# Further Applications of Schur Rings to Produce GRRs for Dihedral Groups.

Jonathan Ebejer, Josef Lauri
University of Malta

Abstract: _This note is a continuation of postgraduate thesis research carried
out by the first author under the supervision of the second author at the
University of Malta. In that research we took a look at several results
relating Schur rings to sufficient conditions for GRRs and then applied those
results to produce numerical methods for constructing trivalent GRRs for
dihedral groups very quickly._

_Here we seek to slightly alter the approach used in the original research to
produce another numerical method for producing other GRRs for dihedral groups.
Moreover, this will hopefully serve as companion literature for the original
research when published._

##  1 Foundations

The background needed to understand the proof below would be quite familiar to
abstract algebraists and algebraic graph theorist and is given a more thorough
treatment in the original postgraduate research [11] and the work which was
uploaded to arXiv [10]. Still, we make a few critical definitions below and
cite theorems and lemmas which will be needed to prove the main theorem of
this work. The proofs for these lemmas and theorems have been omitted but are
available at the cited sources.

Let us start with a few definitions:

######  Definition 1.0.1 (Cayley graphs).

The Cayley graph of a group $\Gamma$ is the graph, denoted by
$\mbox{Cay}(\Gamma,S)$, whose vertex-set is $\Gamma$ and two vertices $u$ and
$v$ are adjacent if $v=us$ where $s\in S$ and $S$ is a subset of $\Gamma$ such
that $1\not\in S$, $S$ generates $\Gamma$, and $S^{-1}=S$. We call the set $S$
the connecting set of the Cayley graph.

$\mathbf{Schur\ Rings}$ A subring $\mathcal{S}$ of the group ring
$\mathbb{Z}[\Gamma]$ is called a Schur ring or an $\mathcal{S}$-ring over
$\Gamma$, of rank $r$ if the following conditions hold:

  * •

$\mathcal{S}$ is closed under addition and multiplication including
multiplication by elements of $\mathbb{Z}$ from the left (i.e. $\mathcal{S}$
is a $\mathbb{Z}[\Gamma]$-module).

  * •

Simple quantities $\overline{B}_{0},\overline{B}_{1},...,\overline{B}_{r-1}$
exist in $\mathcal{S}$ such that every element $\overline{C}\in\mathcal{S}$
has a unique representation:

$\overline{C}=\sum_{i=0}^{r-1}\beta^{i}\overline{B}_{i}$

where $\beta^{i}\geq 0$.

  * •

$\overline{B}_{0}=\overline{1}$.

  * •

$\displaystyle\sum_{i=0}^{r-1}\overline{B}_{i}=\overline{\Gamma}$, that is,
$\\{B_{0},B_{1},...,B_{r-1}\\}$ is a partition of $\Gamma$.

  * •

For every $i\in\\{0,1,2,...,r-1\\}$ there exists a $j\in\\{0,1,2,...,r-1\\}$
such that $\overline{B}_{j}=\overline{B}^{-1}_{i}(=\overline{\\{b^{-1}:b\in
B_{i}}\\})$.

We call the set of simple quantities
$\overline{B}_{0},\overline{B}_{1},\ldots,\overline{B}_{r-1}$the basis of the
Schur ring and we denote it by ${\mathcal{B}}[{\mathcal{S}}]$. Each simple
quantity $\overline{B}_{i}$ of the basis is referred to as a basic element of
the Schur ring. Sometimes we need to refer to the set $B_{i}$ which we call a
basic set. The Cayley graphs formed used the basic sets of a Schur ring as the
connecting sets are referred to as the basic Cayley graphs of the Schur ring.

Let $C$ be a subset of the group $\Gamma$. Then $\langle\langle
C\rangle\rangle$ is the smallest (coarsest) Schur ring of $\Gamma$ containing
$\overline{C}$.

$\mathbf{Structure\ Constants}$ Let $\overline{B}_{i}$ and $\overline{B}_{j}$
be two basis elements of an r-rank Schur ring $\mathcal{S}$. For all values
$i,j,k\in[r]$ there exist non-negative integers $\beta_{ij}^{k}$ called
structure constants, such that

$\overline{B}_{i}\cdot\overline{B}_{j}=\sum_{k=1}^{r}\beta_{ij}^{k}\overline{B}_{k}$

$\mathbf{Automorphism\ groups\ of\ Schur\ rings}$ The automorphism group of a
Schur ring is defined to be the intersection of all automorphism groups of the
basic Cayley graphs of the Schur ring. For any Schur ring $\mathcal{S}$ we
will denote its automorphism group as $Aut(\mathcal{S})$.

The “largest” Schur ring of a group $\Gamma$ is defined to be the finest
partition of $\Gamma$, that is, the basic sets are all the singleton elements
of $\Gamma$. The “smallest” Schur ring is then the coarsest partition, with
basic sets $\\{1\\}$ and $\Gamma-1$

The smallest (coarsest) Schur ring has the largest automorphism group, that
is, the symmetric group $S_{n}$, where $n$ is the order of the group. The
largest Schur ring has the smallest automorphism group, that is, the regular
action of the group on itself.

$\mathbf{Graphical\ Regular\ Representation}$ Let $\Gamma$ be a finite group
and let $G$ be a graph. If $\mbox{Aut}(G)\equiv\Gamma$ and acts regularly on
$V(G)$ then we say that $G$ is a graphical regular representation (GRR)of
$\Gamma$.

The question asking which finite groups have at least one GRR has been
completely solved [13, 14, 15, 16, 17, 18, 23, 24]. Specifically:

  1. 1.

The only abelian groups which have a GRR are $\mathbb{Z}^{n}_{2}$ for $n\geq
5$.

  2. 2.

Except for generalised dicyclic groups and a finite number of known groups,
all non-abelian groups have a GRR.

However, it remains a challenge to discern whether a group known to have GRRs
has GRRs with specific properties, such as being trivalent. This question is
the focus of this note, specifically for dihedral groups.

Below are some results we will be using:

######  Theorem 1.0.1.

$\mbox{Aut}(\mbox{Cay}(\Gamma,C))=\mbox{Aut}(\langle\langle C\rangle\rangle)$
[22]

######  Theorem 1.0.2.

If $\langle\langle C\rangle\rangle$ is trivial then $Cay(\Gamma,C)$ is a GRR
of $\Gamma$. [26]

######  Theorem 1.0.3 (Schur-Wielandt Principle).

Let $r=\sum z_{\gamma}\gamma$ be an element of a Schur ring $\mathcal{S}$ of a
group $\Gamma$. Then, for any integer $k$, the sum $\sum_{z_{\gamma}=k}\gamma$
is also in $\mathcal{S}$. [22]

###  1.1 Main Results

Our main result is the following:

######  Theorem 1.1.

Let $n$ be an odd integer greater than 5 and let $r$, $s$, and $t$ be integers
less than $n$ such that the difference of any two of them is relatively prime
to $n$. If $3r+s=4t\bmod n$, then
$\mbox{Cay}(D_{n},\\{ab^{r},ab^{s},ab^{t}\\})$ is a GRR of $D_{n}$.

To prove this theorem however, we will need the help of the following two
lemmas. We include only the proof of the first one as the second lemma is
proven in the main work on arXiv.

######  Lemma 1.1.

Let $\gamma$ be an element of $\Gamma$ such that $\gamma$ has odd order $n$
and let $r$ and $t$ be two integers such that $r-t$ is relatively prime to
$n$. If $\gamma^{r}+\gamma^{t}$ is an element of some
$\mathcal{B}[\mathcal{S}_{\Gamma}]$ then $\gamma^{s}+\gamma^{-s}$ is also an
element of that same $\mathcal{B}[\mathcal{S}_{\Gamma}]$ for all integers $s$
relatively prime to $n$.

###### Proof.

First we observe that since $\mathcal{S}_{\Gamma}$ is closed then
$(\gamma^{r}+\gamma^{t})^{n}$ is in $\mathcal{S}_{\Gamma}$ whenever
$(\gamma^{r}+\gamma^{t})$ is in $\mathcal{S}_{\Gamma}$. We can use binomial
expansion to get:

$(\gamma^{r}+\gamma^{t})^{n}=\displaystyle\sum_{i=0}^{n}[\binom{n}{i}\gamma^{ri}\gamma^{t(n-i)}]$
which can be re-written as
$\displaystyle\sum_{i=0}^{n}[\binom{n}{i}\gamma^{(r-t)i)}]\gamma^{tn}$.
However, $\gamma$ is of order $n$ and so $\gamma^{tn}\equiv 1$. Therefore, the
sum can once more be rewritten as
$\displaystyle\sum_{i=0}^{n}[\binom{n}{i}\gamma^{(r-t)i)}]$.

Since $r-t$ is relatively prime to $n$ then the summation
$\displaystyle\sum_{i=0}^{n}[\binom{n}{i}\gamma^{(r-t)i)}]$ must include
exactly $n$ terms. In other words every power of $\gamma$ appears in that sum
once and only once. Therefore, the coefficient of every $\gamma^{(r-t)i}$ is
exactly $\binom{n}{i}$. Moreover, we observe that
$\binom{n}{i}=\binom{n}{n-i}$ and so we can re-write this summation as
$\displaystyle\sum_{i=0}^{\frac{n}{2}}[\binom{n}{i}(\gamma^{(r-t)i}+\gamma^{(t-r)i})]$.

This means that for every $i\in[0,\frac{n}{2}]$ either $\gamma^{(r-t)i}$
occurs as a singleton in $\mathcal{B}[\mathcal{S}_{\Gamma}]$ or
$\gamma^{(r-t)i}+\gamma^{(t-r)i}$ occurs in
$\mathcal{B}[\mathcal{S}_{\Gamma}]$ due to the Schur-Wielandt principle.

However, in the cases where $i$ is relatively prime to $n$, $(r-t)i$ would
also be relatively prime to $n$, and so $\gamma^{(r-t)i}$ can generate every
power of $\gamma$. This would mean that should $\gamma^{(r-t)i}$ be a
singleton element of the basis, then every power of $\gamma$ would be a
singleton element of the basis also. However, $\gamma^{r}+\gamma^{t}$ is an
element of $\mathcal{B}[\mathcal{S}_{\Gamma}]$, and so this would imply a
contradiction. Therefore for every $i\in[0,\frac{n}{2}]$ which is relatively
prime to $n$, we have
$\gamma^{(r-t)i}+\gamma^{(t-r)i}\in\mathcal{B}[\mathcal{S}_{\Gamma}]$.

Let us recapitulate that at this point we have shown that if $n$ is the order
of some $\gamma\in\Gamma$, $r$ and $t$ are two integers such that $(r-t)$ are
relatively prime to $n$ and $(\gamma^{r}+\gamma^{t})$ is in
$\mathcal{B}[\mathcal{S}_{\Gamma}]$ then
$\displaystyle\sum_{i=0}^{\frac{n}{2}}[\binom{n}{i}(\gamma^{(r-t)i}+\gamma^{(t-r)i})]$
is in $\mathcal{S}_{\Gamma}$ and that those pairs of summands of the form
$\gamma^{(r-t)i}+\gamma^{(t-r)i}$ appear as a basis element in
$\mathcal{B}[\mathcal{S}_{\Gamma}]$ when $i$ is relatively prime to $n$. All
that is left to show is that every number less than $n$ which is also
relatively prime to it appears in the expression
$\displaystyle\sum_{i=0}^{\frac{n}{2}}[\binom{n}{i}(\gamma^{(r-t)i}+\gamma^{(t-r)i})]$
as a power of $\gamma$ (or, equivalently, a value of $(r-t)i$) and this will
complete the proof.

Consider the fact that
$\displaystyle\sum_{i=0}^{\frac{n}{2}}[\binom{n}{i}(\gamma^{(r-t)i}+\gamma^{(t-r)i})]\equiv\displaystyle\sum_{i=0}^{\frac{n}{2}}[\binom{n}{i}\gamma^{(r-t)i}]+\displaystyle\sum_{i=\frac{n}{2}+1}^{n}[\binom{n}{i}\gamma^{(r-t)i}]$.

Therefore, we can once again re-write our summation as
$\displaystyle\sum_{i\in[n]}[\binom{n}{i}\gamma^{(r-t)i}]$ where $[n]$ is the
set $\\{0,1,2,\dots,n\\}$, and re-write again to

$\displaystyle\sum_{j\in(r-t)[n]}[\binom{n}{i}\gamma^{j}]$

where $(r-t)[n]$ is the set $\\{(r-t)i$ mod $n:i\in[n]\\}$.

However, we now note that the set $(r-t)[n]$ is in fact equivalent to $[n]$.
This is because the modulo multiplication of a set can be represented as a
permutation of a set unto itself, which is a bijective function. Therefore:

$\displaystyle\sum_{i\in[n]}[\binom{n}{i}\gamma^{(r-t)i}]\equiv\displaystyle\sum_{j\in[n]}[\binom{n}{i}\gamma^{j}]$

Obviously every number less than $n$ which is less relatively prime to it
appears as a power of $\gamma$ on the right hand side above, meaning that
those powers of $\gamma$ also appear on the left hand side above.

This means that every $\gamma^{(r-t)i}$ such that $(r-t)i$ is relatively prime
to $n$ appears in the summation
$\displaystyle\sum_{i=0}^{\frac{n}{2}}[\binom{n}{i}(\gamma^{(r-t)i}+\gamma^{(t-r)i})]$
and as we stated above, this completes the proof.

∎

######  Lemma 1.2.

Let $n$ be an odd integer and let $a$ and $b$ be the generators of the group
$D_{n}$ with orders 2 and $n$ respectively. Also let $r$, $s$ and $t$ be
unique integers less than or equal to $n$. The Schur ring $\langle\langle
ab^{r}+ab^{s}+ab^{t}\rangle\rangle$ must be trivial if the following are true:

  1. 1.

$ab^{r}+ab^{s}+ab^{t}\not\in\mathcal{B}[\langle\langle
ab^{r}+ab^{s}+ab^{t}\rangle\rangle]$

  2. 2.

The absolute value of the difference between any two of $r$, $s$ and $t$ is
relatively prime to $n$.

  3. 3.

The sum of any two of $r$, $s$ and $t$ is not equal to twice the third
variable taken modulo $n$.

As mentioned before, the proof for the second theorem will not be included
here as it is included in the main work.

We can now begin on the proof of the main result. This proof is conceptually
identical to the proof for the main theory in the main work.

###### Proof.

We will prove Theorem 1.1 by showing how $ab^{r}+ab^{s}+ab^{t}$ cannot be an
element of $\mathcal{B}[\langle\langle ab^{r}+ab^{s}+ab^{t}\rangle\rangle]$
and then use Lemma 1.2 to meet the sufficient conditions of Theorem 1.0.2 and
so bring about the result.

We substitute $s$ with $4t-3r$ at this point for simplicity’s sake. We will
begin by assuming that $ab^{r}+ab^{s}+ab^{4t-3r}$ is in
$\mathcal{B}[\langle\langle ab^{r}+ab^{s}+ab^{4t-3r}\rangle\rangle]$ and then
show how this must imply a contradiction. We label this assumption as _(Asm)_
for further reference.

So let us begin by considering the following statement, which we label as
_(i)_ for further reference:

$(ab^{r}+ab^{t}+ab^{4t-3r})^{2}=3(1)+b^{r-t}+b^{3(r-t)}+b^{4(r-t)}+b^{-(r-t)}+b^{-3(r-t)}+b^{-4(r-t)}$

Assuming that _(Asm)_ is true then it must be possible to express _(i)_ as the
sum of elements of $\mathcal{B}[\langle\langle
ab^{r}+ab^{s}+ab^{4t-3r}\rangle\rangle]$. However, we shall show that doing so
implies a contradiction which means that _(Asm)_ must be false.

We begin by noting that the group elements on the right hand side of _(i)_
must either be isolated in the basis of the Schur ring or be grouped with
other elements also on the right hand side of _(i)_.

For the sake of brevity let us denote the sum of the elements on the right
hand side as $B_{6}$. We note now that $(B_{6})^{2}$ is equal to

$\displaystyle 6(1)+2B_{6}+b^{6(r-t)}+b^{-6(r-t)}+b^{8(r-t)}+b^{-8(r-t)}+$
$\displaystyle
2(b^{5(r-t)}+b^{-5(s-t)}+b^{7(r-t)}+b^{-7(r-t)})+3(b^{2(r-t)}+b^{-2(r-t)}).$

We notice that the total coefficient of $b^{2(r-t)}+b^{-2(r-t)}$ on the right
hand side is 3 and so, by the Schur-Wielandt principle, either $b^{2(r-t)}$
and $b^{-2(r-t)}$ are isolated elements in the basis of the Schur ring or
$b^{2(r-t)}+b^{-2(r-t)}$ is an element in the basis. Let us first consider the
possibility that $b^{2(r-t)}$ is an isolated element.

Since $r-t$ is relatively prime to $n$ and $n$ is odd then $2(r-t)$ is also
relatively prime to $n$. Therefore if $b^{2(r-t)}$ is isolated then all powers
of $b$ are isolated. This implies that $b^{3(r-t)}$ must also be an isolated
element. Since our assumption is that $ab^{r}+ab^{s}+ab^{4t-3r}$ is in
$\mathcal{B}[\langle\langle ab^{r}+ab^{s}+ab^{3r-2s}\rangle\rangle]$ we can
consider the product of $ab^{r}+ab^{s}+ab^{4t-3r}$ and $b^{3(r-t)}$, which
must be in $\langle\langle ab^{r}+ab^{s}+ab^{4t-3r}\rangle\rangle$.

$(ab^{r}+ab^{t}+ab^{4t-3r})b^{3(t-r)}\equiv ab^{3t-2r}+ab^{4t-3r}+ab^{7t-6r}.$

Note how the term $ab^{4t-3r}$ appears above. Since we are assuming that
$ab^{r}+ab^{s}+ab^{4t-3r}$ is in $\mathcal{B}[\langle\langle
ab^{r}+ab^{s}+ab^{4t-3r}\rangle\rangle]$, $ab^{r}$ and $ab^{t}$ must appear
too. There are only two ways this is possible:

  * •

$3t-2r=t$ and $7t-6r=r$

  * •

$3t-2r=r$ and $7t-6r=s$

All the equations above, however, imply that $r=t$ which is a contradiction.
Therefore it is not possible that $b^{2(r-t)}$ is an isolated element. Let us
consider now the second possibility that $b^{2(r-t)}+b^{-2(r-t)}$ is an
element in the basis.

We note that the difference in powers between $b^{2(r-t)}$ and $b^{-2(r-t)}$
is $\pm 4(r-t)$. Since $r-t$ is relatively prime to $n$ and $n$ is odd then
$\pm 4(r-t)$ is also relatively prime to $n$. Lemma 1.1 therefore gives us
that $b^{r-t}+b^{t-r}$ is in the basis of the Schur ring. Combining this with
_(Asm)_ gives us that $(ab^{r}+ab^{t}+ab^{4t-3r})(b^{r-t}+b^{t-r})$ is in the
Schur ring too. This expression expands to
$ab^{t}+ab^{r}+ab^{2r-t}+ab^{2t-r}+ab^{3t-2r}+ab^{5t-4r}$. Notably the terms
$ab^{t},ab^{r}$ appear here and due to _(Asm)_ this means that $ab^{4t-3r}$
must appear also. There are three ways this could happen:

  * •

$2r-t=4t-3r$ gives us $r=t$

  * •

$2t-r=4t-3r$ gives us $r=t$

  * •

$3t-2r=4t-3r$ gives us $r=t$

  * •

$5t-4r=4t-3r$ gives us $r=t$

But all these ways imply $r=t$ which is a contradiction. Therefore it is also
impossible that $b^{2(r-t)}+b^{-2(r-t)}$ is an element in the basis.

So we have seen that due to _(Asm)_ either $b^{2(r-t)}$ and $b^{-2(r-t)}$ are
isolated elements in the basis of the Schur ring or $b^{2(r-t)}+b^{-2(r-t)}$
is an element in the basis. However, we have also seen that both these
possibilities imply contradictions. Therefore it must be the case that _(Asm)_
is false.

This means that $ab^{r}+ab^{s}+ab^{3r-2s}\not\in\mathcal{B}[\langle\langle
ab^{r}+ab^{s}+ab^{3r-2s}\rangle\rangle]$. Lemma 2.2 gives us that this implies
that $\langle\langle ab^{r}+ab^{s}+ab^{t}\rangle\rangle$ is trivial and so
$\mbox{Aut}(\langle\langle ab^{r}+ab^{s}+ab^{t}\rangle\rangle)\equiv D_{n}$
meaning that $\mbox{Cay}(D_{n},\\{ab^{r},ab^{s},ab^{t}\\})$ is a GRR of
$D_{n}$ as we saw in the first section.

Hence we have proven the theorem. ∎

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# Consistency and Reproducibility of Grades in Higher Education: A Case Study
in Deep Learning

Paul Dubois, Romain Lhotte

###### Abstract

Evaluating the performance of students in higher education is essential for
gauging the effectiveness of teaching methods and achieving greater equality
of opportunities for all. In this study, we investigate the correlation
between two teachers’ grading practices in a deep learning course at the
master’s level, offered at CentraleSupélec. The two teachers, who have
distinct teaching styles, were responsible for marking the final project oral
presentation. Our results indicate a significant positive correlation (0.76)
between the two teachers’ grading practices, suggesting that their assessments
of students’ performance are consistent. Although consistent with each other,
grades do not seem to be fully reproducible from one examiner to the other
suggesting serious drawbacks of only using one examiner for oral projects.
Furthermore, we observed that the maximum difference between the grades
assigned by the two examiners was 12.5%, highlighting the potential impact of
inter-examiner variability on students’ final grades.

##  1 Introduction

In recent years, there has been a surge of interest in the field of deep
learning, with applications ranging from computer vision[10][3] and natural
language processing[9] to bio-informatics[8][2] and medical applications[11].
As a result, there is an increasing demand for high-quality education and
training programs in deep learning[1]. In response to this demand, many
universities and engineering schools have started offering courses and
programs in deep learning at various levels, including undergraduate and
graduate levels.

The evaluation of students’ performance in the course is a critical aspect of
teaching [7][5]. It can be used to measure the effectiveness of the teaching
methods and the quality of the learning outcomes. In this study, we
investigate the correlation between the grading practices of two instructors
who taught the course, and the Kaggle assessments. The Kaggle assessments are
regarded as impartial marking, and considered as a ”ground truth”. Our goal is
to evaluate the reliability and consistency of grading practices in higher
education and to provide insights that can inform efforts to improve the
quality of teaching and learning outcomes.

The findings of this study have implications for educators, administrators,
and policymakers who are interested in improving the quality of education in
deep learning and related fields. The results can also be used to inform the
development of more effective evaluation frameworks and grading practices in
higher education. Overall, this study contributes to the growing body of
knowledge on the pedagogy of teaching scientific courses at advanced
university level.

##  2 Methods

###  2.1 Participants

The participants in this study were 28 students enrolled in a deep learning
course at the master’s level offered at CentraleSupélec during the academic
year 2022/2023. The class was taught by two instructors, both of whom had
distinct teaching styles.

The course was designed to provide students with a comprehensive understanding
of the fundamental concepts, theories, and applications of deep learning. The
course was structured into 10 teaching sessions, each spanning three hours.
Each session comprised one hour of theoretical instruction followed by two
hours of practical exercises. In addition to the classroom instruction, five
Kaggle challenges were assigned to the students, who were expected to work on
them independently and outside of class time. The final component of the
course consisted of a group project, which required all 28 students to form 10
groups of 2-4 individuals. The project served as a comprehensive assessment of
the students’ proficiency in deep learning and required the application of the
concepts and techniques covered in the course. Students were expected to
present their project findings orally and respond to questions from the
instructors.

###  2.2 Evaluation Framework

The evaluation of students’ performance in the course was based on six
components, which accounted for the final grade.

The first five components were Kaggle challenges, each accounting for 6% of
the total mark, adding up to 30% of the final mark. For each challenge, a
marking scheme was established, which was publicly shared with the students:

  * 0/6

no attempt to the challenge or score below a basic attempt of the challenge

  * 3/6

achieved a better score than one typically obtained after a basic attempt of
the challenge

  * 4/6

achieved a better score than one typically obtained after a fair attempt of
the challenge

  * 5/6

achieved a better score than one typically obtained after an advanced attempt
of the challenge

  * 6/6

given to the top 10 students of the class; to obtain 6/6, students also needed
to meet requirement for 5/6 (the class is 28 students, so this score is
reachable for a third of the class)

Details on the exact scores level for each Kaggle challenges are explained on
the official website of the kaggles:

  * Kaggle 1

https://www.kaggle.com/competitions/fitting-a-1d-1d-function-with-deep-
learning

  * Kaggle 2

https://www.kaggle.com/competitions/fitting-a-5d-5d-function-with-deep-
learning

  * Kaggle 3

https://www.kaggle.com/competitions/binary-classification-glasses-no-glasses

  * Kaggle 4

https://www.kaggle.com/competitions/decimal-classification-mics-mnist

  * Kaggle 5

https://www.kaggle.com/competitions/sword-video-classification

The final component was a project, which accounted for the remaining 70% of
the final mark. Students completed the project in groups of 2-4, and the
project was assessed based on an oral presentation of approximately 10-15
minutes, followed by approximately 10 minutes of questions. The final grade
for the project was determined by averaging the independent scores provided by
the two instructors, who evaluated the projects separately and without
knowledge of each other’s assigned scores.

###  2.3 Unbiased versus Biased Evaluation

The Kaggle challenges were employed as an impartial means of evaluating the
students. The five Kaggle challenges were deemed to constitute a
representative measure of the students’ abilities. Teachers may harbor
preferences for certain projects, be subject to halo effect or other types of
biases[4][6]. As a result, the average score attained by the students in the
Kaggle competitions was regarded as the ”ground truth” for grading. It should
be noted that certain students may have invested more time and effort in the
project than in the Kaggle challenges, therefore making the Kaggle metric
imperfect. Such factors, however, lie beyond the scope of this article.

###  2.4 Ethical Considerations

This study was conducted in accordance with the ethical guidelines of
CentraleSupélec, and all participants were were anonymized for confidentiality
purposes.

###  2.5 Normalizing

The Kaggles were originally graded out of 6, the project out of 20. We
normalized all grades to be out of 100 (%), to ba able to compare mean and
standard deviation.

##  3 Results

We rounded all numerical values to $10^{-2}$ for readability purposes.

###  3.1 General results

We obtained the following results ($mean\pm standard\\_deviation$):

  * •

Kaggle 1: $88.89\pm 9.25$

  * •

Kaggle 2: $80.25\pm 26.16$

  * •

Kaggle 3: $89.51\pm 8.20$

  * •

Kaggle 4: $75.31\pm 24.62$

  * •

Kaggle 5: $70.06\pm 31.37$

Kaggle Average: $80.80\pm 13.22$

  * •

Project-Teacher 1: $82.13\pm 8.20$

  * •

Project-Teacher 2: $80.83\pm 11.22$

Project: $81.48\pm 9.13$

The Kaggle and project have similar average. The standard deviation is
slightly lower on the project, but both are still comparable.

###  3.2 Dependence of assessments

Suppose that the score of students on each kaggle is independent. Then
probability theory tells us that the standard deviation of the average kaggle
score should be $9.85$ (using
$\text{Var}\left({\frac{\sum_{i=1}^{5}K_{i}}{5}}\right)=\frac{\sum_{i=1}^{5}\text{Var}\left({K_{i}}\right)}{5^{2}}$).
However, the variance found is $13.22$. This means that the score on kaggle
challenges is not independent of others. It makes sense, since we expect good
students to perform well in all assessments.

Similarly, the variance of the project mark (average of the mark given by
teacher 1 and 2) should be $6.95$ (using
$\text{Var}\left({\frac{P_{1}+P_{2}}{2}}\right)=\frac{\text{Var}\left({P_{1}}\right)+\text{Var}\left({P_{2}}\right)}{2^{2}}$)
if the two teachers’ grades could be treated as independent random variables.
However, we observe a variance of $9.13$, leading to the conclusion that the
two teachers marks cannot be treated as random independent variables. Again,
this is expected, since teachers do not mark randomly.

###  3.3 Distribution of the marks

We can plot the Kernel Density Estimate (KDE) for the distribution of the
marks, both for Kaggle challenges and for the project:

Figure 1: KDE

We observe that the class splits into two groups: one achieving nearly perfect
score, and one getting about $75\%$. This trends can be observed in the
project scores given by teachers 1 and 2, but also on some of the kaggle
challenges: (kaggles 1 and 3 where it’s obvious to observe, and kaggle 4 and 5
where it’s less sharp).

###  3.4 Quantification of the inter-assessments correlation

You may find a complete table of correlation between all scores in the
supplementary material. Here are the most interesting/interpretable ones:

  * •

Kaggles-Project correlation: $0.64$; maximum absolute difference: $24.16\%$

  * •

Inter-examiner correlation: $0.76$; maximum absolute difference: $12.50\%$

  * •

Examiner-Kaggles correlation:
Examiner 1: $0.49$
Examiner 2: $0.69$

  * •

Kaggle-Kaggle average correlation:
Kaggle 1 - Kaggle Avg: $0.64$
Kaggle 2 - Kaggle Avg: $0.53$
Kaggle 3 - Kaggle Avg: $0.33$
Kaggle 4 - Kaggle Avg: $0.71$
Kaggle 5 - Kaggle Avg: $0.83$

##  4 Discussion

The observed higher inter-examiner correlation as compared to the examiner-
Kaggles correlation could potentially suggest that there are inherent
differences in the oral presentation skills of the students being evaluated.
While the examiners demonstrate a higher degree of agreement in their
evaluations of the oral presentations, their scores do not align as closely
with those assigned by the impartial non-oral assessment provided by the
Kaggles. These findings highlight the potential limitations of using oral
assessments for evaluating students. This suggests that there may be
individual differences in the oral presentation proficiency of the students
under evaluation.

The observed low correlation between the Kaggles and their respective average
scores suggests that students may have directed their attention towards
particular Kaggles. This can be explained by two facts: First, students might
have found some Kaggles more interesting than others, ans therefore performed
better on those. Second, since there was an extra point for being in the top
10, students may have focused their attention on one or two Kaggle to get the
extra point. This may have resulted in an uneven distribution of effort across
the different Kaggles. The average of the Kaggle, however, is still a good
unbiased metric.

While the study provides valuable insights about how oral presentations may be
appreciated differently by different examiners, it also has several
limitations.

Firstly, the study only included one class with 28 students, which means that
the findings may not be generalizable to other settings or larger sample
sizes. Secondly, the study only used one project assessment to evaluate
student learning outcomes, which may not provide a comprehensive picture of
student performance or achievement. Thirdly, the study only had two examiners,
which may limit the reliability and validity of the assessment process.

Additionally, it is possible that students may have paid more attention to one
aspect of the course (either the Kaggle competition or the class project) than
the other. This could create of decorrelation between the two marks that is
not due to examiner’s subjective appreciation, but to an objectively
better/worst performance than in the Kaggle competitions.

Further research with larger sample sizes and more diverse student populations
is needed to confirm and extend these findings.

## References

  * [1] Global deep learning market report 2023.
  * [2] Review of the applications of deep learning in bioinformatics.  Current Bioinformatics, 15(8), 2020.
  * [3] Junyi Chai, Hao Zeng, Anming Li, and Eric W.T. Ngai.  Deep learning in computer vision: A critical review of emerging techniques and application scenarios.  Machine Learning with Applications, 6:100134, 2021.
  * [4] Jenny A. Darby.  Are course evaluations subject to a halo effect?  Research in Education, 77(1):46–55, 2007.
  * [5] Sandra Ifeatu Efu.  Exams as learning tools: A comparison of traditional and collaborative assessment in higher education.  College Teaching, 67(1):73–83, 2019.
  * [6] Thomas Hugh Feeley.  Evidence of halo effects in student evaluations of communication instruction.  Communication Education, 51(3):225–236, 2002.
  * [7] Selçuk Karaman.  Examining the effects of flexible online exams on students’ engagement in e-learning.  Educational Research and Reviews, 6:259–264, 04 2011.
  * [8] Seonwoo Min, Byunghan Lee, and Sungroh Yoon.  Deep learning in bioinformatics.  Briefings in Bioinformatics, 18(5):851–869, 07 2016.
  * [9] Xiaofei Sun, Diyi Yang, Xiaoya Li, Tianwei Zhang, Yuxian Meng, Han Qiu, Guoyin Wang, Eduard Hovy, and Jiwei Li.  Interpreting deep learning models in natural language processing: A review, 2021.
  * [10] Athanasios Voulodimos, Nikolaos Doulamis, Anastasios Doulamis, and Eftychios Protopapadakis.  Deep learning for computer vision: A brief review.  Computational Intelligence and Neuroscience, 2018:7068349, Feb 2018\.
  * [11] Stephen Wu, Kirk Roberts, Surabhi Datta, Jingcheng Du, Zongcheng Ji, Yuqi Si, Sarvesh Soni, Qiong Wang, Qiang Wei, Yang Xiang, Bo Zhao, and Hua Xu.  Deep learning in clinical natural language processing: a methodical review.  Journal of the American Medical Informatics Association, 27(3):457–470, 12 2019.

Supplementary Materials

## Complete table of correlation between all scores

| Kaggle 1 | Kaggle 2 | Kaggle 3 | Kaggle 4 | Kaggle 5 | PT 1 | PT 2 | K Avg | PFM
---|---|---|---|---|---|---|---|---|---
Kaggle 1 | 1.00 | 0.56 | 0.23 | 0.30 | 0.30 | 0.09 | 0.36 | 0.64 | 0.26
Kaggle 2 | 0.56 | 1.00 | -0.01 | -0.01 | 0.12 | 0.07 | 0.36 | 0.53 | 0.25
Kaggle 3 | 0.23 | -0.01 | 1.00 | 0.10 | 0.29 | 0.18 | 0.29 | 0.33 | 0.26
Kaggle 4 | 0.30 | -0.01 | 0.10 | 1.00 | 0.61 | 0.56 | 0.47 | 0.71 | 0.54
Kaggle 5 | 0.30 | 0.12 | 0.29 | 0.61 | 1.00 | 0.46 | 0.59 | 0.83 | 0.57
PT1 | 0.09 | 0.07 | 0.18 | 0.56 | 0.46 | 1.00 | 0.76 | 0.49 | 0.92
PT2 | 0.36 | 0.36 | 0.29 | 0.47 | 0.59 | 0.76 | 1.00 | 0.69 | 0.96
K Avg | 0.64 | 0.53 | 0.33 | 0.71 | 0.83 | 0.49 | 0.69 | 1.00 | 0.64
PFM | 0.26 | 0.25 | 0.26 | 0.54 | 0.57 | 0.92 | 0.96 | 0.64 | 1.00
Table 1: Correlation between scores obtained by students.

(All values have been rounded to $10^{-2}$.)

Abbreviation :

  * •

PT 1: Project Teacher 1’s mark

  * •

PT 2: Project Teacher 2’s mark

  * •

K Avg: Kaggle Average

  * •

PFM: Project Final Mark


# Searching for asymmetric and heavily precessing Binary Black Holes in the
gravitational wave data from the LIGO and Virgo third Observing Run

Stefano Schmidt  0000-0002-8206-8089<EMAIL_ADDRESS>Nikhef, Science Park 105,
1098 XG, Amsterdam, The Netherlands  Institute for Gravitational and Subatomic
Physics (GRASP), Utrecht University, Princetonplein 1, 3584 CC Utrecht, The
Netherlands  Sarah Caudill  0000-0002-8927-6673 Department of Physics,
University of Massachusetts, Dartmouth, MA 02747, USA  Center for Scientific
Computing and Data Science Research, University of Massachusetts, Dartmouth,
MA 02747, USA  Jolien D. E. Creighton  0000-0003-3600-2406 Leonard E. Parker
Center for Gravitation, Cosmology, and Astrophysics, University of Wisconsin-
Milwaukee, Milwaukee, WI 53201, USA  Leo Tsukada  0000-0003-0596-5648
Department of Physics, The Pennsylvania State University, University Park, PA
16802, USA  Institute for Gravitation and the Cosmos, The Pennsylvania State
University, University Park, PA 16802, USA  Anarya Ray  0000-0002-7322-4748
Leonard E. Parker Center for Gravitation, Cosmology, and Astrophysics,
University of Wisconsin-Milwaukee, Milwaukee, WI 53201, USA  Shomik Adhicary
Department of Physics, The Pennsylvania State University, University Park, PA
16802, USA  Institute for Gravitation and the Cosmos, The Pennsylvania State
University, University Park, PA 16802, USA  Pratyusava Baral
0000-0001-6308-211X Leonard E. Parker Center for Gravitation, Cosmology, and
Astrophysics, University of Wisconsin-Milwaukee, Milwaukee, WI 53201, USA
Amanda Baylor  0000-0003-0918-0864 Leonard E. Parker Center for Gravitation,
Cosmology, and Astrophysics, University of Wisconsin-Milwaukee, Milwaukee, WI
53201, USA  Kipp Cannon  0000-0003-4068-6572 RESCEU, The University of Tokyo,
Tokyo, 113-0033, Japan  Bryce Cousins  0000-0002-7026-1340 Department of
Physics, University of Illinois, Urbana, IL 61801 USA  Department of Physics,
The Pennsylvania State University, University Park, PA 16802, USA  Institute
for Gravitation and the Cosmos, The Pennsylvania State University, University
Park, PA 16802, USA  Becca Ewing  Department of Physics, The Pennsylvania
State University, University Park, PA 16802, USA  Institute for Gravitation
and the Cosmos, The Pennsylvania State University, University Park, PA 16802,
USA  Heather Fong  Department of Physics and Astronomy, University of British
Columbia, Vancouver, BC, V6T 1Z4, Canada  RESCEU, The University of Tokyo,
Tokyo, 113-0033, Japan  Graduate School of Science, The University of Tokyo,
Tokyo 113-0033, Japan  Richard N. George  0000-0002-7797-7683 Center for
Gravitational Physics, University of Texas at Austin, Austin, TX 78712, USA
Patrick Godwin  LIGO Laboratory, California Institute of Technology, MS
100-36, Pasadena, California 91125, USA  Department of Physics, The
Pennsylvania State University, University Park, PA 16802, USA  Institute for
Gravitation and the Cosmos, The Pennsylvania State University, University
Park, PA 16802, USA  Chad Hanna  Department of Physics, The Pennsylvania State
University, University Park, PA 16802, USA  Institute for Gravitation and the
Cosmos, The Pennsylvania State University, University Park, PA 16802, USA
Department of Astronomy and Astrophysics, The Pennsylvania State University,
University Park, PA 16802, USA  Institute for Computational and Data Sciences,
The Pennsylvania State University, University Park, PA 16802, USA  Reiko
Harada  RESCEU, The University of Tokyo, Tokyo, 113-0033, Japan  Graduate
School of Science, The University of Tokyo, Tokyo 113-0033, Japan  Yun-Jing
Huang  0000-0002-2952-8429 Department of Physics, The Pennsylvania State
University, University Park, PA 16802, USA  Institute for Gravitation and the
Cosmos, The Pennsylvania State University, University Park, PA 16802, USA
Rachael Huxford  Department of Physics, The Pennsylvania State University,
University Park, PA 16802, USA  Institute for Gravitation and the Cosmos, The
Pennsylvania State University, University Park, PA 16802, USA  Prathamesh
Joshi  0000-0002-4148-4932 Department of Physics, The Pennsylvania State
University, University Park, PA 16802, USA  Institute for Gravitation and the
Cosmos, The Pennsylvania State University, University Park, PA 16802, USA
James Kennington  0000-0002-6899-3833 Department of Physics, The Pennsylvania
State University, University Park, PA 16802, USA  Institute for Gravitation
and the Cosmos, The Pennsylvania State University, University Park, PA 16802,
USA  Soichiro Kuwahara  RESCEU, The University of Tokyo, Tokyo, 113-0033,
Japan  Graduate School of Science, The University of Tokyo, Tokyo 113-0033,
Japan  Alvin K. Y. Li  0000-0001-6728-6523 LIGO Laboratory, California
Institute of Technology, Pasadena, CA 91125, USA  Ryan Magee
0000-0001-9769-531X LIGO Laboratory, California Institute of Technology,
Pasadena, CA 91125, USA  Duncan Meacher  0000-0001-5882-0368 Leonard E. Parker
Center for Gravitation, Cosmology, and Astrophysics, University of Wisconsin-
Milwaukee, Milwaukee, WI 53201, USA  Cody Messick  MIT Kavli Institute for
Astrophysics and Space Research, Massachusetts Institute of Technology,
Cambridge, MA 02139, USA  Soichiro Morisaki  0000-0002-8445-6747 Institute for
Cosmic Ray Research, The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa,
Chiba 277-8582, Japan  Leonard E. Parker Center for Gravitation, Cosmology,
and Astrophysics, University of Wisconsin-Milwaukee, Milwaukee, WI 53201, USA
Debnandini Mukherjee  0000-0001-7335-9418 NASA Marshall Space Flight Center,
Huntsville, AL 35811, USA  Center for Space Plasma and Aeronomic Research,
University of Alabama in Huntsville, Huntsville, AL 35899, USA  Wanting Niu
0000-0003-1470-532X Department of Physics, The Pennsylvania State University,
University Park, PA 16802, USA  Institute for Gravitation and the Cosmos, The
Pennsylvania State University, University Park, PA 16802, USA  Alex Pace
Department of Physics, The Pennsylvania State University, University Park, PA
16802, USA  Institute for Gravitation and the Cosmos, The Pennsylvania State
University, University Park, PA 16802, USA  Cort Posnansky  Department of
Physics, The Pennsylvania State University, University Park, PA 16802, USA
Institute for Gravitation and the Cosmos, The Pennsylvania State University,
University Park, PA 16802, USA  Surabhi Sachdev  0000-0002-0525-2317 School of
Physics, Georgia Institute of Technology, Atlanta, GW 30332, USA  Leonard E.
Parker Center for Gravitation, Cosmology, and Astrophysics, University of
Wisconsin-Milwaukee, Milwaukee, WI 53201, USA  Shio Sakon  0000-0002-5861-3024
Department of Physics, The Pennsylvania State University, University Park, PA
16802, USA  Institute for Gravitation and the Cosmos, The Pennsylvania State
University, University Park, PA 16802, USA  Divya Singh  0000-0001-9675-4584
Department of Physics, The Pennsylvania State University, University Park, PA
16802, USA  Institute for Gravitation and the Cosmos, The Pennsylvania State
University, University Park, PA 16802, USA  Urja Shah  0000-0001-8249-7425
School of Physics, Georgia Institute of Technology, Atlanta, GW 30332, USA
Ron Tapia  Department of Physics, The Pennsylvania State University,
University Park, PA 16802, USA  Institute for Computational and Data Sciences,
The Pennsylvania State University, University Park, PA 16802, USA  Takuya
Tsutsui  0000-0002-2909-0471 RESCEU, The University of Tokyo, Tokyo, 113-0033,
Japan  Koh Ueno  0000-0003-3227-6055 RESCEU, The University of Tokyo, Tokyo,
113-0033, Japan  Aaron Viets  0000-0002-4241-1428 Concordia University
Wisconsin, Mequon, WI 53097, USA  Leslie Wade  Department of Physics, Hayes
Hall, Kenyon College, Gambier, Ohio 43022, USA  Madeline Wade
0000-0002-5703-4469 Department of Physics, Hayes Hall, Kenyon College,
Gambier, Ohio 43022, USA

###### Abstract

Leveraging the features of the GstLAL pipeline, we present the results of a
matched filtering search for asymmetric binary black hole systems with heavily
mis-aligned spins in LIGO and Virgo data taken during the third observing run.
Our target systems show strong imprints of precession and current searches
have non-optimal sensitivity in detecting them. After measuring the
sensitivity improvement brought by our search over standard spin-aligned
searches, we report the detection of 30 gravitational wave events already
discovered in the latest Gravitational Wave Transient Catalogues. However, we
do not find any additional significant gravitational wave candidates. Our
results allow us to place an upper limit of
$R_{90\%}=0.28^{+0.33}_{-0.04}\;\;\mathrm{Gpc^{-3}yr^{-1}}$ on the merger rate
of a hypothetical subpopulation of asymmetric, heavily precessing signals, not
identified by other searches. Since our upper limit is consistent with the
latest rate estimates from the LIGO-Virgo-KAGRA collaboration, our findings
rule out the existence of a yet-to-be-discovered population of precessing
binaries.

##  I Introduction

The detection of more than 90 gravitational waves (GW) events [1, 2, 3, 4] by
the LIGO [5], Virgo [6] and KAGRA [7] (LVK) collaboration was made possible
thanks to sophisticated matched-filtering pipelines [8, 9, 10, 11, 12, 13, 14,
15, 16, 17, 18, 19, 20], which correlate a large number of templates with the
interferometric data, looking for a potential match. Besides them, a few
unmodelled searches have been developed to find coherent excess of power
between the output of two detectors [21, 22, 23, 24, 25], with minimal
assumptions about the signal. The sensitivity of matched filtering searches is
close to optimality, however their performance degrades significantly for
signals outside the target region. Conversely, unmodeled searches retain some
sensitivity to a broader variety of signals, though in most cases they are far
from optimal. Moreover, their sensitivity is known to decrease as the duration
of the target signals increases.

In particular, as shown in [26, 27], most of the current searches have a poor
sensitivity for very asymmetric binary black hole (BBH) systems with mis-
aligned spins. Such systems show strong imprints of precession [28, 29, 30,
31], i.e. a time dependent variation of the orientation of the orbital plane,
which manifests itself through the modulation of the signal amplitude and
phase and becomes more detectable as more phase cycles are observed [32].
Targeting these systems with matched filtering is complicated by their great
variability, which makes it challenging to cover the parameter space with
templates [33]. Moreover, also unmodelled searches are suboptimal for
precessing searches, since they tend to have a limited sensitivity for the
signals of length $O($10\text{\,}\mathrm{s}$)$ where the effect of precession
is more pronounced.

Despite such difficulties, precession in BBH is of primary scientific
interests with a profoud impact on GW astronomy, ranging from BBH population
studies [34, 35, 36, 37] to cosmology [38, 39]. While precession has been
observed as a statistical property of the BBH population [40, 41], very few
signals show conclusive evidence for mis-aligned spins [42], as measured by
$\chi_{P}$ [43]. Interesting exceptions include GW190521 [44], GW191109 [45]
and GW200129 [46]. While the lack of strongly precessing signals might be due
to their rarity, it might also be caused by the limited sensitivity of current
searches to such extreme signals. If this is the case, it is important to
explore thoroughly the BBH parameter space, using the most sensitive possible
matched filtering search pipeline. In this spirit, in [47] we described the
updates to the GstLAL search pipeline [48, 49, 50, 51, 52, 53] required to
search for precessing signals and we identified a region of the BBH parameter
space where a sensitivity improvent of up to $120\%$ is within reach. See also
[54] for a promising alternative.

In this Letter, we leverage the results of [47] to search the LIGO-Virgo
pubicly available data from the third observing run O3 [55] for precessing BBH
signals with component masses ${m_{1}\in[15,70]$\text{\,}\mathrm{M}_{\odot}$}$
and ${m_{2}\in[3,10]$\text{\,}\mathrm{M}_{\odot}$}$, with a mass ratio
restricted to ${q=m_{1}/m_{2}\in[5,12]}$. After briefly reviewing our search
method in Sec. II, we report the results of our search in Sec. III. While we
do not detect any novel signal, our results [56] allow us to place an upper
limit on the astrophysical merger rate of a hypothetical subpopulation of
asymmetric, heavily precessing signals, not detected by past searches.

| SNR | $\text{SNR}_{\text{GWTC}}$ | $\textrm{FAR}(1/\textrm{yr})$
---|---|---|---
GW190408_181802 | 14.29 | 14.6 | $<10^{-5}$
GW190412_053044 | 17.82 | 19.8 | $<10^{-5}$
GW190512_180714 | 11.88 | 12.7 | $<10^{-5}$
GW190513_205428 | 11.8 | 12.5 | 0.0008
GW190519_153544 | 12.69 | 15.9 | 0.0001
GW190521_074359 | 22.74 | 25.9 | $<10^{-5}$
GW190527_092055 | 8.79 | 8.0 | 0.1547
GW190706_222641 | 12.54 | 13.4 | 0.0221
GW190707_093326 | 12.68 | 13.1 | $<10^{-5}$
GW190708_232457 | 12.13 | 13.4 | $<10^{-5}$
GW190720_000836 | 10.55 | 10.9 | 0.0012
GW190727_060333 | 11.3 | 11.7 | $<10^{-5}$
GW190728_064510 | 11.92 | 13.1 | $<10^{-5}$
GW190814_211039 | 20.52 | 25.3 | $<10^{-5}$
GW190828_063405 | 15.93 | 16.5 | $<10^{-5}$
GW190828_065509 | 10.56 | 10.2 | 0.001
GW190915_235702 | 12.09 | 13.1 | 0.0015
GW190924_021846 | 12.87 | 12.0 | $<10^{-5}$
GW191109_010717 | 14.19 | 17.3 | $<10^{-5}$
GW191129_134029 | 13.14 | 13.1 | $<10^{-5}$
GW191204_171526 | 16.78 | 17.4 | $<10^{-5}$
GW191215_223052 | 10.36 | 11.2 | 0.0008
GW191216_213338 | 17.52 | 18.6 | $<10^{-5}$
GW191222_033537 | 10.91 | 12.5 | 0.0009
GW200128_022011 | 9.0 | 10.6 | 0.0059
GW200129_065458 | 25.8 | 26.8 | $<10^{-5}$
GW200224_222234 | 17.85 | 20.0 | $<10^{-5}$
GW200225_060421 | 12.1 | 12.5 | 0.0664
GW200311_115853 | 15.69 | 17.8 | $<10^{-5}$
GW200316_215756 | 9.96 | 10.3 | 0.0003
Table 1: Confidents events detected by our search with ${p_{\textrm{astro}}>0.5}$. For each event, we report the event name, together with the SNR and FAR. For reference, we also report the $\text{SNR}_{\text{GWTC}}$ reported in the transient catalogue. GPS time | SNR | $\textrm{FAR}(1/\textrm{yr})$ | $p_{\textrm{astro}}$
---|---|---|---
$1252415231\;\;$ | 8.84 | 2.367 | 0.06
$1252465013\;\;$ | 8.35 | 3.2005 | 0.05
$1253504581\;\;$ | 9.02 | 4.5486 | 0.03
$1267433277\;\;$ | 7.75 | 5.4287 | 0.04
Table 2: Candidates not reported in GWTC, detected by our search with
${\textrm{FAR}<6/\textrm{yr}=1/2\,\textrm{months}}$. For each candidate, we
report the SNR, FAR and $p_{\textrm{astro}}$.  Figure 1: Median masses of the
GWTC candidates as computed with parameter estimation. The GWTC candidates
detected by our search are denoted by large circles, colored according to
their FAR. A black cross marks the GWTC events that our search missed. The
blue dots represent the templates used in our search bank. Figure 2:
Efficiency of the search as a function of the decisive SNR, i.e. the maximum
injected SNR between the two detectors. The efficiency is measured on four
different chunks with GPS start time and length reported on the caption
separated by a dash.

##  II Methods

At its core, the GstLAL matched filtering pipeline correlates the data with
many template signals (stored in a template bank). For a potential candidate,
the GstLAL pipeline records a trigger, notably consisting of Signal-to-noise
ratio (SNR) and signal-consistency test $\xi^{2}$ [48]. The triggers are
ranked according to the likelihood ratio $\mathcal{L}$ [57, 50, 53] (also
called ranking statistics) of their probability of being from astrophysical
origin to their probability of being caused by detector noise. As a final
step, each trigger is assigned a False Alarm Rate (FAR), which amounts to the
rate of occurence of a trigger with a given $\mathcal{L}$ in a search where no
astrophysical signals are present. Finally, we assign to each candidate the
probability $p_{\text{astro}}$ [58, 59] that a signal originates from an
astrophysical source, based on the the Poisson mixture model formalism.

To search for precessing signals, we employ the search technique introduced
and implemented in [47], offering up to 100% sensitive volume increase for
very asymmetric highly precessing BBH sytems, as compared to its aligned-spin
counterpart. The sensitivity improvement is made possible thanks to three main
features:

  * •

New metric template placement algorithm [33]: by reducing the bank generation
time by orders of magnitude, it allows for a fast identification of a region
of the parameter space where a sensitivity improvement can be achieved using
feasible number of templates.

  * •

Updated signal consistency test [60]: it improves the robustness of the signal
identification stage.

  * •

Implementation of a suitable search statistics [61]: it allows for optimal SNR
recovery of the candidates.

Our search employs the “High $q$” $0.9$ match111The choice of using a $0.9$
minimal match, instead of the standard $0.97$, allows us to obtain a bank with
a manageable size. As discussed in [47], it does not seem to have a large
negative impact on the search. precessing template bank introduced in [47]
with 2.3 millions of templates, targeting systems with component masses
${m_{1}\in[15,70]$\text{\,}\mathrm{M}_{\odot}$}$ and
${m_{2}\in[3,10]$\text{\,}\mathrm{M}_{\odot}$}$ and restricted to a mass ratio
${q=m_{1}/m_{2}\in[5,12]}$. Moreover, we only consider templates with
$s_{\text{1y}}=s_{\text{2x}}=s_{\text{2y}}=0$ and zero reference phase $\phi$.
The other spin components are set such that $s_{1}\in[0.5,0.9]$,
$s_{\text{2z}}\in[-0.99,0.99]$ and $\theta_{1}\in[-\pi,\pi]$, with
$\theta_{1}=\arctan\frac{s_{\text{1x}}}{s_{\text{1z}}}$. Such limitation of
the spin degrees of freedom leverages the degeneracies in the spin parameter
space to provide good injection recovery, while limiting the number of
templates required to cover the space [33, 47]. We filter the data in a
frequency range $f\in[15,1024]$\text{\,}\mathrm{H}\mathrm{z}$$ with the
waveform approximant IMRPhenomXP [62]. Note that for our analysis we only
consider simple precession, thus neglecting the effect of transitional
precession [28, 29]. To compare the performance of our precessing search we
use an aligned-spin template bank, also introduced in [47], covering the same
mass range and with $s_{\text{1z}},s_{\text{2z}}\in[-0.99,0.99]$, employing
$2.7\times 10^{4}$ templates.

To estimate the search sensitivity and to place a limit on the merger rate of
precessing signals (see Sec. III.1), we sample a number of injections using
the BBH population results inferred using GWTC-3 [41]. For this purpose, we
use the best fit of the Binned Gaussian Process (BGP) population model [63],
which allows us to easily sample BBH signals in the mass range of
consideration. We sample the two spins isotropically, where the magnitude of
each spin is constrained between $0.5$ and $0.9$. The sources are uniformly
distributed in the volume between
$[30,300]$\text{\,}\mathrm{M}\mathrm{p}\mathrm{c}$$ with uniformly sampled
orientation, and generated with the IMRPhenomXP approximant.

For our search we use publicly available GW data [55] collected by the two
LIGO observatories [5, 64, 65, 66] during the third observing run (O3). Data
collection periods were divided in two halfs: the first half of O3 (O3a)
spanned between 1 April 2019, 1500 UTC and 1 October 2019, 1500 UTC, while the
second half (O3b) happened between 1 November 2019, 15:00 UTC and 27 March
2020, 17:00 UTC. Forced by limited computational resources, we did not use
Virgo data taken in the same period, due to the smaller contribution to the
sensitive volume given by Virgo.

##  III Search results

The search detected 30 events with ${p_{\text{astro}}>0.5}$: they were all
previously reported in the GW transient catalogues GWTC-2 [2, 3] and GWTC-3
[4] released by the LIGO-Virgo-Kagra collaboration. The confident detections
are reported in Tab. 1. In Fig. 1, we summarize the information about our
detected GWTC signals, where we report all the detected events in GWTC-2 and
GWTC-3. Each event not discovered by our search is represented by a cross,
while a large dot, coloured by the detection FAR, represents a detection. Note
that we report events detected by our search up to a
$\text{FAR}=1/\text{day}$, hence some of them do not satisfy our detection
threshold of ${p_{\text{astro}}>0.5}$.

As shown in Fig. 1 and as further confirmed by parameter estimation studies
[2, 4], the GWTC events detected by our search lie mostly outside the region
covered by our template bank. Consequently, we conclude that the sensitivity
of our search extends beyond our intended target, potentially owing to the
large number of templates employed, which enables the capture of events
outside the bank, albeit with suboptimal SNR and $\xi^{2}$ values. The fact
that the detected events are outside the target region also explains the
observation in Tab. 1 that our search measures a systematically lower SNR than
the GWTC searches.

In Fig. 1 we note that two events, GW191113_071753 and GW200210_092254, were
missed by our search, even if they lie close to our target region. The first
one GW191113_071753 was not recorded by the GstLAL pipeline even in the
offline broadband search and for this reason it is not surprising that also
our search failed to detect it. On the other hand, the issue of why
GW200210_092254 was missed requires more investigation. For the moment we can
postulate that this is due to the low mass of the lighter object, which makes
the event at the very edge of our template bank.

Besides the known GW events, our search did not detect any novel GW candidate.
In Tab. 2, we report a list of the candidates observed with
${\text{FAR}<2/\text{year}}$, not included in a previous transient catalogue.

In Fig. 2, we report the efficiency $\epsilon(\text{SNR},\text{FAR})$ of our
search for different bins of decisive SNR, defined as the maximum injected SNR
between the two LIGO detectors. The efficiency amounts to the fraction of
signals detected by the search and it is computed using the BGP injection set
defined above. We perform injections only for two chunks of data in O3a and
O3b rispectively, and we report the results separately for each chunk. The
efficiencies obtained are similar to those reported in [47], however for the
chunk starting at GPS time $1262192988\,$\text{\,}\mathrm{s}$$ we observe a
drop in efficiency of unknown origin.

The efficiency is used to compute the averaged sensitive space-time volume
$\langle VT\rangle$ of the search, defined as:

$\langle
VT\rangle(\text{FAR})=T\int_{0}^{\infty}\text{d}z\,\epsilon(\text{SNR}(z),\text{FAR})\frac{\text{d}V}{\text{d}z}\frac{1}{1+z}$
(1)

where $\frac{\text{d}V}{\text{d}z}\frac{1}{1+z}$ is the comoving volume
element at a given redshit $z$. In Fig. 3, we report the VT as a function of
the parameter space of our search. The efficiency is averaged over the four
chunks for which it was computed and it is assumed constant for all the
observation time. We also report the $\langle VT\rangle$ improvement brought
by our search as compared to an aligned-spin search. For each centroid, we
employ $\sim 3\times 10^{4}$ injections, log-normally distributed around the
center with a variance of $0.1$. All the sensitive volumes are estimated at a
fiducial $\text{FAR}=1/\text{yr}$.

Figure 3: Search sensitivity computed over different regions of the parameter space. Each dot marks the center of a set of log-normally distributed injections and it is colored according to the sensitive space-time volume measured using such injection set. The number overimpressed denotes the ratio between the sensitive space-time volume of our search and the one of an aligned spin search targerting the same space. $\text{FAR}=1/\textrm{yr}$ |
---|---
| $\langle
VT\rangle_{\textrm{O3a}}=4.10^{+0.25}_{-0.26}\;\;\mathrm{Gpc^{3}yr}$
| $\langle VT\rangle_{\textrm{O3b}}=3.2^{+1.5}_{-1.5}\;\;\mathrm{Gpc^{3}yr}$
$\text{FAR}=2.36/\textrm{yr}$ |
| $\langle VT\rangle=8.4^{+1.5}_{-4.6}\;\;\mathrm{Gpc^{3}yr}$
Table 3: Sensitive space-time volume estimated for the BGP population at a
nominal FAR of $1/\text{year}$ and the FAR corresponding to the loudest event
not reported in the GWTC. In the first case, we report the $\langle VT\rangle$
separately for O3a and O3b.

###  III.1 Rate upper limits

The lack of newly detected signals allows to place an upper limit [56, 58, 67,
68] on the presence of a subpopulation of asymmetric, heavily precessing
signals, not identified by other searches. In our analysis, we assume that
such population is well described by the BGP injection set and, unlike
standard works in BBH population analysis [40, 41], we do not attempt to
constrain the distribution of BBH masses or spins.

After removing from consideration all the triggers associated to any GWTC
event, the loudest trigger has a $\overline{\text{FAR}}=2.367/\textrm{yr}$. We
can then use our knowledge of the sensitive space-time volume to place an
upper limit $R_{90\%}$ (90% confidence interval) on the merger rate of the
said population of heavily precessing signals. From this, we can compute the
90% confidence limit for the upper limit to the said population of heavily
precessing signals,

$R_{90\%}=\frac{2.3}{\langle VT\rangle}$ (2)

where the sensitive space-time volume is estimate using the FAR of the loudest
event. Using the VT values for the BGP population reported in Tab. 3, we
obtain an upper limit of

$R_{90\%}=0.28^{+0.33}_{-0.04}\;\;\mathrm{Gpc^{-3}yr^{-1}}$

The rate upper limit we obtain is consistent with the LVK rate estimates for
our sub-population [41]: considering that our target population is only a
small fraction of the total BBH population considered in the analysis, the LVK
results imply a BBH merger rate of
${R_{\text{LVK}}=0.117-0.298\text{Gpc}^{-3}\text{yr}^{-1}}$ for our target
region. Thus, our results do not provide any evidence to support the
hypothesis of a population of asymmetric, heavily precessing signals that have
not been detected by other searches. Searching the data taken during the first
and second observing runs [69] will increase the surveyed sensitive space-time
volume, however it is unlikely that such increase will change the picture
outlined above.

##  IV Conclusion

In this work, we presented the results of a matched-filtering search on the
interferometric data from the LIGO-Virgo-KAGRA third observing run. Our search
targets asymmetric precessing BBHs, with a doubled sensitivity over such
extreme systems: the search method is thoroughly discussed in [47]. We report
the detection of 30 candidates, already announced in the LIGO-Virgo-KAGRA
transient catalogues, but we do not find any new candidate. Our results allow
us to place an upper limit on the merger rate of the binaries targeted by our
search and, as our result is consistent with the rate estimates from the LVK,
we can rule out the existence of a subpopulation of asymmetric precessing BBH
not detected by past searches.

Future work should focus on expanding the parameter space of the search,
possibly targeting Neutron Star- Black Hole systems. This attempt will
struggle with the huge size of the template bank required for this task,
although see [54] which demonstrated that this is possible with only a 3x
increase in template bank size. Moreover, in light of the doubled sensitivity
of our search and of the lack of detections, further work should explore more
deeply the impact of our results on our understanding of the population of
precessing BBH.

###### Acknowledgements.

S.S. is supported by the research program of the Netherlands Organisation for
Scientific Research (NWO). S.C. is supported by the National Science
Foundation under Grant No. PHY-2309332. The authors are grateful for
computational resources provided by the LIGO Laboratory and supported by
National Science Foundation Grants No. PHY-0757058 and No. PHY-0823459. This
material is based upon work supported by NSF’s LIGO Laboratory which is a
major facility fully funded by the National Science Foundation. LIGO was
constructed by the California Institute of Technology and Massachusetts
Institute of Technology with funding from the National Science Foundation
(NSF) and operates under cooperative Agreement No. PHY-1764464. The authors
are grateful for computational resources provided by the Pennsylvania State
University’s Institute for Computational and Data Sciences (ICDS) and the
University of Wisconsin Milwaukee Nemo and support by NSF Grant No.
PHY-2011865, No. NSF OAC-2103662, No. NSF PHY-1626190, No. NSF PHY-1700765,
No. NSF PHY-2207728, No. NSF PHY-2207594 and No. PHY-2309332 as well as by the
research program of the Netherlands Organization for Scientific Research
(NWO). This paper carries LIGO Document No. LIGO-P2400044. This research has
made use of data or software obtained from the Gravitational Wave Open Science
Center (gwosc.org), a service of LIGO Laboratory, the LIGO Scientific
Collaboration, the Virgo Collaboration, and KAGRA. LIGO Laboratory and
Advanced LIGO are funded by the United States National Science Foundation
(NSF) as well as the Science and Technology Facilities Council (STFC) of the
United Kingdom, the Max-Planck-Society (MPS), and the State of
Niedersachsen/Germany for support of the construction of Advanced LIGO and
construction and operation of the GEO600 detector. Additional support for
Advanced LIGO was provided by the Australian Research Council. Virgo is
funded, through the European Gravitational Observatory (EGO), by the French
Centre National de Recherche Scientifique (CNRS), the Italian Istituto
Nazionale di Fisica Nucleare (INFN) and the Dutch Nikhef, with contributions
by institutions from Belgium, Germany, Greece, Hungary, Ireland, Japan,
Monaco, Poland, Portugal, Spain. KAGRA is supported by Ministry of Education,
Culture, Sports, Science and Technology (MEXT), Japan Society for the
Promotion of Science (JSPS) in Japan; National Research Foundation (NRF) and
Ministry of Science and ICT (MSIT) in Korea; Academia Sinica (AS) and National
Science and Technology Council (NSTC) in Taiwan.

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