# Sixteen points in $\mathbb{P}^{4}$ and the inverse Galois problem for del
Pezzo surfaces of degree one

Avinash Kulkarni  Department of Mathematics, Dartmouth College
<EMAIL_ADDRESS>
###### Abstract.

A del Pezzo surface of degree one defined over the rationals has 240
exceptional curves. These curves are permuted by the action of the absolute
Galois group. We show how a solution to the classical inverse Galois problem
for a subgroup of the Weyl group of type $D_{8}$ gives rise to a solution of
the inverse Galois problem for the action of this subgroup on the 240
exceptional curves. A del Pezzo surface of degree one with such a Galois
action contains a Galois invariant sublattice of type $D_{8}$ within its
Picard lattice; this can be characterized in terms of a certain set of sixteen
points in $\mathbb{P}^{4}$.

###### Key words and phrases:

Del Pezzo surface; Inverse Galois Problem

###### 2010 Mathematics Subject Classification:

14J20 (primary), 12F12, 14F22 (secondary)

##  1\. Introduction

If $\widetilde{X}_{2}$ is a del Pezzo surface of degree one, then it is the
double cover of a quadric cone $X_{2}$ branched over a nonsingular genus four
curve $X$ and the vertex of the cone. The quadric cone $X_{2}$ containing $X$
identifies a vanishing even theta characteristic $\theta_{0}$ of $X$. Let
$\kappa$ denote the canonical class of $\widetilde{X}_{2}$. The intersection
pairing on $\widetilde{X}_{2}$ gives $\operatorname{Pic}(\widetilde{X}_{2})$
the structure of a lattice, which admits a decomposition of the form

$\operatorname{Pic}(\widetilde{X}_{2})=\langle\kappa\rangle\oplus\operatorname{Pic}(\widetilde{X}_{2})^{\perp}$

where $\operatorname{Pic}(\widetilde{X}_{2})^{\perp}$ denotes the sublattice
of divisor classes orthogonal to the canonical class under the intersection
pairing. The sublattice $\operatorname{Pic}(\widetilde{X}_{2})^{\perp}$ is a
root lattice of type $E_{8}$, and such a lattice has $135$ sublattices
$\Lambda_{D_{8}}\subset\operatorname{Pic}(\widetilde{X}_{2})^{\perp}$ of type
$D_{8}$. In this article, we consider those del Pezzo surfaces of degree one
whose Picard lattice has a Galois invariant sublattice of type $D_{8}$. The
Galois action on $\operatorname{Pic}(\widetilde{X}_{2})$ permutes the $240$
exceptional curves of $\widetilde{X}_{2}$ and acts on
$\operatorname{Pic}(\widetilde{X}_{2})^{\perp}$ though the Weyl group of type
$E_{8}$, denoted $W_{E_{8}}$. As it turns out, the stabilizer in the Weyl
group $W_{E_{8}}$ of a $D_{8}$-sublattice is of index $135$, and in fact it is
isomorphic to the Weyl group $W_{D_{8}}$. Further details about del Pezzo
surfaces can be found in [Dol2012, Chapter 8]. We show how to transmute the
solution to the classical inverse Galois problem for subgroups of $W_{D_{8}}$
into a solution to the inverse Galois problem for del Pezzo surfaces of degree
one with a Galois invariant sublattice
$\Lambda_{D_{8}}\subset\operatorname{Pic}(\widetilde{X}_{2})$ of type $D_{8}$.

######  Theorem 1.0.1.

Let
$\rho\colon\operatorname{Gal}(\mathbb{Q}^{\mathrm{sep}}/\mathbb{Q})\rightarrow
W_{D_{8}}$ be a continuous homomorphism. If $G:=\operatorname{Im}(\rho)$ is
the Galois group of some irreducible polynomial over $\mathbb{Q}$, then there
exists a nonsingular del Pezzo surface of degree one $\widetilde{X}_{2}$ such
that each $\sigma\in\operatorname{Gal}(\mathbb{Q}^{\mathrm{sep}}/\mathbb{Q})$
permutes the $240$ exceptional curves of $\widetilde{X}_{2}$ as described by
$\rho(\sigma)\subset S_{240}$.

In particular, the subfield of $\mathbb{Q}^{\mathrm{sep}}$ fixed by
$\rho^{-1}(\operatorname{id})$ is the splitting field of the $240$ exceptional
curves. The group $W_{D_{8}}$ admits a transitive permutation representation
on a set of size $16$ (see Section 2). If $\rho^{\prime}\colon
W_{D_{8}}\rightarrow S_{16}$ is the associated morphism and if $G\subseteq
W_{D_{8}}$ is a subgroup for which a solution to the inverse Galois problem is
known, then there exists a (possibly reducible) polynomial $f$ of degree $16$
such that $G\cong\operatorname{Gal}(f)$ and each $\sigma\in G$ permutes the
roots of $f$ as described by $\rho^{\prime}(\sigma)$. A solution to the
inverse Galois problem is known for several subgroups of $W_{D_{8}}$,
including $W_{D_{8}}$ itself.

######  Theorem 1.0.2.

Let $G$ be a group which acts transitively on a set of size at most $16$. Then
there is an irreducible polynomial $f$ over $\mathbb{Q}$ such that
$\operatorname{Gal}(f)\cong G$.

###### Proof.

A list of examples can be obtained from the online database of Klüners and
Malle [KlunersMalleDatabaseArticle, KlunersMalleDatabase2021]. ∎

Our techniques are related to those in the article by Elsenhans and Jahnel
[ElsenhansJahnel2019], where they study plane quartics in terms of a set of
eight points in $\mathbb{P}^{3}$ called a Cayley octad.

In [Corn2007], Corn showed that the Brauer group of a del Pezzo surface is one
of a finite number of explicit possibilities. Specifically, for a del Pezzo
surface $X$ over $\mathbb{Q}$ there is a canonical isomorphism
$\operatorname{Br}X/\operatorname{Br}\mathbb{Q}\cong\operatorname{H}^{1}(\operatorname{Gal}(\mathbb{Q}^{\mathrm{sep}}/\mathbb{Q}),\operatorname{Pic}X^{\mathrm{sep}})$
coming from the Hochschild-Serre spectral sequence. Corn’s result is obtained
by considering the action of an abstract group on the Picard lattice and using
group cohomology to determine the list of possibilities. Corn’s result raises
an interesting question, _which Brauer groups from Corn’s list actually occur
for a del Pezzo surface defined over $\mathbb{Q}$?_ Theorem 1.0.1 allows us to
resolve this question for the groups in Corn’s list [Corn2007, Theorem 4.1] of
exponent $1$, $2$, or $4$; we list these in Table 1.

Degree | Groups of exponent $1,2,4$
---|---
$5\leq d\leq 9$ | $\\{\operatorname{id}\\}$
$d=4$ | any of the groups above, $\mathbb{Z}/2\mathbb{Z},(\mathbb{Z}/2\mathbb{Z})^{2}$
$d=3$ | any of the groups above
$d=2$ | any of the groups above, $(\mathbb{Z}/2\mathbb{Z})^{s}\ \ (3\leq s\leq 6)$, $\mathbb{Z}/4\mathbb{Z}\oplus(\mathbb{Z}/2\mathbb{Z})^{t}\ \ (0\leq t\leq 2)$, $(\mathbb{Z}/4\mathbb{Z})^{2}$
$d=1$ | any of the groups above, $(\mathbb{Z}/2\mathbb{Z})^{s}\ \ (7\leq s\leq 8)$, $\mathbb{Z}/4\mathbb{Z}\oplus(\mathbb{Z}/2\mathbb{Z})^{t}\ \ (0\leq t\leq 4)$,
| $(\mathbb{Z}/4\mathbb{Z})^{2}\oplus(\mathbb{Z}/2\mathbb{Z})^{t}\ \ (1\leq
t\leq 2)$

Table 1. The list of possible $\operatorname{Br}X/\operatorname{Br}\mathbb{Q}$
for del Pezzo surfaces by degree.

######  Corollary 1.0.3.

Any group from row $d$ in Table 1 is of the form
$\operatorname{Br}X/\operatorname{Br}\mathbb{Q}$ for some del Pezzo surface
$X/\mathbb{Q}$ of degree $d$.

###### Proof.

First we consider the case where $d=1$. If $G$ is a subgroup of $W_{D_{8}}$
for which a solution to the inverse Galois problem is known, then the action
of $G\subseteq W_{D_{8}}\subset W_{E_{8}}$ on the exceptional curves of a del
Pezzo surface of degree one can be realized as a Galois action over
$\mathbb{Q}$. We can explicitly compute the group cohomology for the action of
$G$ on the root lattice $\Lambda_{E_{8}}$ of type $E_{8}$ and check for the
desired Brauer group. A script that does this for all of the subgroups of
$W_{D_{8}}$ is available at the link in Subsection 1.1. (Of course, Corn
carries out a similar calculation to prove [Corn2007, Theorem 4.1].) For each
of the groups $M$ in Table 1, our computation shows there is a subgroup
$G\subset W_{D_{8}}$ of order at most $8$ such that
$M\cong\operatorname{H}^{1}(G,\Lambda_{E_{8}})$. In particular, a solution to
the inverse Galois problem for such $G$ is known.

For del Pezzo surfaces of degree $2\leq d\leq 4$, we may apply the result of
[ElsenhansJahnel2019]. Alternatively, we search for a $G$-action on
$\operatorname{Pic}(\widetilde{X}_{2})^{\perp}$ such that there is an
isomorphism of $G$-lattices

$\operatorname{Pic}(\widetilde{X}_{2})^{\perp}\cong\Lambda\oplus\langle
e_{1}\rangle\oplus\cdots\oplus\langle e_{d-1}\rangle$

where $e_{j}^{2}=-1$, the summands are orthogonal, and where
$\operatorname{H}^{1}(G,\Lambda)$ is as desired. A del Pezzo surface of degree
one with such a Picard lattice has a set of $d-1$ pairwise orthogonal
exceptional curves defined over $\mathbb{Q}$; these can be blown-down to
obtain a del Pezzo surface of degree $d$ with Brauer group
$\operatorname{H}^{1}(G,\Lambda)$. ∎

###  1.1. Software

The computations for this paper were performed using the magma computer
algebra system [Magma]. The scripts are available at

https://github.com/a-kulkarn/SixteenPointsScripts.git

The total computation time to run all of the scripts is approximately $1$ hour
on commercial hardware (processor: AMD Ryzen${}^{\text{tm}}$
Threadripper${}^{\text{tm}}$ 2970WX).

### Acknowledgements

I would like to thank Eran Assaf for comments on the early drafts of this
article. The author has been supported by the Simons Collaboration on
Arithmetic Geometry, Number Theory, and Computation (Simons Foundation grant
550033).

##  2\. Background

Let $K/\mathbb{Q}$ be an étale algebra, and let $K=\prod_{j}K_{j}$ be a
decomposition into simple factors. If $K$ is simple, then the _norm_ of an
element $\alpha\in K$ is the usual field norm $N_{K/\mathbb{Q}}$. Otherwise,
we define the norm of $\alpha$ by
$N_{K/\mathbb{Q}}(\alpha):=\prod_{j}N_{K_{j}/\mathbb{Q}}(\alpha_{j})$.

###  2.1. Étale algebras with a $W_{D_{8}}$ action

The (abstract) group $W_{D_{8}}$ is isomorphic to an index $2$ subgroup of the
wreath product $S_{2}\wr S_{8}$. Specifically, it is the extension of $S_{8}$
by the subgroup of elements of $\mu_{2}^{8}$ whose entries multiply to $1$. We
have the exact sequence

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
5.5pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
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29.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\mu_{2}^{7}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
68.32547pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
68.32547pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{W_{D_{8}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
115.55069pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
115.55069pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{S_{8}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
155.05899pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
155.05899pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{1}$}}}}}}}\ignorespaces}}}}\ignorespaces.$

Essentially by definition, the wreath product $S_{2}\wr S_{8}$ admits an
action on a set of $16$ elements $\Omega$ (the eight elements on which $S_{8}$
acts endowed with signs).

######  Construction 2.1.1.

Given
$\rho\colon\operatorname{Gal}(\mathbb{Q}^{\mathrm{sep}}/\mathbb{Q})\rightarrow
W_{D_{8}}$ a continuous homomorphism with image $G$, we describe how to
construct an étale algebra $L$ of degree $16$ such that
$\operatorname{Hom}(L,\mathbb{Q}^{\mathrm{sep}})$ is isomorphic to $\Omega$ as
a $G$-set. We may choose a set of orbit representatives
$\beta_{1},\ldots,\beta_{r}$ such that $\Omega=G\cdot\beta_{1}\cup\cdots\cup
G\cdot\beta_{r}$. Functoriality between étale algebras and continuous Galois
actions on a finite set allows us to identify an étale algebra $\widehat{L}$
from $\rho$ [milneFT, Chapter 8]. Functoriality also allows us to define an
étale subalgebra $L$ of degree $16$ as the subalgebra element-wise fixed by
$\operatorname{Stab}(G;\beta_{1},\ldots,\beta_{r})$.

The $16$ elements of $\Omega$ are identified with the $16$ homomorphisms
$L\rightarrow\mathbb{Q}^{\mathrm{sep}}$ via the $\operatorname{Hom}$ functor.
The natural quotient of $G$-sets $\Omega\rightarrow(\Omega/{\pm})$ induces an
inclusion of étale algebras $K\lhook\joinrel\relbar\joinrel\rightarrow L$,
where the degree of $K$ is equal to $8$. Up to isomorphism we may write
$K=\prod_{j}K_{j}$ and

($\dagger$) $L\cong\prod_{j}K_{j}[x]/(x^{2}-\alpha_{j})$

for some finite field extensions $K_{j}/\mathbb{Q}$ and some $\alpha_{j}\in
K_{j}$ with the property that
$\prod_{j}N_{K_{j}/\mathbb{Q}}(\alpha_{j})\in\mathbb{Q}^{\times 2}$. We call
$K$ the _distinguished subalgebra_.

###  2.2. Zariski density of generators

We denote by $\mathrm{Pol}^{8}$ the scheme whose set of $R$-valued points is
the set of monic polynomials of degree $8$ with coefficients in $R$, where $R$
is a commutative ring containing $\mathbb{Q}$. Of course,
$\mathbb{A}^{8}_{\mathbb{Q}}\cong\mathrm{Pol}^{8}$. If $K$ is an étale
$\mathbb{Q}$-algebra of dimension $8$, we may endow $K$ with the structure of
an affine scheme, specifically the affine space $\mathbb{A}^{8}_{\mathbb{Q}}$.
We denote this scheme by $\mathbb{A}^{\\!K}$. The functor of points is given
by $\mathbb{A}^{\\!K}\\!(R)=K\otimes_{\mathbb{Q}}R$. The scheme
$\mathbb{A}^{\\!K}$ is simply the Weil restriction
$\mathrm{Res}_{K/\mathbb{Q}}\mathbb{A}^{1}_{K}$. If $R$ is a commutative ring
containing $\mathbb{Q}$, then we denote by $\mathbb{A}^{\\!K}_{R}$ the scheme
$\mathbb{A}^{\\!K}\times_{\operatorname{Spec}\mathbb{Q}}\operatorname{Spec}R$.
If $\beta\in\mathbb{A}^{\\!K}\\!(R)$, we denote by $[\beta]$ the endomorphism
of $R\otimes_{\mathbb{Q}}K$ defined by multiplication-by-$\beta$. There is a
canonical morphism of schemes

$\begin{tabular}[]{cccc}\chi^K\colon&\mathbb{A}^{\\!K}&\rightarrow&\mathrm{Pol}^8\\\
&\beta&\mapsto&\mathrm{coeffs}(\det(xI - [\beta]))\end{tabular}.$

As an example, if $K=\prod_{j=1}^{8}\mathbb{Q}$ is the split étale
$\mathbb{Q}$-algebra of dimension $8$, then $\chi^{K}$ is simply the map
defined by the elementary symmetric functions.

######  Lemma 2.2.1.

The set $\chi^{K}(\mathbb{A}^{\\!K}\\!(\mathbb{Q}))$ is Zariski dense in
$\mathrm{Pol}^{8}$.

###### Proof.

Let ${\widehat{K}}$ be a splitting field for $K$. Since $\mathbb{A}^{\\!K}$ is
just affine space, $\mathbb{A}^{\\!K}\\!(\mathbb{Q})$ is dense in
$\mathbb{A}^{\\!K}_{\widehat{K}}$. On the other hand, if $A:={\mathbb{Q}}^{8}$
is the split étale ${\widehat{K}}$ algebra, diagonalization gives an
isomorphism
$\psi\colon\mathbb{A}^{\\!K}_{\widehat{K}}\rightarrow\mathbb{A}^{\\!A}_{\widehat{K}}$.
Diagonalization does not change the characteristic polynomial, so the diagram

---
$\textstyle{\mathbb{A}^{\\!K}_{\widehat{K}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\chi^{K}_{\widehat{K}}}$$\scriptstyle{\psi}$$\textstyle{\mathbb{A}^{\\!A}_{\widehat{K}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\chi^{A}_{\widehat{K}}}$$\textstyle{\mathrm{Pol}^{8}_{\widehat{K}}}$

of morphisms over ${\widehat{K}}$ commutes. But $\chi^{A}$ is just the
morphism defined by elementary symmetric functions, so it is surjective as a
morphism of schemes, and thus so too is $\chi^{K}$. ∎

######  Corollary 2.2.2.

Given an étale algebra $L$ arising from Construction 2.1.1, with distinguished
subalgebra $K$, the set

$\left\\{(\alpha_{1},\ldots,\alpha_{r})\in\prod_{j}K_{j}:L\cong\prod_{j}{K_{j}[x]}/{\langle
x^{2}-\alpha_{j}\rangle},\ K_{j}\cong\mathbb{Q}(\alpha_{j})\right\\}$

has a dense image under $\chi^{K}$ in $\mathrm{Pol}^{8}$.

###### Proof.

Choose a presentation for $L$ as in Equation ($\dagger$ ‣ 2.1). Since the set
$\alpha\cdot K^{\times 2}$ is Zariski dense in $\mathbb{A}^{\\!K}$ for any
unit $\alpha\in K^{\times}$, and contains a dense subset of primitive
elements, the result follows. ∎

###  2.3. Tensors

We denote by $\operatorname{Sym}_{2}\mathbb{Q}^{m+1}$ the space of
$(m+1)\times(m+1)$ symmetric matrices over $\mathbb{Q}$ and by
$\operatorname{Sym}^{2}\mathbb{Q}^{n+1}$ the space of quadratic forms over
$\mathbb{Q}$ in $n+1$ variables. Let
$\mathcal{A}\in\mathbb{Q}^{n+1}\otimes\operatorname{Sym}_{2}\mathbb{Q}^{m+1}$
be a tensor, symmetric in the last two entries. We view $\mathcal{A}$ as an
$(n+1)\times(m+1)\times(m+1)$ array of elements of $\mathbb{Q}$. We may think
of such an array as being an ordered collection $A_{0},\ldots,A_{n}$ of
symmetric $(m+1)\times(m+1)$ matrices, the _slices_ of $\mathcal{A}$. We
denote the contraction of $\mathcal{A}$ along a vector $v\in\mathbb{Q}^{n+1}$
by $\mathcal{A}(v,\cdot,\cdot)$. More generally, we will contract along an
element of $R^{n}$, where $R$ is a $\mathbb{Q}$-algebra. Similarly, we will
denote the contraction by an element $y\in R^{m+1}$ by
$\mathcal{A}(\cdot,y,\cdot)$ or $\mathcal{A}(\cdot,\cdot,y)$, depending along
which axis we contract.

Denote $\mathbb{P}^{n}:=\operatorname{Proj}\mathbb{Q}[x_{0},\ldots,x_{n}]$,
$\mathbb{P}^{m}:=\operatorname{Proj}\mathbb{Q}[y_{0},\ldots,y_{m}]$, and
denote the dual projective space of $\mathbb{P}^{n}$ by
$\widehat{\mathbb{P}}^{n}$. We will also denote
$\mathbf{x}:=(x_{0},\ldots,x_{n})$ and $\mathbf{y}:=(y_{0},\ldots,y_{m})$. If
$x\in\mathbb{Q}^{n+1}$, the contraction $\mathcal{A}(x,\mathbf{y},\mathbf{y})$
is the quadratic form

$\mathbf{y}^{T}\left(x_{0}A_{0}+\ldots+x_{n}A_{n}\right)\mathbf{y}.$

If $x\in\mathbb{P}^{n}(\mathbb{Q})$ is a point, then the contraction
$\mathcal{A}(x,\cdot,\cdot)$ is a symmetric $(m+1)\times(m+1)$ matrix with
entries in $\mathbb{Q}$, well-defined up to scaling. Write
$(q_{0},\ldots,q_{n})$ for the quadrics defined by the $n+1$ slices of
$\mathcal{A}$. The $q_{i}$ define the rational map

$\begin{tabular}[]{cccc}\psi\colon&\mathbb{P}^m&\dashrightarrow&\widehat{\mathbb{P}}^n\\\
&y&\mapsto&\mathcal{A}(\cdot, y, y) = (q_0(y), \ldots, q_n(y))\end{tabular}.$

##  3\. Proof of the main result

###  3.1. Construction of $16$ points in $\mathbb{P}^{4}$

Given a hyperelliptic genus $3$ curve $Y$ of the form

$Y\colon y^{2}=f(z_{0},z_{1})$

such that $f(z_{0},z_{1})$ is homogeneous, square-free, and
$f(0,1),f(1,0)\in\mathbb{Q}^{\times 2}$, we describe a method to construct
$16$ points in $\mathbb{P}^{4}$ based on the construction in
[KulkarniVemulapalli, Section 6].

Define $K:=\mathbb{Q}[z]/(f(z,1))$ and $L:=\mathbb{Q}[z]/(f(z^{2},1))$. An
elementary calculation shows that the conditions on $f(z_{0},z_{1})$ allow us
to write

$f(z_{0},z_{1})=b(z_{0},z_{1})^{2}-z_{0}z_{1}^{3}c(z_{0},z_{1})=-\det\begin{bmatrix}z_{0}z_{1}^{3}&b(z_{0},z_{1})\\\
b(z_{0},z_{1})&c(z_{0},z_{1})\end{bmatrix}$

for some homogeneous polynomials $b(z_{0},z_{1}),c(z_{0},z_{1})$ of degree
$4$. If $\alpha_{1},\ldots,\alpha_{8}$ are the roots of $f(z,1)$, then the set

$\mathfrak{C}:=\left\\{(\pm\sqrt{\alpha_{j}}:(\pm\sqrt{\alpha_{j}})^{-1}\cdot
b(\alpha_{j},1):\alpha_{j}^{2}:\alpha_{j}:1)\ :\ 1\leq j\leq 8\right\\}$

of points in $\mathbb{P}^{4}$ is split over $\widehat{L}$, the Galois closure
of $L/\mathbb{Q}$. One can check that when the $\alpha_{j}$ are distinct, the
Vandermonde matrix of degree $2$ forms evaluated at the points of
$\mathfrak{C}$ has corank $5$, so $\mathfrak{C}$ is contained in the
intersection of $4$ quadrics. Generically, $\mathfrak{C}$ is a complete
intersection of $4$ quadrics in $\mathbb{P}^{4}$. If $L$ is presented as in
Equation ($\dagger$ ‣ 2.1) such that $\alpha_{1},\ldots,\alpha_{r}$ are
primitive elements of $K_{1},\ldots,K_{r}$ (respectively), then we have that
$\\{\pm\sqrt{\alpha_{j}}:1\leq j\leq
8\\}=\operatorname{Hom}_{\mathbb{Q}}(L,\mathbb{Q}^{\mathrm{sep}})$. Thus, the
permutation action of
$\operatorname{Gal}(\mathbb{Q}^{\mathrm{sep}}/\mathbb{Q})$ on the $16$ points
of $\mathfrak{C}$ is identical to the action of
$\operatorname{Gal}(\mathbb{Q}^{\mathrm{sep}}/\mathbb{Q})$ on $\Omega$.

We can explicitly describe the linear space of quadrics containing
$\mathfrak{C}$. Write
$\mathbb{P}^{4}:=\operatorname{Proj}(\mathbb{Q}[y_{0},y_{1},y_{2},y_{3},y_{4}])$,
which admits a natural projection to
$\mathbb{P}^{2}:=\operatorname{Proj}(\mathbb{Q}[y_{2},y_{3},y_{4}])$. Under
the Veronese embedding

ν_2: |  P^1 | → | Proj(Q[y_2, y_3, y_4])
---|---|---|---
| (z_0:z_1) | ↦ | (z_0^2 : z_0z_1 : z_1^2)

the quartic forms $z_{0}z_{1}^{3},b(z_{0},z_{1}),c(z_{0},z_{1})$ can be
identified with the three quadratic forms $y_{3}y_{4}$,
$b(y_{2},y_{3},y_{4})$, $c(y_{2},y_{3},y_{4})$; the identified quadric forms
are unique modulo $y_{3}^{2}-y_{2}y_{4}$. The determinantal representation for
$f(z_{0},z_{1})$ becomes a quadratic determinantal representation, i.e., is a
tensor in
$\operatorname{Sym}^{2}\mathbb{Q}^{3}\otimes\operatorname{Sym}_{2}\mathbb{Q}^{2}$.

######  Construction 3.1.1.

Let $\mathbb{P}^{2}:=\operatorname{Proj}(k[y_{2},y_{3},y_{4}])$. Given a conic
$C\subset\mathbb{P}^{2}$ and a tensor
$\mathcal{B}\in\operatorname{Sym}^{2}k^{3}\otimes\operatorname{Sym}_{2}k^{2}$
representing a $2\times 2$ matrix of quadratic forms, we can construct a
tensor $\mathcal{A}\in
k^{4}\otimes(\operatorname{Sym}_{2}k^{2}\oplus\operatorname{Sym}_{2}k^{3})$ as
follows: Writing $\mathcal{B}(\mathbf{y},\cdot)=\begin{bmatrix}a&b\\\
b&c\end{bmatrix}$, we have that

$\displaystyle a(\mathbf{y})=\sum_{2\leq i,j\leq 4}a_{ij}y_{i}y_{j},\quad
b(\mathbf{y})=\sum_{2\leq i,j\leq 4}b_{ij}y_{i}y_{j},\quad
c(\mathbf{y})=\sum_{2\leq i,j\leq 4}c_{ij}y_{i}y_{j}.$

We may write the defining equation for $C$ as $\sum_{2\leq i,j\leq
4}d_{ij}y_{i}y_{j}$. (The terms with $i\neq j$ appear twice.) Define the
slices of $\mathcal{A}$ to be

$\displaystyle A_{0}:=\begin{bmatrix}-1&0&0&0&0\\\ 0&0&0&0&0\\\
0&0&a_{22}&a_{23}&a_{24}\\\ 0&0&a_{32}&a_{33}&a_{34}\\\
0&0&a_{42}&a_{43}&a_{44}\\\ \end{bmatrix},\quad$ $\displaystyle
A_{1}:=\begin{bmatrix}0&-\frac{1}{2}&0&0&0\\\ -\frac{1}{2}&0&0&0&0\\\
0&0&b_{22}&b_{23}&b_{24}\\\ 0&0&b_{32}&b_{33}&b_{34}\\\
0&0&b_{42}&b_{43}&b_{44}\\\ \end{bmatrix},$ $\displaystyle
A_{2}:=\begin{bmatrix}0&0&0&0&0\\\ 0&-1&0&0&0\\\ 0&0&c_{22}&c_{23}&c_{24}\\\
0&0&c_{32}&c_{33}&c_{34}\\\ 0&0&c_{42}&c_{43}&c_{44}\\\ \end{bmatrix},\quad$
$\displaystyle A_{3}:=\begin{bmatrix}0&0&0&0&0\\\ 0&0&0&0&0\\\
0&0&d_{22}&d_{23}&d_{24}\\\ 0&0&d_{32}&d_{33}&d_{34}\\\
0&0&d_{42}&d_{43}&d_{44}\\\ \end{bmatrix}.$

The tensor $\mathcal{A}$ generically defines a genus $4$ curve via an
intersection of the two symmetric determinantal varieties
$X_{2}:=Z(4x_{0}x_{2}-x_{1}^{2})$ and
$X_{3}:=Z\left(\det\mathcal{A}^{(2)}(\mathbf{x},\cdot,\cdot)\right)$, where

$\mathcal{A}^{(2)}(\mathbf{x},\cdot,\cdot):=x_{0}\begin{bmatrix}a_{22}&a_{23}&a_{24}\\\
a_{32}&a_{33}&a_{34}\\\ a_{42}&a_{43}&a_{44}\\\
\end{bmatrix}+x_{1}\begin{bmatrix}b_{22}&b_{23}&b_{24}\\\
b_{32}&b_{33}&b_{34}\\\ b_{42}&b_{43}&b_{44}\\\
\end{bmatrix}+x_{2}\begin{bmatrix}c_{22}&c_{23}&c_{24}\\\
c_{32}&c_{33}&c_{34}\\\ c_{42}&c_{43}&c_{44}\\\
\end{bmatrix}+x_{3}\begin{bmatrix}d_{22}&d_{23}&d_{24}\\\
d_{32}&d_{33}&d_{34}\\\ d_{42}&d_{43}&d_{44}\\\ \end{bmatrix}.$

From the four symmetric matrices $A_{0},\ldots,A_{3}$ we obtain an
intersection of four quadrics in $\mathbb{P}^{4}$.

######  Example 3.1.2.

Let

$f(z_{0},z_{1}):=(z_{0}^{2}-z_{1}^{2})(z_{0}^{2}-4z_{1}^{2})(z_{0}^{2}-9z_{1}^{2})(z_{0}^{2}-16z_{1}^{2})\in\mathbb{F}_{101}[z_{0},z_{1}].$

Choosing

$b(z_{0},z_{1}):=z_{0}^{4}+86z_{0}^{2}z_{1}^{2}+24z_{1}^{4}\sim
y_{2}^{2}+86y_{3}^{2}+24y_{4}^{2},\qquad c(z_{0},z_{1}):=-z_{0}z_{1}^{3}\sim-
y_{3}y_{4},$

we have that $f(z_{0},z_{1})=b(z_{0},z_{1})^{2}-z_{0}z_{1}^{3}c(z_{0},z_{1})$.
The tensor given by Construction 3.1.1, represented as a matrix of linear
forms, is

$\left(\begin{array}[]{rrrrr}-x_{0}&-51x_{1}&0&0&0\\\ -51x_{1}&-x_{2}&0&0&0\\\
0&0&x_{1}&0&-51x_{3}\\\ 0&0&0&86x_{1}+x_{3}&51(x_{0}-x_{2})\\\
0&0&-51x_{3}&51(x_{0}-x_{2})&24x_{1}\end{array}\right)$

and the resulting genus $4$ curve in $\mathbb{P}^{3}$ is defined by the common
vanishing locus of

$\displaystyle q(x)$ $\displaystyle:=x_{0}x_{2}-x_{1}^{2},$ $\displaystyle
g(x)$
$\displaystyle:=25x_{0}^{2}x_{1}+51x_{0}x_{1}x_{2}+44x_{1}^{3}+24x_{1}^{2}x_{3}+25x_{1}x_{2}^{2}+29x_{1}x_{3}^{2}+25x_{3}^{3}.$

Computer algebra can be used to verify that the genus $4$ curve is
nonsingular. (See subsection 1.1.)

######  Lemma 3.1.3.

Let $L/\mathbb{Q}$ be an étale algebra arising from Construction 2.1.1, let
$K$ be the distinguished subalgebra, and let

$U:=\left\\{(\alpha_{1},\ldots,\alpha_{r})\in\prod_{j}K_{j}:L\cong\prod_{j}{K_{j}[x]}/{\langle
x^{2}-\alpha_{j}\rangle},\ K_{j}\cong\mathbb{Q}(\alpha_{j})\right\\}.$

Let $V\subseteq U$ be the subset such that for any $\alpha\in V$, Construction
3.1.1 applied to the characteristic polynomial of $\alpha$ produces a
nonsingular genus $4$ curve and a base locus $\mathfrak{C}$ of dimension $0$
and degree $16$ of the linear space of quadrics. Then $V$ is non-empty.

###### Proof.

Example 3.1.2 shows that there is a non-empty open subscheme of
$\mathrm{Pol}^{8}$ where the constructed curve is nonsingular. The result
follows from Corollary 2.2.2. ∎

###  3.2. Cycles on double covers

In [Reid1972quadrics], Reid showed that the algebraic cycles on the complete
intersection of 3 quadrics correspond to algebraic cycles within the Prym
variety of the natural double cover of the degeneracy locus. An alternative
reference for this construction is [Tyurin1975]. In this section, we show how
a similar construction allows us to identify pairs of points on the
intersection of $4$ quadrics in $\mathbb{P}^{4}$ with the $112$ exceptional
curves on a del Pezzo surface of degree one.

The following lemma contains some results we will freely use about quadrics.

######  Lemma 3.2.1.

Let $k$ be a field of characteristic not equal to $2$, let
$A\in\operatorname{Sym}_{2}k^{m+1}$ be of rank $r>0$, and let
$Q:=Z(\mathbf{y}^{T}A\mathbf{y})\subset\mathbb{P}_{k}^{m}$ be the associated
quadric. Then:

  1. (a)

Any singular quadric is a cone over a non-singular quadric.

  2. (b)

The singular locus of $Q$ is contained in every maximal isotropic subspace. In
other words, every maximal isotropic subspace is a cone over an isotropic
subspace of the nonsingular part. Furthermore, the singular locus of $Q$ is
the linear subspace $\mathbb{P}(\ker A)\subset\mathbb{P}^{m}$.

  3. (c)

Over $\bar{k}$, the dimension of a maximal isotropic subspace is
$\left\lfloor\frac{r}{2}\right\rfloor+\dim(\ker A)-1$.

  4. (d)

If $r$ is even, there are two distinct families of maximal isotropic subspaces
of $Q$. If $r$ is odd, then there is a unique family of maximal isotropic
subspaces.

  5. (e)

If $r$ is even, and $V,W$ are maximal, then $V,W$ are contained in the same
family if and only if $\dim(V)-\dim(V\cap W)\equiv 0\pmod{2}$.

###### Proof.

See [GriffithsHarris1994]. ∎

If $\mathcal{L}$ is a maximal isotropic subspace of a quadric $Q$, we will
denote by $[\mathcal{L}]$ the family of maximal isotropic subspaces containing
$\mathcal{L}$.

######  Proposition 3.2.2.

Let $S_{r}^{m}$ be the space of quadrics of even rank $r$. Let $x\in
S_{r}^{m}$ and $[\mathcal{L}]$ denote a family of maximal isotropic subspaces
of $x$. Then the choice of generator $(x,[\mathcal{L}])\mapsto x$ defines a
nontrivial double cover branched over $S_{r-1}^{m}$.

###### Proof.

See [Tyurin1975, Section 5]. ∎

######  Definition 3.2.3.

The _generator double cover_ is the morphism
$\operatorname{gen}\colon(x,[\mathcal{L}])\mapsto x$ given by Proposition
3.2.2.

The generator double cover allows us to identify the secants between $16$
points in $\mathbb{P}^{4}$ with exceptional curves of a del Pezzo surface of
degree $1$. In our particular case of a block-diagonal tensor $\mathcal{A}\in
k^{4}\otimes(\operatorname{Sym}_{2}k^{2}\oplus\operatorname{Sym}_{2}k^{3})$,
the locus of quadrics of rank at most $4$ is
$X_{5}:=Z(\det\mathcal{A}(\mathbf{x},\cdot,\cdot))$, which is the union of the
quadric cone $X_{2}$ where the first block degenerates and the symmetroid
cubic $X_{3}$ where the second block degenerates. The generator double cover
$\operatorname{gen}\colon\widetilde{X}_{5}\rightarrow X_{5}$ is branched over
the locus of quadrics of rank at most $3$, which consists of the genus four
curve $X=X_{2}\cap X_{3}$ as well as the singularities of $X_{2}$ and $X_{3}$.
Generically, the web of quadrics defined by $\mathcal{A}$ does not contain any
quadrics of rank $2$. We obtain by restriction a double cover
$\operatorname{gen}\colon\widetilde{X}_{2}\rightarrow X_{2}$ branched along a
genus $4$ curve and the vertex of the cone; in other words,
$\widetilde{X}_{2}$ is a del Pezzo surface of degree one.

Let $\mathfrak{C}$ be the intersection of all the quadrics in the web defined
by $\mathcal{A}$, which we saw before is generically a complete intersection
of four quadrics in $\mathbb{P}^{4}$. If $\ell$ is a secant of $\mathfrak{C}$
and $V_{\ell}\subset k^{5}$ is the affine cone over $\ell$, then we define a
subscheme of $\widetilde{X}_{2}$ by

$\tau(\ell):=\left\\{(x,[\mathcal{L}])\in\widetilde{X}_{2}:\mathcal{L}:=V_{\ell}+\ker\mathcal{A}(x,\cdot,\cdot)\text{
is a maximal isotropic subspace of
}\mathcal{A}(x,\mathbf{y},\mathbf{y})\right\\}.$

Observe that $\tau(\ell)$ is well-defined; the kernel of a quadric $Q$ of rank
$4$ generically does not meet $V_{\ell}$, so the space $V_{\ell}+\ker Q$ is a
maximal isotropic subspace of $Q$. Notice that the automorphism
$\eta\colon\mathbb{P}^{4}\rightarrow\mathbb{P}^{4}$ defined by
$\eta\colon(y_{0}:y_{1}:y_{2}:y_{3}:y_{4})\mapsto(-y_{0}:-y_{1}:y_{2}:y_{3}:y_{4})$
acts invariantly on all of the quadratic forms in the web defined by
$\mathcal{A}$.

If $\mathcal{A}(x,\mathbf{y},\mathbf{y})$ is a quadratic form of rank $4$
vanishing on $\ell$, then it must also vanish on $\eta(\ell)$. Furthermore,
the intersection of the maximal isotropic subspaces
$V_{\ell}+\ker\mathcal{A}(x,\cdot,\cdot)$ and
$\eta(V_{\ell})+\ker\mathcal{A}(x,\cdot,\cdot)$ is generically
$\ker\mathcal{A}(x,\cdot,\cdot)$, and thus the two maximal isotropic subspaces
lie in opposite families. In other words, the curves
$\tau(\ell),\tau(\eta\ell)$ on $\widetilde{X}_{2}$ only intersect along the
branch locus and are exchanged by
$\operatorname{Aut}(\widetilde{X}_{2}/X_{2})$.

The last result we need is a theorem from [KulkarniVemulapalli]. To clarify
the statement of the theorem, any nonhyperelliptic genus $4$ curve $X$ with a
vanishing even theta characteristic $\theta_{0}$ has a unique vanishing even
theta characteristic. This allows us to partition the $2$-torsion classes of
the Jacobian variety into one of two types:

  * •

(odd) $\epsilon\in\operatorname{Jac}(X)[2]$ is of the form
$[\theta-\theta_{0}]$ for some odd theta characteristic of $X$.

  * •

(even) $\epsilon\in\operatorname{Jac}(X)[2]$ is not even.

There are $120$ odd $2$-torsion classes and $135$ nontrivial even $2$-torsion
classes.

######  Theorem 3.2.4 ([KulkarniVemulapalli, Theorem 1.1.3]).

Let $k$ be a field of characteristic not $2$ or $3$. Then:

  1. (a)

There is a canonical bijection between:

$\left\\{\begin{array}[]{c}k\text{-isomorphism classes of tuples
}(X,\epsilon,\theta_{0}),\\\ \text{where }X\text{is a smooth genus $4$ curve
with vanishing even theta}\\\ \text{characteristic }\theta_{0}\text{ with a
rational divisor class defined over }k,\\\ \text{and }\epsilon\text{ is a
nontrivial even $2$-torsion
class}\end{array}\right\\}\longleftrightarrow\left\\{\begin{array}[]{c}\text{nondegenerate
orbit classes of }\\\
k^{4}\otimes\left(\operatorname{Sym}_{2}k^{2}\oplus\operatorname{Sym}_{2}k^{3}\right)\\\
\text{under the action of }\\\
\operatorname{GL}_{4}(k)\times\operatorname{GL}_{2}(k)\times\operatorname{GL}_{3}(k)\end{array}\right\\}$

Let $\mathcal{A}\in
k^{4}\otimes(\operatorname{Sym}_{2}k^{2}\oplus\operatorname{Sym}_{2}k^{3})$ be
a nondegenerate tensor and let $\theta_{0}$ and $\epsilon$ be the associated
line bundles on $X$. Then:

  1. (b)

The images of the $120$ secants of $\mathfrak{C}$ under $\psi$ define $56+8$
tritangent planes of $X$. Viewing $X\cap H$ as a divisor of $X$, eight of
these tritangents satisfy $X\cap H=2D$ where $D\in|\theta_{0}|$. The other
$56$ tritangents satisfy $X\cap H=2D$, where $D$ is the effective
representative of an odd theta characteristic of $X$.

  2. (b)

Let $e_{2}$ denote the Weil pairing on $\operatorname{Jac}(X)[2]$. Then the
$56$ distinct odd theta characteristics constructed from the secants of
$\mathfrak{C}$ are precisely the odd theta characteristics $\theta$ such that
$e_{2}(\theta\otimes\theta_{0}^{\vee},\epsilon)=0$.

The inclusion $X\lhook\joinrel\relbar\joinrel\rightarrow\widetilde{X}_{2}$
induces a restriction morphism of divisor classes. We denote by
$\operatorname{Pic}(\widetilde{X}_{2})^{\perp}$ the divisor classes orthogonal
to the canonical class under the intersection pairing. There is an exact
sequence

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{2\operatorname{Pic}(\widetilde{X}_{2})^{\perp}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Pic}(\widetilde{X}_{2})^{\perp}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Pic}(X)[2]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

Additionally, the anti-canonical class of $\widetilde{X}_{2}$ restricts to the
unique vanishing even theta characteristic $\theta_{0}$ of $X$. If
$e\in\operatorname{Pic}(\widetilde{X}_{2})$ is an exceptional curve and
$\kappa$ is the canonical class, then
$e+\kappa\in\operatorname{Pic}(\widetilde{X}_{2})^{\perp}$, the restriction of
$e$ to $X$ defines an odd theta characteristic, and the sublattice generated
by

$R_{D_{8}}:=\\{e+\kappa\in\operatorname{Pic}(\widetilde{X}_{2}):e^{2}=-1,\ \
e_{2}((e+\kappa)|_{X},\epsilon)=0\\}$

is a root lattice of type $D_{8}$; the set of roots of this lattice is
precisely $R_{D_{8}}$.

######  Proposition 3.2.5.

Let $\ell$ be a secant of $\mathfrak{C}$. Then the curve $\tau(\ell)$ is one
of the $112$ exceptional curves of $\widetilde{X}_{2}$ with the property that

$e_{2}((\tau(\ell)+\kappa)|_{X},\epsilon)=0.$

Furthermore, each of the $112$ exceptional curves of this type is of the form
$\tau(\ell)$ for some secant of $\mathfrak{C}$.

###### Proof.

Follows immediately from Theorem 3.2.4 and the discussion above. ∎

The automorphism $\eta$ restricts to an automorphism of $\mathfrak{C}$. A pair
of geometric points of $\mathfrak{C}$ is either:

  * •

Type $8$: a pair of the form $\\{p,\eta(p)\\}$, or

  * •

Type $112$: a pair not of the form $\\{p,\eta(p)\\}$.

We see that any $G$-action preserves the type of a pair $\\{p,q\\}$. In
particular, Proposition 3.2.5 shows that the $G$-set of pairs of type $112$ is
isomorphic (as a $G$-set) to the $112$ roots of the $D_{8}$ lattice identified
in Proposition 3.2.5.

###  3.3. Proof of the main theorem

Let
$\rho\colon\operatorname{Gal}(\mathbb{Q}^{\mathrm{sep}}/\mathbb{Q})\rightarrow
W_{D_{8}}$ be a continuous homomorphism with image $G$. By Construction 2.1.1,
we construct an étale algebra $L$ of degree $16$ on which Galois acts via $G$,
as well as its distinguished subalgebra $K$ of degree $8$.

We may choose an element $(\alpha_{1},\ldots,\alpha_{r})\in V\subset K$ as in
Lemma 3.1.3 and let $f$ be its characteristic polynomial over $\mathbb{Q}$.
Construction 3.1.1 allows us to construct a locus of $16$ points with a
$G$-action from $f$, as well as a nonsingular genus $4$ curve $X$ contained in
a quadric cone $X_{2}$. The domain of the generator double cover
$\operatorname{gen}\colon\widetilde{X}_{2}\rightarrow X_{2}$ is a del Pezzo
surface of degree one, and the Galois action on the $112$ roots of the $D_{8}$
sublattice from Proposition 3.2.5 has the Galois action prescribed by $\rho$.

## References


# TaylorAECNet: A TAYLOR STYLE NEURAL NETWORK FOR FULL-BAND ECHO CANCELLATION

###### Abstract

This paper describes aecX team’s entry to the ICASSP 2023 acoustic echo
cancellation (AEC) challenge. Our system consists of an adaptive filter and a
proposed full-band Taylor-style acoustic echo cancellation neural network
(TaylorAECNet) as a post-filter. Specifically, we leverage the recent advances
in Taylor expansion based decoupling-style interpretable speech enhancement
[1] and explore its feasibility in the AEC task. Our TaylorAECNet based
approach achieves an overall mean opinion score (MOS) of 4.241, a word
accuracy (WAcc) ratio of 0.767, and ranks 5th in the non-personalized track
(track 1).

Index Terms—  Acoustic echo cancellation, noise suppression, Taylor expansion

##  1 Introduction

The acoustic echo cancellation (AEC) challenge series has provided a common
platform to benchmark modern AEC techniques. The fourth edition of this
challenge held in ICASSP2023 particularly addresses full-band signals with
general AEC track and personalized AEC track. We submit a hybrid system to the
general AEC track, which integrates a DSP based adaptive filter with a neural
network based post filter.

Recently, there has been a trend in interpretable speech enhancement by
decoupling the difficult task into easier interpretable subtasks. Following
this direction, TaylorSENet [1] imitates the form of Taylor expansion to split
the denoising task into a combination of zero-order derivatives (in the real-
valued domain) and multiple higher-order derivatives (in the complex-valued
domain), achieving promising denoising performance. Inspired by this work, we
propose a full-band Taylor-style acoustic echo cancellation neural network
TaylorAECNet as a post-filter cascaded with an adaptive filter to solve the
full-band echo cancellation task. To reduce complexity, we design the Taylor-
style decoupling network with only zero-order and first-order modules and made
the following modifications to better fit the full-band residual echo removal
task.

  * •

In order to utilize phase information for the zero-order module, a gated
version of the phase encoder [2] is used instead of modulo operation to get
the magnitude feature as zero-order module input.

  * •

Pseudo quadrature mirror filter bank (PQMF) is used to reduce the complexity
caused by full-band signal processing.

  * •

Temporal and frequency convolution module (TFCM) [2] is introduced to improve
the receptive field. Similar to [3], auxiliary voice activity detection (VAD)
task is added and echo weighted loss is introduced.

According to the challenge results, our system ranked 5th in the general AEC
track.

##  2 Proposed Method

The AEC task can be described as:

$d(n)=s(n)+h(n)\ast x(n)+v(n).\vspace{-3pt}$ (1)

The near-end microphone signal $d(n)$ is composed of the near-end speech
$s(n)$, the background noise $v(n)$, and the echo signal $h(n)\ast x(n)$. The
echo signal is generated by far-end signal $x(n)$ propagated through echo path
$h(n)$, where $n$ denotes the sample index. The AEC task is to cancel
$h(n)\ast x(n)$ from $d(n)$ given $x(n)$.

Fig. 1: System diagram.

As shown in Fig. 1, our system consists of a time delay estimation (TDE)
module, an adaptive filter, and a full-band Taylor-style acoustic echo
cancellation neural network (TaylorAECNet) as a post-filter. $d(n)$ and $x(n)$
are first aligned using GCC-PHAT as the TDE module to obtain the time-aligned
reference signal $x^{\prime}(n)$. The error signal $e(n)$ is generated using
$x^{\prime}(n)$ and $d(n)$ by an adaptive filter which is a partitioned-block-
based frequency domain Kalman filter (PBFDKF). Finally, $d(n)$, $e(n)$ and
$x^{\prime}(n)$ are stacked and fed to the TaylorAECNet post-filter.

###  2.1 TaylorAECNet post-filter

TaylorAECNet consists of three modules, zero-order module (ZOM), first-order
module (FOM) and voice activity detection (VAD) module shown as Fig. 2(a).

Fig. 2: The detailed design of TaylorAECNet.

As shown in Fig. 2(b), ZOM is designed as a classical UNet-style structure and
it accepts the magnitude feature from the gated phase encoder. The encoder is
a stack of Convolution-TFCM Block (CTBlocks), and each CTBlock contains a
convolution layer followed by a batch normalization layer, leakyReLU layer and
TFCM [2]. The decoder has the same structure as the encoder, but the
convolution layer is replaced by the transpose convolution layer. We use dual
path RNN (DPRNN) and squeezed temporal convolution module (STCM) [4] to
further improve ZOM performance.

Fig. 2(c) shows the detail of FOM which consists of several STCM layers and
two convolution layers – a complex-to-real convolution layer and a complex-to-
imaginary convolution layer.

The structure of the gated phase encoder is shown in Fig. 2(e), which is an
improved version of phase encoder[2] with an additional gated convolution.
Specifically, the gated phase encoder has three complex-valued convolution
layers to receive $D(t,f)$, $E(t,f)$, and $X(t,f)$ respectively. The complex-
to-real layer consists of a complex-valued convolution layer and a modulo
operation. We use a gated phase encoder instead of directly performing modulo
operations on $D(t,f)$, $E(t,f)$, and $X(t,f)$.

We introduce a voice activity detection (VAD) module to improve the echo
suppression performance. The VAD module is the same as that in [3]. By
learning the VAD state of the microphone signal, the model will be more
inclined to preserve near-end speech.

###  2.2 Loss function

We use echo weighted loss [3] with an extra asymmetric loss
$\mathcal{L}_{\text{asym}}$ [5]. The final loss function is

$\mathcal{L}=\mathcal{L}_{\text{echo-
weighted}}+\mathcal{L}_{\text{asym}}+0.2*\mathcal{L}_{\text{mask}}+0.1*\mathcal{L}_{\text{vad}}.\vspace{-4pt}$
(2)

The definition of ${L}_{\text{mask}}$, ${L}_{\text{echo-weighted}}$, and
$\mathcal{L}_{\text{vad}}$ remains the same as [3]. $\mathcal{L}_{\text{echo-
weighted}}$ makes the model pay more attention to the suppression of echoes by
echo weighting but may cause distortion of the near-end speech. So, we add
$\mathcal{L}_{\text{asym}}$ to alleviate the distortion problem and constrain
the spectrum.

##  3 Experiments

###  3.1 Dataset

We use the clean speech provided by the 5th deep noise suppression (DNS5) as
the near-end signal and the reference signal. The noise set in DNS5 is used as
the noise signal. For the echo signal, we use all the synthetic echo signals
and real far-end single-talk recordings provided by the AEC challenge, which
covers a variety of voice devices and echo signal delay. Furthermore, we also
use speech in the DNS5 dataset to simulate echo data by convolving with
100,000 simulated room impulse responses (RIR) generated by the image method.

The training set has 600 hours of data in total, consisting of 300 hours of
data with simulated echo and 300 hours of data with real-recorded echo. The
development and test sets share the same aforementioned generation method,
which contains 10 and 5 hours of data, respectively.

###  3.2 Experimental setup

The proposed model uses a 20ms window length with a 10ms frame shift for STFT.
Noam is adopted as the learning rate decreasing method described as

$lr=d^{-0.5}\cdot\min(\text{step}^{-0.5},\text{step}\cdot\text{warmup\\_step}^{-1.5})\vspace{-4pt}$
(3)

where $d=1e-3$ and warm_up=5000. All the models are trained with Adam
optimizer for 10 epochs. The PQMF splits the signal into 4 sub-band signals.
Each TFCM has 6 layers. The STCM in ZOM has 1 layer and 128 hidden channels
while the STCM in FOM has 2 layers and 64 hidden channels.

###  3.3 Results and analysis

Table 1 shows the experimental results on the simulated test set. Ablation is
studied on models trained using a subset with 100 hours of simulated data.
Here base-TaylorAECNet is a simplified version for ablation, which removes
TFCM and gate PE parts. When adding TFCM to base-TaylorAECNet, all our metrics
on the test set are improved. Further adding the gated phase encoder, which
results in the complete TalyorAECNet, the metrics for ST-FE and DT are
improved, but the metric for ST-NE is slightly decreased.

We finally train the complete TalyorAECNet using the entire 600 hours of data
and the metrics on the simulated test set are shown in the last row of Table
1. We use this model to process the blind test clips and the challenge results
are shown in Table 2. Our proposed TaylorAECNet approach surpasses the
baseline by a large margin in both Overall MOS and WAcc. The Final Score of
our system is 0.803 which outperforms the baseline with a 0.067 gain, leading
to the 5th in the general AEC track. We perform ONNX implementation to the
post filter for speed-up, the RTF of our proposed system is 0.224 tested on
Intel(R) Xeon(R) Gold 5218 CPU @ 2.30GHz and the number of its parameters is
19.18M.

Table 1: Ablation on the simulated test set.

Model | Para. (M) |  ST-FE (ERLE) |  ST-NE (WB-PESQ) |  DT (WB-PESQ) | Data
---|---|---|---|---|---
Noisy |  | 0 | 2.62 | 1.95 |
base-TaylorAECNet | 15.2 | 51.43 | 3.08 | 2.68 |
+TFCM | 19.1 | 62.84 | 3.05 | 2.69 | 100h
+gated phase encoder | 19.2 | 63.33 | 3.01 | 2.72 |
TaylorAECNet | 19.2 | 66.39 | 3.09 | 2.81 | 600h

Table 2:  System performance on the blind test set of the challenge.

Model |  Overall MOS |  WAcc |  Final Score
---|---|---|---
baseline |  4.013 |  0.649 |  0.736
TaylorAECNet |  4.241 |  0.767 |  0.803

##  4 Conclusions

In this paper, we introduce our system for ICASSP 2023 AEC Challenge. We
combine a partitioned-block-based frequency domain Kalman filter (PBFDKF) with
a proposed TaylorAECNet to suppress echo and noise. Gated phase encoder and
TFCM are used for better latent feature extraction. A VAD module and echo-
weighted loss are introduced to suppress residual echo and preserve near-end
speech quality as well. According to the final results, the proposed system
ranked 5th in the general AEC track.

## References

  * [1] A. Li, S. You, G. Yu, C. Zheng, and X. Li,  “Taylor, can you hear me now? a taylor-unfolding framework for monaural speech enhancement,”  2022\.
  * [2] G. Zhang, L. Yu, C. Wang, and J. Wei,  “Multi-scale temporal frequency convolutional network with axial attention for speech enhancement,”  in ICASSP. IEEE, 2022.
  * [3] S. Zhang, Z. Wang, J. Sun, Y. Fu, B. Tian, Q. Fu, and L. Xie,  “Multi-task deep residual echo suppression with echo-aware loss,”  in ICASSP. IEEE, 2022.
  * [4] A. Li, W. Liu, C. Zheng, C. Fan, and X. Li,  “Two heads are better than one: A two-stage complex spectral mapping approach for monaural speech enhancement,”  IEEE/ACM, 2021.
  * [5] S. Braun and M. Valero,  “Task splitting for dnn-based acoustic echo and noise removal,”  in IWAENC. IEEE, 2022.


# Text Simplification for Comprehension-based Question-Answering

Tanvi Dadu
NSIT, Delhi

Ferdous Ahmed Barbhuiya
IIIT, Guwahati

Kartikey Pant 11footnotemark: 1 22footnotemark: 2
IIIT, Hyderabad

Kuntal Dey 22footnotemark: 2
Accenture Tech Labs
Seema Nagar
IBM Research India
The first two authors contributed equally to the work. This work was done when
Kartikey Pant was affiliated to IIIT Hyderabad, Tanvi Dadu was affiliated to
NSIT Delhi, and Kuntal Dey was affiliated to IBM.

###### Abstract

Text simplification is the process of splitting and rephrasing a sentence to a
sequence of sentences making it easier to read and understand while preserving
the content and approximating the original meaning. Text simplification has
been exploited in NLP applications like machine translation, summarization,
semantic role labeling, and information extraction, opening a broad avenue for
its exploitation in comprehension-based question-answering downstream tasks.
In this work, we investigate the effect of text simplification in the task of
question-answering using a comprehension context. We release Simple-SQuAD, a
simplified version of the widely-used SQuAD dataset.

Firstly, we outline each step in the dataset creation pipeline, including
style transfer, thresholding of sentences showing correct transfer, and offset
finding for each answer. Secondly, we verify the quality of the transferred
sentences through various methodologies involving both automated and human
evaluation. Thirdly, we benchmark the newly created corpus and perform an
ablation study for examining the effect of the simplification process in the
SQuAD-based question answering task. Our experiments show that simplification
leads to up to $2.04\%$ and $1.74\%$ increase in Exact Match and F1,
respectively. Finally, we conclude with an analysis of the transfer process,
investigating the types of edits made by the model, and the effect of sentence
length on the transfer model.

##  1 Introduction

Text simplification is the task of modifying the structure of a text to make
it easier to read and comprehend while preserving the content and
approximating the original meaning. Linguistically, simple sentences are
defined as having only one subject and one verb or predicate. Therefore, a
complex sentence can be rewritten into multiple simpler sentences while
retaining the same meaning. When simplifying texts, a myriad of rewriting
transformations have been explored, ranging from replacing complex words or
phrases for simpler synonyms, to changing the syntactic structure of the
sentence. Moreover, modern automated text simplification approaches are data-
driven, attempting to simplify sentences using parallel corpora of aligned
complex-simplified sentences.

Text simplification allows humans to read texts more efficiently and faster,
empowering them to take part in society and become more active in their daily
actions and healthcare Dannells (2020). Another vital application of text
simplification is reading assistance it provides, especially for people with
reading disabilities Carroll and Macdonald (1998); Inui et al. (2003), low-
literacy readers Watanabe et al. (2009), or non-native speakers Siddharthan
(2002).

Original SQuAD Sentence | Transferred Simple-SQuAD Sentence
---|---
| Clark also claimed that Abdul gave him preferential
---
treatment on the show due to their affair.
| Clark also claimed that Abdul gave him preferential
---
treatment on the show. This was due to their affair.
| In his Prices and Production (1931), Hayek argued
---
that the business cycle resulted from the central bank’s
inflationary credit expansion and its transmission over
time, leading to a capital misallocation caused by the
artificially low interest rates.
| In his Prices and Production (1931), Hayek argued
---
that the business cycle resulted from the central bank’s
inflationary credit expansion and its transmission over
time. This led to a capital misallocation caused by the
artificially low interest rates.

Table 1: Examples of Simple and Complex sentences, from our proposed Simple-
SQuAD dataset and the SQuAD dataset. Underlined parts denotes the splitting
point.

There has been considerable work done where NLP applications exploit the
benefits of text simplification. Long sentences with complex syntax or those
laden with long-distance dependencies often pose difficulties for various
downstream tasks. Text simplification has been used to improve the performance
for these tasks including machine translation Hasler et al. (2017),
summarization Silveira and Branco (2012), semantic role labeling Vickrey and
Koller (2008); Evans and Orasan (2019), and information extraction Evans and
Orasan (2019). This opens up the avenue for exploring simplification for
question-answering. To the best of our knowledge, text simplification for
comprehension-based question-answering has not been explored yet.

In this work, we make the following contributions:-

  * •

We propose a transformers-based text-simplification pipeline that splits and
rephrases a complex sentence into its simpler constituent sentences. We
outline each step in the dataset creation pipeline, including data
preprocessing, performing text simplification, thresholding the simplified
sentence based on the quality of the transfer, and offset finding of answers
from each question-answer pair present in the dataset.

  * •

We propose a new dataset Simple-SQuAD by converting each context in the SQuAD
dataset using the proposed text-simplification pipeline. 111Made available at
the following Github repository: https://github.com/kartikeypant/text-
simplification-qa-www2021.

  * •

We perform automated and human evaluation to determine the quality of the text
simplification model.

  * •

We perform event based analysis and sentence-length based transfer analysis to
give deeper insights into the transfer process.

  * •

We then benchmark both Simple-SQuAD and SQuAD for predictive performance in
the Simple-SQuAD question-answering task.

##  2 Related Works

###  2.1 Text Simplification

Text simplification has attracted a great deal of attention due to its
potential impact on society. Prior explorations on text simplification contain
a myriad of approaches, which include from the use of hand-crafted syntactic
rules Carroll and Macdonald (1998); Vickrey and Koller (2008), statistical
simplification model Zhu et al. (2010), quasi-synchronous grammar Woodsend and
Lapata (2011) and the semantic hierarchy of simplified sentences for
recursively splitting and rephrasing complex sentences Niklaus et al. (2019).
Recently, Machine Translation, both statistical and neural, has also been used
for the task of text simplification Narayan and Gardent (2014); Nisioi et al.
(2017).

The process of text simplification has been observed to help in the
performance of multiple downstream tasks. Silveira et al. Silveira and Branco
(2012) explored the use of a sentence simplification module in summarization
systems, concluding that the simplification module removes expendable
information that helps in accommodating relevant data in a summary. Hasler et
al. Hasler et al. (2017) showed that the performance of source simplification
could improve translation quality in machine translation systems. Evans et al.
Evans and Orasan (2019) proved the efficacy of integrating a text
simplification step for improving the predictive performance of semantic role
labeling and information extraction methodologies.

Narayan et al. Narayan et al. (2017) introduced a new text simplification
task, named Split-and-Rephrase on their proposed WebSplit dataset, containing
$1,066,115$ parallel instances of complex and sequence of simple sentences
having a similar meaning. The goal of the task is to split a complex input
sentence into shorter sentences while preserving the meaning. In this task,
the emphasis is on sentence splitting and rephrasing, with no deletion and no
lexical or phrasal simplification. They further proposed five models, ranging
from vanilla sequence-to-sequence to semantically-motivated models to
benchmark the proposed task. Aharoni and Goldberg Aharoni and Goldberg (2018)
and Botha et al. Botha et al. (2018) extended the work by introducing new
datasets, with a more extensive vocabulary and split examples to improve the
efficacy of the prior benchmarks. We use WikiSplit, introduced by Botha et al.
Botha et al. (2018), for training our sentence simplification module.

###  2.2 Style Transfer

In recent works, textual style transfer has been shown to produce
grammatically fluent, and information-preserved texts with fairly accurate
target attributes. For the task in a semi-supervised setting, various
methodologies are exploited, including back-translation Prabhumoye et al.
(2018), back-translation with attribute conditioning Pant et al. (2020),
specialized transfer methodologies like Delete, Retrieve, Generate Li et al.
(2018). However, in the presence of a large parallel corpus, sequence-to-
sequence models perform competitively. Aharoni and Goldberg Aharoni and
Goldberg (2018) exploited a copy-mechanism based sequence-to-sequence model
with attention Bahdanau et al. (2015) for the text simplification transfer.

Transformers Vaswani et al. (2017) have been shown to perform robust language
modeling, given enough data, helping in various downstream tasks. Due to the
parallelized nature of the architecture, it is possible to train using much
larger datasets than recurrent neural networks. However, it becomes necessary
to optimize hyperparameters carefully while ensuring scalability Popel and
Bojar (2018). Transformer-based methodologies for the task of sentence
simplification have been explored by Marayuma and Yamamoto Maruyama and
Yamamoto (2019) for Japanese and Zhao et al. Zhao et al. (2018) for a smaller
Wikipedia-based dataset. However, to the best of our knowledge, there has been
no work done for supervised style transfer exploiting transformer-based models
for the significantly large WikiSplit dataset.

##  3 Corpus Creation

###  3.1 Preliminaries

####  3.1.1 SQuAD

For exploring the effect of text simplification in the question-answering
downstream task, we use Stanford Question Answering Dataset (SQuAD
comprehension), a reading comprehension dataset released by Rajpurkar et al.
Rajpurkar et al. (2016). It consists of $100,000+$ questions from $536$
articles posed by crowdworkers on a set of Wikipedia articles, where the
answer to each question is a segment of text from the corresponding reading
passage. Unlike other datasets, SQuAD does not provide a list of answer
choices for each question; instead, systems must select the answer from all
possible spans in the passage. Though the system must deal with a large number
of candidate answers, yet span-based answers are easier to evaluate than free-
form answers. Further, it is diverse in terms of answer types, containing a
significant percentage of dates, numeric data, adjective phrases, verb
phrases, clauses as answers. The predictive performance of models in the SQuAD
dataset is evaluated using Exact-match and F1-score.

####  3.1.2 WikiSplit

For the task of text simplification, we use the WikiSplit corpus Botha et al.
(2018), a parallel corpus consisting of complex sentences and its consequent
sequence of simple sentences having similar meaning for the Split and Rephrase
task. It contains a set of one million naturally occurring sentence rewrites
mined from English Wikipedia, providing $60$ times more examples and $90$
times more vocabulary as compared to the WebSplit corpus introduced by Narayan
et al. Narayan et al. (2017). Using a larger dataset for the task of text
simplification increases the efficacy of the model and the quality of the
transferred data Botha et al. (2018); Popel and Bojar (2018).

Figure 1: Style Transfer Architecture

###  3.2 Approach

####  3.2.1 Style Transfer

This subsection outlines the process of textual style transfer using
transformers for converting the complex contexts in the SQuAD dataset into
their simpler counterparts. Figure 1 illustrates the process for a sample
context and question-answer pair from the SQuAD dataset.

We use spaCy’s Sentencizer 222https://spacy.io/api/sentencizer to tokenize the
contexts into their respective constituent sentences. This process enables us
to perform the sentence-level transfer, thus ensuring higher degrees of
overall transfer quality. We tokenize each sentence using a SentencePiece
tokenizer, trained on $3.24$ GBs English Wikipedia . While training the
SentencePiece tokenizer, we mark the numerical tokens as custom-defined ones,
ensuring no tokenization happens for such tokens. This step helps in
preserving the numerical tokens in the transfer process by the transfer model.

We transfer each tokenized sentence using a transformers based machine
translation model. We use the OpenNMT-py Klein et al. (2017) toolkit for the
implementation process. The model consists of $6$-layered transformers
architecture with $8$ self-attention heads and $1028$ sized hidden feed-
forward layer. We trained the model for $20000$ training steps with a dropout
of $0.1$ and a batch size of $2048$. For optimization, we used Adam optimizer
with a beta2 value of $0.998$, accumulating the gradient twice. We use an
initial learning rate of $2$ with noam decay method and $800$ ($4\%$ of total)
warmup steps.

####  3.2.2 Thresholding

In this section, we outline the thresholding techniques used in the generated
sequence of simple sentences to maintain their quality. We execute the
following post-editing steps to reduce noise from the dataset:

  1. 1.

Perplexity : We use open GPT-2 Radford et al. (2018) as the language model to
assign perplexity to the generated sentences. We threshold the transferred
sequence of sentences that have the perplexity between $50$ and $600$ and
discard all the remaining sequences. This thresholding ensures that the
sequence of sentences is fluent, as measured by the language model. Table 2
shows that $84951$ of $91757$ sentences are within the fluency thresholding,
comprising of $~{}93\%$ of the total sentences.

  2. 2.

Length of the original sentence: We weed out all sentences whose original
length was shorter than five tokens. This heuristic is based on the condition
that sentences lesser than five tokens are highly likely to be simple,
reducing the number of false positives to a significant extent efficiently.
Table 2 shows that the thresholding process removes $15.16\%$ of the already
thresholded sentences leaving $~{}78.5\%$ of the total sentences.

  3. 3.

Redundancy in sequence of the transferred sentences: We remove all transferred
sentences that display a redundancy behavior, which is defined to be one in
which two or more sentences in the sequence are the same. We introduce this
step after a careful analysis of the transferred sentences. Table 2
illustrates that the thresholding removes $1.57\%$ of the already thresholded
sentences leaving $~{}77.31\%$ of the total sentences.

| Thresholding
---
Step
| Percentage of
---
Total Sentences
| Thresholded
---
Sentences
Step 1: Perplexity | 92.58% | 84,951
|  Step 2: Original
---
Length Thresholding
78.5% | 72,075
|  Step 3: Redundancy
---
Thresholding
77.31% | 70,944

Table 2: Sentences left after each thresholding process step

####  3.2.3 Automated Evaluation

We perform an automated sentence-level analysis of the style transfer model.
We determine the extent of content preservation, lexical simplicity, and
reading ease of the generated sentences using the following three metrics:

  1. 1.

BLEU (Papineni et al. Papineni et al. (2002)): We use the self-BLEU score
taking the original sentence as the reference to measure the extent of content
preservation. Table 3 shows that our model achieves a mean high self-BLEU of
$75.66$, implying that the transferred sentences display a high level of
content-based similarity with the original sentence

  2. 2.

SARI (Xu et al. Xu et al. (2016)): We use the SARI metric to measure the
quality of lexical simplicity in the transferred sentences. It analyzes the
words added, deleted, and retained by a simplification model. Our version of
SARI compares the model’s output to the original sentence. There is a high
correlation with human judgments of simplicity gain and the SARI metric (Xu et
al., 2016). In Table 3, we observe a mean SARI value of $30.08$ with a low
standard deviation of $2.21$, denoting most sentences to have a high level of
lexical simplicity.

  3. 3.

Flesch–Kincaid Grade level (Kincaid et al. Kincaid et al. (1975)): Flesch–
Kincaid Grade Level (FKGL) is a widely-used metric for text readability. It
represents the corresponding U.S. grade level, whose education is appropriate
for understanding the text. As illustrated in Table 3, the Flesch-Kincaid
Grade Level for the sentences decreases by $5.43$ grade levels on average.
Thus, the transferred sentences have much higher readability than the original
sentences.

Metric | Mean | Standard Deviation
---|---|---
BLEU | 75.66 | 13.68
SARI | 30.08 | 2.21
FKGL(Original) | 13.29 | 5.74
FKGL(Transferred) | 7.86 | 3.94

Table 3: Automated analysis of transferred sentences

####  3.2.4 Human Evaluation

Although we use commonly used metrics for the evaluation, automated evaluation
of generative models of text is still an open research problem Hu et al.
(2017). We perform a human evaluation to analyze the quality of the
transferred sentences accurately.

In the human evaluation, we randomly sampled $50$ sentence pairs from the
thresholded dataset consisting of both transferred sentences and their
original variants. The human evaluators were asked to rate each sentence pair
on a $1$-$5$ Likert scale on the following metrics:

  1. 1.

Fluency: A high Fluency score, $4$ or $5$, denotes that the transferred
sentence is well constructed. A medium score of $3$ denotes that the sentence
contains lexical errors, while a low score of $2$ or $1$ denotes major errors
and extremely poor constructions respectively.

  2. 2.

Relative Simplicity: A sentence pair is given a high Relative Simplicity
score, $4$ or $5$, if the transferred sentence is significantly simplified as
compared to the original sentence. If the simplicity remains the same, it is
given a medium score of $3$ score. Moreover, a low score represents that the
transferred sentence is more complex than the original sentence.

  3. 3.

Content Preservation: A high Content Preservation score of $4$ and $5$
indicates that the content of the original sentence was well-preserved in the
transferred sentence. A medium score of $3$ denotes that there were minor
differences between the transferred sentences and the original sentences. A
low score of $2$ and $1$ denotes major violation of content preservation.

To verify the inter-rater agreement, we perform a Krippendorff’s alpha
analysis across all three metrics. Our analysis shows that we obtain an
averaged Krippendorff’s alpha inter-rater agreement of $0.63$ over all
metrics, denoting reasonable agreement between the raters.

Metric | Average Value
---|---
Fluency | 4.26
Relative Simplicity | 3.91
Content Preservation | 4.10
Table 4: Human Evaluation Results

Table 4 illustrates the results of the performed human evaluation. We observe
that the transferred sentences were judged to be fluent, significantly
simplified, and content-preserved compared to their original counterparts.

####  3.2.5 Offset Finding and Dataset Finalization

In this subsection, we outline the process of finding the offsets for each
question-answer pair and reconstructing our proposed simplified version of
SQuAD, Simple-SQuAD, from the thresholded sentences.

Firstly, we preprocess the sentences, which involves removing the split
sentence indicator and reconstructing context texts from both thresholded and
original sentences. If a sentence is not simplified, we use the original
sentence itself, ensuring minimal loss of information due to the transfer
process.

Secondly, for every question-answer pair, we calculate the answer’s character
offset by using exact matching in the reconstructed context. If we do not find
an exact match, we use case-insensitive pattern matching to calculate the
offset. We found the offsets for $88,690$ questions in total.

Finally, we create the following two datasets: Simple-SQuAD and Original. The
Simple-SQuAD dataset contains the reconstructed context with the calculated
character offsets for each answer. The Original dataset contains the original
context itself but with only those questions whose answers are present in the
Simple-SQuAD dataset. We create the new Original dataset to ensure an equal
number of question-answer pairs, leading to a fair comparison. This Original
dataset represents the SQuAD candidate for the benchmarking process.

##  4 Benchmarking Experiments

This section highlights the models and experiments performed for benchmarking
the Simple-SQuAD. We compare the results obtained for Simple-SQuAD against the
SQuAD dataset.

###  4.1 Model Used

For benchmarking Simple-SQuAD, we use two different variations of RoBERTa as
introduced by Liu et al. Liu et al. (2019). RoBERTa is a replication study of
BERT pretraining, which is trained on more extensive training data with bigger
batches, longer sequences, and dynamically changing masking patterns.
Consequently, RoBERTa achieves better results over BERT and attains state-of-
the-art results on GLUE, RACE, and SQuAD.

###  4.2 Experimental Settings

We perform a dataset-based ablation study, experimenting with multiple
variants of input datasets for each model. Firstly, we finetune the model on
the SQuAD and the Simple-SQuAD dataset separately for $2$ epochs. We then
finetune the Simple-SQuAD trained model on the SQuAD dataset and the SQuAD-
trained model on the Simple-SQuAD dataset for $2$ epochs each. We benchmark
the results for each of these combinations of the dataset input to better
infer the effect of simplifying sentences in the original dataset.

For benchmarking, we use $442$ training articles containing $78,810$ questions
and $48$ development articles containing $9880$ questions. Thus, we have a
$90$:$10$ and an approximate $89$:$11$ train-test split based on the number of
articles and the number of questions, respectively. We used $10\%$ of the
training examples as a validation set for both our models.

For both $RoBERTa_{Base}$ and $RoBERTa_{Large}$, we use a maximum sequence
length of $380$, a stride of $128$, a maximum query length of $64$, and a
maximum answer length of 30. We use a learning rate of $1*10^{-5}$ with a
weight decay of $0.01$.

###  4.3 Results

This section outlines the result of the ablation study to determine the effect
of text simplification on the question answering downstream task in the SQuAD
dataset. We observe that text simplification improves the predictive
performance of both $RoBERTa_{base}$ and $RoBERTa_{large}$.

Model | Input | Exact | F1
---|---|---|---
$RoBERTa_{Base}$ | SQuAD | 0.787 | 0.863
$RoBERTa_{Base}$ | Simple-SQuAD | 0.786 | 0.866
$RoBERTa_{Base}$ | Simple-SQUAD $\to$ SQuAD | 0.799 | 0.876
$RoBERTa_{Base}$ | SQuAD $\to$ Simple-SQuAD | 0.803 | 0.878

Table 5: Benchmarking Results for $RoBERTa_{Base}$.

Table 5 illustrates the results for the $RoBERTa_{Base}$ model. We observe an
increase of $2.03\%$ in Exact Match and $1.74\%$ in F1 when fine-tuning it
with SQuAD followed by Simple-SQuAD, in contrast with the model when trained
on only SQuAD.

Model | Input | Exact | F1
---|---|---|---
$RoBERTa_{Large}$ | SQuAD | 0.835 | 0.905
$RoBERTa_{Large}$ | Simple-SQuAD | 0.852 | 0.917
$RoBERTa_{Large}$ | Simple-SQUAD $\to$ SQuAD | 0.838 | 0.907
$RoBERTa_{Large}$ | SQuAD $\to$ Simple-SQuAD | 0.836 | 0.908

Table 6: Benchmarking Results for $RoBERTa_{Large}$.

Table 6 illustrates the results for the $RoBERTa_{Large}$ model. Similar to
$RoBERTa_{Base}$, we observe an increase of $2.04\%$ in Exact Match and
$1.33\%$ in F1 when fine-tuning it with Simple-SQuAD when compared to model
when trained on SQuAD.

Category | Original | Transferred
---|---|---
| Inter-Event
---
Splitting
| Although his administrative abilities had been noticed,
---
on the eve of the U.S. entry into World War II he had
never held an active command above a battalion and
was far from being considered by many as a potential
commander of major operations.
| Although his administrative abilities had been noticed
---
on the eve of the U.S. entry into World War II he had
never held an active command above a battalion.
He was far from being considered by many as a
potential commander of major operations.
| Intra-Event
---
Splitting
| Clark also claimed that Abdul gave him preferential
---
treatment on the show due to their affair.
| Clark also claimed that Abdul gave him preferential
---
treatment on the show. This was due to their affair.

Table 7: Examples of event-based splitting.

(a) BLEU

(b) SARI

(c) FKGL

Figure 2: Plots illustrating sentence-length based transfer analysis

##  5 Discussion

###  5.1 Edit Analysis

We conduct an event-based analysis of the edits performed by the transfer
model to convert the originally complex sentences into simpler forms. We use
the definition of a linguistic event as defined in Pustejovsky Pustejovsky
(1991). We perform the analysis on $50$ parallel sentences in their pre and
post simplification forms, classifying each edit into the following two
classes: Inter-Event Splitting and Intra-Event Splitting.

Inter-Event Splitting denotes the type of edit in which the model splits two
different events. As illustrated in Table 7, the events of notice and consider
are split into two different sentences, thus simplifying the complex sentence
containing the two events. On the other hand, Intra-Event Splitting is the
type of edit in which all the simplified sentences contain the same event as
the original sentence, as is illustrated for the event claim in Table 7.

In our analysis, we found $32\%$ of the instances to show Inter-Event
Splitting, showing that our model can capture event boundaries. On the other
hand, $60\%$ of the total instances show successful Intra-Event Splitting,
illustrating that the model can capture intra-event detailing boundaries.
Interestingly, $8\%$ of the total instances displayed unsuccessful attempts of
Intra-Event Splitting, which can be improved in future work.

###  5.2 Transfer Analysis

For transfer analysis, we divide all the sentences in Simple-SQuAD into four
buckets based on the original sentence length split equally in terms of word-
level tokenization ( $0-20$ tokens, $20-27$ tokens, $27-36$ tokens and
$36-432$ tokens). We then compute the following three metrics on a sentence
level: BLEU score, SARI scores, and FKGL scores. In Figure 2, we observe that
the performance of the model varies with the sentence length.

  1. 1.

BLEU: Sentence-preservation, measured through average BLEU score, is directly
proportional to the sentence length. However, the standard deviation of the
BLEU scores first decreases then increases as we increase the sentence length.

  2. 2.

SARI: Lexical simplicity, measured through average SARI score, first increases
then decreases as we increase the sentence length. However, the standard
deviation of the SARI scores first decreases then increases as sentence length
increases.

  3. 3.

FKGL: Text readability, measured via sentence-level FKGL score, was computed
for both transferred and original sentences. We observe that sentence-level
FKGL score of transferred sentences is directly proportional to the sentence
length. Whereas sentence-level FKGL scores for original sentences first
decreases then increases with sentence length. Moreover, the mean sentence-
level FKGL score for the transferred sentences was always lower than that for
the original sentences regardless of the sentence length.

##  6 Conclusion

In this work, we study the effect of text simplification in the comprehension
based question-answering downstream task using the SQuAD dataset. For Simple-
SQuAD corpus creation, we use a transformers based style transfer model to
transfer complex sentences to sequences of simple sentences while retaining
the original meaning. We further use post-editing techniques to reduce noise
from the dataset, followed by the use of heuristics to find required offsets
for an answer in each question-answer pair. We prove the efficacy of our model
using automated evaluation as well as human evaluation. We then benchmark
Simple-SQuAD using two different variants of RoBERTa and perform an ablation
study to investigate the effects of text simplification using four different
variations of input. We prove that text simplification in the question-
answering downstream task increases the predictive performance of the models.
We further conduct edit-type analysis and sentence-length analysis to give
insights about the transfer process. Future work may include improving style
transfer performance using a more extensive corpus for text simplification and
exploring effects of text simplification for other downstream tasks like text
summarization, sentiment analysis.

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