# Kinks in cuscuton-like models with two scalar fields

I. Andrade  Departamento de Licenciaturas em Ciências Naturais/Biologia,
Universidade Federal do Maranhão, 65400-000 Codó, MA, Brazil  R. Menezes
Departamento de Ciências Exatas, Universidade Federal da Paraíba, 58297-000
Rio Tinto, PB, Brazil

###### Abstract

This work deals with the presence of localized structures in relativistic
systems described by two real scalar fields in two-dimensional spacetime. We
consider the usual two-field model with the inclusion of the cuscuton term,
which couples the fields regardless the potential. First we follow the steps
of previous work to show that the system supports a first-order framework,
allowing us to obtain the energy of solutions without knowing their explicit
form. The cuscuton term brings versatility into the first-order equations,
which gives rise to interesting modifications in the profiles of topological
configurations, such as the smooth control over their slope and the internal
structure of the energy density.

##  I Introduction

Topological defects are of interest in physics in general, playing an
important role in formation of structures in the early Universe and have been
studied in a diversity of scenarios involving the cosmic evolution and other
issues in high energy physics rajaraman ; Wilets ; Vilenkin  and several areas
of non-linear science Whitham ; Eschenfelder ; Davidov ; Murray ; Walgraef .
Kinks are solutions of the equations of motion that connect two minima of a
potential and arise in models with one or more fields in (1,1) spacetime
dimensions. Due to their simplicity, kinks are often used to describe specific
behaviors in many physical situations, see the Refs. Toharia ; Bischof ; Leon
; Martin .

Recent investigations of scalar fields with non-canonical Lagrangians have
also been considered to generate topological structures. These models are
motivated by the current studies in cosmology. In particular, kinematic
modifications of scalar fields have been proposed as an attempt to explain the
accelerated expansion of the Universe in1 , and as possible solutions to
cosmic problems co1 ; co2 . A special non-dynamical class of generalized
models is the cuscuton one, which was introduced as a field with infinite
speed of propagation of sound cuscuton1 ; cuscuton2 . The cuscuton does not
possess dynamical degrees of freedom associated with its temporal evolution,
as it does not contribute to temporal derivative terms in the equation of
motion. There have been several papers published over the years that discuss
the cuscuton term, as referenced in cuscuton3 ; cuscuton4 ; cuscuton5 ;
cuscuton6 ; cuscuton7 ; cuscuton8 .

For topological and non-topological defects, the presence of non-linear
interactions in the dynamics of scalar field models has been studied in Refs.
Babichev:2006cy ; trilogia1 ; Adam:2007ag ; Bazeia:2016xrf ; Babichev:2007tn ;
Adam:2008rf ; Casana:2015cla ; Atmaja:2015umo ; Bazeia:2017nas , within the
context of discrete (kinks) and continuous (Q-balls, vortices, and monopoles)
symmetries. The possibility of such extensions of canonical models leads to
interesting features in the solutions and their properties. For instance, a
distinct character that may appear in non-canonical models is the twinlike
character. In Ref. Andrews:2010eh , it was introduced a generalized model that
engender a mass parameter which supports the same solutions and energy density
of the standard model; see also Ref. Bazeia:2011sb . In this scenario, the
generalized and standard models are called twins. Notwithstanding that, they
can be distinguished by their linear stability, whose properties differ from
one another.

In Ref. cuscutonigor , the cuscuton term was added to the canonical kinematics
in the Lagrangian density of a single real scalar field,

${\cal
L}=\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi+\frac{f(\phi)\partial_{\mu}\phi\partial^{\mu}\phi}{\sqrt{|\partial_{\mu}\phi\partial^{\mu}\phi|}}-V(\phi),$
(1)

with $f(\phi)$ and $V(\phi)$ being arbitrary functions of the field. It is
possible to rewrite the middle term in the form
$f(\phi)\,\mbox{sgn}(\partial_{\mu}\phi\partial^{\mu}\phi)\sqrt{|\partial_{\mu}\phi\partial^{\mu}\phi|}$,
where $\mbox{sgn}(z)$ is the signal function with argument $z$. The properties
of this model were studied and kink solutions for the static case
($\phi_{s}(x)$) were obtained analytically. The way the additional term was
introduced leads to two characteristics that we emphasize: i) the cuscuton
term does not contribute to the equation of motion; ii) although it
contributes to the linear stability equation, it does not appear in terms
involving derivatives of the perturbation $\eta$, where
$\phi(x,t)=\phi_{s}(x)+\eta(x,t)$. Here, we will study how the presence of
another scalar field affects these features.

The paper is organized as follows. In Sec. II we introduce a new class of non-
canonical models and analyze their properties. We also investigate the first-
order formalism and explore the peculiarities arising from the inclusion of a
new term, as well as the constraints considered. Next, in Sec. III, we provide
three explicit examples in detail to illustrate the procedure. Kink profiles
and their energy densities were obtained analytically. We conclude our work in
Sec. IV, where we comment on the main results and discuss possible
perspectives. For simplicity, we consider all quantities to be dimensionless
and set $c=\hbar=1$.

##  II The model

We start by considering a Lagrangian density for two real scalar fields,
$\phi$ and $\chi$, in the two-dimensional $(1,1)$ Minkowski spacetime, with
$\eta_{\mu\nu}={\rm diag}(+,-)$,

${\cal
L}=\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi+\frac{1}{2}\partial_{\mu}\chi\partial^{\mu}\chi+\frac{f(\chi)\partial_{\mu}\phi\partial^{\mu}\phi}{\sqrt{|\partial_{\mu}\phi\partial^{\mu}\phi|}}-V(\phi,\chi).$
(2)

This model is a generalization for two fields of the one considered in Ref.
cuscutonigor . In addition to the interaction between the fields $\phi$ and
$\chi$ through the potential term, there is a non-standard coupling between
the fields in which the function $f(\chi)$ plays the role of a coupling
parameter for the cuscuton term.

The equations of motion derived from the above model are

$\partial_{\mu}\left(\left(1+\frac{f(\chi)}{\sqrt{|\partial_{\nu}\phi\partial^{\nu}\phi|}}\right)\partial^{\mu}\phi\right)=-V_{\phi}\,\,\,\,\,\,\,\,{\rm
and}\,\,\,\,\,\,\,\,\Box\chi=\frac{f_{\chi}\partial_{\mu}\phi\partial^{\mu}\phi}{\sqrt{|\partial_{\mu}\phi\partial^{\mu}\phi|}}-V_{\chi},$
(3)

where we use the notation $V_{\phi}=\partial V/\partial\phi$,
$V_{\chi}=\partial V/\partial\chi$ and $f_{\chi}=df/d\chi$. The energy-
momentum tensor $T_{\mu\nu}$ for this generalized model is given by

$T_{\mu\nu}=\left(1+\frac{f(\chi)}{\sqrt{|\partial_{\lambda}\phi\partial^{\lambda}\phi|}}\right)\partial_{\mu}\phi\partial_{\nu}\phi+\partial_{\mu}\chi\partial_{\nu}\chi-\eta_{\mu\nu}\left(\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi+\frac{1}{2}\partial_{\mu}\chi\partial^{\mu}\chi+\frac{f(\chi)\partial_{\mu}\phi\partial^{\mu}\phi}{\sqrt{|\partial_{\mu}\phi\partial^{\mu}\phi|}}-V(\phi,\chi)\right).$
(4)

Following Ref. trilogia1 , we constrain ourselves to obey the null energy
condition (NEC), that imposes $T_{\mu\nu}n^{\mu}n^{\nu}\geq 0$, where
$n^{\mu}$ is an arbitrary null vector ($\eta_{\mu\nu}n^{\mu}n^{\nu}=0$). For
the above energy-momentum tensor, this condition leads to

$\left(\left(1+\frac{f(\chi)}{\sqrt{|\partial_{\lambda}\phi\partial^{\lambda}\phi|}}\right)\partial_{\mu}\phi\partial_{\nu}\phi+\partial_{\mu}\chi\partial_{\nu}\chi\right)n^{\mu}n^{\nu}\geq
0,$ (5)

that depends on $\chi$ and the derivative of both fields.

Since we are interested in topological structures, we consider static
solutions $\phi=\phi(x)$ and $\chi=\chi(x)$, with boundary conditions
$\phi(x\to\pm\infty)=v_{\pm}$ and $\chi(x\to\pm\infty)=w_{\pm}$, where
$v_{\pm}$ and $w_{\pm}$ are constants that represent the asymptotic values of
the fields $\phi$ and $\chi$. In this situation, the equations of motion (3)
reduce to

$\displaystyle\phi^{\prime\prime}$
$\displaystyle=V_{\phi}-\frac{f_{\chi}\chi^{\prime}\phi^{\prime}}{|\phi^{\prime}|},$
(6a) $\displaystyle\chi^{\prime\prime}$
$\displaystyle=V_{\chi}+f_{\chi}|\phi^{\prime}|,$ (6b)

Here, the prime denotes derivative with respect to $x$. In contrast to the
one-field model investigated in Ref. cuscutonigor , the cuscuton term
contributes to the equations above due to the presence of the function
$f(\chi)$. However, this contribution is of first order in the derivative of
the fields, and there are no second-order terms. This is one of the
characteristics of a Cuscuton term.

The non-zero components of the energy-momentum tensor in Eq. (4) are given by

$\displaystyle\rho=\frac{1}{2}{\phi^{\prime}}^{2}+\frac{1}{2}{\chi^{\prime}}^{2}+f(\chi)|\phi^{\prime}|+V(\phi,\chi),$
(7a)
$\displaystyle\tau=\frac{1}{2}{\phi^{\prime}}^{2}+\frac{1}{2}{\chi^{\prime}}^{2}-V(\phi,\chi).$
(7b)

The energy and stress densities are denoted by $\rho$ and $\tau$,
respectively. To ensure that the energy $E=\int^{\infty}_{-\infty}dx\rho$ is
finite, the function $f(\chi)$ must be chosen properly. The component related
to the stress of the solution does not depend on the function $f(\chi)$.

Multiplying the Eq. (6a) by $\phi^{\prime}$ and the Eq. (6b) by
$\chi^{\prime}$, it is possible to integrate the sum of them to obtain

$\frac{1}{2}{\phi^{\prime}}^{2}+\frac{1}{2}{\chi^{\prime}}^{2}=V(\phi,\chi),$
(8)

where we have taken the constant of integration as zero to obtain the
stressless solutions ($\tau=0$) with finite energy trilogia1 ; trilogia2 . In
contrast to the one field situation trilogia1 , the stressless condition is
not sufficient to obtain a first order formalism here. Substituting the
stressless constraint (8) in the energy density (7a), we get

$\rho(x)=\left(1+\frac{f(\chi)}{|\phi^{\prime}|}\right){\phi^{\prime}}^{2}+{\chi^{\prime}}^{2}.$
(9)

The condition imposed in Eq. (5) ensures the positivity of energy density.
Following the lines of Ref. trilogia2 , we construct a first order formalism
by introducing the new functions $P(\phi)$ e $Q(\chi)$ as

$\displaystyle\phi^{\prime}$
$\displaystyle=\pm\left(1+\frac{f(\chi)}{|\phi^{\prime}|}\right)^{-1}P_{\phi},$
(10a) $\displaystyle\chi^{\prime}$ $\displaystyle=\pm Q_{\chi}.$ (10b)

We rewrite the energy density (9) in the form

$\rho(x)=\rho_{1}(x)+\rho_{2}(x),\quad{\rm
with}\quad\rho_{1}(x)=\pm\frac{dP}{dx}\quad\text{and}\quad\rho_{2}(x)=\pm\frac{dQ}{dx}.$
(11)

The signals in Eqs. (10) and (11) are independent and must be chosen based on
the values of the functions $P(\phi)$ and $Q(\chi)$ for the boundary
conditions $\phi(x\to\pm\infty)=v_{\pm}$ and $\chi(x\to\pm\infty)=w_{\pm}$.
Thus the energy can be expressed as the variation of $P$ and $Q$ and we can
show that $E=E_{1}+E_{2}$, where $E_{1}=|P(v_{+})-P(v_{-})|$ and
$E_{2}=|Q(w_{+})-Q(w_{-})|$. It is possible to calculate the energy without
knowing the solutions, depending only on their respective boundary conditions
in the functions $P$ and $Q$. Finally, we note that the function of the
cuscuton term does not contribute to the energy.

To simplify the system of Eqs. (10), we focus our attention on monotonic
topological configurations. If $P$ and $Q$ are chosen to be monotonically non-
decreasing functions and $f(\chi)$ as an even function, these equations become

$\displaystyle\phi^{\prime}$ $\displaystyle=P_{\phi}-f(\chi),$ (12a)
$\displaystyle\chi^{\prime}$ $\displaystyle=Q_{\chi}.$ (12b)

These equations hold for the kink-kink situation ($\phi_{k}^{\prime}$ and
$\chi_{k}^{\prime}$ non-negative), since $P_{\phi}\geq f(\chi)$. Other
situations can be obtained by changing the sign of $x\to-x$ in the above first
order equations. For instance, to obtain the pair kink-antikink we change the
(12b) for $-\chi^{\prime}=Q_{\chi}$, keeping the other equation unchanged; and
so on.

From the stressless constraint in Eq. (8), we can write the potential in the
form

$V(\phi,\chi)=\frac{1}{2}{(P_{\phi}-f(\chi))}^{2}+\frac{1}{2}Q_{\chi}^{2}.$
(13)

It is clear to see that the above potential is non-negative, i.e,
$V(\phi,\chi)\geq 0$. Topological solutions connect potential minima, which
correspond to constant values $(\phi,\chi)=(v_{i},w_{i})$ such that
$V(v_{i},w_{i})=0$. As we focus on kink-kink configurations, we study the
cases where $v_{+}>v_{-}$ and $w_{+}>w_{-}$, and seek solutions that connect
the minima $(v_{-},w_{-})$ to $(v_{+},w_{+})$.

As the Eq. (12b) does not depend of $\phi$, we can obtain the solution
$\chi(x)$ by evaluating the integral

$x-x_{0}=\int^{\chi}\frac{d\tilde{\chi}}{Q_{\tilde{\chi}}}=h(\chi)\quad\quad\text{to
get}\quad\quad\chi(x)=h^{-1}(x-x_{0}),$ (14)

for a specific choice of $Q(\chi)$. The constant of integration $x_{0}$
determines the center of the solution. For the symmetric configuration, the
kink center is at the origin, so $x_{0}=0$.

By using the form of $\chi=\chi(x)$, we can rewrite the first equation (12a)
as $\phi^{\prime}=P_{\phi}-f(\chi(x))$, which explicitly depends on the
spatial coordinate. The presence of the function $f(\chi(x))$ modifies the
profile of $\phi(x)$ resulting in interesting configurations.

For the solutions of the Eqs. (12), we can separate the energy density (11)
into two contributions, $\rho(x)=\rho_{1}(x)+\rho_{2}(x)$, where

$\displaystyle\rho_{1}(x)$ $\displaystyle=P_{\phi}\,(P_{\phi}-f(\chi)),$ (15a)
$\displaystyle\rho_{2}(x)$ $\displaystyle=Q_{\chi}^{2}.$ (15b)

The second contribution depends solely on the field $\chi$, with energy
$E_{2}=Q(w_{+})-Q(w_{-})$, while the first one depends on both fields ($\phi$
and $\chi$), with energy $E_{1}=P(v_{+})-P(v_{-})$. Although the energy
density $\rho_{1}$ depends on the fields $\phi$ and $\chi$, its energy depends
only on the function $P(\phi)$ evaluated at the extrema of solution $\phi(x)$.

We can also study linear stability of the model. Indeed, by taking small
fluctuations around the static fields in the form
$\phi(x,t)=\phi(x)+\sum_{n}\eta_{n}(x)\cos(\omega_{n}t)$ and
$\chi(x,t)=\chi(x)+\sum_{n}\xi_{n}(x)\cos(\omega_{n}t)$, one can substitute
them in Eq. (3) to obtain

$\displaystyle-\eta_{n}^{\prime\prime}-\frac{f_{\chi}\phi^{\prime}}{|\phi^{\prime}|}\xi_{n}^{\prime}+V_{\phi\phi}\eta_{n}+\left(V_{\phi\chi}-\left(\frac{f_{\chi}\phi^{\prime}}{|\phi^{\prime}|}\right)^{\prime}\right)\xi_{n}=\omega_{n}^{2}\left(1+\frac{f(\chi)}{|\phi^{\prime}|}\right)\eta_{n},$
(16a)
$\displaystyle-\xi_{n}^{\prime\prime}+\frac{f_{\chi}\phi^{\prime}}{|\phi^{\prime}|}\eta_{n}^{\prime}+V_{\chi\phi}\eta_{n}+\big{(}V_{\chi\chi}+f_{\chi\chi}|\phi^{\prime}|\big{)}\xi_{n}=\omega_{n}^{2}\xi_{n}.$
(16b)

Despite the intricated form, we observe that, unlike the case of a single
scalar field studied in cuscutonigor , the cuscuton contributes to terms
involving first derivatives of the perturbation $\eta^{\prime}_{n}$ and
$\xi^{\prime}_{n}$. However, its contributions with terms involving second
derivatives remain absent. This will be investigated in another work.

In more general situations, the dynamical term
$\partial_{\mu}\phi\partial^{\mu}\phi$ can vanish, leading to a divergence in
the terms where this quantity is in the denominator, both in the equation of
motion and in the energy-momentum tensor in Eqs. (3) and (4). This also occurs
when studying the cuscuton term in cosmology, where the scalar field depends
on the temporal coordinate $\phi=\phi(t)$. For this specific case, there is a
cancellation due to other contributions that also vanish. In the case of
static solutions, the behavior is similar. As we have seen, there is no
divergence when considering fields dependent only on the spatial coordinate.
One may wonder if the fluctuations $\eta_{n}(x)$ and $\xi_{n}(x)$ added in the
expressions above Eq. (16) lead to divergencies. We have considered them to be
sufficiently small to avoid this issue. Despite that, in scenarios where
temporal and spatial dependencies freely compete, such as in the study of kink
collisions and the formation of defect networks in higher dimensions, this
model needs to be further investigated. One possibility is to introduce a
generalization where, in a certain limit of a given parameter, we obtain the
model studied here. With this, it would be possible to regularize and
understand the behavior of possible divergences. This type of approach was
used in Refs. deSouzaDutra:2005cc ; Simas:2017fjt  in the context of divergent
solutions associated to vacuumless potentials.

##  III Examples

In this section, we discuss some examples that allow us to obtain analytical
solutions of the model introduced in this paper. For this purpose, we choose
the following functions

$P(\phi)=\lambda\left(\phi-\frac{1}{3}\phi^{3}\right)\quad\text{and}\quad
Q(\chi)=\chi-\frac{1}{3}\chi^{3},$ (17)

with $\lambda>0$. As $P$ and $Q$ are odd functions, the models studied in this
section present $Z_{2}\times Z_{2}$ symmetry. The kink solution of the $\chi$
field can be obtained using Eq. (14)

$\chi(x)=\tanh(x),$ (18)

connecting $w_{-}=-1$ to $w_{+}=1$. The second contribution of the energy
density (15b) is written as $\rho_{2}(x)=\mbox{sech}^{4}(x)$, that can be
integrated all over space to determine $E_{2}=4/3$. The presence of the term
$f(\chi)$ leads to an explicit dependence on the spatial coordinate in Eq.
(12a), so we get

$\phi^{\prime}=\lambda\left(1-\phi^{2}\right)-f(\tanh(x)).$ (19)

To write the energy density $\rho_{1}(x)$ in Eq. (15a), we must solve the
equation above. Of course that, in the uncoupled case ($f(\chi)=0$), the above
equation has the solution

$\phi(x)=\tanh(\lambda\,x),$ (20)

whose asymptotic behavior is $\phi(\pm\infty)=\pm 1$. We take three specific
functions that preserve these boundary conditions,

$\displaystyle f_{I}(\chi)=\beta\left(1-\chi^{2}\right),$ (21a) $\displaystyle
f_{II}(\chi)=\beta\,\chi^{2}\\!\left(1-\chi^{2}\right),$ (21b) $\displaystyle
f_{III}(\chi)=\beta\left(1-a\chi^{2}\right)^{2}\left(1-\chi^{2}\right).$ (21c)

These functions correspond to type-I, type-II, and type-III models, where
$\beta$ and $a$ are real parameters. As seen, the energy is independent of the
choice of the function $f(\chi)$ and we obtain $E_{1}=4\lambda/3$. However,
the explicit expression for the energy density $\rho_{1}(x)$ depends on the
solution $\phi(x)$, which we determine in the following subsections.

###  III.1 Type-I models

We start by considering the function $f_{I}(\chi)$ from Eq. (21a). This
function is proportional to $Q_{\chi}$ and, therefore, vanishes at $\chi=\pm
1$ and have maxima in $\chi=0$, with $f(0)=\beta$, reaching a maximum at
$\chi=0$ with $f(0)=\beta$. The first-order equation (19) becomes

$\phi^{\prime}=\lambda\left(1-\phi^{2}\right)-\beta\,\mbox{sech}^{2}(x),$ (22)

where $\lambda$ and $\beta$ are the parameters that control the model. The
kink solution for the above equation is

$\phi(x)=\frac{1}{2\lambda}\left(\sqrt{1+4\beta\lambda}+1\right)\tanh(x)+\frac{1}{2\lambda}\left(\sqrt{1+4\beta\lambda}+1-2\lambda\right)\left(\frac{P_{+}(x)+C_{0}Q_{+}(x)}{P_{-}(x)+C_{0}Q_{-}(x)}\right).$
(23)

Here, $C_{0}$ is an integration constant that arises in the process and the
functions $P_{\pm}(x)$ and $Q_{\pm}(x)$ are given by

$\displaystyle
P_{\pm}(x)=P_{\nu_{\pm}}^{\lambda}(\tanh(x))\,\,\,\,\,\,\,\,{\rm
and}\,\,\,\,\,\,\,\,Q_{\pm}(x)=Q_{\nu_{\pm}}^{\lambda}(\tanh(x)),$ (24a)

with

$\nu_{\pm}=\frac{1}{2}\left(\sqrt{1+4\beta\lambda}\,\pm 1\right).$ (25)

$P_{\nu}^{\mu}(z)$ and $Q_{\nu}^{\mu}(z)$ are the associated Legendre
functions of the first and second kind with argument $z$ and parameters $\mu$
and $\nu$. We choose the constant of integration $C_{0}$, such that
$\phi(0)=0$. This leads us

$C_{0}=-\frac{Q_{\nu_{+}}^{\lambda}(0)}{P_{\nu_{+}}^{\lambda}(0)}.$ (26)

For $\beta=0$, the solution in Eq. (23) reduces to the kink solution given in
Eq. (20).

According to the Eq. (23), for $\lambda\geq 1$, the range of the parameter
$\beta$ is $\beta\in[0,\lambda]$, that is always non-negative; while for
$\lambda<1$ it is $\beta\in[\lambda-1,\lambda]$, allowing negative values. In
particular, when $\beta=\lambda-1$, the solution for $\phi(x)$ is identical to
the solution for the field $\chi$, given by Eq. (18). For the upper limit,
$\beta=\lambda$, there is a plateau in the solution at $x=0$; for this reason,
we denote $\beta_{c}=\lambda$ as critical value. To analyze the behavior of
Eq. (23) near the origin, we can use a power series expansion, which yields

$\phi(x)=(\beta_{c}-\beta)\,x+\left(\beta-\beta_{c}(\beta_{c}-\beta)^{2}\right)\frac{x^{3}}{3}+\mathcal{O}\\!\left(x^{5}\right).$
(27)

For the critical value $\beta=\beta_{c}$, the linear contribution vanishes and
the leading term becomes $\phi(x\approx 0)\approx(\beta_{c}/3)\,x^{3}$. This
behavior is shown in Fig. 1, where we plot the solutions (23) for
$\lambda=1/2$ and $1$, and some values of $\beta$. The parameter $\beta$
smoothly modifies the slope of the solutions at the origin: as it approaches
its critical value ($\beta_{c}$), the slope decreases until zero. The
thickness of these configurations depends on $\lambda^{-1}$, as usual.

An analytical expression for the contribution $\rho_{1}(x)$ to the energy
density (15a) can be obtained, but it is cumbersome and therefore was omitted
here. We can investigate this quantity near the origin, similarly to what was
done for the solution,

$\rho_{1}(x)=\beta_{c}(\beta_{c}-\beta)+\beta_{c}\left(\beta-(2\beta_{c}-\beta)(\beta_{c}-\beta)^{2}\right)x^{2}+\mathcal{O}\\!\left(x^{4}\right).$
(28)

Note that the intensity of the energy density at the origin is
$\rho_{1}(0)=\beta_{c}(\beta_{c}-\beta)$, which can be a maximum or a minimum
depending on the values of $\lambda$ and $\beta$. For the critical value
$\beta=\beta_{c}$, the quantity $\rho_{1}(0)$ vanishes, and the energy density
is symmetrically divided into two portions. This behavior is illustrated in
Fig. 2, where we display this quantity for the same values of $\lambda$ and
$\beta$ as in the previous figure.

Figure 1: The solution $\phi(x)$ in Eq. (23). In left panel, it is displayed
for $\lambda=1/2$, and $\beta=0$ (dotted line), $-1/2,-1/4,1/4$ (solid lines)
and $1/2$ (dashed line). In the middle panel, it is shown for $\lambda=1$, and
$\beta=0$ (dotted line), $1/4,1/2,3/4$ (solid lines) and $1$ (dashed line). In
the right panel, it is depicted for $\lambda=2$, and $\beta=0$ (dotted line),
$1/2,1,3/2$ (solid lines) and $2$ (dashed line).

Figure 2: Contribution $\rho_{1}(x)$ to the energy density (15a) for the
solutions (23) with the same line styles and values of the parameters
$\lambda$ and $\beta$ in Fig. 1.

###  III.2 Type-II models

Now we consider novel configurations that arise with the function
$f_{II}(\phi)$ given by Eq. (21b). In this model, we take $\beta\geq 0$. The
first order equation (12) becomes

$\phi^{\prime}=\lambda\left(1-\phi^{2}\right)-\beta\,\mbox{sech}^{2}(x)\tanh^{2}(x).$
(29)

For $\beta=0$, the solution is given by (20), and for $\beta>0$, it is given
by

$\phi(x)=\frac{d}{dx}\left(\ln^{\frac{1}{\lambda}}\left(\mbox{sech}^{\lambda}(x)\,\text{HeunC}\\!\left(0,-\frac{1}{2},\lambda,-\frac{\beta\lambda}{4},\frac{\lambda^{2}+1}{4};\tanh^{2}(x)\right)\right)\right),$
(30)

where $\text{HeunC}(\alpha,\gamma,\delta,\epsilon,\eta;z)$ are the confluent
Heun functions with the parameters $\alpha,\gamma,\delta,\epsilon,\eta$ and
argument $z$. We fix the constant of integration to obtain $\phi(0)=0$. The
parameters $\lambda$ and $\beta$ control the behavior of the solution. As in
the previous model, the thickness of the solution is controlled by $\lambda$,
and $\beta$ modifies the slope at points outside the origin. Instead, it
changes the slope at symmetrical points about the origin. There is a maximum
value $\beta$, which we define as $\beta_{c}$. For this value, the solution
has null derivative at two points, $x_{\pm}\approx\pm 0.8813$, leading to the
presence of two plateaus. In Fig. 3 we can see this behavior for
$\lambda=1/2,1$ and $2$, and some values of $\beta$. As $\beta$ increases, the
inclination of the solution decreases until two plateaus arise at $x_{\pm}$.
For $\lambda=1/2,1$ and $2$ the critical values of $\beta$ are
$\beta_{c}=1.9170,3.3639$ and $3.8227$, respectively. It is noteworthy that
the parameter $\lambda$, in addition to modify the thickness of the solutions,
shifts the value of the $\phi(x_{\pm})$, which has zero derivative.

The demeanor of the contribution $\rho_{1}(x)$ to the energy density (15a) for
this solutions is shown in Fig. 4 for several values of $\lambda$ and $\beta$.
One can see that always has a maximum at the center with intensity
$\rho_{1}(0)=\lambda^{2}$. As $\beta$ increase, local minima and maxima
appear. For $\beta=\beta_{c}$, the minima turn global minima at $x_{\pm}$,
with $\rho_{1}(x_{\pm})=0$. At this critical value, the contribution
$\rho_{1}(x)$ splits in one central and two outer portions, whose ratio
between then depends on $\lambda$. as can see in figure.

Figure 3: The solution $\phi(x)$ in Eq. (30). In the left panel, it is
depicted for $\lambda=1/2$, and $\beta=0$ (dotted line), $1/2,1,3/2$ (solid
lines) and $=1.9170$ (dashed line). In the middle panel, it is displayed for
$\lambda=1$, $\beta=0$ (dotted line), $1,2,3$ (solid lines) and $3.3639$
(dashed line). The right panel, it is shown for $\lambda=2$, $\beta=0$ (dotted
line), $1,2,3$ (solid lines) and $3.8227$ (dashed line).

Figure 4: Contribution $\rho_{1}(x)$ of the energy density (15a) for the
solutions (30) with the same line styles and values of the parameters
$\lambda$ and $\beta$ in Fig. 3.

###  III.3 Type-III models

We now consider another class of models described by the function
$f_{III}(\chi)$ in the Eq. (21c). The first order equation (12) becomes

$\phi^{\prime}=\lambda\left(1-\phi^{2}\right)-\beta\,\mbox{sech}^{2}(x)(1-a\tanh^{2}(x))^{2},$
(31)

with $\beta\in[0,\beta_{c}]$ and $a\in[-1/2,a_{c}]$, where $\beta_{c}=\lambda$
and $a_{c}$ is determined for specific values of $\lambda$ and $\beta$. When
$a=0$, the model reduces to the type-I model discussed in Sec. III.1. The
solution is given by

$\displaystyle\phi(x)$
$\displaystyle=\frac{d}{dx}\left(\ln^{\frac{1}{\lambda}}\left(\mbox{sech}^{\lambda}(x)\,\text{HeunC}\\!\left(a\sqrt{\beta\lambda},-\frac{1}{2},\lambda,-\frac{\beta\lambda
a(a-2)}{4},\frac{1+\lambda^{2}-\beta\lambda}{4};\tanh^{2}(x)\right)\right)\right)$
(32)
$\displaystyle+a\sqrt{\frac{\beta}{\lambda}}\,\mbox{sech}^{2}(x)\tanh(x),$

where we have fixed the constant of integration to obtain $\phi(0)=0$. The two
parameters provide new features for the kink solution. As in the type-I model,
$\beta$ modifies the inclination at the origin, and becomes null at its upper
value, $\beta=\beta_{c}$. This behavior can be seen by analyzing the solution
around $x=0$,

$\phi(x)=A\,x+B\,x^{3}+C\,x^{5}+\mathcal{O}\\!\left(x^{7}\right),$ (33)

where $A=\beta_{c}-\beta,B=-\beta_{c}A^{2}/3+\beta(1+2a)/3$ and
$C=-2\beta_{c}AB/5-\beta\left(2+10a+3a^{2}\right)\\!/15.$ In addition with
respect to the type-I model, the parameter $a$ also contributes to modifying
this feature. For $\beta=\beta_{c}$ and $a\neq-1/2$ the linear contribution
disappears ($A=0$), giving rise to a plateau where the cubic dependence is
dominant, similar to the aforementioned model. When $\beta=\beta_{c}$ and
$a=-1/2$, the linear and cubic terms vanishes ($A=0,B=0$), making the plateau
wider; for this case, we have $\phi(x\approx 0)\approx(3\lambda/20)\,x^{5}$.
Besides contributing to the behavior at $x=0$, the parameter $a$ also plays a
role in modifying the slope of the solution at symmetrical points outside the
center ($x\neq 0$). When $a=a_{c}$, the solution gets inflection points with
null derivative, leading to two additional plateaus. To demonstrate this new
feature, we plot the solution (32) in Fig. 5, where we take $\beta=\beta_{c}$,
$\lambda=1/2,1$ and $2$, and some values of $a$, including
$a_{c}=3.8723,3.5156$ and $2.4471$. For these choices, we can see that the
solution with one plateau gets two additional plateaus, resulting a three-
plateau configuration for $a=a_{c}$. As we increase $\lambda$, the symmetric
plateaus get closer to the tail of the solution.

Analogously to the previous examples, we omit the expression of the energy
density $\rho_{1}(x)$ given by Eq. (15a), but we depict it in Fig. 6 for same
choices of parameters utilized in Fig. 5. An analysis close to the origin
reveals that

$\rho_{1}(x)=\beta_{c}A+\beta_{c}\left(3B-A^{2}\right)x^{2}+5\beta_{c}\left(C-A^{2}B\right)x^{4}+\mathcal{O}\\!\left(x^{6}\right),$
(34)

where the dominant contributions depend on the chosen parameters. It is clear
that for $\beta=\beta_{c}$ and $a\neq-1/2$ the intensity at the origin is
null, so $\rho_{1}(x\approx 0)\approx\beta_{c}^{2}(1+2a)\,x^{2}$. Finally, in
the additional case $\beta=\beta_{c}$ and $a=-1/2$, we have $\rho_{1}(x\approx
0)\approx(3\beta_{c}^{2}/4)\,x^{4}$. Notice that besides the hole in the
center, there are also two symmetric others. As it was seen, this happens when
we take $\beta=\beta_{c}$ and $a=a_{c}$. In this situation, $\rho_{1}(x)$
splits in four portions whose ratios depend on $\lambda$. For example, for
$\lambda=2$, the contribution of the peripheral pieces is much smaller than
that of the others.

Figure 5: The solutions (32) for $\beta=\lambda$ and some values of $\lambda$
and $a$. In the left panel, it is shown for $\lambda=1/2$, and $a=0$ (dotted
line), $1,2,3$ (solid lines) and $3.8723$ (dashed line). In the middle panel,
it is depicted for $\lambda=1$, $a=0$ (dotted line), $1,2,3$ (solid lines) and
$3.5156$ (dashed line). In the right panel, it is displayed for $\lambda=2$,
$a=0$ (dotted line), $1/2,1,2$ (solid lines) and $2.4471$ (dashed line).

Figure 6: Contribution $\rho_{1}(x)$ to the energy density (15a) for the
solutions (23) with the same line styles and values of the parameters
$\lambda$, $\beta$ and $a$ in Fig. 5. In the right panel, in the inset, we
enlarged the tail to better visualize its peripheral portion.

##  IV End Comments

In this work we have investigated a class of models of two scalar fields in
the presence of a new coupling inspired by cuscuton dynamics. Unlike the case
of a single field model investigated in Ref. cuscutonigor , here, this
addition contributes in the equations of motion for the static case, without
introducing second-derivative terms of the field. To find the solutions, one
must solve the equations of motion. Since they are of second order with
couplings between the fields, we have developed a formalism by introducing
smooth functions that lead to first-order equations for kink solutions,
providing a simple and direct way to obtain their energy. We introduced three
classes of examples from which we obtain analytical solutions with distinct
behaviors

Using the aforementioned procedure, we have developed a first-order equation
for the second field $\chi$ that only depends on $Q(\chi)$. Then, we have
fixed the same $\chi$ solution for the all examples and focused our attention
on how the field $\phi$ behaves for specific choices of $f(\chi)$. The right-
hand side of the first-order equation (12a) contains two terms, with one of
them depending on both $\phi$ and $\chi$ and competing with each other. We
have shown that this new interaction leads to meaningful modulations in the
kink profile. With appropriate choices for the parameters, it is possible to
control the slope of the $\phi$ field and the internal structure of its energy
density, which has plateaus and null intensity, respectively, for the critical
situations.

The profile of the solutions described in the present work reminders us of the
ones found in Ref. liao ; Sebastian:2023dds , where the authors have dealt
with the problem through a coupling in the usual dynamics term. This coupling
leads to first order equations that differ from those presented here. In that
work, they also investigated other models, which lead to asymmetric
configurations that have not been considered here. An interesting direction
for future research would be to explore this possibility. Other perspectives
include the study of the model in the presence of gravity, in the five-
dimensional braneworld scenario, with a single extra dimension brana1 ; brana2
; brana3 ; cuscutondouglas . Another interesting issue concerns extending the
above results for relativistic field theories that support vortex solutions,
where a complex scalar field is coupled with a gauge field under the action of
the local $U(1)$ gauge symmetry vortexM1 ; vortexM2 ; vortexCS1 ; vortexCS2 .
We hope to report on these and other related issues in the near future.

###### Acknowledgements.

We would like to thank D. Bazeia and M.A. Marques for the discussions that
have contributed to this work. We acknowledge the financial support from the
Brazilian agencies Conselho Nacional de Desenvolvimento Científico e
Tecnológico (CNPq), grant No. 306504/2018-9 (RM), and Paraíba State Research
Foundation (FAPESQ-PB), grant No. 0003/2019 (RM).

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# Visually Aware Skip-Gram for Image Based Recommendations

Parth Tiwari  IIT Kharagpur1 Thørväld CircleKharagpurIndia
<EMAIL_ADDRESS>,  Yash Jain  IIT Kharagpur1 Thørväld
CircleKharagpurIndia<EMAIL_ADDRESS>,  Shivansh Mundra  IIT
Kharagpur1 Thørväld CircleKharagpurIndia<EMAIL_ADDRESS>,  Jenny
Harding  Loughborough UniversityLoughboroughUnited Kingdom and  Manoj Kumar
Tiwari  National Institute of Industrial EngineeringMumbaiIndia

(2020)

###### Abstract.

The visual appearance of a product significantly influences purchase decisions
on e-commerce websites. We propose a novel framework VASG (Visually Aware
Skip-Gram) for learning user and product representations in a common latent
space using product image features. Our model is an amalgamation of the Skip-
Gram architecture and a deep neural network based Decoder. Here the Skip-Gram
attempts to capture user preference by optimizing user-product co-occurrence
in a Heterogeneous Information Network while the Decoder simultaneously learns
a mapping to transform product image features to the Skip-Gram embedding
space. This architecture is jointly optimized in an end-to-end, multitask
fashion. The proposed framework enables us to make personalized
recommendations for cold-start products which have no purchase history.
Experiments conducted on large real world datasets show that the learned
embeddings can generate effective recommendations using nearest neighbour
searches.

Recommender Systems, Skip-Gram, Representation learning, Multitask learning,
Cold-Start, Image Features

††copyright: acmcopyright††journalyear: 2020††conference: Under review; 2020;
Earth††price: 15.00††ccs: Computing methodologies Image representations††ccs:
Applied computing Online shopping††ccs: Computing methodologies Unsupervised
learning††ccs: Computing methodologies Multi-task learning

##  1\. Introduction

Recommender Systems have been described as “an intuitive line of defense
against consumer over-choice” (Zhang et al., 2019). With the rapid growth in
the number of products available on e-commerce websites, this problem of
”over-choice” is becoming more and more significant, especially in the case of
clothing related products. In fact, fashion recommendation has attracted
considerable attention in recent literature. Studies tackling this problem
have shown improved performance by incorporating visual information into their
recommendation procedure. Hence it is safe to conclude that the visual
appearance of a product plays a significant role while making purchase
decisions.

Image-based recommendation systems can be broadly classified into two groups -
(i) Systems which incorporate images into a product rating/rank prediction
function. These systems build upon well known Latent Factor models like
Singular Value Decomposition (SVD) or Bayesian Personalized Ranking (BPR).
(Chen et al., 2019; He and McAuley, 2016; He et al., 2016b, a; Chen et al.,
2017) (ii) Systems which learn latent representations (a.k.a embeddings) using
image features and subsequently use embedding distance for making
recommendations. These systems rely on extracting fine-grained embeddings
which can capture user preference and/or product similarity (Guo et al., 2018;
McAuley et al., 2015; Ying et al., 2018; Shankar et al., 2017; Jiang et al.,
2016). Beyond images, information coming from product metadata (Vasile et al.,
2016), user-product information networks (Shi et al., 2019), review-text (Guo
et al., 2018; Cheng et al., 2019) etc. has also been leveraged for learning
effective representations.

Making recommendations in the cold-start setting is a key area where addition
of visual information has shown significant performance improvement (He and
McAuley, 2016). However, we observe that there is considerable variability in
the definition of cold-start setting across literature. Some studies describe
cold-start products as items that have fewer instances of positive feedback in
the training set (He and McAuley, 2016; He et al., 2016b; Shi et al., 2019;
Chen et al., 2017; Liu et al., 2020), whereas (Vasile et al., 2016) describes
its cold-start scenario when certain product-pairs have zero co-occurrences
while training. Recommending completely new products (i.e. products which are
unseen during the training step) is a tougher problem which is studied less
frequently (Schein et al., 2002; Saveski and Mantrach, 2014; Gantner et al.,
2010).

Systems falling under category (i) (as described above) rely on user and
product latent factors coming from interaction records, hence, they are
incapable of handling new products unless product latent factors are estimated
by other methods. Systems falling under category (ii) show better promise of
handling this problem. Methods like (Shankar et al., 2017; McAuley et al.,
2015), which make recommendations based on product-product similarity are
capable of extracting latent representations for a new product. However, here
user preference is ignored while feature extraction hence making personalized
recommendations is not possible. Methods which attempt to capture preferences
for individual users are better suited for handling this problem.

To this end, we propose a novel framework which jointly learns (i) user and
product embeddings in a common latent space and (ii) a mapping function which
transforms product image features to the same latent space. In specific, we
augment the Skip-Gram model (Mikolov et al., 2013b) with a decoder
architecture where the decoder uses the product Skip-Gram embeddings to
reconstruct their corresponding image features. Here Skip-Gram maximizes the
probability of co-occurrence of users and products in a Heterogeneous
Information Network (HIN) while the decoder minimizes the image feature
reconstruction loss. This decoder serves two purposes- (i) it incorporates
visual information into the skip gram embeddings and (ii) it is later used for
learning a mapping function which transforms product image features to the
Skip-Gram embedding space. The learning objective of this architecture is
dependent on the genre of input (users or products) observed while training.
The Skip-Gram model is optimized in case of both users and products whereas
the decoder is updated only for products in a multitask fashion. We term this
model as VASG (Visually Aware Skip-Gram).

The mapping from the product image feature space to VASG embedding space is
subsequently learned and can be used for finding effective embeddings for
cold-start products. VASG embeddings reflect user preference while
simultaneously capturing a product’s purchase history and visual appearance.
Since users and products are represented in the same latent space,
personalized recommendations can be made directly by searching for products in
the user’s vicinity. Our contributions can be summarized as -

  * •

We propose a novel multitasking Skip-Gram architecture, VASG, which is trained
on two different objectives depending on the genre of the input (users or
products). This architecture enables us to learn a mapping which transforms
product image features to the learned embedding space.

  * •

Extensive experiments are performed on real world datasets and VASG embeddings
are compared to state-of-the-art recommendation systems. The performance for
unseen cold start products is also studied.

  * •

We analyze the learned embeddings in detail to shed some light on the
information captured by them.

Figure 1. The VASG framework. (a) User-Product interactions are represented in
a HIN. (b) Sequences are generated by performing random walks on the HIN. The
Skip-Gram context window is used to describe the neighbourhood of a node. (c)
The Skip-Gram model is trained on the generated sequences. (d) The decoder
$\mathcal{M}^{-1}(.)$ attempts to reconstruct product image features from the
Skip-Gram embeddings. (e) An autoencoder is trained to estimate the mapping
$\mathcal{M}(.)$

##  2\. Related Work

Our work relies on several advancements in image feature extraction methods,
representation learning techniques in graphs and multitask learning.

With advancements in deep learning, VBPR (He and McAuley, 2016) established
that the addition of visual signals can be useful for making recommendations.
VBPR adds image features extracted from a pre-trained convolutional neural
network to the BPR framework. Similarly, (Chen et al., 2017) extracts more
fine grained image features using attention modules and incorporates them in
collaborative filtering. Images have also been proven to be useful for Point-
of-Interest recommendations in (Wang et al., 2017) and for tag recommendations
in (Rawat and Kankanhalli, 2016).

More recently, there has been a surge of methods which combine information
coming from different modalities and metadata with images (Cheng et al., 2019;
He et al., 2016b; Guo et al., 2018; He et al., 2016a; Chen et al., 2019).
Attention mechanisms which utilize textual reviews have been used for finding
specific parts of an image where users are most interested in (Chen et al.,
2019). Reviews and image features have also been used for learning user
preference which is then integrated into a rating-based matrix factorization
model (Cheng et al., 2019). Pre-defined product categories and hierarchy trees
have been leveraged for learning hierarchical product embeddings (He et al.,
2016b).

Recommendation Systems which rely on product image similarity; for example
methods which address the Street-to-shop recommendation problem (Liu et al.,
2012); have received a tremendous boost with improvements in metric learning
methods (Gajic and Baldrich, 2018; Jiang et al., 2016; Shankar et al., 2017) .
These methods use triplet loss to learn embeddings from product images which
are robust to change in background, pose, lighting conditions etc. and hence
can be used for making cross-domain recommendations. However, learning both
user and product representations in a joint latent space is less frequently
explored in existing literature. (Gong et al., 2020) embeds users and topics
to the same low dimensional space to capture of their mutual dependency while
(Guo et al., 2018) attempts to bring users, products and product search
queries to the same latent space using both textual and visual modalities.

Heterogeneous Information Networks (HINs) can naturally model complex user and
product interactions. Traditionally, meta-path based similarity and link
predictions have been used for making recommendations in networks. With the
development of graph representation learning methods (Perozzi et al., 2014;
Grover and Leskovec, 2016; Dong et al., 2017), network embedding based
recommendation systems have shown improved performance. The study (Shi et al.,
2019) learns user and product representations from a HIN and incorporates them
into the SVDFeature (Chen et al., 2012) framework. This idea of learning
network representations has also been extended by using multitask learning for
jointly optimizing the tasks of recommendation and link prediction in a HIN
(Li et al., 2020). Multitask learning is a training regime where multiple
objectives are optimized simultaneously. It has proven to provide performance
boosts in works like (Gidaris et al., 2019; Xu et al., 2020).

A few studies which are relevant to our proposed approach should be discussed
here. Learning attribute-to-feature mappings has been proposed in (Gantner et
al., 2010). Here k-nearest neighbours, least squares approximation is used for
approximating latent factors for new users or products. A large scale
recommender system which uses images along with the HIN structure is proposed
in (Ying et al., 2018). Here highly efficient Graph-Convolutional Neural
Networks are used on random walks generated from the HIN and recommendations
are then made using similarity between the learned embeddings. Contrastive
Predictive Coding (Oord et al., 2018) (CPC) is a generic representation
learning regime which uses a similar training procedure as the proposed
framework. CPC uses encoders to map raw data to latent representations and
subsequently trains auto-regressive models using a contrastive loss. VASG
differs from CPC in terms of usage of a decoding architecture instead of an
encoder; usage of Skip-Gram instead of an auto-regressive model; and training
using auxiliary supervision instead of self-supervision.

To the best of our knowledge, the model architecture proposed in this paper
has not been studied before. In addition, we also make a novel attempt to
bring unseen cold start products to the same latent space as existing users
and products using product image features.

##  3\. VASG Framework

Consider a set of users $\mathcal{U}$ and a set of products $\mathcal{P}$
where each user $u\in\mathcal{U}$ has interacted with a subset of products
$\mathcal{P}\textsubscript{\it u}\subset\mathcal{P}$. Each product
$p\in\mathcal{P}$ is associated with a visual feature
$f\textsubscript{p}\in\mathbb{R\textsuperscript{I}}$ which is extracted from
its image using a Deep Convolutional Neural Network. Only a subset of products
$\mathcal{P}\textsubscript{warm}\subset\mathcal{P}$ is observed during
training and is termed as the warm start set. The set of products which is
never observed during training is termed as completely cold set
$~{}\mathcal{P}\textsubscript{cold}\subset\mathcal{P}$ . It is to be noted
that
$\mathcal{P}\textsubscript{cold}\cap\mathcal{P}\textsubscript{warm}=\phi$.
Our objective is to learn (i) a D dimensional latent representation
$\mathcal{X}\in\mathbb{R}\textsuperscript{\it D}$ for each $u\in\mathcal{U}$
and $p\in\mathcal{P}\textsubscript{warm}$; and (ii) a mapping function
$\mathcal{M}(f\textsubscript{p}):\mathbb{R}\textsuperscript{\it
I}\rightarrow\mathbb{R}\textsuperscript{\it D}$. This function is used for
finding embeddings of products that have no purchase history. The learned
embeddings are expected to represent characteristic features of users and
products which can be exploited for making representations. The VASG framework
achieves the above mentioned objectives in two steps-

  * •

The first step involves training a Skip-Gram model on sequences of users and
products. Here, the Skip-Gram architecture is modified by adding an auxiliary
objective of learning an inverse mapping function
$\mathcal{M}\textsuperscript{-1}(\mathcal{X}\textsubscript{p}):\mathbb{R}\textsuperscript{\it
D}\rightarrow\mathbb{R}\textsuperscript{\it I}$. The D dimensional embeddings
corresponding to users $u\in\mathcal{U}$ and products
$p\in\mathcal{P}\textsubscript{warm}$ are learned during this step.

  * •

The second step involves learning the mapping function
$\mathcal{M}(f\textsubscript{p}):\mathbb{R}\textsuperscript{\it
I}\rightarrow\mathbb{R}\textsuperscript{\it D}$. This is achieved by training
a deep encoder-decoder architecture using the inverse mapping
$\mathcal{M}\textsuperscript{-1}(.)$ learned in the previous step.

We describe the network structure used for representing user-product
interactions before describing the model architecture.

###  3.1. Heterogeneous Information Network

Skip-Gram (Mikolov et al., 2013a) is an unsupervised representation learning
model for words in a text corpus. The representation corresponding to each
word is learned by maximizing the probability of correctly predicting the
context surrounding the given word. The proposed framework builds upon the
Skip-Gram architecture, hence, it requires sequences of user and products.
Taking inspiration from (Shi et al., 2019), we generate these sequences by
performing random walks on a bipartite Heterogeneous Information Network (HIN)
of users and products.

Formally, our HIN is a graph $G=(V,E)$ where
$V=\mathcal{U}\cup\mathcal{P}\textsubscript{warm}$ is the set of nodes.
$E=\\{(u,p)|p\in\mathcal{P}\textsubscript{u}\cap\mathcal{P}\textsubscript{warm}~{};~{}u\in\mathcal{U}\\}$
is the set of edges connecting the nodes where each edge connects a user to
the products purchased by him. Sequences are generated in the following manner
- Starting from a node $\mathrm{v}\in V$, $n$ number of random walks, each of
length $wl$ are performed on the HIN. Every subsequent node in a walk is
chosen randomly from the direct neighbours of the current node. For example,
consider the sample HIN shown in Figure 1(a). A sequence starting from node u1
could be u1 p2 u2 p2 u3 … repeated for $wl$ nodes. Here, $n$ and $wl$ are
hyper-parameters which determine the size of the corpus generated for
training. It is to be noted that the bipartite structure of the HIN ensures
that users and products appear in an alternating fashion in each random walk.
The neighbouring context of a node, $\mathcal{N}(\mathrm{v})$ is determined by
sliding a fixed size context window over the generated sequences. We discuss
the effect of the context window size on the learned representations in
Section 6.

Table 1. Dataset statistics (after preprocessing) Dataset | #products | #users | #ratings
---|---|---|---
Women | 116,971 | 14,110 | 307,863
Men | 62,376 | 19,618 | 187,181
Shoes | 71,219 | 22,066 | 211,818
Jewelry | 62,958 | 13,975 | 161,439

###  3.2. Visually Aware Skip-Gram

Consider an HIN node $\mathrm{v}\in V$ and its neighbouring context
$\mathcal{N}(\mathrm{v})$. We can learn the the representations corresponding
to each node in the HIN by optimizing the Skip-Gram objective -

(1) $\underset{\theta}{arg\,max}\displaystyle\sum_{\mathrm{v}\in
V}\displaystyle\sum_{c\in N(\mathrm{v})}log(p(c|v;\theta))$

Here $p(c|v;\theta)$ is the commonly used softmax probability, given by:
$p(c|v,\theta)=\frac{e^{\mathcal{X}\textsubscript{c}\mathcal{X}\textsubscript{v}}}{\sum_{n\in
V}e^{\mathcal{X}\textsubscript{\it n}\mathcal{X}\textsubscript{v}}}$ where
$\mathcal{X}\textsubscript{v}\in\mathbb{R}\textsuperscript{D}$ is the $D$
dimensional latent representation corresponding to node $\mathrm{v}$. Mikolov
et al. (Mikolov et al., 2013b) introduces negative sampling for efficiently
estimating the softmax probabilities however, generating effective negative
samples can be an expensive process (Jain et al., 2019). We expect the
negative samples to highlight the contrast in a product’s visual appearance
and a user’s purchase preference simultaneously. In our experiments, we are
able to obtain meaningful results without negative sampling, hence we
approximate the softmax probability as:
$\sigma(\mathcal{X}\textsubscript{c}~{}.~{}\mathcal{X}\textsubscript{v})$
where $\sigma$ is the $sigmoid$ function, given by:
$\sigma(x)=\frac{1}{1+e^{-x}}$.

We add the auxiliary objective of learning
$\mathcal{M}\textsuperscript{-1}(\mathcal{X}\textsubscript{p}):\mathbb{R}\textsuperscript{\it
D}\rightarrow\mathbb{R}\textsuperscript{\it I}$ to the Skip-Gram objective
whenever $\mathrm{v}\in\mathcal{P}\textsubscript{warm}$. The function
$\mathcal{M}\textsuperscript{-1}(.)$ attempts to reconstruct the DeepCNN image
feature $f\textsubscript{p}$ of a product from its latent representation
$\mathcal{X}\textsubscript{p}$. This inverse mapping function is approximated
by training a decoder architecture with the objective of minimizing the mean
squared error of image feature reconstruction -

(2)
$\underset{\mathcal{M}^{-1}}{arg\,min}\displaystyle\sum_{\mathrm{v}\in\mathcal{P}\textsubscript{warm}}||f\textsubscript{v}-f^{{}^{\prime}}_{\textsubscript{p}}||^{2}_{2}$

Where
$f^{{}^{\prime}}_{\textsubscript{p}}=\mathcal{M}\textsuperscript{-1}(\mathcal{X}\textsubscript{p})$.
It is to be noted that the auxiliary task is optimized only when
$\mathrm{v}\in\mathcal{P}\textsubscript{warm}$. This implies that users have a
single loss function while products have two different loss functions which
are optimized simultaneously in a multi-task learning fashion. When a node
$\mathrm{v}$ is observed in the training corpus, the loss can be written as -

(3)
$\mathcal{L}(\mathrm{v})=\begin{cases}-log(\sigma(\mathcal{X}\textsubscript{c}\mathcal{X}\textsubscript{v}))&\text{
$\mathrm{v}\in\mathcal{U}$}\\\
w\textsubscript{1}(-log(\sigma(\mathcal{X}\textsubscript{c}\mathcal{X}\textsubscript{v})))+w\textsubscript{2}||f\textsubscript{v}-f^{{}^{\prime}}_{\textsubscript{p}}||^{2}_{2}&\text{$\mathrm{v}\in\mathcal{P}\textsubscript{warm}$}\par\end{cases}$

Here $w\textsubscript{1}$ and $w\textsubscript{2}$ are trainable weights used
for combining the Skip-Gram and image reconstruction losses. We follow the
approach introduced in (Kendall et al., 2017) for learning these weights.

Once the embeddings and the inverse mapping $\mathcal{M}^{-1}(.)$ have been
learned, the function $\mathcal{M}(.)$ is approximated by training a deep
autoencoder for reconstructing the product image features for each product
$p\in\mathcal{P}\textsubscript{warm}$. The autoencoder has
$\mathcal{M}^{-1}(.)$ as the decoder while the encoder attempts to approximate
$\mathcal{M}(.)$. Weight updates are not allowed in the decoder, therefore
$\mathcal{M}^{-1}(.)$ remains unchanged during the training process. An
overview of the proposed architecture is presented in Figure 1.
Implementation Details: We implemented our model in Pytorch. The decoder
architecture $\mathcal{M}^{-1}(.)$ consists of 5 fully connected layers with
[256, 512, 1024, 2048, $I$] neurons respectively ($I=4096$). Each layer has
reLu activation and dropout regularization with dropout probability set to
0.5. The encoder $\mathcal{M}(.)$ follows the same architecture. Adam
optimizer with learning rate 1e-3 is used for parameter updates. To fully
utilize GPU acceleration, we use a batch wise implementation and ensure each
batch homogeneously comprises either users or products as we iterate over the
Skip-Gram corpus. Training on our largest dataset requires around 30 minutes
on a Nvidia Quadro P5000 GPU. We provide our implementation here111Link to
repository. Will be released at the time of publication.

##  4\. Experiments

We perform experiments on real-world datasets to evaluate the VASG embeddings.
Our experiments attempt to answer the following research questions - (i) Can
the embeddings generate personalized recommendations for each user? (ii) Can
the embeddings identify product to product relationships? (iii) What are the
properties displayed by the learned embedding space? (vi) How do different
components and hyper-parameters of the proposed framework affect its
performance?

###  4.1. Datasets

Four subcategories under the ”Clothing, Shoes & Jewelry” dataset from Amazon
Product Reviews (McAuley et al., 2015) are used for our experiments. The
selected subcategories are - Men, Women, Shoes and Jewelry. The dataset is
pre-processed to remove users that do not have sufficient purchase history.
Users with $|\mathcal{P}\textsubscript{u}|<5$ are discarded from the datasets
. Users’ rating history and product image features are used in our framework.
Product metadata is also available, however it not utilized for learning
embeddings. Only image URLs are used for retrieving images for the purpose of
visualization. Product image features were extracted using a pre-trained Caffe
reference model (Jia et al., 2014), with 5 convolutional layers followed by 3
dense layers. The feature vector is obtained by taking the output of the
second dense layer (FC7) of this network and has length $I=4096$. The details
of the datasets are specified in Table 1.

The training and test sets are created by following the leave one out protocol
described by (He and McAuley, 2016). For each user $u$, one random product
rated by the user is used for testing, while the remaining products are used
for training. Two subsets of the test set are created for testing in warm
start and completely cold start setting:
$\mathcal{T}\textsubscript{warm}=\\{(u,p)\mid
u\in\mathcal{U}~{};~{}p\in\mathcal{P}\textsubscript{warm}\\}$ and
$\mathcal{T}\textsubscript{cold}=\\{(u,p)\mid
u\in\mathcal{U}~{};~{}p\in\mathcal{P}\textsubscript{cold}\\}$. Here
$\mathcal{T}\textsubscript{warm}$ and $\mathcal{T}\textsubscript{cold}$ are
approximately equal in size.

Table 2. AUC values on the test sets. The best scores are boldfaced. Refer Section 4.2 Dataset | Setting | RAND | WBOI | VBPR | VASG
---|---|---|---|---|---
Women | Warm Products | 0.4998 | 0.6134 | 0.7982 | 0.8964
| Cold Products | 0.5001 | 0.5912 |  | 0.7596
Men | Warm Products | 0.4897 | 0.6468 | 0.7753 | 0.8758
| Cold Products | 0.4954 | 0.6094 |  | 0.7061
Shoes | Warm Products | 0.5023 | 0.6318 | 0.8116 | 0.8972
| Cold Products | 0.4995 | 0.5846 |  | 0.7261
Jewelry | Warm Products | 0.4987 | 0.6087 | 0.7629 | 0.8801
| Cold Products | 0.5003 | 0.5704 |  | 0.7134

###  4.2. Making Personalized Recommendations

VASG embeddings can be used for generating recommendations by searching for
relevant products in the vicinity of corresponding users in the learned latent
space. We evaluate this ability to generate personalized recommendations in a
Bayesian Personalized Ranking (BPR) scenario. BPR based methods (He and
McAuley, 2016; Rendle et al., 2012) learn a rating prediction function by
optimizing pairwise rankings of products w.r.t. to users. Predicted rankings
are then evaluated using the well known AUC (Area Under ROC Curve) metric:

(4)
$AUC=\frac{1}{|\mathcal{T}\textsubscript{test}|}\sum_{{}u\in\mathcal{T}\textsubscript{test}}\frac{1}{|\mathcal{P}-\mathcal{P}\textsubscript{u}|}\sum_{{}p\textsubscript{j}\in\mathcal{P}-\mathcal{P}\textsubscript{u}}\textbf{1}((u,p\textsubscript{test})>(u,p\textsubscript{j}))$

where $\mathcal{T}\textsubscript{test}$ corresponds to
$\mathcal{T}\textsubscript{warm}$ or $\mathcal{T}\textsubscript{cold}$ and
$u,p\textsubscript{test}\in\mathcal{T}\textsubscript{test}$. Here,
$\textbf{1}(.)$ is an indicator function which evaluates if
$p\textsubscript{test}$ has been ranked higher than $p\textsubscript{j}$ for
user $u$. BPR based methods rank products using the learned rating prediction
function. We evaluate VASG embeddings for ranking products using the cosine
similarity between corresponding user and product embeddings.

####  4.2.1. Baselines

We compare the proposed embeddings against the raw image features and against
a state-of-the-art visually aware BPR method:

  * •

RAND (Random) - Product rankings are decided randomly for all users.

  * •

WBOI (Weighted Bag Of Images) - User embeddings are directly computed using
the image features corresponding to the products purchased by him. The rating
weighted mean of image features is used as the user embedding and ranks are
computed using cosine similarity as described above.

  * •

VBPR (Visual Bayesian Product Ranking) - Introduced by (He and McAuley, 2016),
VBPR incorporates visual factors to the BPR framework by using product image
DeepCNN features in its rating prediction function.

####  4.2.2. Results

The entire pipeline for VASG with $D=100$ is run 5 times and the averaged
results are reported in Table 2. The results for VBPR are calculated using 20
latent factors and 100 visual factors. Results are reported on the set
$\mathcal{T}\textsubscript{warm}$ (Warm Products) and for on
$\mathcal{T}\textsubscript{cold}$ (Cold Products) separately. VASG embeddings
for cold start products are found using the mapping $\mathcal{M}(.)$. Since
latent factors for cold start products are not available in VBPR, computing
its results on the set $\mathcal{T}\textsubscript{cold}$ is infeasible.

VASG embeddings show an average improvement of 3.84% over VBPR in the warm
start setting. The AUC scores drop in the cold start setting, however they are
significantly better than the WBOI baseline. This drop in performance can be
explained by imperfections in the approximation of the mapping function
$\mathcal{M}(.)$.

###  4.3. Identifying Product Relationships

We expect the embeddings corresponding to co-purchased products to be similar.
The evaluation strategy used by McAuley et al. (IBR) (McAuley et al., 2015) is
followed for identifying product relationships using VASG embeddings. IBR uses
product DeepCNN image features to learn a parametric distance function by
maximizing the probability of correctly identifying co-purchased product
pairs. Formally, two products $p\textsubscript{i}$ and $p\textsubscript{j}$
share a relation $R\textsubscript{ij}$ if they have been purchased by the same
user. This relation is termed as an ”also bought” relation in IBR. We attempt
to correctly identify co-purchased product pairs using the cosine similarity
between the VASG embeddings. The test set is restructured in the following
manner for reporting the results - A set of positive relation pairs
$\mathcal{R}=\\{(p\textsubscript{i},p\textsubscript{j})\mid\exists~{}R\textsubscript{ij}\\}$
and a set of negative relation pairs
$\mathcal{Q}=\\{(p\textsubscript{i},p\textsubscript{j})\mid\nexists~{}R\textsubscript{ij}\\}$
is constructed such that
$|\mathcal{R}|=|\mathcal{Q}|~{};~{}p\textsubscript{i}\in\mathcal{P}$. A
positive relation is identified when
$cosine~{}similarity(\mathcal{X}\textsubscript{i},\mathcal{X}\textsubscript{j})>t$,
where $\mathcal{X}\textsubscript{i}$ is the VASG embedding corresponding to
$p\textsubscript{i}$ and $t$ is a manually assigned threshold. The accuracy of
identifying co-purchased product pairs is reported on the combined set
$\mathcal{R}\cup\mathcal{Q}$.

Table 3. Accuracy of predicting product relations. Refer Section 4.3 Dataset | INN | IBR | VASG |  | VASG
---
vs INN
| VASG
---
vs IBR
Women | 67.82% | 87.43% | 79.59% | 17% | -10%
Men | 66.23% | 87.13% | 80.12% | 21% | -9%
Shoes | 69.34% | 89.67% | 81.56% | 18% | -10%
Jewelry | 65.42% | 83.25% | 79.36% | 21% | -5%
Figure 2. 2D visualizations of VASG embeddings(a) PCA projections of 100
dimensional embeddings for each dataset. Here 200 randomly selected products
are plotted (b) t-SNE projections of 100 dimensional embeddings for each
dataset. Here products purchased by 4 randomly selected users are plotted.
Products corresponding to the same user have the same frame colour

####  4.3.1. Baselines

Since $|\mathcal{R}|=|\mathcal{Q}|$, random classification is 50% accurate. In
addition, the performance of VASG embeddings is compared to the following
methods:

  * •

INN (Image Nearest Neighbours) - Cosine similarity between the DeepCNN image
features $f\textsubscript{i},f\textsubscript{j}$ are used for identifying
relationships in the same manner as described above.

  * •

IBR (Image Based Recommendations) - As described above, this method is trained
on all co-purchased product pairs occurring in the training set.

####  4.3.2. Results

The entire pipeline for VASG with $D=100$ is run 5 times and the averaged
results are reported in Table 3. For IBR, the rank of the Mahalanobis
transform is set to 100 for fair comparison. Here, VASG embeddings show an
average improvement of 19% over INN, however they do not outperform IBR which
shows significantly better results across all datasets. However it should be
noted that IBR is trained for specifically identifying all product-product
relations (McAuley et al., 2015) whereas VASG embeddings are not. Our method
learns embeddings by capturing co-occurrence of users and products in a small
network neighbourhood which is more suited for making personalised
recommendations which is shown by the results in Section 4.2

##  5\. Analysis of Embeddings

Embedding Visualization. We visualize the learned embeddings in lower
dimensions to understand the information captured. We plot 2D projections of a
random set of products from each dataset in Figure 2(a). Here dimensions are
reduced using Principal Component Analysis (PCA). A variance in the visual
appearance of the products can be seen when we travel along the axes in these
plots. For example in the Men dataset, the x-axis is populated with watches
while the y-axis contains clothing related items like t-shirts. A similar
trend can be observed for the Women and Shoes dataset, however, the plot for
the Jewelry dataset looks noisy. Next, we visualize groups of products which
have been purchased by specific users in Figure 2(b). Here we select 4 random
users and plot all the products purchased by them. Dimensions are reduced
using t-SNE with the perplexity parameter set to 30. Products corresponding to
a specific user are marked by the colour of the frame around them. In case of
co-purchased products, the frame colour is chosen randomly. It can be seen
that products purchased by the same user have been clustered together. These
plots show that the leaned embeddings can simultaneously capture the visual
properties and the purchase history of a product.

Figure 3. Examples of recommendations generated by exploiting linear
relationships in the embeddings

A woman and a girl in white dresses sit in an open car.

Linear relationships. Skip-Gram word embeddings are known to follow a linear
algebraic structure where embeddings display associative properties (shown by
the famous example vector(”King”) - vector(”Man”) +
vector(”Woman”)$\approx$vector(”Queen”) (Mikolov et al., 2013a)). We attempt
to exploit a similar structure in the VASG embeddings for recommending
products. Consider a pair: $\\{(u1,p1),p1\in\mathcal{P}\textsubscript{u1}\\}$
and a user $u2;u2\neq u1$, keeping $(u1,p1)$ and $u2$ fixed, if we can a find
a product $p\in\mathcal{P}$ such that

$cosine~{}similarity(\mathcal{X}\textsubscript{p1}-\mathcal{X}\textsubscript{u1}+\mathcal{X}\textsubscript{u2},\mathcal{X}\textsubscript{p})\approx
1$

Then we say that $p$ is a user specific recommendation generated for the user
$u2$ having a query product $p1$. To test this methodology, we sample 10,000
pairs of $u1,u2$ and find the nearest neighbour of the point
$\mathcal{X}\textsubscript{p1}+\mathcal{X}\textsubscript{u2}-\mathcal{X}\textsubscript{u1}$,
using cosine distance, from the set $\mathcal{P}$ and treat them as
recommendations generated by VASG. We obtained a precision@5 of 89.7% in the
Men dataset. Figure 3. shows some of the recommendations generated by this
methodology. Approximately speaking, here a product $p$ is recommended for
user $u2$ when the similarity between $p,p1$ and between $p,u2$ is high and
when similarity between $u1,p$ is low.
Embedding similarity distribution. This analysis is inspired by (Ying et al.,
2018). Effectiveness of the learned embeddings can be judged by the
distribution of distances between random pairs of embeddings. A wide
distribution indicates that the embedding space has enough “resolution” to
capture relevance of different product pairs. Figure 4 plots the distribution
of cosine similarities between pairs of products from the Men  dataset. The
distribution coming from the VASG embeddings is compared against the
distribution computed from product image features. VASG embeddings show a
wider distribution hence proving their superiority over using raw image
features. The kurtosis of this distribution for VASG embeddings is 0.87,
compared to 3.67 for image features.

Figure 4. Probability density of pairwise cosine similarity obtained from VASG
embeddings and Product (image) Features.

A woman and a girl in white dresses sit in an open car.

##  6\. Sensitivity Analysis

Influence of Embedding size. We study the variation of AUC scores with the
embedding dimension size $D$. Figure 4 shows AUC scores for warm-start
products when evaluating the performance of making personalized
recommendations. The performance improves with increase in dimension size,
however the rate of improvement seems to slow down after $D=70$.
Influence of noise addition. Autoencoders are known to suffer from over
fitting. To overcome this problem, we add a small amount of Gaussian noise to
the product image features while learning the mapping $\mathcal{M}(.)$. We
find that noise addition helps in improving the performance of cold-start
products for making personalized recommendations by an average margin of 1.8%
in AUC score.
Influence of Skip-Gram window Size. The window size determines the
neighbourhood size of a HIN node. For example a window size of 5 restricts the
neighborhood to second degree neighbours. A smaller context window exposes the
embeddings to local HIN structures which is suited for making personalized
recommendations. We experiment with window size [3,5,7,9] and find that a
window size of 7 gives the best results. Smaller window sizes lead to a faster
convergence, but the embeddings over-fit in a small neighbourhood whereas
larger window sizes make the training corpus unnecessarily large.
Influence of the auxiliary decoder. We study how VASG’s performance is
affected by removing the auxiliary decoder. This reduces the proposed
architecture to the general Skip-Gram model. We evaluate the performance of
making personalized recommendations in the set
$\mathcal{T}\textsubscript{warm}$. The performance drops by an average of 2.4%
on the Women, Men and Shoes dataset while the performance remains almost
consistent for the Jewelry dataset. This indicates that addition of visual
features improves the convential Skip-Gram embeddings.

Figure 5. Variation of AUC scores with embedding dimensions

##  7\. Conclusion

In this paper, we proposed a novel embedding learning architecture, VASG,
which incorporates visual features into the Skip-Gram model. This model has
two different loss functions depending on whether a user or a product is
observed during training. VASG embeddings can be used for making personalized
recommendations using nearest neighbour searches in the learned latent space.
In addition, this framework enables us to find embeddings for cold start
products which have never been observed during training. The embeddings are
trained on real world datasets and extensive analysis brings forward several
interesting properties captured by them.

There are multiple potential directions in which this work can be extended.
Use of effective negative samples while training can enable us to learn a more
refined latent space. Adversarial training can be used to reconstruct product
images using Generative Adversarial Networks conditioned on the Skip-Gram
embeddings. Additional auxiliary tasks which bring information from other
modalities can be added to central Skip-Gram architecture. VASG embeddings can
be also be incorporated into frameworks like SVDFeature for predicting
ratings.

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