$k^{2}=\frac{1}{(2,q-1)}q^{\frac{1}{8}n\left(n+2\right)}\prod_{i=1}^{n/2}(q^{i}+(-1)^{i})$
(7.6)

and hence $q^{n/2-1}+1$ divides $k^{2}$. Moreover, (7.6) is not fulfilled for
$(n/2-1,q)=(6,2)$. Then $\Phi_{(n-2)f}^{\ast}(p)\mid k^{2}$, where
$\Phi_{(n-2)f}^{\ast}(p)>1$, by [49], P1.7, and hence
$\Phi_{(n-2)f}^{\ast}(p)^{1/2}\mid\left|G_{B}\right|$. Moreover,
$\left(\Phi_{(n-2)f}^{\ast}(p),\left|X_{x}\right|\right)=1$, since $n/2$ is
odd and $n\geqslant 6$, and
$\left(\Phi_{(n-2)f}^{\ast}(p),\left|\mathrm{Out(}X\mathrm{)}\right|\right)=1$,
as $\Phi_{(n-2)f}^{\ast}(p)\equiv 1\pmod{(n-2)f}$ by [39], Proposition
5.2.1.5.(i). Thus $\left(\Phi_{(n-2)f}^{\ast}(p),\left|G_{x}\right|\right)=1$
and $\left(\Phi_{(n-2)f}^{\ast}(p),\left[G_{B}:X_{B}\right]\right)=1$ as both
$\left[G_{x}:X_{x}\right]$ and $\left[G_{B}:X_{B}\right]$ divide
$\left|\mathrm{Out(}X\mathrm{)}\right|$. Then
$\Phi_{(n-2)f}^{\ast}(p)^{1/2}\mid\left|X_{B}\right|$ and
$\Phi_{(n-2)f}^{\ast}(p)^{1/2}\mid\left[X:X_{B}\right]$ as
$\left|G_{x}\right|=\lambda(k+1)\left|G_{x,B}\right|$ and
$\left|G_{B}\right|=k\left|G_{x,B}\right|$. Therefore
$\Phi_{(n-2)f}^{\ast}(p)^{1/2}$ divides both $\left|X_{B}^{\char
94\relax}\right|$ and $\left[SL_{n}:X_{B}^{\char 94\relax}\right]$. Moreover,
$p$ divides $\left|X_{B}^{\char 94\relax}\right|$ by Corollary 4.10(ii), as
$\left|X_{B}^{\char 94\relax}\right|_{p}=\left|X_{B}\right|_{p}$ and
$n\geqslant 6$. Then $X_{B}$ lies in a parabolic subgroup of type $P_{2}$ or
$P_{1,n-1}$ of $SL_{n}(q)$ by Proposition 5.3(3).

Assume that $X_{B}^{\char 94\relax}$ lies in a parabolic subgroup of type
$P_{2}$ of $SL_{n}(q)$. Then $X_{B}$ fixes a line $\ell$ of $PG_{n-1}(q)$, and
hence $\ell$ is non-degenerate by [39], Propositions 4.1.3.(II) and
4.1.19.(II), since $\Phi_{(n-2)f}^{\ast}(p)^{1/2}\mid\left|X_{B}^{\char
94\relax}\right|$. Also $X_{B}\leqslant SL_{2}(q)\circ Sp_{n-2}(q)$. Since
$\Phi_{(n-2)f}^{\ast}(p)^{1/2}$ divides
$\left|\left(X_{B}\right)_{\ell^{\perp}}^{\ell^{\perp}}\right|$ and
$\left[SL_{n-2}(q):\left(X_{B}\right)_{\ell^{\perp}}^{\ell^{\perp}}\right]$,
where $\left(X_{B}\right)_{\ell^{\perp}}^{\ell^{\perp}}$ denotes the group
induced by $\left(X_{B}\right)_{\ell^{\perp}}$ on $\ell^{\perp}$, it follows
that either $n-2=4$ and $q=7$, or
$\left(X_{B}\right)_{\ell^{\perp}}^{\ell^{\perp}}\leqslant
GL_{1}(q^{n-2})\cdot Z_{n-2}$ and $p\mid n-2$ by Proposition 5.3(1). The
former does not fulfill (7.6), the latter implies
$\left|X_{B}\right|_{p}\leqslant q\cdot(n-2)_{p}$. Then
$q^{\frac{n^{2}-12}{12}}\leqslant q(n-2)$ by Corollary 4.10(ii), which yields
$n=6$ and $q\leqslant 5$, as $n\geqslant 6$. However, none of these values
fulfills (7.6).

Assume that $X_{B}$ lies in a parabolic subgroup of type $P_{1,n-1}$ of
$SL_{n}(q)$. Then $X_{B}$ is a subgroup of $X$ fixing a point $\left\langle
w\right\rangle$ of $PG_{n-1}(q)$. Then $X_{B}\leqslant[q^{n-1}]:Z_{q-1}\circ
Sp_{n-2}(q)$ by [39], Proposition 4.1.19.(II). A similar argument to that used
above shows that $\Phi_{(n-2)f}^{\ast}(p)^{1/2}$ divides both
$\left|\left(X_{B}\right)_{\left\langle w\right\rangle^{\perp}/\left\langle
w\right\rangle}^{\left\langle w\right\rangle^{\perp}/\left\langle
w\right\rangle}\right|$ and
$\left[Sp_{n-2}(q):\left(X_{B}\right)_{\left\langle
w\right\rangle^{\perp}/\left\langle w\right\rangle}^{\left\langle
w\right\rangle^{\perp}/\left\langle w\right\rangle}\right]$. Thus
$\left(X_{B}\right)_{\left\langle w\right\rangle^{\perp}/\left\langle
w\right\rangle}^{\left\langle w\right\rangle^{\perp}/\left\langle
w\right\rangle}\leqslant GL_{1}(q^{n-2})\cdot Z_{n-2}$ and hence
$\left|X_{B}\right|_{p}\leqslant q^{n-1}\cdot(n-2)_{p}$. Then
$q^{\frac{n^{2}-12}{12}}\leqslant q^{n-1}(n-2)$ by Corollary 4.10(ii). Thus
either $n=6$ or $n=10$, as $n/2$ is odd. If $n=6$, then (7.6) becomes

$k^{2}=\frac{q^{6}\left(q^{2}+1\right)\left(q-1\right)^{2}}{(2,q-1)}\left(q^{2}+q+1\right),$

and we reach a contradiction by [49], A7.1.
If $n=10$, then

$k^{2}=\frac{q^{15}\left(q-1\right)\left(q^{2}+1\right)\left(q^{3}-1\right)\left(q^{4}+1\right)\left(q^{5}-1\right)}{(2,q-1)},$

and we obtain a contradiction, as $k+1$ divides $(q+1)^{3}(q^{3}+1)(2,q-1)f$
by Lemma 3.2(3).
Finally, assume that case (3) holds. If
$(X,Y)=(PSp_{4}(7),2^{4}.O_{4}^{-}(2))$, then $\Phi_{4}^{\ast}(7)=25$ is the
highest power of $5$ dividing $\left|X\right|$. Since
$5\mid\left|X_{x}\right|$ and $\left[X:X_{x}\right]$ is a square, it follows
that $5^{2}$ divides $\left|X_{x}\right|$ and hence $\left|Y\right|$, a
contradiction.

If $(X,Y)=(PSp_{4}(3),2^{4}.\Omega_{4}^{-}(2))$, then
$5\mid\left|X_{x}\right|$ and hence $k\mid 2^{3}\cdot 3^{2}$. Then
$k=3,4,8,9$, since $k+1\mid\left|G\right|$, with $G\leqslant
PSp_{4}(3).Z_{2}$, by Lemma 3.2(3). However, $X$ does not have $k^{2}$ as a
transitive permutation representation degree for any such values of $k$ by
[20]. ∎

######  Lemma 7.7.

$X$ is not isomorphic to $PSU_{n}(q)$ for $n\geqslant 3$.

###### Proof.

Assume that $X\cong PSU_{n}(q^{1/2})$, with $n\geqslant 3$. Then $X_{x}$ lies
in a large maximal geometric subgroup $Y$ of $X$ by Theorem 7.1, and hence $Y$
is one of the groups listed in [1], Proposition 4.17. Also
$\left(\Phi_{ef}^{\ast}(p),\left|X_{x}\right|\right)>1$, and hence
$\left(\Phi_{ef}^{\ast}(p),\left|Y\right|\right)>1$, where $e=n$ or $n-1$
according as $n$ is odd or even respectively.

Assume that $n$ is odd. Filtering the list of subgroups in [1], Proposition
4.17, with respect to the property of
$\left(\Phi_{nf}^{\ast}(p),\left|Y\right|\right)>1$, we see that the unique
admissible case is when $Y$ is a $\mathcal{C}_{3}$-subgroup of $X$ of type
$GU_{n/3}(3^{3})$. Then $Y\cong Z_{7}.PSU_{n/3}(3^{3}).Z_{(n/3,7)}.Z_{3}$ by
[39], Proposition 4.3.6(II). It follows from Lemma 4.8(1) that
$\left|X\right|\leqslant\left|\mathrm{Out(}X\mathrm{)}\right|^{2}\cdot\left|Y\right|\cdot\left|Y\right|_{p^{\prime}}^{2}$
which yields inequality (6) in the proof of Proposition 4.3 of [9] for
$t=q=3$. Thus $(n/t,t)=(1,3)$, and hence $k^{2}=\allowbreak 288$, which is a
contradiction.

Assume that $n$ is even. Recall that $n\geqslant 4$ by Remark 4.2. The
constraint $\left(\Phi_{(n-1)f}^{\ast}(p),\left|Y\right|\right)>1$ reduces the
list of the subgroups in [1], Proposition 4.17 to the case
$Y\in\mathcal{C}_{1}(X)$. Then $Y$ is the stabilizer in $X$ of a non-isotropic
point of $PG_{n-1}(q)$ by [39], Propositions 4.1.4.(II) and 4.1.18.(II).
Moreover, $X_{x}=Y$ by Lemma 3.6(1) and by [39], Proposition 4.1.4.(I). So,
$k^{2}=q^{(n-1)/2}\frac{q^{n/2}-1}{(q^{1/2}+1)}$, which is exactly (6.3) in
Lemma 6.3, with $s=q^{1/2}$. Hence, the same argument used in Lemma 6.3 rules
out this case. ∎

######  Lemma 7.8.

$X$ is not isomorphic to $P\Omega_{n}(q)$, with $nq$ odd, for $n\geqslant 5$.

###### Proof.

Assume that $X\cong P\Omega_{n}(q)$ with $nq$ is odd and $n\geqslant 5$. The
group $X_{x}$ lies in a large maximal geometric subgroup $Y$ of $X$ such that
$\left(\Phi_{(n-1)f}^{\ast}(p),\left|Y\right|\right)>1$ by Theorem 7.1. Then
$Y$ is one of the groups listed in [1], Proposition 4.23, and hence
$Y\in\mathcal{C}_{1}(X)$, since
$\left(\Phi_{(n-1)f}^{\ast}(p),\left|Y\right|\right)>1$. Thus either $X\cong
P\Omega_{7}(q)$, with $q=3,5$, and $Y$ of type
$O^{\varepsilon^{\prime}}_{1}(q)\wr S_{7}$, or $Y\in\mathcal{C}_{1}(X)$. The
former is ruled out since $13\mid\left|X\right|$, but
$13^{2}\nmid\left|X\right|$, and $13\nmid\left|X_{x}\right|$. Thus,
$Y\in\mathcal{C}_{1}(X)$ and hence $Y\cong\Omega_{n-1}^{-}(q).Z_{2}$ is the
stabilizer of a non-singular point of $PG_{n-1}(q)$ by [39], Propositions
4.1.6.(II) and 4.1.20.(II). Moreover, $X_{x}=Y$ by Lemma 3.6(1) and by [39],
Proposition 4.1.6.(I). Then $k^{2}=q^{(n-1)/2}\frac{q^{(n-1)/2}-1}{2}$, as
$k^{2}=\left[X:X_{x}\right]$ and $q$ is odd. Hence,

$k^{2}-1=\frac{(q^{(n-1)/2}+1)\left(q^{(n-1)/2}-2\right)}{2}\text{.}$ (7.7)

On the other hand, $X$ has subdegrees $q^{(n-1)}-1$,
$q^{(n-3)/2}\frac{(q^{(n-1)/2}+1)}{2}$ and $(q-3)/2$ times
$q^{(n-3)/2}(q^{(n-1)/2}+1)$ (e.g. see [50], p. 331). Thus
$q^{(n-3)/2}\frac{(q^{(n-1)/2}+1)}{2}$ is a subdegree for $G$ too, and hence
$k+1\mid\frac{(q^{(n-1)/2}+1)}{2}$ by Lemma 3.2(3). So, $q^{(n-1)/2}-2\mid
k-1$ by (7.7), and hence $q=2$, $n=5$ and $k^{2}=6$, which is a contradiction.
∎

We recall some useful facts about the orthogonal group in order to tackle the
remaining cases $X\cong P\Omega_{n}^{\varepsilon}(q)$, where
$\varepsilon=\pm$.

Let $\mathbb{L}=GF(q^{2})$, $\mathbb{K}=GF(q)$, $q$ odd, let $\zeta$ be
generator of $\mathbb{L}^{\ast}$ and let $\omega=\zeta^{\frac{q+1}{2}}$ and
$\epsilon=\zeta^{\frac{q-1}{2}}$. Then $\omega^{2}$ is a generator of
$\mathbb{K}^{\ast}$, $\omega^{q}=-\omega$ and $\epsilon\zeta=\omega$, and
$\left\\{1,\omega\right\\}$ is a basis of $\mathbb{L}$ over $\mathbb{K}$. If
$\theta\in\mathbb{L}$, then $\theta=\alpha+\beta\omega$, with
$\alpha,\beta\in\mathbb{K}$. If we fix a basis
$\left\\{e_{1},...,e_{n/2}\right\\}$ of the $\mathbb{L}$-vector space
$\mathbb{L}^{n/2}$, where $n$ is an even integer, we may consider the
$\mathbb{K}$-isomorphism

$\Phi:\mathbb{L}^{n/2}\longrightarrow\mathbb{K}^{n},(\theta_{1},...,\theta_{n})\longmapsto(\alpha_{1},\beta_{1},...,\alpha_{n},\beta_{n})\text{,}$

where $\theta_{i}=\alpha_{i}+\beta_{i}\omega$ for each $i=1,...,n$. We will
denote both $y$ and $\Phi(y)$ simply by $y$. It will be clear from the context
if we are regarding $y$ as $n/2$-dimensional $\mathbb{L}$-vector, or as a
$n$-dimensional $\mathbb{K}$-vector. Hence, $\Phi(\left\langle
y\right\rangle_{\mathbb{L}})=\left\langle y,\omega
y\right\rangle_{\mathbb{K}}$ is a $2$-dimensional subspace of
$\mathbb{K}^{n}$.

A set of lines of $PG_{n-1}(q)$ partitioning the point set of $PG_{n-1}(q)$ is
said _spread_ of $PG_{n-1}(q)$. The set $\mathcal{S}$ of $1$-dimensional
$\mathbb{L}$-subspaces of $\mathbb{L}^{n/2}$ is mapped by $\Phi$ onto a spread
of $PG_{n-1}(q)$ by [21], Proposition 2.1. As for the vectors, we denote both
$\mathcal{S}$ and its image under $\Phi$ simply by $\mathcal{S}$.
If $Q_{\\#}$ is quadratic form on $\mathbb{L}^{n/2}$ polarized by the
symmetric bilinear form $F_{\\#}$, and
$T:\mathbb{L}\longrightarrow\mathbb{K},t\longmapsto t+t^{q}$ is the trace form
from $\mathbb{L}$ onto $\mathbb{K}$, it follows that $Q=T\circ Q_{\\#}$ is a
quadratic form on $\mathbb{K}^{n}$ polarized by the symmetric bilinear form
$F=T\circ F_{\\#}$. As pointed out in [21], $\Phi$ maps singular
$1$-dimensional subspaces of $\mathbb{L}^{n/2}$ onto totally singular
$2$-dimensional subspaces of $\mathbb{K}^{n}$, non-degenerate $1$-dimensional
subspaces of $\mathbb{L}^{n/2}$ onto non-degenerate $2$-dimensional subspaces
of $\mathbb{K}^{n}$ and provides an embedding of
$O_{n/2}^{\varepsilon^{\prime}}(q^{2})$ in $O_{n}^{\varepsilon}(q)$. The
approach in [21] is geometric, its group-theoretical counterpart is contained
in [39], Propositions 4.3.14, 4.3.16 and 4.3.20.
We are actually interested in
$\left(\varepsilon,\varepsilon^{\prime}\right)=\left(-,-\right)$ for $n/2$
even. Note that, $O_{n/2}^{-}(q^{2}).Z_{2}$ preserves $\mathcal{S}$ by [21],
Proposition 2.4. Then $X_{\mathcal{S}}\cong P\Omega_{n/2}^{-}(q^{2}).Z_{2}$ by
[21], Main Theorem (b), and by [39], Proposition 4.3.16.(II). We use these
information to prove the following lemma, which will play a central role in
excluding some of the cases where $X_{x}$ is a $\mathcal{C}_{3}$-subgroup of
$X$.

######  Lemma 7.9.

If $X\cong P\Omega_{n}^{\varepsilon}(q)$, where $\varepsilon=\pm$, then there
is a collineation $\alpha\in X-X_{\mathcal{S}}$ such that
$\Omega^{-}_{n/2-1}(q^{2})\leqslant X_{\mathcal{S},\mathcal{S}^{\alpha}}$.

###### Proof.

Let $\ell\in\mathcal{S}$ corresponding to a non-degenerate $2$-dimensional
subspace $U$ of $\mathbb{K}^{n}$ of type $-$. Such a subspace does exists.
Indeed, $U$ is either $\left\langle z,\omega z\right\rangle_{\mathbb{K}}$ or
$\left\langle z,\epsilon z\right\rangle_{\mathbb{K}}$ according to whether $q$
is equivalent to $1$ or $3$ modulo $4$ by [21], Proposition 2.3.(c). Moreover,
$D(Q)=\boxtimes$ by [39], Proposition 2.5.10.(ii), as $n/2$ is even. Then
$X_{\ell}\cong\left(\Omega_{2}^{-}(q)\times\Omega_{n-2}^{+}(q)\right).[4]$ by
[39], Proposition 4.1.6.(II). If $\ell^{\perp}$ denotes the
$(n-3)$-dimensional subspace of $PG_{n-1}(q)$ corresponding to $U^{\perp}$, we
see that $X(\ell^{\perp})$, the pointwise stabilizer in $X$ of $\ell^{\perp}$,
contains $\Omega_{2}^{-}(q)$.

The subspace $U$ is the image under $\Phi$ of a non-degenerate $1$-dimensional
subspace of $\mathbb{L}^{n/2}$. Hence, by [39], Proposition 4.1.6.(II), and by
[21], Proposition 2.3.(c), we see that

$X_{\mathcal{S},\ell}\cong\left\\{\begin{split}&\Omega_{n/2-1}(q^{2})\hphantom{.2}\hphantom{oho}\text{
for }q(n-1)/8\text{ odd}\\\ &\Omega_{n/2-1}(q^{2}).2\hphantom{oho}\text{ for
}q(n-1)/8\text{ even.}\\\ \end{split}\right.$

Therefore $X_{\mathcal{S}}(\ell^{\perp})=1$. Thus $X(\ell^{\perp})\neq
X_{\mathcal{S}}(\ell^{\perp})$, and hence there is $\alpha\in
X(\ell^{\perp})-X_{\mathcal{S}}(\ell^{\perp})$ such that
$\Omega^{-}_{n/2-1}(q^{2})\leqslant X_{\mathcal{S},\mathcal{S}^{\alpha}}$. ∎

Let $\mathbb{L}$, $\mathbb{K}$ and $\zeta$ as above, but this time $q$ can
also be even. Then $\left\\{1,\zeta\right\\}$ is a basis of $\mathbb{L}$ over
$\mathbb{K}$. If $\theta\in\mathbb{L}$, then $\theta=\alpha+\beta\zeta$, with
$\alpha,\beta\in\mathbb{K}$. If we fix a basis
$\left\\{e_{1},...,e_{n/2}\right\\}$ of the $\mathbb{L}$-vector space
$\mathbb{L}^{n/2}$, where $n$ is an even integer, we may consider the
$\mathbb{K}$-isomorphism

$\Psi:\mathbb{L}^{n/2}\longrightarrow\mathbb{K}^{n},(\theta_{1},...,\theta_{n})\longmapsto(\alpha_{1},\beta_{1},...,\alpha_{n},\beta_{n})\text{,}$

where $\theta_{i}=\alpha_{i}+\beta_{i}\zeta$ for each $i=1,...,n$. As done for
the map $\Phi$, we will denote both $y$ and $\Psi(y)$ simply by $y$, but
differently from $\Phi$, the map $\Psi$ works also for $q$ even.
Let $\varphi$ be a Hermitian form on $\mathbb{L}^{n/2}$ and let $Q$ be a
quadratic form on $\mathbb{K}^{n}$ defined by $Q(z)=\varphi(z,z)$. Then $\Psi$
maps totally isotropic subspaces of $\mathbb{L}^{n/2}$ onto totally singular
subspaces of $\mathbb{K}^{n}$, and non-degenerate subspaces of
$\mathbb{L}^{n/2}$ onto non-degenerate subspaces of $\mathbb{K}^{n}$ by [29],
Lemma 1.(iii), and provides an embedding of $U_{n/2}(q^{2})$ in
$O_{n}^{\varepsilon}(q)$.

The group $X$ preserves a non-degenerate quadric $\mathcal{Q}$ of
$PG_{n-1}(q)$, which is either elliptic or hyperbolic according to whether
$\varepsilon=-$ or $+$ respectively. A partition of $\mathcal{Q}$ by a set of
skew, singular lines of $PG_{n-1}(q)$ is said _spread of lines_ of
$\mathcal{Q}$. The isotropic $1$-dimensional subspaces of $\mathbb{L}^{n/2}$
are the points of an hermitian variety $\mathcal{H}$ of $PG_{n/2-1}(q^{2})$
and are mapped by $\Psi$ onto a spread of lines of $\mathcal{Q}$ by [29],
Theorem 1. We will denote both $\mathcal{H}$, and its image under $\Psi$,
simply, by $\mathcal{H}$. Then

$X_{\mathcal{H}}\cong\left[\frac{q+1}{\left(q+1,3-\varepsilon\right)}\right].PSU_{n/2}(q).\left[\mu\left(q+1,n/2\right)\right]\text{,}$
(7.8)

where $\mu=1$ or $(n/2,2)$ according as $q$ is odd or even, respectively, by
[29], Theorems 2 and 3, and by [39], Proposition 4.3.18.

######  Lemma 7.10.

If $X\cong P\Omega_{n}^{\pm}(q)$, then there is a collineation $\beta\in
X-X_{\mathcal{H}}$ such that

$\left[X_{\mathcal{H}}:X_{\mathcal{H},\mathcal{H}^{\beta}}\right]\mid
2(q+1,n/2)(q+1,n/2-2)q^{n-3}(q^{n/2}-(-1)^{n/2})(q^{n/2-1}-(-1)^{n/2-1}).$

###### Proof.

Let $w_{1},w_{2}$ be two orthonormal vectors of $\mathbb{L}^{n/2}$ with
respect $\varphi$, let $W=\langle w_{1},w_{2}\rangle$ and let $\ell$ be the
corresponding line of $PG_{n/2-1}(q^{2})$. Since $Q(x)=\varphi(x,x)$, it is
clear that $\langle w_{i}\rangle$, $i=1,2$, is a non-degenerate
$2$-dimensional subspace of type $-$ of $\mathbb{K}^{n}$. Then $W$ is a non-
degenerate $4$-dimensional subspace of type $+$ of $\mathbb{K}^{n}$ by
Proposition 2.5.11.(ii). If $\ell^{\perp}$ denotes the $n-3$ subspace of
$PG_{n-1}(q)$ corresponding to $W^{\perp}$, then $X\left(\ell^{\perp}\right)$
contains a subgroup isomorphic to $\Omega_{4}^{+}(q)$ inducing on
$\ell^{\perp}$ either the identity, or a central involution, by [39],
Proposition 4.1.6.(II). Thus
$\left|X\left(\ell^{\perp}\right)_{p}\right|=q^{2}$. On the other hand,
$\left|X_{\mathcal{H}}(\ell^{\perp})\right|_{p}=q$ by [39], Proposition
4.1.4.(II). Therefore, $X(\ell^{\perp})\neq X_{\mathcal{H}}(\ell^{\perp})$,
and hence there is $\beta\in X(\ell^{\perp})-X_{\mathcal{H}}(\ell^{\perp})$
such that $X_{\mathcal{H}}\left(\ell\right)\leqslant
X_{\mathcal{H},\mathcal{H}^{\beta}}$ by [39], Proposition 4.1.4.(II). Then
$\left[X_{\mathcal{H}}:X_{\mathcal{H}}(\ell)\right]$ divides
$2(q+1,n/2)(q+1,n/2-2)q^{n-3}(q^{n/2}-(-1)^{n/2})(q^{n/2-1}-(-1)^{n/2-1})2$
again by [39], Proposition 4.1.4(II). So does
$\left[X_{\mathcal{H}}:X_{\mathcal{H},\mathcal{H}^{\beta}}\right]$, as
$X_{\mathcal{H}}\left(\ell\right)\leqslant
X_{\mathcal{H},\mathcal{H}^{\beta}}$, which is the assertion. ∎

######  Lemma 7.11.

$X$ is not isomorphic to $P\Omega_{n}^{-}(q)$ for $n\geqslant 6$.

###### Proof.

Assume that $X\cong P\Omega_{n}^{-}(q)$ with $n\geqslant 6$. Actually,
$P\Omega_{6}^{-}(q)$ cannot occur by Lemma 7.7, since $P\Omega_{6}^{-}(q)\cong
PSU_{4}(q)$. Thus $n\geqslant 8$. The group $X_{x}$ lies in a large maximal
geometric subgroup $Y$ of $X$ by Theorem 7.1. Then either
$Y\in\mathcal{C}_{1}(X)$, or $Y$ is a $\mathcal{C}_{3}$-subgroup of $X$ either
of type $O_{n/2}^{-}(q^{2})$ or $GU_{n/2}(q)$ according as $n/2$ is even or
odd, respectively, by [1], Proposition 4.23, since
$\left(\Phi_{nf}^{\ast}(p),\left|Y\right|\right)>1$. Actually, the case
$Y\in\mathcal{C}_{1}(X)$ is ruled out by Theorem 3.5 of [33]. Thus, either
$Y\cong P\Omega_{n/2}^{-}(q^{2}).Z_{2}$ or
$Y\cong\frac{q+1}{\left(q+1,4\right)}.PSU_{n/2}(q).\left[\left(q+1,n/2\right)\right]$
by [39], Propositions 4.3.16.(II) and 4.3.18.(II). Moreover, $X_{x}=Y$ by
Lemma 3.6(1) and by [39], Propositions 4.3.16.(I) and 4.3.18.(I).

Assume that $X_{x}\cong P\Omega_{n/2}^{-}(q^{2}).Z_{2}$. Then

$k^{2}=\frac{1}{2}q^{n^{2}/8}\prod_{i=0}^{n/4-1}(q^{4i+2}-1)\text{.}$ (7.9)

Then $n^{2}/8$ is even, as $n/2$ is even. If $q$ is even, then the highest
power of $2$ dividing $k^{2}$ is $2^{f\frac{n^{2}}{8}-1}$, but this is a
contradiction as $fn^{2}/8-1$ is odd. Therefore, $q$ is odd and hence (7.9)
becomes

$k^{2}=q^{n^{2}/8}\frac{(q^{2}-1)^{n/4}}{2}\prod_{i=1}^{n/4-1}\left(\frac{q^{4i+2}-1}{q^{2}-1}\right)\text{.}$
(7.10)

Moreover, the actions of $X$ on a suitable $X$-orbit of spreads of
$PG_{n-1}(q)$ and on the point set of $\mathcal{D}$ are equivalent (see
above). Thus we may identify the two sets.
Suppose that $n/4$ is even and let $2^{j}$ be the highest power of $2$
dividing $q^{2}-1$. Then $2^{jn/4-1}$ must be a square, as
$q^{n^{2}/8}\prod_{i=1}^{n/4-1}\left(\frac{q^{4i+2}-1}{q^{2}-1}\right)$ is
odd, whereas it is not, as $jn/2-1$ is odd, and so we reach a contradiction.
Thus $n/4$ is odd and hence $n\geqslant 12$, as $n\geqslant 8$.

Since there is $\alpha\in X-X_{x}$ such that $\Omega_{n/2-1}(q^{2})\leqslant
X_{x,x^{\alpha}}$ by Lemma 7.9, it follows that
$\Omega_{n/2-1}(q^{2})\leqslant G_{x,x^{\alpha}}$. Thus $k+1\mid
8q^{n/2-2}(q^{n/2}+1)f$, as $k+1$ divides
$\left[G_{x}:G_{x,x^{\alpha}}\right]$ by Lemma 3.2(3), and hence $k+1\mid
8(q^{n/2}+1)f$, as $p$ divides $k$. So,

$q^{n^{2}/8}(q^{2}-1)^{n/4}<k<64(q^{n/2}+1)^{2}f^{2}<16(q^{n/2}+1)^{2}q^{2}\text{,}$

which has no solutions for $n\geqslant 12$.

Assume that
$X_{x}\cong\frac{q+1}{\left(q+1,4\right)}.PSU_{n/2}(q).\left[\left(q+1,n/2\right)\right]$.
Let $\mathcal{Q}$ be the elliptic quadric of $PG_{n-1}(q)$ preserved by $X$.
Then the actions of $X$ on the sets of the spreads of lines of $\mathcal{Q}$
and on the point set of $\mathcal{D}$ are equivalent. Therefore, we may
identify the points of $\mathcal{D}$ with the spreads of lines of the elliptic
quadric $\mathcal{Q}$. Hence

$k^{2}=q^{n(n-2)/8}\prod_{i=2}^{n/2-1}\left(q^{i}+\left(-1\right)^{i}\right)\text{,}$
(7.11)

where $n\geqslant 10$, as $n/2$ is odd and $n\geqslant 8$.
There is $\beta\in X-X_{x}$ such that $[X_{x}:X_{x,x^{\beta}}]$ divides
$2(q+1,n/2)(q+1,n/2-2)q^{n-3}(q^{n/2}+1)(q^{n/2-1}-1)$ by Lemma 7.10, since
$n/2$ is odd. Then

$k+1\mid f(q+1,n/2)(q+1,n/2-2)(q^{n/2}+1)(q^{n/2-1}-1)$ (7.12)

by Lemma 3.2(3), since $k$ is even and $p$ divides $k$. Now, comparing (7.11)
and (7.12), and using [1], Lemma 4.1(i), we obtain

$q^{n(n-2)/4-1}<k^{2}<q^{2}(q+1)^{4}(q^{n/2}+1)^{2}(q^{n/2-1}-1)^{2}<64q^{2n+4}$

and hence $n=10$, as $n/2$ is odd. Then (7.11) becomes

$k^{2}=q^{10}(q^{2}+1)(q^{3}-1)(q^{4}+1)$ (7.13)

which, together with (7.12), yields $k+1\mid f(q+1,15)(q^{5}+1)(q+1)$. Then
$k^{2}<225q^{2}(q+1)^{2}(q^{5}+1)^{2}$ which compared to 7.13 yields $q=2,3$.
However, these values of $q$ do not fulfill (7.13), and hence this case is
ruled out. This completes the proof. ∎

######  Lemma 7.12.

$X$ is not isomorphic to $P\Omega_{n}^{+}(q)$ for $n\geqslant 4$.

###### Proof.

Assume that $X\cong P\Omega_{n}^{+}(q)$ with $n\geqslant 4$. Actually,
$n\geqslant 8$ by Remark 4.2 and since $P\Omega_{6}^{+}(q)\cong PSL_{4}(q)$ is
ruled out in Lemma 7.2. The group $X_{x}$ lies in a large maximal geometric
subgroup $Y$ of $X$ by Theorem 7.1. Then $Y$ is one of the the groups listed
in [1], Proposition 4.23. Filtering such a list with respect to the constraint
$\left(\Phi_{(n-2)f}^{\ast}(p),\left|Y\right|\right)>1$, we see that either
$Y\in\mathcal{C}_{1}(X)$, or $Y$ is a $\mathcal{C}_{3}$-group of type
$O_{n/2}(q^{2})$ or $GU_{n/2}(q)$ according to whether $n/2$ is odd or even,
respectively, or $(X,Y)=(P\Omega_{8}^{+}(3),O_{1}(3)\wr
S_{8}),(P\Omega_{8}^{+}(3),2^{6}.\Omega_{6}^{+}(2))$. The latter is
immediately ruled out. Indeed, $13\mid\left|X\right|$ but
$13^{2}\nmid\left|X\right|$ and $13\nmid\left|Y\right|$.

Assume that $Y$ is a maximal $\mathcal{C}_{3}$-subgroup of $X$ of type
$O_{n/2}(q^{2})$. Then $X$ has one or two conjugacy classes of subgroups
isomorphic to $Y$ by [39], Propositions 4.3.20(I) and 2.5.10(i), according to
whether $q\equiv 3\pmod{4}$ or $q\equiv 1\pmod{4}$ respectively. In the former
case $X_{x}=Y$ by Lemma 3.6(1), whereas in the latter $G_{x}$ is a type 1
novelty with respect to $Y$ by Lemma 3.6(2). However, this is impossible by
[39], Tables 3.5.H–I. Thus $X_{x}=Y$ in each case, and hence

$k^{2}=q^{\left(n/2-1\right)^{2}}\frac{(q-1)^{2}(q+1)}{(4,q^{n/2}-1)}\cdot\frac{q^{n/2}-1}{q-1}$
(7.14)

by [39], Proposition 4.3.20.(II). Then $\frac{q^{n/2}-1}{q-1}$ is a square,
being coprime to the other factor of $\frac{4k^{2}}{(q+1)^{2}}$ or
$\frac{k^{2}}{(q+1)^{2}}$ according to whether $q\equiv 1\pmod{4}$ or $q\equiv
3\pmod{4}$ respectively. Hence $(n,q)=(10,3)$ by [49], A8.1, since $n/2$ is
odd. However, such a pair does not fulfill (7.14), and hence this case is
excluded.

Assume that $Y$ is a maximal $\mathcal{C}_{3}$-subgroup of $X$ of type
$GU_{n/2}(q)$ with $n/2$ even. Then $X$ has two conjugacy classes of subgroups
isomorphic to $Y$ by [39], Proposition 4.3.18.(I). If $X_{x}<Y$, then $G_{x}$
is a type 1 novelty with respect to $Y$ by Lemma 3.6(2). However, this is
impossible by [39], Tables 3.5.H–I. Thus $X_{x}=Y$ and hence

$k^{2}=\frac{1}{2}q^{n\left(n-2\right)/8}\prod_{i=1}^{n/2-1}(q^{i}+(-1)^{i})$
(7.15)

Arguing as in the $P\Omega^{-}(q)$-case, and bearing in mind that $n/2$ is
even, we see that

$k+1\mid f(q+1,n/2)(q+1,n/2-2)(q^{n/2}-1)(q^{n/2-1}+1)$ (7.16)

and hence

$q^{n(n-2)/4-1}/2<k^{2}<q^{2}(q+1)^{4}(q^{n/2}+1)^{2}(q^{n/2-1}-1)^{2}<64q^{2n+4},$

which yields either $n=8$, or $n=12$ and $q<64$. It is easy to check that,
there are no $q$ less than $64$ fulfilling 7.15 for $n=12$. Thus $n=8$ and
hence (7.16) becomes

$k^{2}=\frac{1}{2}q^{6}\left(q^{2}+1\right)(q-1)^{2}\left(q^{2}+q+1\right).$

So $q^{2}+q+1$ is a square. However, this is impossible by [49], A7.1. Thus
$Y$ is not a $\mathcal{C}_{3}$-subgroup of $X$.

Assume that $Y\in\mathcal{C}_{1}(X)$. Then $Y$ is not a parabolic subgroup of
$X$ by [39], Proposition 4.1.20.(II), since
$\left(\Phi_{(n-2)f}^{\ast}(p),\left|Y\right|\right)>1$. Moreover, by [39],
Propositions 4.1.26.(II)–4.1.7.(II) and 2.5.10.(i), one of the following cases
occurs:

  1. (1)

$Y$ is the stabilizer in $X$ of a non-singular point of $PG_{n-1}(q)$:

    1. (a)

$Y\cong\Omega_{n-1}(q)$ with $q\equiv 1\pmod{4}$, or $q\equiv 3\pmod{4}$ and
$n/2$ even;

    2. (b)

$Y\cong\Omega_{n-1}(q).Z_{2}$, with $q\equiv 3\pmod{4}$ and $n/2$ odd;

    3. (c)

$Y\cong Sp_{n-2}(q)$, with $q$ even.

  2. (2)

$Y$ is the stabilizer in $X$ of a non-singular line of type $-$ of
$PG_{n-1}(q)$:

    1. (a)

$Y\cong\left(Z_{\frac{q+1}{(q+1,2)}}\times\Omega_{n-2}^{-}(q)\right).Z_{2}$,
with $q$ even, or $q\equiv 1\pmod{4}$;

    2. (b)

$Y\cong\left(Z_{\frac{q+1}{2}}\times\Omega_{n-2}^{-}(q)\right).[4]$, with
$q\equiv 3\pmod{4}$ and $n/2$ odd;

    3. (c)

$Y\cong Z_{2}.\left(Z_{\frac{q+1}{4}}\times P\Omega_{n-2}^{-}(q)\right).[4]$,
with $q\equiv 3\pmod{4}$ and $n/2$ even.

The group $X$ has a unique conjugacy class of subgroups isomorphic to $Y$ in
case (1c) and in cases (2a)–(2c) by [39], Propositions 4.1.6.(I)–4.1.7.(I).
Then $X_{x}=Y$ in these cases by Lemma 3.6(1).

In cases (1a) and (1b), $X$ has two conjugacy classes of subgroups isomorphic
to $Y$ again by [39], Proposition 4.1.6.(I). If $X_{x}<Y$, then $G_{x}$ is a
type 1 novelty with respect to $Y$ by Lemma 3.6(2), but this is impossible by
[39] Tables 3.5.H–I. Thus, $Y=X_{x}$ in each case.

Assume that $X_{x}$ is the stabilizer in $X$ of a non-singular point of
$PG_{n-1}(q)$. As pointed in [50], Section 5 and (1.b), the lengths of the
$X_{x}$-orbits distinct from $\left\\{x\right\\}$ are as follows:

$q\equiv 1\pmod{4}$:

one orbit of length $q^{n-2}-1$, one of length $q^{n/2-1}(q^{n/2-1}+1)/2$,
$(q-1)/4$ ones of length $q^{n/2-1}(q^{n/2-1}-1)$, and $(q-5)/4$ ones of
length $q^{n/2-1}(q^{n/2-1}+1)$.

$q\equiv 3\pmod{4}$:

one orbit of length $q^{n-2}-1$, one of length $q^{n/2-1}(q^{n/2-1}-1)/2$,
$(q-3)/4$ ones of length $q^{n/2-1}(q^{n/2-1}-1)$, and $(q-3)/4$ others of
length $q^{n/2-1}(q^{n/2-1}+1)$

$q\equiv 0\pmod{2}$:

one of length $q^{n-2}-1$, $q/2$ ones of length $q^{n/2-1}(q^{n/2-1}-1)$ and
$(q-2)/2$ ones of length $q^{n/2-1}(q^{n/2-1}+1)$.

In each of the three cases the greatest common divisor of the lengths of the
$X_{x}$-orbits is $1$. Thus
$k+1\mid(2,q-1)\left|\mathrm{Out(}X\mathrm{)}\right|$, since
$\frac{k+1}{\left(k+1,\left|\mathrm{Out(}X\mathrm{)}\right|\right)}$ divides
the length of each $X_{x}$-orbit distinct from ${\\{x\\}}$ by Lemma 3.5(2).
Hence, $k<(2,q-1)\cdot(4,q^{n/2}-1)\cdot 2\theta f$, where either $\theta=3$
and $n=8$, or $\theta=1$. Since $k^{2}=[X:X_{x}]$, it follows that
$k^{2}_{p}=q^{(n-2)/2}$, hence $q^{(n-2)/4}\mid k$. Thus
$q^{(n-2)/4}<(2,q-1)\cdot(4,q^{n/2}-1)\cdot 2\theta f$ and hence either $n=8$
and $q=4,9$, or $n=10$ and $q=3$. Therefore, either $X\cong
P\Omega_{8}^{+}(4)$ and $X_{x}\cong Sp_{6}(2)$, or $X\cong P\Omega_{8}^{+}(9)$
and $X_{x}\cong\Omega_{7}(9)$, or $X\cong P\Omega_{10}^{+}(3)$ and
$X_{x}\cong\Omega_{9}(3).Z_{2}$. The first two cases are ruled out, since
$\left[X:X_{x}\right]$ is not a square for any of them, whereas in the latter
$k=99$. Then $b=9900\lambda$, where $\lambda\mid 99$ and $\lambda\geqslant 2$.
Thus $3^{16}\cdot 7\cdot 13\cdot 41\mid\left|X_{B}\right|$ and hence
$\Omega_{9}(3)\unlhd X_{B}\leqslant\Omega_{9}(3).Z_{2}$ by [14], Tables
8.58–8.59 and 8.66–8.67. So, either $[X:X_{B}]=99^{2}$ or
$[X:X_{B}]=2*99^{2}$, and hence $b\mid 8*99^{2}$ since
$\left|\mathrm{Out(}X\mathrm{)}\right|=4$. However, this is impossible since
$100$ divides $b$. So, this case is ruled out.

Finally, assume that $X_{x}$ is the stabilizer in $X$ of a non-singular line
of type $-$ of $PG_{n-1}(q)$. Then
$\left|X_{x}\right|=\left|\Omega_{n-2}^{-}(q)\right|(q+1)\mu$, where

$\centering\mu=\left\\{\begin{split}2&\hphantom{ciao}q\text{ even, or }q\equiv
3\pmod{4}\text{ and }n/2\text{ odd}\\\ 1/2&\hphantom{ciao}q\equiv
3\pmod{4}\text{ and }n/2\text{ even}\\\ 1&\hphantom{ciao}q\equiv
1\pmod{4}\text{.}\\\ \end{split}\right.\@add@centering$ (7.17)

Set $\rho=\frac{\left(q^{n/2-1}+1,4\right)}{\left(q^{n/2}-1,4\right)\mu}$.
Then $\rho$ is $2$ for $q\equiv 3\pmod{4}$ and $n/2$ even, and $1/2$
otherwise, and hence

$k^{2}=q^{n-2}\rho(q-1)^{2}\left[\frac{1}{(q+1)}\left(\frac{q^{n/2}-1}{q-1}\right)\left(\frac{q^{n/2-1}-1}{q-1}\right)\right]\text{.}$
(7.18)

Assume that $n/2$ is odd. Then $\frac{q^{n/2}-1}{q-1}$ is odd and
$\frac{q^{n/2-1}-1}{q^{2}-1}$ is an integer. Moreover, they are coprime. Then
$\frac{q^{n/2}-1}{q-1}$ is a square, since the maximal odd divisor of
$\frac{(q-1)^{2}}{\left(4,q^{n/2}-1\right)}$ is a square too. Then
$(n,q)=(10,3)$ by [49], A7.1 and A8.1, since $n\geqslant 8$. However, such
$(n,q)$ does not fulfill (7.18).

Assume that $n/2$ is even. Arguing as above, we see that
$\frac{q^{n/2-1}-1}{q-1}$ is a square and hence $(n,q)=(12,3)$ by [49], A7.1
and A8.1, since $n\geqslant 8$, but such $(n,q)$ does not fulfill (7.18). This
completes the proof. ∎

Now, the proof of Theorem 1.1 is a summary of the previous results.

###### Proof of Theorem 1.1.

Suppose that $X$ is a simple group. Then $X$ is classical by Theorem 1 of
[44]. Note that $X\cong$ ${}^{2}G_{2}(3)^{\prime}$ is classical, since
${}^{2}G_{2}(3)^{\prime}\cong PSL_{2}(8)$. If $X\cong PSL_{2}(q)$, then
assertions (1) and (2) follow from Proposition 4.1. If $X\ncong PSL_{2}(q)$,
then the followings hold by Remark 4.2:

  1. (1)

$n\geqslant 3$ and $X\cong PSL_{n}(q)$

  2. (2)

$n\geqslant 3$ and $X\cong PSU_{n}(q^{1/2})$ for $q$ square;

  3. (3)

$n\geqslant 4$ and $X\cong PSp_{n}(q)$;

  4. (4)

$n\geqslant 5$ and $X\cong P\Omega_{5}(q)$;

  5. (5)

$n\geqslant 6$ and $X\cong P\Omega_{n}^{-}(q)$;

  6. (6)

$n\geqslant 6$ and $X\cong P\Omega_{n}^{+}(q)$.

Cases (2)–(6) are excluded in Lemmas 7.6, 7.7, 7.8, 7.11 and 7.12,
respectively. Thus $X\cong PSL_{n}(q)$, with $n\geqslant 3$, and hence
assertions (3) and (4) follow from Lemma 7.2 and Proposition 7.5. ∎

## References

  * [1] S. H. Alavi, T. C. Burness, Large subgroups of simple groups, J. Algebra 421 (2015) 187–233.
  * [2] S. H. Alavi , M. Bayat, M. Biliotti, A. Daneshkhah, E. Francot, H. Guan, A. Montinaro, F. Mouseli, P. Rizzo, D. Tian, Y. Wang, X. Zhan, Y. Zhang, S. Zhou, Y. Zhu, Block designs with flag-transitive automorphism groups. submitted
  * [3] S. H. Alavi, Flag-Transitive Block Designs and Finite Simple Exceptional Groups of Lie Type, Graphs and Combinatorics 36 (2020) 1001–1014.
  * [4] S. H. Alavi, A note on two families of 2-designs arose from Suzuki-Tits ovoid, Algebra and Discrete Math. Accepted.
  * [5] S. H. Alavi, M. Bayat, A. Daneshkhah, Symmetric designs admitting flag-transitive and point-primitive automorphism groups associated to two dimensional projective special groups, Designs, Codes and Cryptography 79 (2016) 337–351.
  * [6] S. H. Alavi, M. Bayat, and A. Daneshkhah. Flag-transitive block designs and unitary groups. Monatshefte fur Mathematik, 193 (2020) 535–553.
  * [7] S. H. Alavi, M. Bayat, and A. Daneshkhah. Correction to: Flag-transitive block designs and unitary groups. Monatshefte fur Mathematik, 195 (2021) 371–376.
  * [8] S. H. Alavi, M. Bayat, and A. Daneshkhah, Finite exceptional groups of Lie type and symmetric designs. Submitted arxiv:1702.01257.
  * [9] S. H. Alavi, M. Bayat, and A. Daneshkhah. Flag-transitive block designs and unitary groups. Monatshefte fur Mathematik, 193 (2020) 535–553.
  * [10] S. H. Alavi, M. Bayat, and A. Daneshkhah. Correction to: Flag-transitive block designs and unitary groups. Monatshefte fur Mathematik, 195 (2021) 371–376.
  * [11] S. H. Alavi, A. Daneshkhah, and F. Mouseli. A classification of flag-transitive block designs. Journal of Algebraic Combinatorics, 2021. Accepted.
  * [12] S. H. Alavi, C. E. Praeger, On The Triple Factorisations of Finite Groups, J. Group Theory 14 (2011) 341–360.
  * [13] J. Bamberg, T. Penttila, Overgroups of Cyclic Sylow Subgroups of Linear Groups, Comm. Algebra, 36 (2008) 2503–2543.
  * [14] J. Bray, D. F. Holt, C. M. Roney-Dougal, The maximal subgroups of the low-dimensional finite classical groups. With a foreword by Martin Liebeck. London Mathematical Society Lecture Note Series, 407. Cambridge University Press, Cambridge, 2013.
  * [15] Á. Bereczky. Maximal overgroups of Singer elements in classical groups. Journal of Algebra, 234 (2000) 187–206.
  * [16] M. Biliotti, A. Montinaro, On flag-transitive symmetric designs of affine type. J. Combin. Des. 25 (2017) 85–97.
  * [17] M. Biliotti, A. Montinaro, P. Rizzo, $2$-$(v,k,\lambda)$ designs, with $(r,\lambda)=1$, admitting a solvable flag-transitive automorphism group of affine type. J. Combin. Des 27 (2019) 784–800.
  * [18] F. Buekenhout, A. Delandtsheer, J. Doyen, Finite linear spaces with flag-transitive sporadic groups, Unpublished note.
  * [19] F. Buekenhout, A. Delantsheer, J. Doyen, P. B. Kleidman, M. Liebeck, and J. Saxl, Linear Spaces with flag-transitive automorphism groups, Geom. Dedicata 36 (1990) 89-94.
  * [20] J. H. Conway, R. T. Curtis, R. A. Parker, R. A. Wilson, An Atlas of Finite Groups, Clarendon Press, Oxford, 1985.
  * [21] A. Cossidente, O. H. King, Maximal Subgroups of Finite Orthogonal Groups Stabilizing Spreads of Lines, Communications in Algebra, 34 (2006) 4291-4309.
  * [22] D. Davies, Automorphisms of designs, Phd thesis, University of East Anglia (1987).
  * [23] A. Delandtsheer, Flag-transitive finite simple groups, Arch. Math. (Basel) 47 (1986) 395–400.
  * [24] A. Delandtsheer, Finite Flag-transitive Linear Spaces with Alternating Socle, pp. 79–88 in Algebraic combinatorics and Applications, Eds. A. Betten, A Kohnert, R. Laue, A. Wassermann, Proc.of the Euroconference ALCOMA, held in Gö$\beta$wienstein, Germany, Springer, 1999.
  * [25] P. Dembowski, Finite Geometries, Springer, Berlin, Heidelberg, New York, 1968.
  * [26] A. Deviller, H. Liang, C. E. Praeger, B. Xia, OnGalg-transitive $2$-$(v,k,2)$ designs, J. Combin. Theory Ser. A 177 (2021) 105309.
  * [27] L. E. Dickson, Linear groups with an exposition of the Galois eld theory, Dover Publications, Inc., New York, 1958.
  * [28] J. D. Dixon, B. Mortimer, Permutation Groups. Springer Verlag, New York (1966).
  * [29] R. H. Dye, Maximal subgroups of finite orthogonal groups stabilizing spreads of lines, J. London Math. Soc. 33 (1986) 279–293.
  * [30] R. H. Dye, Spreads and Classes of Maximal Subgroups of $GL_{n}(q)$, $SL_{n}(q)$, $PGL_{n}(q)$ and $PSL_{n}(q)$, Annali di Matematica Pura e Applicata 68 (1991) 33–50.
  * [31] S. P. Galsby, F. Lübeck, A. C. Niemeyer, C. E. Praeger, Primitive prime divisors and the $n$-th cyclotomic polynomial. Journal of Austral. Math. Soc. 102 (2017) 122–135.
  * [32] R. Guralnick, T. Penttila, C. E. Praeger, J. Saxl, Linear groups with orders having certain large prime divisors, Proc. London Math. Soc. 78 (1999) 167–214.
  * [33] C. Hering, Transitive Linear Groups which contain irreducible subgroups of prime order, Geom. Dedicata 2 (1974) 425–460.
  * [34] M. D. Hestenes, Singer Groups, Can. J. Math. 22 (1970) 492–513.
  * [35] D. G. Higman and J. E. McLaughlin, Geometric ABA-groups, Illinois J. Math. 5 (1961) 382–397.
  * [36] D. R. Hughes, F.C. Piper, Projective Planes. Springer, New York, Berlin (1973).
  * [37] B. Huppert, Endliche gruppen I, Die Grundlehren der Mathematischen Wissenschaften, Band 134, Springer-Verlag, Berlin–New York, 1967\.
  * [38] P. B. Kleidman, The finite flag-transitive linear spaces with an exceptional automorphism group, 461 in: Finite geometries and combinatorial designs (Lincoln, NE, 1987), vol. 111 of Contemp. Math., 462 Amer. Math. Soc., Providence, RI, 1990, pp. 117–136.
  * [39] P. Kleidman, M. Liebeck, The subgroup structure of the finite classical groups. London Mathematical Society Lecture Note Series, 129\. Cambridge University Press, Cambridge, 1990.
  * [40] M. W. Liebeck, J. Saxl, The Primitive Permutation Groups of Odd Degree, Journal of the London Mathematical Soc. 31 (1985) 250–264.
  * [41] M. W. Liebeck, The classification of finite linear spaces with flag-transitive automorphism groups of affine type, J. Combin. Theory Ser. A 84 (1998) 196–235.
  * [42] A. Montinaro, M. Biliotti, E. Francot, Classification of $2-(v,k,\lambda)$ designs with $(r,\lambda)=1$ and $\lambda>1$, admitting a non-solvable flag-transitive automorphism group of affine type, J. Algebr. Comb. (2021). https://doi.org/10.1007/s10801-021-01075-1.
  * [43] A. Montinaro, Classification of the $2$-$(k^{2},k,\lambda)$ design, with $\lambda\mid k$, admitting a flag-transitive automorphism group $G$ of affine type. Submitted
  * [44] A. Montinaro, E. Francot, On Flag-transitive $2$-$(k^{2},k,\lambda)$ designs with $\lambda\mid k$, submitted.
  * [45] A. Montinaro, Flag-transitive, point-imprimitive symmetric $2$-$(v,k,\lambda)$ designs with $k>\lambda(\lambda-3)/2$. Submitted.
  * [46] C. E. Praeger, S. Zhou, Imprimitive flag-transitive symmetric designs. J. Comb. Theory Ser. A, 113 (2006) 1381–1395.
  * [47] D. Passman, Permutation groups. W. A. Benjamin, Inc., New York-Amsterdam 1968.
  * [48] C. Praeger, S. Zhou, Imprimitive flag-transitive symmetric designs. J. Comb. Theory Ser. A, 113 (2006) 1381–1395.
  * [49] P. Ribenboim, Catalan’s Conjecture: are $8$ and $9$ the only consecutive powers? Academic Press, Inc. Boston, 1994.
  * [50] J. Saxl. On finite linear spaces with almost simple flag-transitive automorphism groups. J. Combin. Theory Ser. A, 100 (2002) 322–348.
  * [51] G. M. Seitz, Flag-transitive subgroups of Chevalley group, Ann. of Math. (2) 97 (1973) 27–56.
  * [52] D. Tian, S. Zhou, Flag-transitive $2-(v,k,\lambda)$ symmetric designs with sporadic socle, Journal of Combinatorial Designs 23 (2015) 140–150.
  * [53] A. Wagner, On finite affine line transitive planes. Math Z 87 (1965) 1–11.
  * [54] R. Wilson, P. Walsh, J. Tripp, I. Suleiman, R. Parker, S. Norton, S. Nickerson, S. Linton, J. Bray, R. Abbott, An Atlas of Finite Groups Representation- Version 3, available at http://brauer.maths.qmul.ac.uk/Atlas/v3/
  * [55] P. H. Zieschang, Flag transitive automorphism groups of $2$-designs with $(r,\lambda)=1$, Journal of Algebra 118 (1988) 369–375.
  * [56] X. Zhan, S. Ding, Comments on “Flag-transitive block designs and unitary groups”, Monatshefte für Mathematik 195 (2021) 177–180.
  * [57] X. Zhan, S. Zhou, Flag-transitive non-symmetric $2$-designs with $(r,\lambda)=1$ and sporadic socle, Des. Codes Cryptography 81 (2016) 481–487.
  * [58] X. Zhan, S. Zhou, Non-symmetric 2-designs admitting a two-dimensional projective linear group, Designs, Codes and Cryptography 86 (12) (2018) 2765–2773.
  * [59] Y. Zhang, S. Zhou, Flag-transitive non-symmetric $2$-designs with $(r,\lambda)=1$ and Exceptional Groups of Lie type, The Electronic Journal of Combinatorics 27 (2020) P2.9.
  * [60] S. Zhou, Y. Wang, Flag-transitive non-symmetric $2$-designs with $(r,\lambda)=1$ and alternating socle, Electron. J. Combin. 22 (2015) Paper 2.6, 15.
  * [61] Y. Zhu, H. Guan, S. Zhou, Flag-transitive $2$-$(v,k,\lambda)$ symmetric designs with $(r,\lambda)=1$ and alternating socle, Frontiers of Mathematics in China 10 (2015) 1483–1496.
  * [62] The GAP Group, GAP ”Groups, Algorithms, and Programming, Version 4.11.0, 2020, available at http://www.gap.system.org


# Doodle to Search: Practical Zero-Shot Sketch-based Image Retrieval

Sounak Dey, Pau Riba††footnotemark: , Anjan Dutta, Josep Lladós
Computer Vison Center, UAB, Spain
<EMAIL_ADDRESS>These authors contributed equally to
this work. Yi-Zhe Song
SketchX, CVSSP, University of Surrey, UK
<EMAIL_ADDRESS>
###### Abstract

In this paper, we investigate the problem of zero-shot sketch-based image
retrieval (ZS-SBIR), where human sketches are used as queries to conduct
retrieval of photos from unseen categories. We importantly advance prior arts
by proposing a novel ZS-SBIR scenario that represents a firm step forward in
its practical application. The new setting uniquely recognizes two important
yet often neglected challenges of practical ZS-SBIR, (i) the large domain gap
between amateur sketch and photo, and (ii) the necessity for moving towards
large-scale retrieval. We first contribute to the community a novel ZS-SBIR
dataset, QuickDraw-Extended, that consists of $330,000$ sketches and $204,000$
photos spanning across 110 categories. Highly abstract amateur human sketches
are purposefully sourced to maximize the domain gap, instead of ones included
in existing datasets that can often be semi-photorealistic. We then formulate
a ZS-SBIR framework to jointly model sketches and photos into a common
embedding space. A novel strategy to mine the mutual information among domains
is specifically engineered to alleviate the domain gap. External semantic
knowledge is further embedded to aid semantic transfer. We show that, rather
surprisingly, retrieval performance significantly outperforms that of state-
of-the-art on existing datasets that can already be achieved using a reduced
version of our model. We further demonstrate the superior performance of our
full model by comparing with a number of alternatives on the newly proposed
dataset. The new dataset, plus all training and testing code of our model,
will be publicly released to facilitate future
research222http://dag.cvc.uab.es/doodle2search/.

##  1 Introduction

In the context of retrieval, sketch modality has shown great promise thanks to
the pervasive nature of touchscreen devices. Consequently, research on sketch-
based image retrieval (SBIR) has flourished, with many great examples
addressing various aspects of the retrieval process: fine-grained matching
[37, 30, 24], large-scale hashing [17, 16], cross-modal attention [5, 30] to
name a few.

However, a common bottleneck identified by almost all sketch researches is
that of data scarcity. Different to photos that can be effortlessly crawled
for free, sketches have to be drawn one by one by human being. As a result,
existing SBIR datasets suffer in both volume and variety, leaving only less
than thousand of sketches per category, with maximum number of classes limited
to few hundreds. This largely motivated the problem of zero-shot SBIR (ZS-
SBIR), where one wishes to conduct SBIR on object categories without having
the training data. ZS-SBIR is increasingly being regarded as an important
component in unlocking the practical application of SBIR, since million-scale
datasets that have been used to train commercial photo-only systems [4] might
not be feasible.

| Fish | Frog | Crab | Owl | Sea-turtle
---|---|---|---|---|---
Sketchy [27] |  |  |
---|---
|
|  |  |
TUBerlin [6] |  |  |
---|---
|
|  |  |
QuickDraw |  |  |
---|---
|
|  |  |
Figure 1: Qualitative comparison of sketch datasets, columns show examples
belonging the same class. _Sketchy_ , _TUBerlin_ and _QuickDraw_ datasets
orderly contain sketches with increasing level of abstraction. It is worth
noting that despite being the most abstract dataset, _QuickDraw_ sketches can
still be reliably recognised.

The problem of ZS-SBIR is extremely challenging. It shares all challenges laid
out in conventional SBIR: (i) large domain gap between sketch and image, and
(ii) high degree of abstraction found in human sketches as a result of variant
drawing skills and visual interpretations. Additionally, it also needs the
semantic transference from the seen to unseen categories for the purpose of
zero-shot learning. Over and above all, in this paper, we are interested in
moving towards the practical adaptation of ZS-SBIR technology. For that, a
more appropriate dataset that best capture all these challenges is required.

Therefore, our first contribution is a new dataset to simulate the real
application scenario of ZS-SBIR, which should satisfy the following
requirements. First, the dataset needs to mimic the real-world abstraction gap
between sketch and photo. Such amateur sketches are very different from the
ones currently studied by existing datasets, which are either too photo-
realistic [7] or produced by recollection of a reference images [27] (Figure 1
offers a comparative example). Second, in order to learn a reliable cross-
domain embedding between amateur sketch and photo, the dataset much faithfully
capture of a full variety of sketch samples from users having various drawing
skills. Our proposed dataset, QuickDraw-Extended, contains $330,000$ sketches
and $204,000$ photos in total spanning across $110$ categories. In particular,
it includes $3,000$ amateur sketches per category carefully sourced from the
recently released Google Quickdraw dataset [12] – six times more than the next
largest. It also has a search space stretching to $166$million total
comparisons in the test set, compared to _Sketchy-Extended_ and _TUBerlin-
Extended_ with just $10$ million and $1.9$ million, respectively.

This dataset and the real-world scenario it mimics, essentially make the ZS-
SBIR task more difficult. This leads to our second contribution which is a
novel cross-domain zero-shot embedding model that addresses all challenges
posed by this new setting. Our base network is a visually-attended triplet
ranking model that is commonly known in the SBIR community to produce state-
of-the-art retrieval performances [37, 30]. To our surprise, just by adopting
such a triplet formulation, we can already achieve retrieval performances
drastically better than that of the previously reported ZS-SBIR results on
commonly used datasets. We attribute this phenomena to previous datasets being
too simplistic in terms of the cross-domain abstraction gap and the diversity
of sketch samples. This further justifies the necessity of a new practical
dataset like ours. We then propose two novel techniques to help learn a better
cross-domain transfer model. First, a domain disentanglement strategy is
designed to bridge gap between the domains by forcing the network to learn a
domain-agnostic embedding, where a _Gradient Reversal Layer_ (GRL) [8]
encourages the encoder to extract mutual information from sketches and photos.
Second, a novel semantic loss to ensure that semantic information is preserved
in the obtained embedding. By applying a GRL only to the negative samples at
the input of the semantic decoder helps the encoder network to separate the
semantic information of similar classes.

Extensive experiments are first carried out on the two commonly used ZS-SBIR
datasets, TUBerlin-Extended [6] and Sketchy-Extended [27]. The results show
that the even a reduced version of our model can outperform current state-of-
the-arts by a significant margin. The superior performance of the proposed
method is further validated on our own dataset, with ablative studies to draw
insights towards each of the proposed system components.

##  2 Related Work

SBIR Datasets. One of the key barriers towards large-scale SBIR research is
the lack of appropriate benchmarks. The Sketchy dataset [27] is the mostly
used one for this purpose, which contains 75,471 hand-drawn sketches of 12,500
object photos belonging to 125 different categories. Later, Liu  _et al._ [17]
collected 60,502 natural images from ImageNet [4] in order to fit the task of
large-scale SBIR. This dataset having contained highly detailed or less
abstract sketches, models trained on Sketchy have high chance of getting
collapsed in real life scenario. Two more fine-grained SBIR datasets with
paired sketches and images are _shoe_ and _chair_ datasets which were proposed
in [37]. The _shoe_ dataset contains altogether 6648 sketches and 2000 photos,
whereas, the _chair_ dataset altogether contains 297 sketches and photos.
However, being fine-grained pairs these two datasets also have similar
disadvantages as the Sketchy dataset. TU-Berlin [6] being the other popular
dataset originally contains 250 classes of hand-drawn sketches, where each
class roughly contains 80 instances. It was extended with real images by [38]
for SBIR purposes. This dataset has a lot of confusion regarding the class
hierarchy, for an example, swan, seagull, pigeon, parrot, duck, penguin, owl
have substantial visual similarity and commonality with standing bird and
flying bird which are another separate categories of the TU-Berlin dataset. To
obliterate, these difficulties faced by the SBIR works, in this paper, we
introduce QuickDraw-Extended dataset, where we take the sketch classes of the
Google QuickDraw dataset [12] and provide the corresponding set of images to
facilitate the training of large-scale SBIR system.

Sketch-based Image Retrieval (SBIR). The main challenge that most of the SBIR
tasks address is bridging the domain gap between sketch and natural image. In
literature, these existing methods can be roughly grouped into two categories:
_hand crafted_ and _cross-modal deep learning_ methods. The hand-crafted
techniques mostly work with Bag-of-Words representations of sketch and edge
map of natural image on top of some off-the-shelf features, such as, SIFT
[19], Gradient Field HOG [10], Histogram of Edge Local Orientations [25] or
Learned Key Shapes [26]) etc. This domain shift issue is further addressed by
cross-domain deep learning-based methods [27, 37], where they have used
classical ranking losses, such as, contrastive loss, triplet loss [32] or more
elegant HOLEF loss [30] within a siamese like network. Based on the problem at
hand, two separated tasks have been identified: (1) _Fine-grained SBIR_ (FG-
SBIR) aims to capture fine-grained similarities of sketch and photo [15, 27,
37] and (2) _Coarse-grained SBIR_ (CG-SBIR) performs a instance level search
across multiple object categories [38, 10, 11, 31, 38], which has received a
lot of attention due to its importance. Realising the need of large-scale
SBIR, some researchers have proposed a variant of cross-modal hashing
framework for the same [17, 39], which also showed promising results in SBIR
scenario. In contrast, our proposed model overcomes this domain gap by mining
the modality agnostic features using a domain loss along with a GRL.

Zero-Shot Sketch-based Image Retrieval (ZS-SBIR). Early works on zero-shot
learning (ZSL) were mostly focused on attribute based recognition [14], which
is later augmented by another major line that focus on learning a joint
embedding space for image feature representation and class semantic descriptor
[3, 34, 13, 35, 18]. Depending on the selection of joint embedding space and
type of projection function utilised between the visual to semantic space,
existing models can be divided into three groups: (i) projected from visual
feature space to semantic space [14, 21], (ii) projected from semantic space
to the visual feature space [3], and (iii) an intermediate space that both are
simultaneously projected to [40]. In contrast to these existing works, our
model can be seen as a combination of the first and second groups, where the
embedding is on the visual feature space, but asked to additionally recover
its embodied semantics with a decoder.

Although SBIR and ZSL have been extensively studied among the research
community, very few works have studied their combination. Shen _et al._ [28]
propose a multi-modal network to mitigate the sketch-image heterogeneity and
enhance semantic relations. Yelamarthi _et al._ [36] resort to a deep
conditional generative model, where a sketch is taken as input and learned to
generate its photo features by stochastically filling the missing information.
The main motivation behind ZS-SBIR lies with sketches being costly and labour-
intensive to source – sketches need to be individually drawn by hand, other
than crawled for free from the internet. To enable rapid deployment on
categories where training sketches are not readily available, it is important
to leverage on existing sketch data from other categories. The key difference
between ZS-SBIR and other ZS tasks, which is also the main difficulty of the
problem, lies with the additional modality gap between sketch and photo.

##  3 QuickDraw-Extended Dataset

Existing datasets do not cover all the challenges derived from a ZS-SBIR
system. Therefore, we propose a new dataset named _QuickDraw-Extended Dataset_
that is specially designed for this task. First we review the existing
datasets in the literature used for ZS-SBIR and motivate the purpose of the
new dataset. Thus, we provide a large-scale ZS-SBIR dataset that overcomes the
main problems of the existing ones. Existing datasets were not originally
designed for a ZS-SBIR scenario, but they have been adapted by a redefining
the partitions setup. In addition, the main limitations that we overcome with
the new dataset are (i) the large domain gap between amateur sketch and photo,
and (ii) the necessity for moving towards large-scale retrieval.

Sketchy-Extended Dataset [27]: Originally created as a fine-grained
association between sketches to particular photos for fine-grained retrieval.
This dataset has been adapted to the task of ZS-SBIR. On one hand, Shen  _et
al._ [28] proposed to set aside 25 random classes as a test set whereas the
training is performed in the rest 100 classes. On the other hand, Yelamarthi
_et al._ [36] proposed a different partition of 104 train classes and 21 test
classes in order to make sure that test is not present in the 1,000 classes of
ImageNet. Its main limitation for the task of ZS-SBIR is its fine-grained
nature, i.e., each sketch has a corresponding photo that was used as reference
at drawing time. Thus, participants tended to draw the objects in a realistic
fashion, producing sketches resembling that of a true edge-map very well. This
essentially narrows the cross-domain gap between sketch and photo.

TUBerlin-Extended Dataset [6]: It is a dataset that was created for sketch
classification and recognition bench-marking. In this case, drawers were asked
to draw the sketches giving them only the name of the class. This allows a
semantic connection among sketches and avoids possible biases. However, the
number of sketches is scarce, considering the variability among the
observations of a concept in the real world. Also, some of the design
decisions on the selection of object categories prevent it to be adequate for
our zero-shot setting: (i) classes are defined both in terms of a concept and
an attribute (e.g., seagull, flying-bird); (ii) different WordNet levels are
used, _i.e._ there are classes that are semantically included in others (e.g.,
mug, beer-mug).

###  3.1 The Dataset

Taking into account the limitations of the previously described datasets in a
ZS-SBIR scenario, we contribute to the community a novel large-scale dataset,
_QuickDraw-Extended_. We identified the following challenges of a practical
ZS-SBIR, (i) the large domain gap between amateur sketch and photo, and (ii)
the necessity for moving towards large-scale retrieval. According to this, the
new dataset must fulfil the following aspects: (i) to not have a direct one-
to-one correspondence between sketches and images, _i.e._ sketches can be
rough conceptual abstractions of images produced in an amateur drawing style;
(ii) to avoid ambiguities and overlapping classes; (iii) large intra-class
variability provided by the high abstraction level of different drawers.

In order to accomplish these objectives, we took advantage of the Google
Quick, Draw! [12] data which is a huge collection of drawings (50 millions)
belonging to 345 categories obtained from the _Quick, Draw!_
333https://quickdraw.withgoogle.com/ game. In this game, the user is asked to
draw a sketch of a given category while the computer tries to classify them.
The way sketches are collected provides the dataset a large variability,
derived from human abstraction. Moreover, it addresses the large domain gap
between non-expert drawers and photos that is not considered in previous
benchmarks. Hence, we propose to make use of a subset of sketches to construct
a novel dataset for large-scale ZS-SBIR containing 110 categories (80 for
training and 30 for testing). Classes such as circle of zigzag are directly
discarded because they can not be used in an appropriate SBIR. As a retrieval
gallery, we provide images extracted from _Flickr_ tagged with the
corresponding label. Manual filtering is performed to remove outliers.
Moreover, following the idea introduced in [36] for the _Sketchy-Extended_
dateset, we provide a test split which forces that test classes are not
present in ImageNet in case of using pre-trained models. Finally, this dataset
consists of 330,000 sketches and 204,000 photos moving towards a large-scale
retrieval. We consider that this dataset will provide better insights about
the real performance of ZS-SBIR in a real scenario.

Table 1: Dataset comparison in terms of their size. Partition is presented in terms of number of classes used for each set, moreover, # Comparisons stands for the number of comparisons sketch-image performed in test. |  Sketchy [27] |  TUBerlin [6] | QuickDraw
---|---|---|---
| Partition
---
(tr+va, te)
$(104,21)$ | $(220,30)$ | $(80,30)$
# Sketch/class | $500$ | $80$ | $3,000$
# Image/class |  $600$-$700$ |  $\sim 764$***Extremely imbalanced | $\sim 1,854$
# Comparisons |  $\sim 10$Mill. |  $\sim 1.9$Mill. |  $\sim 166$Mill.

Table 1 provides a comparison of the three benchmarks for the task of ZS-SBIR.
To the best of our knowledge, this is the first time that a real large-scale
problem is addressed providing 6 times more sketches and more than the double
of photos per each class. Qualitatively QuickDraw-Extended provides a high
abstraction level than previous benchmarks as it is shown in Figure 2.

| Fish | Frog | Crab | Owl
---|---|---|---|---
Sketchy [27] |  |  |  |
|  |  |
TUBerlin [6] |  |  |  |
|  |  |
QuickDraw |  |  |  |
|  |  |
Figure 2: Qualitative comparison of the datasets. The different levels of
abstraction in the sketches can be appreciated. From the top to the bottom,
the figure also shows the decrease in the alignment between sketches and
images.

##  4 A ZS-SBIR framework

###  4.1 Problem Formulation

Let $\mathcal{C}$ be the set of all possible categories in a given dataset;
$\mathcal{X}=\\{x_{i}\\}_{i=1}^{N}$ and $\mathcal{Y}=\\{y_{i}\\}_{i=1}^{M}$ be
the set of photos and sketches respectively;
$l_{x}:\mathcal{X}\rightarrow\mathcal{C}$ and
$l_{y}:\mathcal{Y}\rightarrow\mathcal{C}$ be two labelling functions for
photos and sketches respectively. Such that give an input sketch an optimal
ranking of gallery images can be obtained. In a _zero-shot_ framework,
training and testing sets are divided according to _seen_
$C^{s}\subset\mathcal{C}$ and _unseen_ $C^{u}\subset\mathcal{C}$ categories,
where $\mathcal{C}^{s}\cap\mathcal{C}^{u}=\varnothing$. Thus, the model needs
to learn an aligned space between sketches and photos to perform well on test
data whose classes have never been used in training. We define the set of
_seen_ and _unseen_ photos as
$\mathcal{X}^{s}=\\{x_{i};l_{x}(x_{i})\in\mathcal{C}^{s}\\}_{i=1}^{N}$ and
$\mathcal{X}^{u}=\mathcal{X}\setminus\mathcal{X}^{s}$. We define analogously
the _seen_ and _unseen_ sets for sketches, denoted as $\mathcal{Y}^{s}$ and
$\mathcal{Y}^{u}$.

Figure 3: Proposed architecture for ZS-SBIR which maps sketches and photos in
a common embedding space. It combines three losses: (i) triplet loss, to learn
a ranking metric; (ii) domain loss to merge images and sketches to an
indistinguishable space making use of a GRL; (iii) semantic loss forces the
embeddings to contain semantic information by reconstructing the word2vec
embedding of the class. It also helps to distinguish semantically similar
classes by means of a GRL on the negative example (best viewed in color).

The proposed framework is divided in two main components. The encoder
transforms the input image to the corresponding embedding space. The second
component is the cost function which guides the learning process to provide
the embedding with the desired properties. Figure 3 outlines the proposed
approach.

###  4.2 Encoder Networks

Given a distance function $d(\cdot,\cdot)$, the aim of our framework is to
learn two embedding functions $\phi:\mathcal{X}\rightarrow\mathbb{R}^{D}$ and
$\psi:\mathcal{Y}\rightarrow\mathbb{R}^{D}$ which respectively map the photo
and sketch domain into a common embedding space. Later, these embedding
functions are used in the retrieval task during the test phase, therefore,
they should possess a ranking property related to the considered distance
function. Hence, given two photos $x_{1},x_{2}\in\mathcal{X}$ and a sketch
$y\in\mathcal{Y}$, we expect the embedding fulfils the following condition:
$d(\phi(x_{1}),\psi(y))<d(\phi(x_{2}),\psi(y))$, when $l_{x}(x_{1})=l_{y}(y)$
and $l_{x}(x_{2})\neq l_{y}(y)$. In a retrieval scenario, our system is able
to provide a ranked list of images by the chosen distance function. In this
framework, $d$ has been set as $\ell_{2}$-distance. During training, the two
embedding $\phi(\cdot)$ and $\psi(\cdot)$ are trained with multi-modal
information, therefore they presume to learn a modality free representation.

Our embedding functions $\phi(\cdot)$ and $\psi(\cdot)$ are defined as two
CNNs with attention where the last fully-connected layer has been replaced to
match the desired embedding size $D$. The _attention_ [33] mechanism helps our
system to localise the important features in both modalities. Soft-attention
is the widely used one because it is differentiable, and hence it can be
learned end-to-end with the rest of the network. Our soft-attention model
learns an attention mask which assigns different weights to different regions
of an image given a feature map. These weights are used to highlight important
features, therefore, given an attention mask $att$ and a feature map $f$, the
output of the attention module is computed by $f+f\cdot att$. The attention
mask is computed by means of $1\times 1$ convolution layers applied on the
corresponding feature map.

###  4.3 Learning objectives

The learning objective of the proposed framework combines: (i) _Triplet Loss_
; (ii) _Domain Loss_ , (iii) _Semantic Loss_. These objective functions
provide visual and semantic information to the encoder network. Let us
consider a triplet $\\{a,p,n\\}$ where $a\in\mathcal{Y}^{s}$,
$p\in\mathcal{X}^{s}$ and $n\in\mathcal{X}^{s}$ are respectively the _anchor_
, _positive_ and _negative_ samples during the training. Moreover,
$l_{x}(p)=l_{y}(a)\text{ and }l_{x}(n)\neq l_{y}(a)$.

Triplet Loss: This loss aims to reduce the distance between embedded sketch
and image if they belong to the same class and increase it if they belong to
different classes. For simplicity, if we define the distances between the
samples as $\delta_{+}=\left\lVert\psi(a)-\phi(p)\right\rVert_{2}$ and
$\delta_{-}=\left\lVert\psi(a)-\phi(n)\right\rVert_{2}$ for the _positive_ and
_negative_ samples respectively, then, the ranking loss for a particular
triplet can be formulated as
$\lambda(\delta_{+},\delta_{-})=\max\\{0,\mu+\delta_{+}-\delta_{-}\\}$ where
$\mu>0$ is a margin parameter. Batch-wise, the loss is defined as:

$\mathcal{L}_{t}=\frac{1}{N}\sum_{i=1}^{N}\lambda(\delta_{+}^{i},\delta_{-}^{i}).$
(1)

This loss measures the violation of the ranking order of the embedded
features. Therefore, the order aimed by this loss is
$\delta_{-}>\delta_{+}+\mu$, if this is the case, the network is not updated,
otherwise, the weights of the network are updated accordingly. Triplet loss
provides a metric space with ranking properties based on visual features.

Domain Loss: Triplet loss mentioned above does not explicitly enforce the
mapping of sketch and image samples to a common space. Therefore, at this end,
to ensure that the obtained embedding belong to the same space, we propose to
use a domain adaptation loss [8]. The basic idea of this loss is to obtain a
domain-agnostic embedding that does not contain enough information to decide
whether it comes from a sketch or photo. Given the embedding $\phi(\cdot)$ and
$\psi(\cdot)$, we make use of a Multilayer Perceptron (MLP) as a binary
classifier trying to predict which was the initial domain. Purposefully, in
order to create indistinguishable embedding we use a _GRL_ defined as
$R_{\lambda}(\cdot)$, which applies the identity function during the forward
pass $R_{\lambda}(x)=x$, whereas during the backward pass it multiplies the
gradients by the meta-parameter $-\lambda$,
$\frac{\operatorname{d}R_{\lambda}}{\operatorname{d}x}=-\lambda I$. This
operation reverses the sign of the gradient that flows through the CNNs. In
this way, we encourage our encoders to extract the shared representation from
sketch and photo. For this loss, we define a meta-parameter $\lambda_{d}$ that
changes from $0$ (only trains the classifier but does not update the encoder
network) to $1$ during the training according to a defined function. In our
case it is defined according to the iteration $i$ as
$z_{\lambda}(i)=(i-5)/20$. Following the notation,
$f:\mathbb{R}^{D}\rightarrow[0,1]$ be the MLP and $e\in\mathbb{R}^{D}$ an
embedding coming from the encoders network. Then we can define the binary
cross entropy of one of the samples as
$l_{t}(e)=t\log(f(R_{\lambda_{d}}(e)))+(1-t)\log(1-f(R_{\lambda_{d}}(e)))$,
where $e$ is the embedding obtained by the encoder network and $t$ is $0$ and
$1$ for sketch and photo domains respectively. Hence, the domain loss is
defined as:

$\mathcal{L}_{d}=\frac{1}{3N}\sum_{i=1}^{N}\left(l_{0}(\psi(a_{i}))+l_{1}(\phi(p_{i}))+l_{1}(\phi(n_{i}))\right)$
(2)

Semantic Loss: A decoder network trying to reconstruct the semantic
information of the corresponding category from the generated embedding is
proposed. This reconstruction forces that the semantic information is encoded
in the obtained embedding. In this case, we propose to minimise the cosine
distance with the reconstructed feature vector and the semantic representation
of the category. Inspired by the idea presented by Gonzalez  _et al._ [9] for
cross-domain disentanglement, we propose to exploit the negative sample to
foster the difference between similar semantic categories. Hence, we apply a
GRL $R_{\lambda_{s}}(\cdot)$ to the negative sample at the input of the
semantic decoder and we train it to reconstruct the semantics of the positive
example. The idea is to help the encoder network to separate the semantic
information of similar classes. In this case, we decided to keep the meta-
parameter $\lambda_{s}$ to a fixed value among all the training, in
particular, it was set to $0.5$.

Let $c\in\mathcal{C}^{s}$ be the corresponding category of the anchor $a$. The
semantics of this category are obtained by the _word2vec_ [20] embedding
trained on part of Google News dataset ($\sim$ 100 billion words), _GloVe_
[23] and _fastText_ [1] (more results are available in supplementary materials
). Let $g:\mathbb{R}^{D}\rightarrow\mathbb{R}^{300}$ be the semantic
reconstruction network and $s={\rm embedding}(c)\in\mathbb{R}^{300}$ be the
semantics of the given category. Hence, given an image embedding
$e\in\mathbb{R}^{D}$ the cosine loss is defined as
$l_{c}(e,s)=\frac{1}{2}\left(1-\frac{g(e)s^{t}}{||g(e)||\cdot||s||}\right)$.
The semantic loss is defined as follows:

$\mathcal{L}_{s}\negmedspace=\negmedspace\frac{1}{3N}\negmedspace\sum_{i=1}^{N}\left(l_{c}(\psi(a_{i}),s_{i})+l_{c}(\phi(p_{i}),s_{i})\right.\\\
\left.+l_{c}(R_{\lambda_{s}}(\phi(n_{i})),s_{i})\right)$ (3)

Therefore, the whole network will be trained by a combination of three
proposed loss functions.

$\mathcal{L}=\alpha_{1}\mathcal{L}_{t}+\alpha_{2}\mathcal{L}_{d}+\alpha_{3}\mathcal{L}_{s},$
(4)

where the weighting factors $\alpha_{1}$, $\alpha_{2}$ and $\alpha_{3}$ are
equal in our model. Algorithm 1 presents the training algorithm followed in
this work. $\Gamma(\cdot)$ denotes the optimiser function.

Input: Photo-Sketch data $\\{\mathcal{X},\mathcal{Y}\\}$; Class semantics
$\mathcal{S}$;
$\lambda_{s}=0.5$ and max training iterations $T$
Output:  Encoder networks parameters $\\{\Theta_{\phi},\Theta_{\psi}\\}$.

1:repeat

2: Get a random mini-batch
$\\{y_{i},x_{i}^{p},x_{i}^{n},s_{i}\\}_{i=1}^{N_{B}}$; where

3: $y_{i}$, $x_{i}^{p}$ belong to the same class and $x_{i}^{n}$ does not.

4:
$\lambda_{d}\leftarrow\operatorname{clip}(z_{\lambda}(\cdot),\operatorname{min}=0,\operatorname{max}=1)$

5: $\mathcal{L}\leftarrow$ Eq. 4

6: $\Theta\leftarrow\Theta-\Gamma(\nabla_{\Theta}\mathcal{L})$

7:until Convergence or max training iterations $T$

Algorithm 1 Training algorithm for the proposed model . Table 2: Comparison against the state-of-the-art with that of the proposed model. Note: the same train and test split are used for all experiments on CVAE [36] and ours. ZSIH [28] did not report the specific details on their split (other than 25 classes were used for testing), and we could not produce their results on _QuickDraw-Extended_ due to the lack of publicly available code. Method |  Sketchy-Extended [27] |  |  TUBerlin-Extended [6] |  | QuickDraw-Extended
---|---|---|---|---|---
mAP | mAP@200 | P@200 |  | mAP | mAP@200 | P@200 |  | mAP | mAP@200 | P@200
ZSIH [28] |  $0.2540$†††Using a random partition of 25 test categories following the setting proposed in [26], we obtained 0.3521 for our model. | $-$ | $-$ |  | $\boldsymbol{0.2200}$ | $-$ | $-$ |  | Not able to produce
CVAE [36] | $0.1959$ | $0.2250$ | $0.3330$ |  | $0.0050$ | $0.0090$ | $0.0030$ |  | $0.0030$ | $0.0060$ | $0.0030$
Ours | $\boldsymbol{0.3691}$ | $\boldsymbol{0.4606}$ | $\boldsymbol{0.3704}$ |  | $0.1094$ | $\boldsymbol{0.1568}$ | $\boldsymbol{0.1208}$ |  | $\boldsymbol{0.0752}$ | $\boldsymbol{0.0901}$ | $\boldsymbol{0.0675}$
Table 3: Ablation study for the proposed model. As baseline, the triplet loss is used and the different modules are incrementally added. Attn. | Dom. | Sem. |  Sketchy-Extended [27] |  |  TUBerlin-Extended [6] |  | QuickDraw-Extended
---|---|---|---|---|---|---|---
mAP | mAP@200 | P@200 |  | mAP | mAP@200 | P@200 |  | mAP | mAP@200 | P@200
- | - | - | $0.3020$ | $0.3890$ | $0.3091$ |  | $0.0590$ | $0.1040$ | $0.0682$ |  | $0.0354$ | $0.0546$ | $0.0454$
✓ | - | - | $0.3207$ | $0.4150$ | $0.3342$ |  | $0.0729$ | $0.1141$ | $0.1002$ |  | $0.0456$ | $0.0635$ | $0.0496$
✓ | ✓ | - | $0.3256$ | $0.4113$ | $0.3444$ |  | $0.0845$ | $0.1264$ | $0.1080$ |  | $0.0651$ | $0.0881$ | $0.0615$
✓ | - | ✓ | $0.3392$ | $0.4146$ | $0.3586$ |  | $0.1055$ | $0.1496$ | $0.1115$ |  | $0.0693$ | $0.0896$ | $0.0625$
✓ | ✓ | ✓ | $\boldsymbol{0.3691}$ | $\boldsymbol{0.4606}$ | $\boldsymbol{0.3704}$ |  | $\boldsymbol{0.1094}$ | $\boldsymbol{0.1568}$ | $\boldsymbol{0.1208}$ |  | $\boldsymbol{0.0752}$ | $\boldsymbol{0.0901}$ | $\boldsymbol{0.0675}$

##  5 Experimental Validation

This Section experimentally validates the proposed ZS-SBIR approach on three
benchmarks _Sketchy-Extended_ , _TUBerlin-Extended_ and _QuickDraw-Extended_ ,
highlighting the importance of the newly introduced dataset which is more
realistic for practical SBIR purpose. A detailed comparison with the state-of-
the-art is also presented.

###  5.1 Zero-shot Experimental Setting

Implementation details: Our CNN-based encoder networks $\phi(\cdot)$ and
$\psi(\cdot)$ make use of a ImageNet pre-trained VGG-16 [29] architecture.
This can be replaced by any model to enhance the extracted feature quality.
Both, domain classifier $f(\cdot)$ and semantic reconstruction $g(\cdot)$ of
the proposed model makes use of 3 fully connected layers with ReLU activation
functions. The whole framework was implemented with PyTorch [22] deep learning
tool and is trainable on single Pascal Titan X GPU card.

Training setting: Our system uses triplets to utilise the inherent ranking
order. The training batches are constructed in a way so that it can take the
advantage of the semantic information in order to mine hard negative samples
for a given anchor class. This implies that semantically closer classes will
have a higher probability to be used during training and thus they are likely
to be disjoint in the final embedding. We trained our model following an early
stopping strategy in validation to provide the final test result. The model is
trained end-to-end using the SGD [2] optimiser. The learning rate used
throughout is $1e-4$. The epochs required to train the model on different
dataset is around $40$.

Evaluation protocol: The proposed evaluation uses the metrics used by
Yelamarthi  _et al._ [36]. Therefore, the evaluation is performed taking into
account the top 200 retrieved samples. Moreover, we also provide metrics on
the whole dataset. Images labelled with the same category as that of the query
sketch, are considered as relevant. Note that this evaluation does not
consider visually similar drawings that can be considered correct by human
users. For the existing datasets, we used the proposed splits in [36, 28].

|  Sketchy [27] |  | QuickDraw
---|---|---|---
| Query | Top-8 retrieved candidates |  | Query | Top-8 retrieved candidates
| CVAE
---
[36]
|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
Ours | pear |  |  |  |  |  |  |  |  |  | skyscraper |  |  |  |  |  |  |  |
| CVAE
---
[36]
|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
Ours | door |  |  |  |  |  |  |  |  |  | helicopter |  |  |  |  |  |  |  |
Figure 4: Top 8 image retrieval examples given a query sketch. All the
examples correspond to a zero-shot setting, _i.e._ no example have been seen
in training. First row provides a comparison with CVAE [36] method against our
pipeline. Note that in some retrieval cases, for instance, door is confused
with window images which can be true even for humans. Green and Red stands for
correct and incorrect retrievals. (Better viewed in pdf)

###  5.2 Model Discussion

This section presents a comparative study with the state-of-the-art followed
by a discussion on the _TUBerlin-Extended_ results and finally the ablative
study. As mentioned, our model is build on top of a triplet network. We take
this as a baseline and study the importance of the different components of the
full model which includes the _attention mechanism_ , the _semantic loss_ and
the _domain loss_.

Comparison: Table 2 provides comparisons of our full model results against
those of the state-of-the-art. We report a comparative study with regard to
two methods presented in Section 2, namely ZSIH [28] and CVAE [36]. Note that
we have not been able to reproduce the ZSIH model due to lack of technical
implementation details and the code being unavailable. Hence, the results on
_QuickDraw-Extended_ dataset nor an evaluation using the top 200 retrieval
could be computed. The last row of the Table 2 shows the result of our full
model. From the Table 2 the results suggest the limitation of the previous
models regarding their ability in an unconstrained domain where sketches have
higher level of abstractions. The CVAE [36] method trained with sketch-image
correspondence has difficulties to capture the intra-class variability, the
domain gap and also the ability to infer unseen classes. The following
conclusions are drawn: (i) our base model outperforms all the state-of-the-art
methods in _Sketchy-Extended_ Dataset; (ii) our model performs the best
overall on each metric and on almost all the datasets; (iii) the gap between
our model and the state-of-the-art datasets is almost double in _Sketchy-
Extended_ Dataset; (iv) the difference in the result in previous dataset
points out the need of a new well structured dataset for ZS-SBIR (v) the new
benchmark also provides the different aspects (_i.e_ of semantics, mutual
information) that can play important role in a real ZS-SBIR scenario; (vi) the
evaluation shows the importance of going towards large-scale ZS-SBIR where the
retrieval search space is in the range of $166$ million comparisons ($16$
times of the current largest dataset).

Discussion on _TUBerlin-Extended_ : As stated in Section 3, the results could
be heavily affected by the chosen classes for experiments. Since [28] did not
report specific details on their train and test split, we can not offer a fair
comparison on TUBerlin-Extended. Instead, for both [36] and ours, we resort to
the commonly accepted median over random splits setting. And it shows our
method favourably beats [36] by a clear margin. We did however observe a high
degree of fluctuation over the different splits on TUBerlin-Extended, which
re-affirms our speculation on how the categories included in TUBerlin-Extended
might not be optimal for the zero-shot setting (see Section 3). This could
explain the superior performance of [28], yet more experiments are needed to
confirm such suspicion. Unfortunately, again such experiments would not be
possible without details on their train and test split.

Ablation study: Here, we investigate the contribution of each component to the
model, as well as other issues of the architecture. The first 5 rows of Table
3 present a study of the contribution of each component to the whole proposed
model. From this Table we can draw the following conclusions: (i) attention
plays a major role in improving the baseline result; (ii) the domain loss is
able to alleviate to some extend the domain gap, this is more remarkable in
those datasets where sketches are more abstract; (iii) as the difficulty of
the dataset increases, the semantic and the domain losses start playing a
major role in improving the baseline result; (iv) semantics provide better
extrapolation to unseen data than domain loss which shows that either the
mutual information is very less or that the semantic information is really
needed in this extrapolation; (v) the poor performance in the _QuickDraw-
Extended_ dataset shows that the practical problem of ZS-SBIR is still indeed
unsolved. It should be noted, that the best model makes use of the three
losses.

Qualitative: Some retrieval results are shown in Figure 4 for _Sketchy-
Extended_ and _QuickDraw-Extended_. We also provide a qualitative comparison
with CVAE proposed by Yelamarthi  _et al._ [36]. The qualitative results
reinforce that the combination of semantic, domain and triplet loss fairs well
in a dataset with substantial variances on visual abstraction. We would also
like to point out that the retrieved results for the class skyscraper show
high visual shape similarity with rectangle _i.e._ door and saw. The retrieved
circular saw could also might be retrieved because of the semantic rather than
the visual similarity. Similar visual correspondences can also be noticed
between the query sketch helicopter and the retrieved result windmill.

##  6 Conclusions

This paper represents a first step towards a practical ZS-SBIR task. Previous
works on this task do not address some of the important challenges that appear
when moving to an unconstrained retrieval and do not tackle with the large
domain gap between amateur sketch and photo. In this scenario, to overcome the
lack of proper data, we have contributed to the community a specifically
designed large-scale ZS-SBIR dataset, _QuickDraw-Extended_ which provides
highly abstract amateur sketches collected with the Google Quick, Draw! game.
Then, we have proposed a novel ZS-SBIR system that combines visual as well as
semantic information to generate an image embedding. We experimentally show
that this novel framework overcomes recent state-of-the-art methods in the ZS-
SBIR setting.

## Acknowledgements

Work supported by EU’s MSC grant No. 665919, Spanish grants FPU15/06264 and
TIN2015-70924-C2-2-R; and the CERCA Program/Generalitat de Catalunya. The
Titan X was donated by NVIDIA. This work was carried out during research stay
at SketchX Lab in QMUL.

## References

  * [1] Piotr Bojanowski, Edouard Grave, Armand Joulin, and Tomas Mikolov.  Enriching word vectors with subword information.  TACL, 2017.
  * [2] Léon Bottou.  Large-scale machine learning with stochastic gradient descent.  In COMPSTAT. 2010.
  * [3] Soravit Changpinyo, Wei-Lun Chao, and Fei Sha.  Predicting visual exemplars of unseen classes for zero-shot learning.  In ICCV, 2017.
  * [4] J. Deng, W. Dong, R. Socher, L. Li, Kai Li, and Li Fei-Fei.  Imagenet: A large-scale hierarchical image database.  In CVPR, 2009.
  * [5] Sounak Dey, Anjan Dutta, Suman Kumar Ghosh, Ernest Valveny, Josep Lladós, and Umapada Pal.  Learning cross-modal deep embeddings for multi-object image retrieval using text and sketch.  In ICPR, 2018.
  * [6] Mathias Eitz, James Hays, and Marc Alexa.  How do humans sketch objects?  In SIGGRAPH, 2012.
  * [7] Mathias Eitz, Kristian Hildebrand, Tamy Boubekeur, and Marc Alexa.  Sketch-based image retrieval: Benchmark and bag-of-features descriptors.  IEEE transactions on VCG, pages 1624–1636, 2011.
  * [8] Yaroslav Ganin and Victor Lempitsky.  Unsupervised domain adaptation by backpropagation.  In ICML, 2015.
  * [9] Abel Gonzalez-Garcia, Joost van de Weijer, and Yoshua Bengio.  Image-to-image translation for cross-domain disentanglement.  NeurIPS, 2018.
  * [10] Rui Hu and John Collomosse.  A performance evaluation of gradient field hog descriptor for sketch based image retrieval.  CVIU, 117(7):790–806, 2013.
  * [11] Stuart James, Manuel J Fonseca, and John Collomosse.  Reenact: Sketch based choreographic design from archival dance footage.  In ICMR, 2014.
  * [12] Jonas Jongejan, Henry Rowley, Takashi Kawashima, Jongmin Kim, and Nick Fox-Gieg.  The quick, draw! - a.i. experiment.  https://quickdraw.withgoogle.com, 2016.
  * [13] Nour Karessli, Zeynep Akata, Bernt Schiele, Andreas Bulling, et al.  Gaze embeddings for zero-shot image classification.  In CVPR, 2017.
  * [14] Christoph H Lampert, Hannes Nickisch, and Stefan Harmeling.  Attribute-based classification for zero-shot visual object categorization.  IEEE PAMI, 36(3):453–465, 2014.
  * [15] Yi Li, Timothy M Hospedales, Yi-Zhe Song, and Shaogang Gong.  Fine-grained sketch-based image retrieval by matching deformable part models.  In BMVC, 2014.
  * [16] Yanan Li, Donghui Wang, Huanhang Hu, Yuetan Lin, and Yueting Zhuang.  Zero-shot recognition using dual visual-semantic mapping paths.  In CVPR, 2017.
  * [17] Li Liu, Fumin Shen, Yuming Shen, Xianglong Liu, and Ling Shao.  Deep sketch hashing: Fast free-hand sketch-based image retrieval.  In CVPR, 2017.
  * [18] Yang Long, Li Liu, Yuming Shen, Ling Shao, and J Song.  Towards affordable semantic searching: Zero-shot. retrieval via dominant attributes.  In AAAI, 2018.
  * [19] David G Lowe.  Object recognition from local scale-invariant features.  In ICCV, 1999.
  * [20] Tomas Mikolov, Kai Chen, Greg Corrado, and Jeffrey Dean.  Efficient estimation of word representations in vector space.  In ICLR, 2013.
  * [21] Mohammad Norouzi, Tomas Mikolov, Samy Bengio, Yoram Singer, Jonathon Shlens, Andrea Frome, Greg S Corrado, and Jeffrey Dean.  Zero-shot learning by convex combination of semantic embeddings.  In ICLR, 2014.
  * [22] Adam Paszke, Sam Gross, Soumith Chintala, and Gregory Chanan.  Pytorch, 2017.
  * [23] Jeffrey Pennington, Richard Socher, and Christopher Manning.  Glove: Global vectors for word representation.  In EMNLP, pages 1532–1543, 2014.
  * [24] Yonggang Qi, Yi-Zhe Song, Honggang Zhang, and Jun Liu.  Sketch-based image retrieval via siamese convolutional neural network.  In ICIP, 2016.
  * [25] Jose M Saavedra.  Sketch based image retrieval using a soft computation of the histogram of edge local orientations (s-helo).  In ICIP, 2014.
  * [26] Jose M Saavedra, Juan Manuel Barrios, and S Orand.  Sketch based image retrieval using learned keyshapes (lks).  In BMVC, 2015.
  * [27] Patsorn Sangkloy, Nathan Burnell, Cusuh Ham, and James Hays.  The sketchy database: Learning to retrieve badly drawn bunnies.  SIGGRAPH, 2016.
  * [28] Yuming Shen, Li Liu, Fumin Shen, and Ling Shao.  Zero-shot sketch-image hashing.  In CVPR, 2018.
  * [29] K. Simonyan and A. Zisserman.  Very deep convolutional networks for large-scale image recognition.  arXiv, abs/1409.1556, 2014.
  * [30] Jifei Song, Qian Yu, Yi-Zhe Song, Tao Xiang, and Timothy M Hospedales.  Deep spatial-semantic attention for fine-grained sketch-based image retrieval.  In ICCV, 2017.
  * [31] Fang Wang, Le Kang, and Yi Li.  Sketch-based 3d shape retrieval using convolutional neural networks.  In CVPR, 2015.
  * [32] Jian Wang, Feng Zhou, Shilei Wen, Xiao Liu, and Yuanqing Lin.  Deep metric learning with angular loss.  In ICCV, 2017.
  * [33] Kelvin Xu, Jimmy Ba, Ryan Kiros, Kyunghyun Cho, Aaron Courville, Ruslan Salakhudinov, Rich Zemel, and Yoshua Bengio.  Show, attend and tell: Neural image caption generation with visual attention.  In ICML, 2015.
  * [34] Xing Xu, Fumin Shen, Yang Yang, Dongxiang Zhang, Heng Tao Shen, and Jingkuan Song.  Matrix tri-factorization with manifold regularizations for zero-shot learning.  In CVPR, 2017.
  * [35] Meng Ye and Yuhong Guo.  Zero-shot classification with discriminative semantic representation learning.  In CVPR, 2017.
  * [36] Sasikiran Yelamarthi, Shiva Krishna Reddy M, Ashish Mishra, and Anurag Mittal.  A zero-shot framework for sketch based image retrieval.  In ECCV, 2018.
  * [37] Qian Yu, Feng Liu, Yi-Zhe Song, Tao Xiang, Timothy M Hospedales, and Chen-Change Loy.  Sketch me that shoe.  In CVPR, 2016.
  * [38] H. Zhang, S. Liu, C. Zhang, W. Ren, R. Wang, and X. Cao.  Sketchnet: Sketch classification with web images.  In CVPR, 2016.
  * [39] Jingyi Zhang, Fumin Shen, Li Liu, Fan Zhu, Mengyang Yu, Ling Shao, Heng Tao Shen, and Luc Van Gool.  Generative domain-migration hashing for sketch-to-image retrieval.  In ECCV, 2018.
  * [40] Ziming Zhang and Venkatesh Saligrama.  Zero-shot learning via semantic similarity embedding.  In ICCV, 2015.