# Field-free anomalous junction and superconducting diode effect in spin split
superconductor/topological insulator junctions

T.H. Kokkeler<EMAIL_ADDRESS>Donostia International Physics Center
(DIPC), 20018 Donostia–San Sebastián, Spain
Interfaces and Correlated Electron Systems, Faculty of Science and Technology,
University of Twente, Enschede, The Netherlands
A.A. Golubov  Interfaces and Correlated Electron Systems, Faculty of Science
and Technology, University of Twente, Enschede, The Netherlands
F. S. Bergeret  Centro de Física de Materiales (CFM-MPC) Centro Mixto CSIC-
UPV/EHU, E-20018 Donostia-San Sebastián, Spain
Donostia International Physics Center (DIPC), 20018 Donostia–San Sebastián,
Spain

###### Abstract

We study the transport properties of a diffusive Josephson junction between
two spin-split superconductors made of superconductor-ferromagnetic insulator
bilayers (FIS) on top of a 3D topological insulator (TI). We derive the
corresponding Usadel equation describing the quasiclassical Green’s functions
in these systems and first solve the equation analytically in the weak-
proximity case. We demonstrate the appearance of an anomalous phase in the
absence of an external magnetic field. We also explore non-reciprocal
electronic transport. Specifically, we calculate the junction’s diode
efficiency $\eta$ by solving the Usadel equation. We obtain a sizable diode
effect even at zero applied magnetic field. We discuss how the diode
efficiency $\eta$ depends on the different parameters and find a non-monotonic
behavior of $\eta$ with temperature.

††preprint: APS/123-QED

##  I Introduction

Recent advances attracting attention in superconductivity research are effects
related to non-reciprocal charge transport [1, 2, 3, 4, 5, 6, 7, 8, 9, 10],
particularly the superconducting diode effect [11, 12, 13, 14, 15, 16, 17, 18,
19, 20, 21, 22, 23, 24, 25, 26, 27]. A mesoscopic superconducting junction is
called a superconducting diode if the critical current is different for
opposite current directions. That is, for a superconducting diode the minimum,
$I_{-}=\min_{\phi}{I_{s}}(\phi)$, and maximum,
$I_{+}=\max_{\phi}{I_{s}}(\phi)$, of the current phase relation (CPR) are
unequal in magnitude. If a current $I$ flows in such a junction, with
$\min(|I_{-}|,I_{+})<I<\max(|I_{-}|,I_{+})$, in one direction it is a
supercurrent, whereas in the other direction the current dissipates. In a
conventional Josephson junction the critical current is the same in both
directions, the CPR has the following symmetry: $I(-\phi)=-I(\phi)$ [28]. This
symmetry holds if either time reversal symmetry or inversion symmetry is
present in the system [29, 30].

The superconducting diode effect can thus only be obtained if both time
reversal symmetry and inversion symmetry are broken [11, 12]. Time reversal
symmetry breaking can be achieved by a magnetic field. On the other hand
inversion symmetry can be broken intrinsically, such as in topological
insulators [31, 32, 33] or superconductors with Rashba spin orbit coupling
[34, 35, 36, 37, 38, 39, 40, 41]. Inversion symmetry can also be broken by
using an asymmetric junction geometry [42] or asymmetry of the device
originated in the fabrication [16].

If both time-reversal symmetry and inversion-symmetry are broken, in general
$I(-\phi)\neq-I(\phi)$ and thus possibly $I(\phi=0)\neq 0$. Such junctions,
for which the current vanishes at a nonzero phase difference, are called
$\phi_{0}$-junctions [35]. In weak coupling Josephson junctions, the CPR is
proportional to $\sin(\phi+\phi_{0})$; hence, in this regime, one cannot
observe the diode effect. However, if higher harmonics contribute to the
current, in general $I_{+}\neq|I_{-}|$, and the diode effect can be observed
[15, 13, 17, 18].

One way of breaking time-reversal symmetry without an external magnetic field
is to attach a ferromagnetic insulator (FI) to the superconductor. FI-S
systems have been discussed extensively in the literature [43, 44, 45, 46].
The exchange interaction between the localized magnetic moments of the FI and
the itinerant electrons in the SC leads to a spin split in the density of
states of the latter. Spin split superconductors form an active field of
research with varying directions [47, 48, 49, 50, 51, 52, 53].

In this work, we investigate the diode effect in a Josephson junction made of
spin-split superconducting electrodes on the 2D surface of a disordered 3D
topological insulator (TI). Specifically, our setup consists of two spin-split
superconductors (FIS) placed on top of a topological insulator, see Fig. 1a.
The spin-splitting in the superconductor breaks time-reversal symmetry,
whereas inversion symmetry is broken because we consider only the top surface
of the topological insulator. Thus, the conditions to have a $\phi_{0}$-effect
are fulfilled even without an external magnetic field. Using the linearized
Usadel equation, we first show analytically that such CPR exhibits the
$\phi_{0}$-effect. Going beyond the linear regime we compute the diode
efficiency. Even in the case of low transmission FIS/TI interfaces we obtain
an efficiency of 1%. By increasing the interface transmission the efficiency
can reach values larger than 7%. We also find that for short junctions the
efficiency is maximized at a finite temperature independent of the strength of
the exchange field.

The article is structured as follows. In section II we introduce the setup and
the basic equations. We derive the Usadel equation for a diffusive topological
insulator in proximity with a spin-split superconductor. In section III, we
focus on analytical results that can be obtained by linearizing the Usadel
equation, which is valid under the assumption that the proximity effect is
small. Even though in this limit the CPR contains only the first harmonic, and
thus has no diode effect, we can determine the condition for observation of
the anomalous Josephson currents. The latter is the precursor of a diode
effect in the non-linearized equation. We also show that the $\phi_{0}$\-
effect is suppressed by impurity scattering. In section IV we go beyond the
linear regime and present our numerical results for the nonlinear equation and
the diode efficiency as a function of the temperature for different values of
the exchange field, length of the junction, and transparency of the
interfaces. Finally in section V we conclude and propose real material
combinations to test our predictions. Throughout the paper it is assumed that
$\hbar=k_{B}=1$.

##  II The system and basic equations

We consider the FIS-TI-FIS system shown in Fig. 1a. A FIS can be realized by
placing a ferromagnetic insulator on top of a superconducting film (SC) with a
conventional s-wave pair potential. The thickness of the latter is assumed to
be small compared to its coherence length, so that we can assume a homogeneous
splitting field in the SC induced by the magnetic proximity effect [54]. We
neglect the inverse proximity effect of the topological insulator on the
superconductor. In this setup a current can flow through the top surface of
the TI from one FIS to the other. Because the system is finite in $x$
direction, no current can flow at $x=\pm L/2$, where $L$ is the length of the
TI. We denote by $L_{1}$ the length of the TI between the two FIS electrodes.

(a)

(b)

Figure 1: (a) Sketch of the junction under consideration. (b) The CPR of the
FIS-TI-FIS junction for different temperatures. The parameters chosen are
$\frac{\gamma_{0}}{E_{\text{Th}}}=\frac{\Delta_{0}}{E_{\text{Th}}}=\frac{5h}{2E_{\text{Th}}}=25$,
$\frac{L_{1}}{L}=\frac{1}{10}$ and $\frac{l}{L}=0.08\ll 1$. Inset: zoom in of
the CPR around $\phi=0$.

We assume that the transport at the TI surface is diffusive and can be
described by the Usadel equation: the derivation of this equation for our
system closely resembles the derivation of the Usadel equation for a TI in an
exchange field as presented in Refs. [55, 56]. However, whereas
superconductivity in the systems discussed in these papers is introduced as an
effective pair potential, for the spin split superconductor a different
approach is taken.

We incorporate the effect of the spin-split superconductor as a self-energy
term $\bar{\Sigma}_{s}$. For the self-energy we follow the approach similar to
[57, 58], in which the self-energy, up to second order in the tunneling
parameter $T_{1}$ between the TI and the superconductor, is introduced as

$\displaystyle\bar{\Sigma}_{s}=T_{1}^{2}\rho_{S}\check{\tau}_{3}\sigma_{0}\bar{G^{\prime}}_{S}\sigma_{0}\check{\tau}_{3}=T_{1}^{2}\rho_{S}\bar{G}_{S},$
(1)

where $\rho_{S}$ is the density of states in the superconductor, $\sigma_{0}$
is the identity matrix in spin space and $\check{\tau}_{3}$ is the third Pauli
matrix in electron-hole space. $\bar{G^{\prime}}_{S}$ is the momentum
integrated Green’s function in the spin-split superconductor and
$\bar{G}_{S}=\check{\tau}_{3}\sigma_{0}\bar{G^{\prime}}_{S}\sigma_{0}\check{\tau}_{3}$
is introduced to shorten notation. Note that the only effect of this
transformation by $\check{\tau}_{3}\sigma_{0}$ is to negate the pair
amplitudes. The self-energy term appears as an added term in the commutator on
the left hand side of the Eilenberger equation. The Eilenberger equation thus
reads, using the same presentation in spin-Nambu space as in [55]:

$\displaystyle\frac{v_{F}}{2}\\{\\{\eta_{j},\nabla_{j}\check{g}(1+\vec{n}_{F}\cdot\vec{\eta})\\}\\}$
$\displaystyle=[\check{g}(1+\vec{n}_{F}\cdot\eta),\omega_{n}\check{\tau}_{3}+\Gamma(x)\bar{G}_{S}+\frac{\langle\check{g}(1+\vec{n}_{F}\cdot\vec{\eta})\rangle}{2\tau}],$
(2)

where $\\{\cdot,\cdot\\}$ denotes the anticommutator and $[\cdot,\cdot]$ the
commutator. In our notation $\check{g}$ is the quasiclassical Green’s
function, $\omega_{n}$ is the nth Matsubara frequency, $\vec{n}_{F}$ is the
direction of the momentum at the Fermi surface, $v_{F}$ is the magnitude of
the Fermi velocity, $\tau$ is the collision time, $\check{\tau}_{3}$ is the
third Pauli matrix in Nambu space, $\mu$ is the Fermi energy and
$\vec{\eta}=(-\sigma_{2},\sigma_{1},0)$. The tunneling parameter $T_{1}$ is
nonzero only in FIS regions. To reflect this we introduce the boundary
parameter $\Gamma(x)$:

$\Gamma(x)=\gamma_{0}\Theta(|x|-\frac{L_{1}}{2})\Theta(\frac{L}{2}-|x|),$ (3)

where $\Theta$ denotes the Heaviside function and
$\gamma_{0}=T_{1}^{2}\rho_{S}$.

The Green’s function is written as
$\check{g}\frac{1}{2}(1+\vec{n}_{F}\cdot\vec{\eta})$ to reflect the strong
coupling between spin and direction of momentum in a topological insulator. In
this work a 2D surface of a 3D TI is studied. Therefore scattering is not
prohibited, unlike in a 1D edge of a 2D TI. We assume the junction is in the
dirty limit, that is, the inverse scattering time $\frac{1}{\tau}$ is much
larger than any energy scale other than the chemical potential $\mu$. In that
case the Green’s function $g$ is almost isotropic. Thus, it is a good
approximation to keep only the zeroth and first term in the expansion in
angular momentum, that is,

$\check{g}\approx\check{g}_{s}+\vec{n}_{F}\cdot\vec{\check{g}}_{a},$ (4)

where the zeroth order $\check{g}_{s}$ and the first order angular momentum
$\vec{\check{g}}_{a}$ satisfy $\check{g}_{s}^{2}=\mathbf{1}$ and
$\check{g}_{s}\vec{\check{g}}_{a}+\vec{\check{g}}_{a}\check{g}_{s}=\vec{0}$ in
order to satisfy the normalisation condition $\bar{g}^{2}=\mathbf{1}$ up to
second order in $\tau$. The Green’s functions $\check{g}_{s}$ and
$\vec{\check{g}}_{a}$ do not have any degrees of freedom in momentum space nor
in spin space and are thus functions which map position into the space of 2 by
2 matrices. Using the expansion in angular momentum, the Usadel equation can
be derived. The strategy followed to derive the Usadel equation for our
structure is very similar to the strategy used in [55], [56]. To this end, we
first write

$\bar{G}_{S}=\check{G}_{S0}\sigma_{0}+\vec{\check{G}}_{S1}\cdot\vec{\sigma},$
(5)

where $\check{G}_{S0}$ and $\vec{\check{G}}_{S1}$ are matrix functions without
spin degrees of freedom, and $\vec{\sigma}$ is the vector of Pauli matrices in
spin space. It is assumed that the exchange field in the FI always points in
the same direction, so that there are no domain walls which may affect the
density of states significantly [53]. The position independent Green’s
function in the spin-split superconductor is

$\displaystyle\bar{G}_{S}$
$\displaystyle=\frac{1}{2}(1+\vec{b}\cdot\vec{\sigma})\check{G}_{\uparrow}+\frac{1}{2}(1-\vec{b}\cdot\vec{\sigma})\check{G}_{\downarrow}$
(6) $\displaystyle\check{G}_{\uparrow,\downarrow}$
$\displaystyle=g_{\uparrow,\downarrow}\tau_{3}+f_{\uparrow,\downarrow}(\cos{\phi}\tau_{1}+\sin\phi\tau_{2}),$
(7)

where $g_{\uparrow,\downarrow}=(\omega_{n}\pm ih)/{\sqrt{(\omega_{n}\pm
ih)^{2}+\Delta^{2}}}$ are the normal parts and
$f_{\uparrow,\downarrow}=\Delta/{\sqrt{(\omega_{n}\pm ih)^{2}+\Delta^{2}}}$
are the anomalous parts [59]. The + sign is used for the spin-up component and
the - sign for the spin down component, $\tau_{1,2,3}$ are the Pauli matrices
in Nambu space, $h$ is the magnitude of the exchange field $\vec{h}=h\vec{b}$,
$\Delta$ is the superconducting potential calculated self-consistently and
$\phi$ is the phase of the superconductor. Combining Eq. (5) with Eqs. (6) and
(7), we may write

$\displaystyle\check{G}_{S0,1}$
$\displaystyle=g_{S0,1}\tau_{3}+f_{S0,1}(\cos{\phi}\tau_{1}+\sin{\phi}\tau_{2}),$
(8) $\displaystyle\vec{\check{G}}_{S1}$
$\displaystyle=\check{G}_{S1}\vec{b}\cdot\vec{\sigma},$ (9)

where $g_{s0,1}=(g_{\uparrow}\pm g_{\downarrow})/2$ and
$f_{S0,1}=(f_{\uparrow}\pm f_{\downarrow})/2$. The component $f_{S0}$ of the
condensate is the usual singlet component, whereas $f_{S1}$ is the odd-
frequency triplet component [59].

The Usadel equation for the angular averaged Green’s function $g_{s}$ without
spin degrees of freedom in the TI is obtained analogous to the approach laid
out in [55]. We combine the spin-trace of the equation obtained by angular
averaging and the equation obtained by angular averaging after multiplication
by $\vec{n}_{F}$. The resulting Usadel equation is

$D\hat{\nabla}(\check{g}_{s}\hat{\nabla}\check{g}_{s})=[\omega_{n}\check{\tau}_{3}+\frac{\Gamma(x)}{2}\check{G}_{S0},\check{g}_{s}],$
(10)

with $D=v_{F}^{2}\tau$. Eq. (10) is similar to the Usadel equation in normal
metals. However, the derivative is replaced by a generalized derivative:

$\displaystyle\hat{\nabla}=\nabla+\frac{\Gamma(x)}{2v_{F}}[\cdot,\check{G}_{S1y}]e_{x}-\frac{\Gamma(x)}{2v_{F}}[\cdot,\check{G}_{S1x}]e_{y},$
(11)

For a spin split superconductor this becomes

$\displaystyle\hat{\nabla}=\nabla+\frac{\Gamma(x)}{2v_{F}}(b_{y}e_{x}-b_{x}e_{y})[\cdot,\check{G}_{S1}].$
(12)

This derivative is similar to the derivative presented in [56] for a
topological insulator with an exchange field, in fact, Eq. (10) reduces to
this expression if $\check{G}_{S1}=h\check{\tau_{3}}$. Throughout this work we
will assume that the magnetic field is oriented perpendicular to the current
direction, so that $b_{y}=1$ and $b_{x}=0$. The equation is accompanied by the
boundary conditions

$\displaystyle\hat{\nabla}G_{S1}(x=\pm\frac{L}{2})=0.$ (13)

In this paper it is assumed that the ferromagnetic insulator is either very
thin or very thick, so that there is no y nor z-dependence in the problem. As
a consequence the equation becomes effectively one-dimensional. From the
solutions of Eqs. (10),(12) and (13) one can determine the current:

$\displaystyle
I=\frac{\sigma_{N}}{2}T\sum_{n}\text{Tr}\Big{(}\tau_{3}\bar{G}(x^{*},\omega_{n})\hat{\nabla}\bar{G}(x^{*},\omega_{n})\Big{)}\;,$
(14)

where $\sigma_{N}$ is the normal state conductance, $T$ is the temperature
entering the Matsubara frequencies $\omega_{n}=(2n+1)\pi T$, and $x^{*}$ is
any position for which $\Gamma(x^{*})=0$. The quantities of interest in this
article are the maximum supercurrents $I_{c}^{+}$ and $|I_{c}^{-}|$ in both
directions, and the diode efficiency, defined by

$\displaystyle\eta=\frac{I_{c}^{+}-|I_{c}^{-}|}{I_{c}^{+}+|I_{c}^{-}|}.$ (15)

Before showing the numerical solution of the above boundary problem, in the
next section, we study the linearized equation in the limit of a small
proximity effect. As discussed in the introduction, in this limiting case the
diode effect vanishes, but the anomalous phase $\phi_{0}$ , can be studied
analytically. The anomalous current is a strong indication for the diode
effect to appear.

##  III  Linearized case: The $\phi_{0}$-junction

To get an understanding of the physics behind the new Usadel equation, Eq.
(10), we focus first on the case of a weak proximity effect. In this case the
anomalous parts of the GF are much smaller than the normal one, that is,
$\text{Tr}(\tau_{1,2}G)\ll\text{Tr}(\tau_{3}G)$, and $G$ can be approximated
by $G(i\omega_{n})\approx\begin{bmatrix}\text{sgn}(\omega_{n})&F\\\
\tilde{F}&-\text{sgn}(\omega_{n})\end{bmatrix}$, where $|F|,|\tilde{F}|\ll 1$.
Using this approximation the Usadel equation reduces to a linear equation. We
assume here that the system in Fig. 1a is infinite in $x$ direction. The
superconductor is only absent in the region
$(-\frac{L_{1}}{2},\frac{L_{1}}{2})$ and present everywhere outside this
region. In this case Eq. (10) can be written in the three separate spatial
regions:

$\displaystyle\begin{cases}D\partial_{xx}F=2|\omega_{n}|F-\gamma_{0}f_{S0}e^{-i\frac{\phi}{2}}&x<-\frac{L_{1}}{2}\\\
D\partial_{xx}F=2|\omega_{n}|F&|x|<\frac{L_{1}}{2}\\\ \end{cases}$ (16)

For $x\xrightarrow{}\pm\infty$ the Green’s function should commute with
$\omega\tau_{3}+\check{G}_{S0}$. This implies that the pair potential is given
by

$\displaystyle\lim_{x\xrightarrow{}\pm\infty}F=\frac{\gamma_{0}f_{S0}}{2|\omega_{n}|}e^{\pm
i\frac{\phi}{2}}.$ (17)

These are exactly the same equations as for the conventional SNS junction.
However, the equations which joint the solutions at $x=\pm\frac{L_{1}}{2}$,
are different compared to the conventional SNS junction. Requiring continuity
of both the Green’s function and the current through the junction yields

$\displaystyle
F\left(\pm\frac{L_{1}}{2}+0^{+}\right)=F\left(\pm\frac{L_{1}}{2}+0^{-}\right)$
(18)
$\displaystyle\derivative{F}{x}\left(\pm\frac{L_{1}}{2}+0^{\pm}\right)+\frac{\gamma_{0}}{v_{F}}f_{S1}\text{sgn}(\omega_{n})e^{\pm
i\frac{\phi}{2}}=\derivative{F}{x}\left(\pm\frac{L_{1}}{2}+0^{\mp}\right).$
(19)

This expression differs from the expression for the SNS junction in the
appearance of the $f_{S1}$-term on the right hand side of Eq. (19). The CPR
following from these equations is

$\displaystyle I(\phi)$
$\displaystyle=\sum_{n}\frac{1}{4}e^{-2\sqrt{\frac{2|\omega_{n}|}{D}}}\text{Im}((A_{n}-iB_{n})e^{i\frac{\phi}{2}})^{2}$
(20)
$\displaystyle=\frac{1}{4}\sum_{n}e^{-2\sqrt{\frac{2|\omega_{n}|}{D}}}\left(A_{n}^{2}-B_{n}^{2}\sin{\phi}+2A_{n}B_{n}\cos{\phi}\right)$
(21)

where $A_{n}$ and $B_{n}$ are real coefficients given by
$A_{n}=\frac{\gamma_{0}f_{S0}}{2|\omega_{n}|}$ and
$B_{n}=-i\gamma_{0}f_{S1}\sqrt{\frac{D\omega_{n}}{2}}\frac{1}{|\omega_{n}|}$.
This implies that

$\displaystyle\phi_{0}=\arctan
2\frac{\sum_{n=-\infty}^{\infty}A_{n}B_{n}e^{-2\sqrt{\frac{2|\omega_{n}|}{D}}}}{\sum_{n=-\infty}^{\infty}(A_{n}^{2}-B_{n}^{2})e^{-2\sqrt{\frac{2|\omega_{n}|}{D}}}}$
(22)

The $\phi_{0}$-effect is largest if $A_{n}=B_{n}$ and small if
$|\frac{B_{n}}{A_{n}}|$ is not close to 1. This ratio is given by

$\displaystyle|\frac{B_{n}}{A_{n}}|$
$\displaystyle=\frac{\gamma_{0}|f_{S1}|\sqrt{\frac{D}{2|\omega_{n}|}}2|\omega_{n}|}{\gamma_{0}f_{S0}v_{F}}=\frac{|f_{S1}|}{f_{S0}}\frac{1}{v_{F}}\sqrt{2|\omega_{n}|D}$
(23)

Recall that the diffusion constant is given by $D=v_{F}^{2}\tau$. This means
that $\frac{1}{v_{F}}\sqrt{2|\omega_{n}|D}=O(\sqrt{|\omega_{n}|\tau})$, which
is small in the diffusive regime. In fact, in the derivation of the Usadel
equation, it is assumed that $\frac{1}{\tau}\gg|\Delta|$ and thus
$\frac{1}{\tau}\gg|\omega_{n}|$ for every $n$ that contributes significantly
to the current. This means the effect can only be large if $f_{S1}\gg f_{S0}$.
This constraint can only be satisfied if,
$h\gg\sqrt{\omega_{n}^{2}+\Delta^{2}}$ for all Matsubara frequencies that have
a significant contribution to the critical current. However, in the setup
discussed in this paper the condition $h\gg\Delta$ can not be realized, since
the magnetisation is induced via the superconductor and a high magnetisation
destroys the superconductivity. The $\phi_{0}$-effect is suppressed by a
factor $\sqrt{|\omega_{n}|\tau}$ in the linearized case.
Next the temperature dependence is discussed. Because the $\phi_{0}$-effect is
small we can simplify the equation for $\phi_{0}$ to

$\displaystyle\phi_{0}\approx
2\frac{\sum_{n=-\infty}^{\infty}A_{n}B_{n}e^{-2\sqrt{\frac{2|\omega_{n}|}{D}}}}{\sum_{n=-\infty}^{\infty}(A_{n}^{2})e^{-2\sqrt{\frac{2|\omega_{n}|}{D}}}}.$
(24)

Now, consider the behaviour at low temperatures. Since the triplet component
$f_{S1}$ is odd in frequency, whereas the singlet component $f_{S0}$ is even
in frequency, $|\frac{B_{n}}{A_{n}}|$ is small for small Matsubara
frequencies. As the temperature is decreased these terms become more and more
dominant in the sum. Thus, at low temperatures, the $\phi_{0}$-effect is
increases with increasing temperature. On the other hand, for large Matsubara
frequencies the ratio between triplet and singlet components
$|\frac{f_{s1}}{f_{s0}}|=O(\frac{h}{|\omega_{n}|})$. This means that at high
temperatures the $\phi_{0}$-effect decreases. Therefore, the $\phi_{0}$-effect
must be non-monotonic as a function of temperature, it attains a maximum.
Moreover, since $\sqrt{\tau}$ is an $\omega_{n}$-independent prefactor it can
not determine the maximum. The temperature at which the maximum is attained is
determined by only two dimensionless quantities,
$\frac{\Delta}{E_{\text{Th}}}$ and $\frac{h}{\Delta}$.
An interesting limit is the limit in which $\sqrt{\frac{2\pi T}{D}}L_{1}\ll 1$
and $h\ll\Delta$ so that the exponential suppression can be ignored to first
order in $\sqrt{\frac{2\pi T}{D}}L_{1}$. The following expression is obtained
111Strictly speaking, the linearized case leads to a divergency as T goes to
0. We therefore replace $\Delta/|\omega_{n}|$ by
$\Delta/\sqrt{\omega^{2}+\gamma_{0}^{2}}$, based on the non-linearized case:

$\displaystyle\phi_{0}\approx$
$\displaystyle\frac{\sum_{n=0}^{\infty}A_{n}B_{n}}{\sum_{n=0}^{\infty}A_{n}^{2}}=h\sqrt{\tau}\frac{\sum_{n=0}^{\infty}\frac{1}{(\omega_{n}^{2}+\Delta^{2})^{2}\sqrt{\omega_{n}}}}{\sum_{n=0}^{\infty}\frac{1}{\omega_{n}^{4}+\Delta^{2}\omega_{n}^{2}}}$
(25)

The multiplication of the sum with $\sqrt{\tau}$ signals the dirty limit
suppression of the $\phi_{0}$-effect. The resulting expression is evaluated
numerically as a function of temperature. Numerical evaluation confirmed the
non-monotonicity, see Fig. 2.

Figure 2: The $\phi_{0}$-effect as a function of temperature as calculated
using Eq. (25). The $\phi_{0}$-effect is suppressed at low temperatures. The
$\phi_{0}$-effect is given in units of the small quantity
$h\sqrt{\tau/\Delta}$.

In the following sections we discuss the full non-linear equation, and we show
that the diode-effect is non-monotonic with temperature for short junctions.

##  IV Non-linearized case: The superconducting diode effect

To investigate the diode effect one needs to go beyond the linear approach and
solve numerically the Usadel equation. In this section we present our
numerical results for the supercurrent in the FIS-TI-FIS junction. As a first
step it is convenient to write the Usadel equation, Eq. (10), in dimensionless
form, normalising $x$ by the total length of the junction. The obtained
equation is

$\displaystyle\hat{\nabla}(\check{G}\hat{\nabla}\check{G})=[\frac{\omega_{n}}{E_{\text{Th}}}\check{\tau}_{3}+\frac{\Gamma(x)}{2E_{\text{Th}}}\check{G}_{S0},\check{G}],$
(26)
$\displaystyle\hat{\nabla}=\derivative{x}+\frac{\Gamma(x)L}{2v_{F}}\tilde{b}_{y}[\check{G}_{S1},\cdot]$
(27)

Thus, all energies are given in units of the Thouless energy
$E_{\text{Th}}=\frac{D}{L^{2}}$, whereas lengths are given in units of $L$.
The strength of the proximity effect is described by the dimensionless
parameter
$\frac{\gamma_{0}L}{v_{F}}=\frac{\gamma_{0}}{E_{\text{Th}}}\frac{l}{L}$, where
$l=v_{F}\tau$ is the mean free path, which in the diffusive limit must be the
shortest length involved in the problem. This puts a constraint on the
magnitude of the new dimensionless quantity, it must be much smaller than
$\frac{\gamma_{0}}{E_{\text{Th}}}$. However, without this term the equations
reduce to the equations for the SNS junction, which is known not to have a
diode effect [28], it has no time-reversal symmetry breaking. Thus, if the new
quantity is very small, the diode effect is very small. Therefore,
$\frac{\gamma_{0}}{E_{\text{Th}}}$ must be chosen large, that is, the contact
between the superconductor and the TI must be good to have a large
$\gamma_{0}$.

To solve the non-linearized Usadel equation, Eq. (10), the so-called Riccati
parametrisation is used,

$\check{G}=\frac{1}{1+\bar{\gamma}\tilde{\gamma}}\begin{bmatrix}1-\bar{\gamma}\tilde{\gamma}&2\gamma\\\
2\tilde{\gamma}&-1+\bar{\gamma}\tilde{\gamma}\end{bmatrix},$ (28)

where $\bar{\gamma}$ and $\tilde{\gamma}$ are the Riccati parameters.

In principle, the pair potential $\Delta$ has to be determined self-
consistently since it is suppressed by the exchange field [61]. In the
numerical calculations we choose values of the exchange field smaller than
$\frac{h}{\Delta_{0}}=\frac{2}{5}$. Other parameters are set as follows:
$\frac{\gamma_{0}}{E_{\text{Th}}}=25$, whereas $\frac{L_{1}}{L}$ is chosen to
be $\frac{1}{10}$ and $\frac{l}{L}=0.08\ll 1$.

Fig. 1b shows our numerical results for the CPR obtained from the non-
linearized Eq. (10). One can see a finite current value at $\phi=0$ associated
with the appearance of the anomalous phase $\phi_{0}$. Moreover, even though
small, there is a difference in the absolute value of the maximum and minimum
of the current. This asymmetry reflects the diode effect. By increasing the
temperature both the current at zero phase and the critical current decrease.

We now study the temperature dependence of the diode effect for different
exchange fields and sizes of the junction. The numerical results for the diode
efficiency are shown in Fig. 3. Interestingly, we find a non-monotonic
behaviour with a maximum efficiency at a finite temperature, $T_{d}$. It is
important to notice, that by computing $\eta$ in Fig. 3, the self-consistency
of the pair potential is ignored, because of the reduction of computational
costs. However, we verify that for all values of $h$ considered here, the
self-consistency leaves the magnitude of the gap almost unchanged for
temperatures of the order of $T=T_{d}$. The critical temperatures for all
cases shown in Fig. 3a lie right of the dashed vertical line.

If the exchange field is increased, Fig. 3a, the diode efficiency becomes
larger. $\eta$ increases approximately linearly with $h$. The temperature at
which the diode efficiency is maximal is almost independent of the exchange
field, with $T_{d}\approx 0.18$.

We also investigate the influence of the distance between the leads, $L_{1}$
on $\eta$, see Fig. 3b. Specifically, $E_{Th1}=D/L_{1}^{2}$ is varied, whereas
the quantities $\frac{l}{L}$ and $\frac{L_{1}}{L}$ are held constant. As the
Thouless energy is decreased, the diode efficiency decreases. Moreover, the
temperature $T_{d}$ at which the diode effect is maximal decreases with the
length of the junction. For enough long junctions the dependence of $\eta$ on
temperature becomes monotonic.

(a)

(b)

Figure 3: The temperature dependence of the diode efficiency $\eta$ for
different values of the exchange field strength $h$ (a), and Thouless energy
$E_{\text{Th}1}=\frac{D}{L_{1}^{2}}$ of the TI part (b). The critical
temperature is for all magnitudes of the exchange field considered here larger
than $0.9T_{c}$, highlighted by the black dotted line.

So far, we have considered disordered systems with low transparent S/FI
interfaces. One can, though, increase the $\phi_{0}$ and diode effects by
relaxing these conditions. On the one hand, our analytical results in section
III indicate that the $\phi_{0}$-effect can be increased by increasing $\tau$,
see for example Eq. (25). We also verified numerically that the diode effect
increases if the degree of disorder decreases.

On the other hand, we also investigated the effect of increasing the ratio
$\frac{L_{1}}{L}$. Our numerical calculations demonstrate that whereas
$I(\phi=0)/I_{c}$ increases, the diode effect decreases with increasing
$\frac{L_{1}}{L}$. This can be explained as follows. By increasing the
distance $L_{1}$ between the electrodes the CPR becomes more sinus-like. To be
precise, when increasing $\frac{L_{1}}{L}$ from $\frac{1}{10}$ to
$\frac{1}{2}$, the ratio between the magnitudes of the second and first
harmonics decreases from $\approx\frac{1}{6}$ to $\approx\frac{1}{10}$. As we
discussed before, besides the breaking of time-reversal and inversion
symmetries, the diode effect relies crucially on the contribution of higher
harmonics to the CPR.

A way to increase the higher harmonics contribution, is to increase the
coupling between the superconducting correlations from the left and right
electrodes. This can be achieved by increasing the transparency of the FIS/TI
interfaces, as shown in Fig. 4.

Figure 4: The diode efficiency as a function of temperature for different
values of $\gamma_{0}$.

Finally, another way to increase the diode effect is by increasing the
exchange field, as shown in Fig. 3. In our junction, however, the value of $h$
is limited by the critical field of the superconductor. To increase the
strength of the exchange field without suppressing superconductivity in the S
electrodes one could add an additional ferromagnetic insulating layers
directly on top of the TI between the two superconductors, similar to the
situation investigated in Refs. [55, 62, 63]. In that case the exchange field
can be larger than the superconducting gap and the diode effect may increase.

##  V Conclusions

We present a study of the $\phi_{0}$ and diode effects in a FIS-TI-FIS
Josephson junction. Though disorder tend to suppress them [40], we found, even
in the diffusive limit, sizable effects without applying any external field.
We found that by increasing the FIS/TI interface transparency and the magnetic
field one can increase the diode effect. For short junctions the diode effect
is non-monotonic as a function of temperature. By increasing the distance
between the electrodes the $\phi_{0}$-effect is enhanced, however the diode
effect is suppressed due to the loss of higher harmonics.

From the point of view of materials the proposed structure can be fabricated
with well studied material combinations. On the one hand the use of
topological insulators in Josephson junctions is well understood [64, 65, 66,
67, 68, 69, 70, 71, 72]. On the other hand, spin-split superconductivity has
been measured in several experiments on ferromagnetic insulator/superconductor
bilayers, as for example EuS/Al structures [73, 74, 51, 54]. Moreover, good
interfaces between TI and FI has been reported in Ref. [75].

## Acknowledgements

We thank Stefan Ilic for useful discussions. We acknowledge financial support
from Spanish AEI through project PID2020-114252GB-I00 (SPIRIT). FSB
acknowledges financial support from the European Union’s Horizon 2020 Research
and Innovation Framework Programme under Grant No. 800923 (SUPERTED), the A.
v. Humboldt Foundation, and the Basque Government through grant IT-1591-22.

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# One-class Recommendation Systems with the Hinge Pairwise Distance Loss and
Orthogonal Representations

Ramin Raziperchikolaei  Young-joo Chung

###### Abstract

In one-class recommendation systems, the goal is to learn a model from a small
set of interacted users and items and then identify the positively-related
user-item pairs among a large number of pairs with unknown interactions. Most
previous loss functions rely on dissimilar pairs of users and items, which are
selected from the ones with unknown interactions, to obtain better prediction
performance. This strategy introduces several challenges such as increasing
training time and hurting the performance by picking ”similar pairs with the
unknown interactions” as dissimilar pairs. In this paper, the goal is to only
use the similar set to train the models. We point out three trivial solutions
that the models converge to when they are trained only on similar pairs:
collapsed, partially collapsed, and shrinking solutions. We propose two terms
that can be added to the objective functions in the literature to avoid these
solutions. The first one is a hinge pairwise distance loss that avoids the
shrinking and collapsed solutions by keeping the average pairwise distance of
all the representations greater than a margin. The second one is an
orthogonality term that minimizes the correlation between the dimensions of
the representations and avoids the partially collapsed solution. We conduct
experiments on a variety of tasks on public and real-world datasets. The
results show that our approach using only similar pairs outperforms state-of-
the-art methods using similar pairs and a large number of dissimilar pairs.

##  1 Introduction

Feedback in recommendation systems (RSs) can be explicit or implicit. In the
case of explicit feedback [20, 24, 35, 12], users provide rating values after
interacting with the items, which shows how satisfied they are. In the case of
implicit feedback [20, 11, 15], which is a common scenario in the real world,
we only know if the user interacted with an item or not (such as clicked,
purchased, etc.). The goal of one-class recommendation systems is to solve the
implicit feedback prediction problem. It’s called a one-class problem since a
”no interaction” between a user and an item in the training set does not
necessarily mean the user does not like that item. It just means we do not
have enough information about their interaction. That’s because the set of the
items in RSs is huge and the users can’t see all the items, i.e., users can
only see a small subset of the items and then interact with a few of them.

There are three main steps in learning a RS model to predict the implicit
feedback. The first step is to learn user and item representations. This can
be done by learning user and item embedding matrices from their IDs [24, 35,
12, 8, 9], learning user and item multi-layer perceptrons (MLPs) from the
interaction vectors [21, 30, 5] or side information [23, 34, 4], or learning
graph neural networks from the bipartite user-item graph [25, 31, 32, 10].

The second step is to model the interaction score from the user and item
representations. The common functions are dot product [11, 15, 24, 12, 4, 32,
10, 18], cosine similarity [30, 31], and neural networks over the concatenated
user and item representations [8, 9, 5, 17, 16].

The third step is to optimize a loss function, which gets smaller values as
the model gives a larger score to the similar pairs of the user and items
compared to the dissimilar ones. Different types of loss functions have been
used in the literature. Mean squared error (MSE) loss [11, 15, 24, 12, 4, 32,
17, 18] and Binary cross-entropy (BCE) [30, 8, 5] directly minimize the
difference between the predicted and the actual scores. The Bayesian
personalized rank (BPR) loss [19, 9, 10] tries to make the interaction score
of the similar pairs greater than the dissimilar ones, instead of directly
mapping them to the actual scores. The contrastive learning loss tries to put
representations of the similar pairs close to each other and put the
dissimilar ones far away [18, 14, 29].

All the above loss functions need both similar and dissimilar pairs of the
users and items to learn a model. If we train these loss functions only using
the similar pairs, we get a collapsed solution: all representations will be
mapped to the same point in the latent space and the model predicts the same
interaction score for all the pairs. The performance of the collapsed solution
is as bad as assigning random representations to the users and items. So the
dissimilar sets are essential in training RS models to avoid the collapsed
solution.

In one-class recommendation systems, we only have access to the implicit
(known) interactions, and the rest of the interactions are unknown. To create
a dissimilar set of users and items, the common approach is to randomly select
a set of user and item pairs with the unknown interactions and consider them
dissimilar [30, 8, 5, 4, 32, 18]. Another strategy is to find out the hard-
negatives: the pairs with the unknown interactions that the model has
difficulty with classifying as dissimilar [33, 2, 3].

Creating dissimilar pairs from unknown ones is problematic for two main
reasons. First, as we show in experiments, we might need a large number of
dissimilar pairs to achieve reasonable results, which makes the training slow.
Second, using a large number of dissimilar pairs increases the chance of
converting a ”similar pair with an unknown interaction” to a dissimilar pair,
which hurts the performance. This issue is more severe in the case of the hard
negative approach, since ”similar pairs with unknown interactions” are by
definition difficult to be classified as dissimilar, and will be mistakenly
taken as hard negatives.

The main goal of this paper is to propose a new objective function that avoids
the collapsed solution without using the dissimilar pairs. As we define it
mathematically in Section 3, at the collapsed solution, the average pairwise
distance between all the user-user, item-item, and user-item representations
is $0$. To avoid the collapsed solution, we propose a hinge pairwise distance
loss which keeps the pairwise distance above a margin. We show that this new
loss function alone is not enough to generate high-quality representations, as
the solution might converge to a ”partially collapsed solution”, where all
users and items converge to exactly two unique representations. To avoid this
solution, which performs as poorly as the collapsed solution, we propose to
minimize the correlation between the dimensions of representations by making
them orthogonal.

We conduct extensive experiments on several real-world and public datasets in
both warm-start and cold-start settings. The experimental results in Section 4
show that our approach, which only uses similar pairs, has a better prediction
performance than state-of-the-art methods, which use both similar and
dissimilar pairs.

#### Notations.

We denote the sparse interaction matrix by $\mathbf{R}\in\mathbb{R}^{m\times
n}$, where $m$ and $n$ are the number of users and items, respectively,
$R_{jk}>0$ is the interaction value of the user $j$ on the item $k$, and
$R_{jk}=0$ means the interaction is unknown. The goal is to predict the
unknown interactions in $\mathbf{R}$. The $i$th row of a matrix $\mathbf{H}$
is shown by $\mathbf{H}_{i,:}$ and the $j$th column is shown by
$\mathbf{H}_{:,j}$. The $d$-dimensional representations of the all users and
all items are denoted by $\mathbf{Z}^{u}\in\mathbb{R}^{m\times d}$ and
$\mathbf{Z}^{i}\in\mathbb{R}^{n\times d}$, respectively. The representation of
the $j$th user and $k$th item are denoted by
$\mathbf{z}_{j}^{u}=\mathbf{Z}_{j,:}^{u}$ and
$\mathbf{z}_{k}^{i}=\mathbf{Z}_{k,:}^{i}$, respectively.

##  2 Related works

Since our main contribution is proposing a new objective function, we review
the different loss functions used in the RS literature. Most of these loss
functions are defined based on the predicted user-item interactions from the
representations. The most common mappings from representations to predicted
interactions are dot product [11, 15, 24, 12, 4, 32, 10, 18], cosine
similarity [30, 31], and neural networks [8, 9, 5, 17, 16]:

$\text{dot
product:}\hat{R}_{jk}=(\mathbf{z}_{j}^{u})^{T}\mathbf{z}_{k}^{i},\quad\text{cosine:}\hat{R}_{jk}=\frac{(\mathbf{z}_{j}^{u})^{T}\mathbf{z}_{k}^{i}}{||\mathbf{z}_{j}^{u}||||\mathbf{z}_{k}^{i}||},\quad\text{NNs:}\hat{R}_{jk}=h([\mathbf{z}_{j}^{u},\mathbf{z}_{k}^{i}])$
(1)

where $[\cdot,\cdot]$ merges the two vectors and $h()$ is a neural network. In
[8, 5],$[\cdot,\cdot]$ is concatenation and $h()$ is an MLP. In [9],
$[\cdot,\cdot]$ is outer product and $h()$ is a convolutional network.

To define the loss functions in RSs, previous works use sets of similar and
dissimilar pairs of the users and items. The similar set $S^{+}$ contains all
the users and items that interacted with each other, i.e., $(j,k)\in S^{+}$ if
$R_{jk}=1$. The dissimilar set $S^{-}$ contains a subset of the users and
items with unknown interactions, i.e., $R_{jk}=0$.

Two popular loss functions in the literature are mean squared error (MSE) [11,
15, 24, 12, 4, 32, 17, 18] and binary cross-entropy (BCE) [30, 8, 5], which
directly minimize the difference between the predicted and actual
interactions:

$\displaystyle
l_{\text{BCE}}(\mathbf{z}_{j}^{u},\mathbf{z}_{k}^{i})=-(R_{jk}\log\hat{R}_{jk}+(1-R_{jk})\log(1-\hat{R}_{jk})),$
$\displaystyle
l_{\text{MSE}}(\mathbf{z}_{j}^{u},\mathbf{z}_{k}^{i})=(\hat{R}_{jk}-R_{jk})^{2},$
(2)

where $j,k\in S^{+}\cup S^{-}$. On the other hand, the BPR loss [19, 9, 10] is
defined based on the difference between the predicted interactions of the
similar and dissimilar pairs:

$l_{\text{BPR}}(\mathbf{z}_{j}^{u},\mathbf{z}_{k}^{i})=-\ln{\sigma(\hat{R}_{jk}-\hat{R}_{jl})},$
(3)

where $j,k\in S^{+}$ and $j,l\in S^{-}$, i.e., user $j$ is similar to item $k$
and dissimilar to item $l$. Note that the above loss functions are optimized
over the user and item representations, $\mathbf{z}_{j}^{u}$ and
$\mathbf{z}_{k}^{i}$, which are used to generate the predicted interaction.

Contrastive loss functions [6] are also used in recommendation systems [18,
14, 29]. The idea is to map representations of similar pairs of the users and
items close to each other and the dissimilar ones far away:

$l_{\text{contrastive}}(\mathbf{z}_{j}^{u},\mathbf{z}_{k}^{i})=R_{jk}||\mathbf{z}^{u}_{j}-\mathbf{z}^{i}_{k}||^{2}+(1-R_{jk})\lambda\max{(0,m_{d}-||\mathbf{z}^{u}_{j}-\mathbf{z}^{i}_{k})||)}^{2},$
(4)

where $m_{d}$ is a margin and $j,k\in S^{+}\cup S^{-}$.

As we explain in the next section, without the dissimilar sets, all the above
loss functions will converge to a collapsed solution, which performs as bad as
a random assignment of the representations. The goal of this paper is to
introduce new terms to let the above loss functions be trained only on similar
sets and still avoid the collapse solution.

##  3 Proposed method

As mentioned in the introduction, there are several issues in using the
dissimilar pairs in one-class RSs. First, we are never sure whether the
selected pairs with the unknown interactions are dissimilar or not. A ”no
interaction” does not necessarily mean that the user did not like the item.
Selecting similar pairs and using them as dissimilar pairs can hurt the
performance. Second, as we show in experiments, previous works use a large
number of dissimilar pairs to get reasonable performance, which will increase
the training time. Our goal is to learn a recommendation system model by
discarding the dissimilar set of users and items, training only on the similar
pairs, avoiding the collapsed solution, and achieving state-of-the-art
results.

In this section, we first explain how the previous objective functions
converge to a collapsed solution when they only use similar pairs. Then, we
explain our proposed method by considering the
$E_{\text{cont}}(\mathbf{Z}^{u},\mathbf{Z}^{i})$ in Eq. (LABEL:e:con_reg) as
the main term of the objective function. We show how we avoid the collapsed,
partially collapsed, and shrinking solutions by proposing new terms. Then, we
will explain how our objective function works with other loss functions.
Finally, we go over the computational complexity of the proposed objective
function.

###  3.1 The collapsed solution without the dissimilar set

The loss functions introduced in Section 2 use both similar and dissimilar
pairs to predict the actual representations. Let us explain what happens if we
only use the similar pairs and discard the dissimilar ones in two cases: 1)
using the loss function $l_{\text{MSE}}$ in Eq. (2) and dot product mapping
function in Eq. (1), and 2) using the constrastive loss defined in Eq. (4):

$\begin{split}&E_{\text{MSE}}(\mathbf{Z}^{u},\mathbf{Z}^{i})=\sum_{j,k\in
S^{+}}((\mathbf{z}^{u}_{j})^{T}\mathbf{z}^{i}_{k}-1)^{2},\\\
&E_{\text{cont}}(\mathbf{Z}^{u},\mathbf{Z}^{i})=\sum_{j,k\in
S^{+}}||\mathbf{z}^{u}_{j}-\mathbf{z}^{i}_{k}||^{2}.\end{split}$ (5)

Both objective functions above are defined on similar pairs $S^{+}$ and
optimizing them will lead to a collapsed solution, where all the users and
item representations are mapped to a certain point in the latent space. The
optimal solution of the $E_{\text{MSE}}$ is achieved by mapping all the user
and item representations to any $d$-dimensional vector with a unit L2 norm.
That’s because the dot product of all pairs becomes $1$, i.e.,
$\hat{R}_{jk}=(\mathbf{z}^{u}_{j})^{T}\mathbf{z}^{i}=1$, which makes the loss
value $0$ for all the terms. In the case of $E_{\text{cont}}$, the optimal
(and collapsed) solution is achieved by mapping all the user and item
representations to any $d$-dimensional vector.

At the collapsed solution, which is the result of removing dissimilar pairs
from the loss functions, the model always returns the same prediction, no
matter what the input pairs are. This works as poorly as a random model.

Note that different combinations of mapping functions and loss functions can
give us different objective functions. In all cases, without dissimilar pairs,
the result is a collapsed solution.

###  3.2 Avoiding the collapsed solution: hinge pairwise distance loss

Let us assume the $d$-dimensional representations of the users and items is
denoted by $\mathbf{Z}^{u}\in\mathbb{R}^{m\times d}$ and
$\mathbf{Z}^{i}\in\mathbb{R}^{n\times d}$, respectively. The joint user-item
representation is achieved by vertically concatenating the user and item
representations,
$\mathbf{Z}=[\mathbf{Z}^{u},\mathbf{Z}^{i}]\in\mathbb{R}^{(m+n)\times d}$. The
pairwise distance between all the representations in $\mathbf{Z}$ is computed
as:

$d_{p}=E_{\text{cont}}(\mathbf{Z},\mathbf{Z})=\frac{1}{(m+n)^{2}}\sum_{l,s=1}^{m+n}||\mathbf{z}_{l}-\mathbf{z}_{s}||^{2}.$
(6)

Note that $d_{p}$ computes distance between all the user-user, item-item, and
user-item representations, which is different from
$E_{\text{cont}}(\mathbf{Z}^{u},\mathbf{Z}^{i})$ that computes the distance
between similar pairs of the users and items.

Mathematically, at the collapsed solution, we have $d_{p}=0$. To avoid the
collapsed solution, we propose a hinge pairwise distance loss that keeps the
average pairwise distance $d_{p}$ greater than a margin $m_{p}$. The new
objective function can be written as:

$E=E_{\text{cont}}(\mathbf{Z}^{u},\mathbf{Z}^{i})+E_{d_{p}}(\mathbf{Z})=\sum_{j,k\in
S^{+}}||\mathbf{z}^{u}_{j}-\mathbf{z}^{i}_{k}||^{2}+\max{(0,m_{p}-d_{p})}^{2}.$
(7)

Note that $d_{p}$ involves computing the distances between all the pairs,
which could be very time-consuming. As we show below, $d_{p}$ is equivalent to
the summation of the variance of each dimension, which can be computed way
faster.

Let us denote the $q$th dimension of the $l$th representation as $z_{l,q}$,
and the pairwise distance of the $q$th dimension as $d_{p}^{q}$. Then, $d_{p}$
in Eq. (6) can be separated over the $d$ dimensions:

$d_{p}=\sum_{q=1}^{d}d_{p}^{q}=\frac{1}{(m+n)^{2}}\sum_{q=1}^{d}\sum_{l,s=1}^{m+n}(z_{lq}-z_{sq})^{2}.$
(8)

We can rewrite $d_{p}^{q}$ as:

$d_{p}^{q}=\frac{1}{(m+n)^{2}}\sum_{l,s=1}^{m+n}(z_{lq}-z_{sq})^{2}=\frac{1}{(m+n)^{2}}\sum_{l,s=1}^{m+n}z_{lq}{{}^{2}}+z_{sq}{{}^{2}}-2z_{lq}z_{sq}=\\\
\frac{2}{(m+n)^{2}}\sum_{l,s=1}^{m+n}z_{lq}{{}^{2}}-\frac{2}{(m+n)^{2}}\sum_{l=1}^{m+n}z_{lq}\sum_{s=1}^{m+n}z_{sq}=\frac{2}{(m+n)}\sum_{l=1}^{m+n}z_{lq}{{}^{2}}-2\bar{\mathbf{Z}}_{:,q}^{2}=2\text{var}(\mathbf{Z}_{:,q}).$
(9)

So twice the variance of a dimension is equivalent to the pairwise distances
of the representations in that dimension. This means that at the collapse
solution, the variance of each dimension is $0$, and to avoid this solution,
we just need the summation of the variance of the dimensions to be greater
than a margin. At the end of this section, we explain how the new formulation
helps the computational complexity.

###  3.3 Avoiding the partially collapsed solution: orthogonality term

While the objective function of Eq. (7) avoids the collapsed solution, it
might give us low-quality representations by converging to a ”partially
collapsed solution”. This solution returns only two representations for the
whole set of users and items: a portion of the users and items are mapped to
one representation and the rest of them are mapped to a second representation.
Below, we explain when the objective function converges to this solution and
how we avoid it by proposing an orthogonality term.

To explain the partially collapsed solution, we consider the user-item
bipartite graph. This graph contains two disjoint sets of nodes: set $I$
contains the items and set $U$ contains the users. The edges are determined by
$S^{+}$, i.e., there is an edge between nodes of user $j$ and item $k$ if
$(j,k)\in S^{+}$. Now, let us assume the graph can be partitioned into $V>1$
disjoint components $\\{C_{v}\\}_{v=1}^{V}$, where the edges in each component
are denoted by $S_{v}^{+}$. Since it’s a disjoint partition, each node from
$U$ and $I$ appears in exactly one component, $\cup\\{S_{v}^{+}\\}=S^{+}$, and
there is no edge across the nodes of the two different components. The
interaction matrix is more than $99\%$ sparse in real world scenarios, which
can easily lead to having multiple disjoint partitions.

The optimal solution of our objective function in the setting above can be
achieved by assigning representation $\mathbf{p}_{1}$ to all the nodes in some
of the components and $\mathbf{p}_{2}$ to all the nodes in the rest of the
components.

First, let us show how the above solution makes makes $E_{\text{cont}}=0$.
Here, for simplicity, let us assign $\mathbf{p}_{1}$ to the components
$1,\dots,\frac{v}{2}$ and assign $\mathbf{p}_{2}$ to the components
$\frac{v}{2}+1,\dots,V$. Then we have:

$E_{\text{con}}=\sum_{j,k\in
S^{+}}||\mathbf{z}^{u}_{j}-\mathbf{z}^{i}_{k}||^{2}=\sum_{v=1}^{V}\sum_{j,k\in
S_{v}^{+}}||\mathbf{z}^{u}_{j}-\mathbf{z}^{i}_{k}||^{2}=\\\
\sum_{v=1}^{\frac{V}{2}}\sum_{j,k\in
S_{v}^{+}}||\mathbf{z}^{u}_{j}-\mathbf{z}^{i}_{k}||^{2}+\sum_{v={\frac{V}{2}+1}}^{V}\sum_{j,k\in
S_{v}^{+}}||\mathbf{z}^{u}_{j}-\mathbf{z}^{i}_{k}||^{2}=\\\
\sum_{v=1}^{\frac{V}{2}}\sum_{j,k\in
S_{v}^{+}}||\mathbf{p}_{1}-\mathbf{p}_{1}||^{2}+\sum_{v={\frac{V}{2}+1}}^{V}\sum_{j,k\in
S_{v}^{+}}||\mathbf{p}_{2}-\mathbf{p}_{2}||^{2}=0.$ (10)

We can easily prove that $E_{\text{con}}$ becomes $0$ for the other ways of
assigning the two representations to the components.

To make the hinge pairwise distance loss $E_{d_{p}}=0$, we just need to put
the two points $\mathbf{p}_{1}$ and $\mathbf{p}_{2}$ far away from each other,
which makes the average variance of the dimensions greater than the margin
$m_{d}$. In the extreme case, consider each dimension of $\mathbf{p}_{1}$ goes
to $+\infty$ and each dimension of $\mathbf{p}_{2}$ goes to$-\infty$.

The partially collapsed solution is (almost) as bad as the collapsed solution.
It assigns the same representation to all the nodes in the same component
which makes them inseparable, while the nodes in each component should be
separated based on their interactions with the other nodes.

To avoid the partially collapsed solution, we look at the correlation
coefficients between the dimensions of the representations. If we have only
two unique representations in the user-item representations matrix
$\mathbf{Z}$, then there is a linear relationship between the dimension of the
$\mathbf{Z}$, and by definition the correlation between the dimensions becomes
maximum ($-1$ or $+1$). To avoid this, we add an orthogonality term to the
objective function to make the representations orthogonal:

$E_{ours}=\lambda_{1}E_{\text{cont}}(\mathbf{Z}^{u},\mathbf{Z}^{i})+\lambda_{2}E_{d_{p}}(\mathbf{Z})+\lambda_{3}E_{\text{orth}}(\mathbf{Z}),$
(11)

where $E_{\text{cont}}(\mathbf{Z}^{u},\mathbf{Z}^{i})$ and
$E_{d_{p}}(\mathbf{Z})$ are defined in Eq. (7) and we defined
$E_{\text{orth}}(\mathbf{Z})$ as follows:

$E_{\text{orth}}(\mathbf{Z})=\sum_{q=1}^{d}\sum_{s=q+1}^{d}\hat{\mathbf{Z}}_{:,q}^{T}\hat{\mathbf{Z}}_{:,s},$
(12)

where $\hat{\mathbf{Z}}$ is achieved by subtracting the mean of each dimension
from $\mathbf{Z}$, and $\hat{\mathbf{Z}}_{:,q}$ is the $q$th column of
$\hat{\mathbf{Z}}$. The orthogonality term let the objective avoid the
partially collapsed solution by making the off-diagonal values of the
covariance matrix small, which consequently makes the correlation of the
dimensions of the representations small.

###  3.4 Shrinking solution: why we need both terms

As explained above, if we only optimize the hinge pairwise distance loss
$E_{d_{p}}(\mathbf{Z})$ and $E_{\text{cont}}(\mathbf{Z}^{u},\mathbf{Z}^{i})$
together, without using the orthogonality term $E_{\text{orth}}(\mathbf{Z})$,
it will converge to a partially collapsed solution. On the other hand,
optimizing the orthogonality term $E_{\text{orth}}(\mathbf{Z})$ and
$E_{\text{cont}}(\mathbf{Z}^{u},\mathbf{Z}^{i})$, without
$E_{d_{p}}(\mathbf{Z})$, is enough to avoid both the collapsed and partially
collapsed solutions.

Then, a question arises here: do we need both hinge pairwise distance loss and
the orthogonality term in the same objective function? Or is it enough to only
enforce representation orthogonality to make the correlation small? In other
words, where does the solution converge if we set $\lambda_{2}=0$ in Eq. (11)?

To answer this question, let us consider a trivial (but not collapsed)
solution, where all users’ and items’ d-dimensional representations are
generated from a multivariate uniform distribution. More specifically, each of
the $d$ dimensions is drawn from the interval $[-r,r]$, independently from
other dimensions. In this scenario, the dimensions are uncorrelated, so the
orthogonality term $E_{\text{orth}}(\mathbf{Z})$ is almost $0$ in practice.
The maximum distance between any two representations is $4dr^{2}$, which is
the upper-bound for $E_{cont}(\mathbf{Z}^{u},\mathbf{Z}^{i})$. So by setting
$r$ to an extremely small number (but larger than $0$), we can make
$E_{cont}(\mathbf{Z}^{u},\mathbf{Z}^{i})$ extremely small. So this solution,
which generates low-quality random representations from a multivariate uniform
distribution, can make the objective function of Eq. (11) very small if we
don’t include the hinge pairwise distance loss. That’s because as we make $r$
smaller, the space of the representation shrinks, and the variance of each
dimension becomes very small. The hinge pairwise distance loss makes sure that
the average variance is larger than a margin and avoids this shrinking
solution.

###  3.5 Other objective functions in RSs

In this section, we explore the impact of the proposed terms on the different
objective functions. We explain what happens if we change the objective from
$E_{\text{cont}}$ to another one, achieved by combining loss functions in Eq.
(2) and mappings in Eq. (1). Is it necessary to use both $E_{d_{p}}$ and
$E_{\text{orth}}$ with the objective functions other than $E_{\text{cont}}$?

Let us explain what happens if we replace $E_{\text{cont}}$ by
$E_{\text{MSE}}$, defined in Eq. (LABEL:e:con_reg). By optimizing
$E_{\text{MSE}}$ on the similar pairs, the collapsed solution is achieved by
assigning all users and items to any point $\mathbf{p}\in\mathbb{R}^{d}$ such
that $||\mathbf{p}||_{2}=1$. Let us assume the user-item graph can be
partitioned into $V$ components. The partially collapsed solution is achieved
by assigning all representations to any two points
$\mathbf{p}_{1}\in\mathbb{R}^{d}$ and $\mathbf{p}_{2}\in\mathbb{R}^{d}$, where
$||\mathbf{p}_{1}||_{2}=1$, $||\mathbf{p}_{2}||_{2}=1$, such that all the
users and items in the same component get the same representation.

Unlike the previous case where we used $E_{\text{cont}}$, we do not get the
shrinking solution if we draw each dimension of each representation from a
multivariate uniform distribution in $[-r,r]$. That’s because as we make $r$
smaller, the absolute value of the dimensions of the representations
decreases, the dot product of the representations becomes smaller and further
away from $1$, and the loss function $E_{\text{MSE}}$ increases. So the loss
function $E_{\text{MSE}}$ itself avoids the shrinking solution.

Since we can prevent the collapsed and partially collapsed solutions using the
orthogonality term, without converging to a shrinking solution, we cannot gain
much by adding $E_{d_{p}}$. The results on CiteULike dataset in our
experimental results section confirm this.

For other objective functions in the literature, one needs to first verify the
existence of the collapsed, partially collapsed, and shrinking solutions and
then decide which terms to use to prevent them.

###  3.6 Computational complexity and batch-wise training

Here, we analyze the computational complexity of each term in our objective
function in Eq. (11). The time complexity of $E_{\text{cont}}$ and
$E_{\text{orth}}$ are $O(|S^{+}|d)$ and $O((m+n)d)$, respectively. For
$E_{d_{p}}$, if we use Eq. (6), the complexity is $O((m+n)^{2}d)$. If we use
Eq. (9) to compute the variance, then the time complexity of computing
$E_{d_{p}}$ will decrease to $O((m+n)d)$, which is linear in the total number
of users and items.

In our training, we use neural networks of the previous works to extract user
and item representations. Since we use batches to train the model and update
the representations, the three terms are computed on a batch and will have a
much smaller time complexity, which depends on the number of users and items
in the batch.

##  4 Experiments

###  4.1 Experimental setup

We conduct experiments in both warm-start and item cold-start settings. In the
warm-start setting, the interaction feedback of the test users and items is
available in training data and no new user or item will be added to the system
at test time. In the item cold-start setting, there is no feedback history for
the test items and we only have access to their side information.

In the cold-start setting, we use the following two datasets:

  * •

CiteULike [27]. In this dataset, each user has a set of saved articles and the
goal is to recommend new (cold-start) articles to the users. We use the subset
provided by [26], which contains $5\,551$ users, $16\,980$ articles, and
$204\,986$ user-article pairs. The interaction matrix $\mathbf{R}$ is $99.8$%
sparse and $R_{jk}=1$ means user $j$ saved article $k$. A set of $3\,396$
articles are removed from the training data and used as the test cold-start
articles.

  * •

Ichiba10m. This dataset contains $10$ million purchases from a specific
category of the Rakuten Ichiba 111www.rakuten.co.jp. There are around
$844\,000$ items, $1.4$M users, and $2\,000$ cold-start items. The dataset is
$99.9991$% sparse. The model uses the user interaction vector and item side
information, which includes item’s category, price, and title. The train/test
split is done based on the time period. The cold start items are the items
that didn’t appear during the training period.

In Ichiba10m dataset, the goal is to find interested users for a newly
released (cold-start) item. In CiteULike dataset, the goal is to assign a set
of cold-start articles to each user. In both datasets, we report recall to
evaluate the performance of the methods, which shows what portion of the
retrieved items/articles is relevant to the users. Since ranking all users is
time-consuming in Ichiba, we randomly sample 1 000 unobserved interactions for
each item and report recall by returning top 50 users with highest score.

We use the following two datasets in the warm-start setting:

  * •

Lastfm222https://www.last.fm/. The
dataset333http://ocelma.net/MusicRecommendationDataset/lastfm-1K.html contains
$69\,149$ interactions from $1\,741$ users on $2\,665$ songs. The dataset is
$98.5$% sparse.

  * •

AMusic [7]. The dataset contains $1\,700$ users, $13\,000$ musics, and
$46\,000$ interactions. The dataset is $99.8$% sparse.

We use the pre-processed datasets provided by Dong et al. [5], which is
publicly available for non-commercial
use444https://github.com/familyld/DeepCF.

We report Hit Ratio (HR) to evaluate the implicit feedback prediction
performance. We follow the same protocols as [5, 8] and truncate the rank list
at $10$ for both metrics. HR measures whether the actual test item exists in
the top-ranked list.

#### Baselines.

We compare our method with the state-of-the-art methods in both cold-start and
warm-start settings:

  * •

Shared [18]. It uses a shared item network to learn both user and item
representations. The item network utilizes side information to create item
representations. The user representations are generated from the
representations of the items purchased by the user.

  * •

Shared attention [18]. Same the the Shared model above, except that it uses
attention mechanism to combine the item representations.

  * •

DropoutNet-WMF and DropoutNet-CDL [26]. The key idea is to apply input dropout
to the neural network model such that it can handle the missing input data in
the cold-start setting.

  * •

ACCM [22]. It applies the attention mechanism to hybrid RSs to adjust the
importance of the input sources.

  * •

BUIR-ID [13]. This is the first work to utilize the stop-gradient approach [1]
in RSs. Similar to SimPDO, BUIR-ID only uses positive pairs. The difference
from SimPDO is that BUIR-ID has two encoders and uses the momentum mechanism
to update the parameters of one encoder and to avoid the collapse solution.

  * •

CFNet [5]. It uses interaction vector as the input and learns two user and two
item representations by two neural network models, which are then combined and
mapped to the final interaction by an MLP.

  * •

NeuMF [8]. It uses IDs as the input and learns two user and two item
representations, which are then combined by an MLP to predict the final
interaction.

  * •

DMF [30]. It learns user and item representations by training a neural
networks which uses the explicit feedback as the input and the dot-product as
the mapping function.

Ichiba10m
---
|
CiteULike
|
Figure 1: Impact of each term of our objective function on the results in
Ichiba10m (top row) and CiteULike (bottom row) datasets. We report the average
variance of each dimension, the average correlation between the dimensions,
and recall, as we train the models. The model that uses all three terms
achieves the best results.

#### RSs models to learn the user and item representations.

The specific models to map users and items to the low-dimensional
representations are taken from previous works since the novelty of our work is
in designing a new objective function, which works no matter what the model
is. In the cold-start setting, we use the Shared model proposed by
Raziperchikolaei et al. [18] and in the warm-start setting we use the neural
collaborative framework proposed by Wang et al. [28].

#### Implementation details.

We implemented our method using Keras with TensorFlow 2 backend. We ran all
the experiments on one Nvidia Tesla V100-SXM2 32GB GPU in the internal
cluster. We use grid search to set the hyper-parameters using a subset of
training set and a small validation set. We set the maximum number of epochs
to $50$. We use SGD with the learning rate of $0.5$ for our method in all
experiments. We set the batch size to $32$ in CiteULike and $128$ in other
datasets. We set $\lambda_{1}=0.01$, $\lambda_{1}=1$, and $\lambda_{3}=1$ in
all datasets. The margin $m_{p}$ is $0.1$ in the CiteULike and $0.01$ in all
other datasets. The dimension of the user and item embeddings are $100$ in the
CiteULike and Ichiba10m dataset and $1\,000$ in AMusic and Lastfm.

Figure 2: SimPDO vs Shared [18] as we change the number of training pairs from
$1$M to $10$M and report recall@50 in Ichiba10m dataset. SimPDO achieves
significantly better results when we use a smaller number of training pairs.

###  4.2 Experimental results

Our proposed objective function uses Similar pairs, Pairwise Distance loss,
and Orthogonality loss, and is denoted by SimPDO.

#### Impact of each term of SimPDO.

Our proposed objective function has three terms. The first term,
$E_{\text{cont}}$ or $E_{\text{MSE}}$, tries to generate a high interaction
score for the similar pairs, either by putting their representations close to
each other or by maximizing the dot product of the representations. The second
term, $E_{d_{p}}$ defined in Eq. (6), tries to maximize the pairwise distance
of all the pairs (or equivalently maximize the variance) and avoid the
collapsed and shrinking solutions. The third term, $E_{\text{orth}}$ in Eq.
(11), avoid the partially collapsed solution by making the representations
orthogonal.

In Fig. 1, we investigate the impact of each term by reporting: 1) the average
variance of the dimensions of all the representations, 2) the average
correlation between the dimensions of the representations, and 3) the recall,
as we train the models.

In the top row, we report the results on Ichiba10m dataset. We can see that
the method with all three terms achieves the maximum recall because: 1)
without $E_{\text{cont}}$, the similarity between the similar pairs will not
be preserved, 2) without $E_{d_{p}}$, as we can see in the second column, the
variance drops significantly towards $0$, which is a sign of the shrinking or
collapsed solution, and 3) without $E_{\text{orth}}$, as we can see in the
third column, the average correlation increase, which is a sign of the
partially collapsed solution.

In the bottom row of the Fig. 1, we show the results on CiteULike, where we
use $E_{\text{MSE}}$ instead of the $E_{\text{cont}}$. The results on the
CiteULike confirm that the model with all three terms performs best. Note that
in CiteULike the curve without $E_{d_{p}}$ performs almost as well as using
all the terms. In addition, note that even without $E_{d_{p}}$, the variance
does not drop significantly and it is a bit smaller compared to the other
curves. That’s because $E_{\text{orth}}$ avoids the collapsed and partially
collapsed solution and the shrinking solution does not happen when we use
$E_{\text{MSE}}$.

#### Training only on similar pairs: better results with fewer training
points.

In Ichiba10m dataset, we fix the test set to $2\,000$ cold-start items, change
the number of training pairs from $1$M to $10$M, and report the recall of our
method (SimPDO) and Shared model [18] in Fig. 2. SimPDO only uses similar
pairs in training and optimizes the objective function of Eq. (11). Shared
model [18] uses both similar and dissimilar pairs and optimizes the objective
function of Eq. (4).

Table 1: Comparison with the state-of-the-art cold-start methods on CiteULike dataset. The * in front of the methods’ name means we report the results from the Shared model paper [18]. Our method outperforms the competitors. We run SimPDO three times and report the mean and standard deviation of the results. method | recall@100
---|---
SimPDO |  $\mathbf{66.8}\pm 0.11$%
Shared* [18] |  $65.7$%
Shared attention* [18] |  $66.4$%
DropoutNet-WMF* [26] |  $65.2$%
DropoutNet-CDL* [26] |  $62.9$%
ACCM [22] |  $63.1$%

In SimPDO, all training pairs are similar pairs so we have no dissimilar
pairs. In Shared, the training pairs are divided between the similar and
dissimilar pairs in different portions. In Fig. 2, we considered two cases,
where the number of dissimilar pairs is equal to (first case) and three times
more than (second case) the number of similar pairs.

There are three remarkable points about the results of Fig. 2. First, SimPDO
performs better than the Shared model no matter how many training pairs are
used, which shows the advantage of training only on similar pairs. Second, the
gap between the Shared and SimPDO becomes smaller as we increase the portion
of the dissimilar pairs compared to the similar ones, which shows the
importance of using a large number of dissimilar pairs. Third, SimPDO is
significantly better than the Shared model using a smaller number of training
pairs. This is a big advantage of SimPDO when the datasets have billions of
pairs and training on all of them is time-consuming: SimPDO can be trained on
a smaller training set and still achieve reasonable results.

Table 2: Comparison with the state-of-the-art warm-start methods on AMusic and Lastfm datasets. The * in front of the methods’ name means we report the results from the DeepCF paper [5]. Our method achieves competitive results using smaller number of training pairs. We put $-$ for HR@5 of some of the methods since only HR@10 is reported in [5]. We run SimPDO, BUIR-ID, and CFNet three times and report the mean and standard deviation of the results. | AMusic | Lastfm
---|---|---
method | HR@5 | HR@10 | HR@5 | HR@10
SimPDO | $\mathbf{32.66\pm 0.4}$ | $\mathbf{43.47\pm 0.09}$ | $75.4\pm 0.1$ | $\mathbf{89.6\pm 0.18}$
BUIR-ID [13] | $25.76\pm 0.2$ | $36.5\pm 0.7$ | $70.16\pm 0.6$ | $84.15\pm 0.9$
CFNet [5] | $29.7\pm 0.08$ | $38.4\pm 0.003$ | $\mathbf{77.1\pm 0.17}$ | $88.6\pm 0.02$
CFNet* [5] | - | $41.2$ | - | $89.9$
CFNet-ml* [5] | - | $40.7$ | - | $88.3$
CFNet-rl* [5] | - | $39.4$ | - | $88.4$
NeuMF* [8] | - | $38.9$ | - | $88.6$
DMF* [30] | - | $37.4$ | - | $88.4$
Table 3: We report number of similar and dissimilar pairs to train different methods and training time per epoch of different methods. SimPDO and BUIR-ID use a smaller number of pairs and can be trained much faster than the other methods. | AMusic | Lastfm | CiteULike
---|---|---|---
method | numS | numD | training time | numS | numD | training time | numS | numD | training time
SimPDO |  $46$K | $0$ |  $13$s |  $69$K | $0$ |  $15$s |  $200$K |  $0$M |  $123$s
BUIR-ID |  $46$K | $0$ |  $8$s |  $69$K | $0$ |  $11$s |  $200$K |  $0$M |  $123$s
CFNet |  $46$K |  $184$K |  $36$s |  $69$K |  $276$K |  $36$s | - | - | -
Shared | - | - | - | - | - | - |  $200$K |  $2$M |  $1871$s
Shared_attention | - | - | - | - | - | - |  $200$K |  $2$M |  $1963$s

#### Comparison with the state-of-the-art methods.

In Table 1, we compare the methods on CiteULike dataset in the cold-start
setting. The dataset and the Shared code are available
online555https://github.com/rakutentech/shared_rep. We use the code and
processed data provided in this repository to train our model. Note that our
method uses the same user and item models as the shared model, but uses the
new objective function. This makes it clear that the better performance of the
SimPDO comes from the proposed objective function that is only using similar
pairs.

In Table 2, we compare the methods on AMusic and Lastfm datasets in the warm-
start setting. The dataset and the DeepCF code are available
online666https://github.com/familyld/DeepCF. We use the code and processed
data provided in this repository to train our model. We also trained CFNet and
BUIR-ID from the publicly available code and put the results in the second and
third row of Table 2.

Results of Table 1 and Table 2 show that SimPDO outperforms most state-of-the-
art methods and achieves very competitive results. Note that SimPDO and BIUR
use a significantly smaller number of training pairs compared to the other
methods, such as Shared and CFNet. We have put the number of similar and
dissimilar pairs used in training the methods with the best results in Table
3. SimPDO and BUIR-ID use a smaller number of pairs and can be trained much
faster than the other works.

##  5 Conclusion

In this paper, we proposed SimPDO, a new objective function that enables
training of one-class recommendation system models without dissimilar pairs.
We showed that by only using similar pairs, the optimal solution of existing
objective functions becomes a collapsed solution, where every representation
is mapped to the same point in the latent space. We avoided the collapsed
solution by providing a hinge loss for pairwise distance which is equivalent
to the average variance of each dimension of representations. We also showed
that we need an orthogonality term to avoid collapsed and partially collapsed
solutions, where the optimal solution under the pairwise loss returns only two
representations. The orthogonality term minimizes the correlation between each
dimension of representations and forces the objective function to return
various representations. Finally, we showed that both terms are necessary to
learn meaningful representations. The results demonstrated that SimPDO
outperformed the existing RS objective functions without using dissimilar
pairs. We also showed that SimPDO can be trained more efficiently with a
smaller number of training pairs.

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