)

and

$h_{2}=\frac{F(0.5)-F(0)}{F(1)}=c\,2^{-\alpha}$ (15)

and $h_{1}$ and $h_{2}$ are only weakly correlated (see also Short et al. 36).

We are not aware of proposed re-parameterisations for any of the other laws, a
point also noted by Czismadia 37.

## Data for numerical experiments

It is desirable to avoid strong correlations between parameters when fitting
the light curves of EBs and TEPs. We therefore chose to re-parameterise the
two-parameter LD laws with coefficients that are less strongly correlated. As
multiple differing options have been published for the quadratic law, and none
for any of the other laws (except power-2), we decided to determine our own.
The most straightforward way to do this is via numerical experiments.

We identified a set of five EBs and TEPs with a variety of properties and for
which excellent light curves exist. The rationale for these choices is that we
expected the correlations between LD coefficients to depend on the physical
attributes of a given system so needed to include objects with a range of
characteristics, and that very high-quality photometry is needed to fit for
both LD coefficients in a given system.

The first object we analysed was the EB IT Cas, which was chosen because it
shows deep V-shaped eclipses which arise from two very similar stars with an
orbital inclination near 90∘, and thus should sample the full range of $\mu$
values on the stellar discs. For this we used the Simple Aperture Photometry
(SAP) from sector 17 of the Transiting Exoplanet Survey Satellite (TESS)
downloaded from the Mikulski Archive for Space Telescopes
(MAST†††https://mast.stsci.edu/portal/Mashup/Clients/Mast/Portal.html). We
used only data with a QUALITY flag of zero, ignored the data errors as they
were too small, and rejected all data more than one eclipse duration from the
midpoint of an eclipse in order to save computing time. A detailed analysis of
this system is in preparation and will be presented in due course as part of
the Rediscussion of Eclipsing Binaries project 38.

FIG. 1:  TESS short-cadence SAP photometry of the three EBs analysed in the
current work. The primary eclipses are shown in the left panels and the
secondary eclipses in the right panels. The names are labelled on the panels.
FIG. 2:  Light curves of the two TEPs analysed in the current work from the
New Technology Telescope (WASP-50) and Kepler (HAT-P-7).

For our second object we chose WASP-50, a TEP for which an extremely high-
quality transit light curve is available from a ground-based telescope 39.
These data proved to be useful but of insufficient quality to reliably measure
two LD coefficients. We therefore chose a third object, the TEP system HAT-P-7
40 for which an extraordinarily good light curve is available from the Kepler
satellite 41. We used the same data as in Southworth 27, which comprise the
first 59 transits observed, all in short cadence mode 42.

We also added a fourth object, the totally-eclipsing binary YZ Cas 43, for
which we used the sector 19 data from TESS. Finally, after inspection of the
preliminary results, we added WW Aur 44 as it shows deep V-shaped eclipses
similar to those of IT Cas but has a circular orbit. For WW Aur we used the
sector 45 data from TESS. For both objects the TESS data were prepared in the
same way as for IT Cas. The light curves of the five objects are shown in
Figs. 1 and 2. We would have liked to extend this to stars hotter than YZ Cas
A but were unable to identify a suitable candidate: all options we explored
had either shallow eclipses, large fractional radii, pulsations, or no high-
quality light curve.

The light curve of each object was modelled using jktebop and a two-parameter
LD law, with both LD coefficients fitted. Once a good fit was obtained, we ran
a set of 1000 Monte Carlo simulations 45; 25, which comprised the generation
and then least-squares fit of 1000 synthetic datasets with the same timestamps
as the original data and brightness measurements taken from the original best-
fitting model with Gaussian noise applied. This was performed for the
quadratic, logarithmic, square-root and cubic LD laws. We did not consider the
linear LD law, because it only has one coefficient so is not affected by
correlations between coefficients, or the power-2 law, as the $h_{1}$ and
$h_{2}$ approach was judged to be already satisfactory. Conversely, the four-
parameter law exhibits such strong correlations between its coefficients that
we considered it to be a lost cause so made no attempt to re-parameterise it.

Table I:  Linear Pearson correlation coefficients between the $u$ and $v$ coefficients of the two-parameter LD laws, assessed using Monte Carlo simulations as implemented in jktebop, for each of the five objects included in the numerical experimentation. Object | quadratic law | logarithmic law | square-root law | cubic law
---|---|---|---|---
IT Cas |  $-$0.982 |  $+$0.998 |  $-$0.999 |  $-$0.952
WASP-50 |  $-$0.951 |  $+$0.994 |  $-$0.996 |  $-$0.605
HAT-P-7 |  $-$0.992 |  $+$0.992 |  $-$0.999 |  $-$0.973
YZ Cas |  $-$0.978 |  $+$0.987 |  $-$0.999 |  $-$0.914
WW Aur |  $-$0.995 |  $+$0.997 |  $-$0.999 |  $-$0.985

## New re-parameterisations

We first assessed the linear Pearson correlation between the two LD
coefficients in each Monte Carlo simulation, using the correlate function in
IDL‡‡‡http://www.harrisgeospatial.com/SoftwareTechnology/IDL.aspx. The results
are given in Table I and support several conclusions. First, the correlations
between $u$ and $v$ are in general horrendous. Second, we notice that the
correlations are at their worst when the data are of the highest quality.
Third, the coefficients of the square-root law exhibit almost perfect
correlations so should never be fitted together. Fourth, the coefficients of
the cubic LD law have the lowest correlations, supporting the expectation
mentioned above.

Table II:  Values of $x$ which minimise the correlation between $u^{\prime}$ and $v^{\prime}$, for each LD law and each object studied. Object | quadratic law | logarithmic law | square-root law | cubic law
---|---|---|---|---
IT Cas | 0.44 | 0.75 | 0.57 | 0.19
WASP-50 | 0.59 | 0.57 | 0.51 | 0.29
HAT-P-7 | 0.62 | 0.60 | 0.62 | 0.39
YZ Cas | 0.63 | 0.64 | 0.60 | 0.33
WW Aur | 0.58 | 0.62 | 0.62 | 0.35
Adopted value | 0.6 | 0.6 | 0.6 | 0.3

We next sought alternative parameterisations that would reduce these
correlations. We chose a functional form that is similar to that of Pál 30 but
simpler:

$u^{\prime}=u+x\,v$ (16) $v^{\prime}=u-x\,v$ (17)

where the quantity $x$ can be chosen to minimise the correlation between
$u^{\prime}$ and $v^{\prime}$ for each LD law. The implementation of this in
jktebop was done by modifying the input and output sections but converting the
LD to the original parameterisations when calculating a model datapoint. This
meant that we needed only the inverse transforms, which can easily be shown to
be:

$u=\frac{u^{\prime}+v^{\prime}}{2}$ (18) $v=\frac{u^{\prime}-v^{\prime}}{2x}$
(19)

independently of the LD law.

We then determined the value of $x$, for each LD law and for each object, that
minimised the correlation between $u^{\prime}$ and $v^{\prime}$. This was done
by manual iteration and was restricted to two significant figures in $x$ both
for convenience and to avoid unnecessary precision. These values are given in
Table II and show that the best value of $x$ depends on both the object and
the LD law, as expected. The results are highly consistent, with the exception
of IT Cas for which significantly different $x$ values are found in some
cases. A plausible explanation for this is that IT Cas is the only object with
an eccentric orbit, and the inclusion of $e\sin\omega$ as a fitted parameter
has modified the correlations between the LD coefficients. However, an
exploratory Monte Carlo simulation with $e\sin\omega$ fixed showed the same
result so this supposition was not confirmed.

Table III:  Linear Pearson correlation coefficients between $u^{\prime}$ and $v^{\prime}$ in our new LD law parameterisations, calculated for each of the five objects included in the numerical experimentation using Monte Carlo simulations. Object | quadratic law | logarithmic law | square-root law | cubic law
---|---|---|---|---
IT Cas |  $-$0.860 |  $+$0.969 |  $-$0.888 |  $-$0.842
WASP-50 |  $-$0.034 |  $-$0.375 |  $-$0.890 |  $-$0.028
HAT-P-7 |  $+$0.206 |  $-$0.022 |  $+$0.704 |  $-$0.222
YZ Cas |  $-$0.207 |  $+$0.383 |  $-$0.064 |  $-$0.745
WW Aur |  $-$0.363 |  $+$0.374 |  $-$0.711 |  $-$0.688

Given this relatively good consistency in $x$, we chose suitable values for
implementation in jktebop for general use: 0.3 for the cubic law and 0.6 for
the other three laws. For clarity, here are the revised versions of the LD
laws we propose:

$u^{\prime}_{\rm quad}=u_{\rm quad}+0.6\,v_{\rm quad}$ (20) $v^{\prime}_{\rm
quad}=u_{\rm quad}-0.6\,v_{\rm quad}$ (21)

for the quadratic law,

$u^{\prime}_{\rm log}=u_{\rm log}+0.6\,v_{\rm log}$ (22) $v^{\prime}_{\rm
log}=u_{\rm log}-0.6\,v_{\rm log}$ (23)

for the logarithmic law,

$u^{\prime}_{\rm sqrt}=u_{\rm sqrt}+0.6\,v_{\rm sqrt}$ (24) $v^{\prime}_{\rm
sqrt}=u_{\rm sqrt}-0.6\,v_{\rm sqrt}$ (25)

for the square-root law, and

$u^{\prime}_{\rm cub}=u_{\rm cub}+0.3\,v_{\rm cub}$ (26) $v^{\prime}_{\rm
cub}=u_{\rm cub}-0.3\,v_{\rm cub}$ (27)

for the cubic law.

We assessed the correlation between $u^{\prime}$ and $v^{\prime}$ for each of
these laws and for each of the five objects to gauge the improvement brought
by the revised laws. These are given in Table III and show a clear improvement
in all cases. There are nevertheless still some strong correlations,
particularly for the logarithmic and square-root laws. We recommend that these
laws are not used when attempting to fit both coefficients of a two-parameter
LD law. As an example, in Figs. 3 and 4 we show scatter plots of the Monte
Carlo simulation output for WW Aur, for the LD laws in their original form,
for the lowest correlation for this object, and for the recommended re-
parameterisations.

FIG. 3:  Scatter plots of the LD coefficients for the quadratic and
logarithmic laws obtained from fitting the light curve of WW Aur and then
performing 1000 Monte Carlo simulations. The correlation coefficient is
printed in each panel.

Several published re-parameterisations of the quadratic LD law 31; 32; 30 were
quoted above. We checked these against each of our five objects (allowing for
values between 35∘ and 40∘ for the functional form proposed by Pál 30) and
found that they all yielded significantly stronger correlations than the re-
parameterisa- tions proposed in the current work. Finally, we did not attempt
to compare the coefficients to theory in order to avoid “mission creep”.

FIG. 4:  As Fig. 3 but for the square-root and cubic LD laws.

## Testing the new LD laws

Table IV:  Selected results from fitting the TESS light curve of WW Aur with one or two LD coefficients fitted, for all possible versions of the one- and two-parameter laws. $N_{\rm cof}$ is the number of LD coefficients fitted. The bracketed quantities indicate uncertainties in the final digit of the preceding values LD approach | $N_{\rm cof}$ | rms (mmag) | $r_{\rm A}$ | $r_{\rm B}$ |  $i$ (∘)
---|---|---|---|---|---
Linear law | 1 | 0.350 | $0.15958~{}(4)$ | $0.15121~{}(4)$ | $87.550~{}(2)$
Quadratic law | 1 | 0.343 | $0.15973~{}(4)$ | $0.15148~{}(4)$ | $87.497~{}(2)$
Logarithmic law | 1 | 0.352 | $0.15957~{}(4)$ | $0.15118~{}(4)$ | $87.555~{}(2)$
Square-root law | 1 | 0.341 | $0.15973~{}(4)$ | $0.15140~{}(4)$ | $87.508~{}(2)$
Cubic law | 1 | 0.341 | $0.15973~{}(4)$ | $0.15138~{}(4)$ | $87.510~{}(2)$
Power-2 law | 1 | 0.341 | $0.15971~{}(4)$ | $0.15138~{}(4)$ | $87.512~{}(2)$
Quadratic re-par | 1 | 0.342 | $0.15972~{}(4)$ | $0.15146~{}(4)$ | $87.501~{}(2)$
Logarithmic re-par | 1 | 0.647 | $0.16019~{}(7)$ | $0.15254~{}(7)$ | $87.300~{}(4)$
Square-root re-par | 1 | 0.341 | $0.15973~{}(4)$ | $0.15140~{}(4)$ | $87.508~{}(2)$
Cubic re-par | 1 | 0.348 | $0.15960~{}(4)$ | $0.15124~{}(4)$ | $87.543~{}(2)$
Power-2 ($h_{1}$ and $h_{2}$) | 1 | 0.342 | $0.15970~{}(4)$ | $0.15136~{}(4)$ | $87.517~{}(2)$
Quadratic law | 2 | 0.342 | $0.15969~{}(4)$ | $0.15141~{}(4)$ | $87.510~{}(3)$
Logarithmic law | 2 | 0.341 | $0.15972~{}(4)$ | $0.15141~{}(4)$ | $87.508~{}(3)$
Square-root law | 2 | 0.341 | $0.15973~{}(4)$ | $0.15140~{}(4)$ | $87.508~{}(3)$
Cubic law | 2 | 0.341 | $0.15974~{}(4)$ | $0.15139~{}(4)$ | $87.507~{}(3)$
Power-2 law | 2 | 0.341 | $0.15974~{}(4)$ | $0.15141~{}(4)$ | $87.507~{}(3)$
Quadratic re-par | 2 | 0.342 | $0.15969~{}(4)$ | $0.15141~{}(5)$ | $87.510~{}(3)$
Logarithmic re-par | 2 | 0.341 | $0.15972~{}(4)$ | $0.15141~{}(4)$ | $87.508~{}(3)$
Square-root re-par | 2 | 0.341 | $0.15973~{}(4)$ | $0.15140~{}(5)$ | $87.508~{}(3)$
Cubic re-par | 2 | 0.341 | $0.15973~{}(4)$ | $0.15140~{}(4)$ | $87.508~{}(3)$
Power-2 ($h_{1}$ and $h_{2}$) | 2 | 0.341 | $0.15974~{}(4)$ | $0.15141~{}(5)$ | $87.507~{}(3)$
Four-parameter | 1 | 0.341 | $0.15970~{}(4)$ | $0.15136~{}(4)$ | $87.517~{}(2)$
Four-parameter | 2 | 0.341 | $0.15972~{}(4)$ | $0.15139~{}(4)$ | $87.509~{}(3)$

Now we had re-parameterisations of the LD laws and implemented them into
jktebop, we proceeded to test the code and assess the effect of the revised LD
laws. To limit the computational load of this work we analysed only one
object, WW Aur, and fitted only the data near eclipse in the first half of the
light curve from TESS sector 45. Best fits and 1000 Monte Carlo simulations
were performed for the linear LD law, for all two-parameter laws in their
original form, for the re-parameterisations presented here, and for the
$h_{1}$ and $h_{2}$ approach for the power-2 law. Initial or fixed LD
coefficients were set to values for the Cousins $R$ passband from Claret &
Hauschildt 46, with the exception of the power-2 law for which we used the
TESS passband predictions from Claret & Southworth 47. We also ran two fits
using the four-parameter LD law: one with coefficient $u_{2}$ fitted and one
with $u_{2}$ and $u_{4}$ fitted. The values of the fixed coefficients were
taken from Claret 48.

We report only the most relevant results from this work: the r.m.s. scatter
around the best fit, the fractional radii ($r_{\rm A}$ and $r_{\rm B}$), and
the orbital inclination ($i$). These are given in Table IV with errorbars
assessed using the Monte Carlo simulations. The errorbars are not true
uncertainties, as Monte Carlo simulations are only one of the tools typically
deployed in our error analyses 49, and are almost certainly too small 50.
Extensive comparisons between the results from different LD laws can also be
found in the supplementary material to our Homogeneous Studies papers 25; 26;
27; 28 for 94 TEPs.

Based on experience, Table IV and the Homogeneous Studies supplementary
material, we draw the following conclusions. First, the linear LD law is too
simplistic and gives slightly different results to those from all other LD
laws. It should not be used except for convenience in cases where the data
quality is low. Second, the re-parameterised laws give results that are
consistent with the original laws. Third, fitting for both LD coefficients
yielded comparable results to fitting for one coefficient in the case of WW
Aur, for which the data are of extremely high quality. Fourth, the anomalously
poor solution in Table IV for the re-parameterised square-root law suggests
that the re-parameterised laws risk giving bad results if only one LD
coefficient is fitted and the other coefficient is fixed at a bad value.

## Summary

A profusion of LD laws have been proposed, many of which have two
coefficients. All of these suffer from strong correlations between the two
coefficients which hinders the modelling process of observed light curves when
both coefficients are fitted parameters. We have proposed a re-
parameterisation of the quadratic, logarithmic, square-root and cubic laws and
performed numerical simulations to calibrate the re-parameterisations. This
was done considering three EBs and two TEPs with a variety of light curve
shapes.

We give the following recommendations:

  1. 1.

Light curves of low quality can be modelled using either the linear law for
simplicity, or one of the two-parameter laws with one or both coefficients
fixed.

  2. 2.

Light curves of medium quality should be modelled using one of the standard
two-parameter laws, with one coefficient fitted and one fixed.

  3. 3.

Light curves of high quality should be modelled including two LD coefficients
as fitted parameters. In this case the re-parameterised laws should be used,
to avoid the strong correlations found with the standard two-parameter laws.
This is particularly important for sampling agorithms such as Markov chain
Monte Carlo (MCMC), to avoid long autocorrelation lengths in the Markov
chains.

  4. 4.

If you are unsure whether a light curve is of low, medium or high quality, you
should try two or all three options and decide which is best based on the
values of and uncertainties in the fitted LD coefficients (and other system
parameters).

  5. 5.

The linear LD law should be avoided when performing any detailed analysis.

  6. 6.

The quadratic LD law should be avoided as it does not match theoretical LD
predictions well 47; 51.

  7. 7.

The power-2 LD law should be adopted as the default law because it _does_
match theoretical LD predictions well 35; 47; 52.

  8. 8.

The four two-coefficient LD laws give highly consistent results when treated
in the same way, so the choice between them is not important.

  9. 9.

The best re-parameterisation of a given LD law varies slightly between light
curves. If this is an issue, or if you want to avoid parameter correlations as
much as possible, you should use principal component analysis (PCA) to
orthogonalise the model parameters in the course of obtaining a least-squares
solution to a given light curve.

All LD laws and re-parameterisations have been implemented in version 43 of
the jktebop code, which is freely available for download from the author’s
website. The choice of which LD function to adopt is left to the user.

## Acknowledgements

We thank Drs. Pierre Maxted and Antonio Claret for comments on a draft of this
manuscript. This paper includes data collected by the Kepler and TESS missions
and obtained from the MAST data archive at the Space Telescope Science
Institute (STScI). Funding for the Kepler and TESS missions is provided by the
NASA’s Science Mission Directorate. STScI is operated by the Association of
Universities for Research in Astronomy, Inc., under NASA contract NAS 5–26555.
The following resources were used in the course of this work: the NASA
Astrophysics Data System; the SIMBAD database operated at CDS, Strasbourg,
France; and the ar$\chi$iv scientific paper preprint service operated by
Cornell University.

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