# Anisotropic phase diagram of the frustrated spin dimer compound Ba3Mn2O8

E. C. Samulon1, K. A. Al-Hassanieh2, Y.-J. Jo3, M. C. Shapiro1, L. Balicas3,
C. D. Batista2, I. R. Fisher1 1Geballe Laboratory for Advanced Materials and
Department of Applied Physics, Stanford University, Stanford, California
94305, USA  2Theoretical Division, Los Alamos National Laboratory, Los Alamos,
New Mexico 87545, USA  3National High Magnetic Field Laboratory, Florida State
University, Tallahassee, Florida 32306, USA

###### Abstract

Heat capacity and magnetic torque measurements are used to probe the
anisotropic temperature-field phase diagram of the frustrated spin dimer
compound Ba3Mn2O8 in the field range from 0T to 18T. For fields oriented along
the $c$ axis a single magnetically ordered phase is found in this field range,
whereas for fields oriented along the $a$ axis two distinct phases are
observed. The present measurements reveal a surprising non-monotonic evolution
of the phase diagram as the magnetic field is rotated in the [001]-[100]
plane. The angle dependence of the critical field ($H_{c1}$) that marks the
closing of the spin gap can be quantitatively accounted for using a minimal
spin Hamiltonian comprising superexchange between nearest and next nearest Mn
ions, the Zeeman energy and single ion anisotropy. This Hamiltonian also
predicts a non-monotonic evolution of the transition between the two ordered
states as the field is rotated in the $a$-$c$ plane. However, the observed
effect is found to be significantly larger in magnitude, implying that either
this minimal spin Hamiltonian is incomplete or that the magnetically ordered
states have a slightly different structure than previously proposed.

###### pacs:

75.30.Kz, 75.40.-s, 75.30.-m, 75.45.+j

††preprint: APS/123-QED

##  I Introduction

Ba3Mn2O8 is a novel layered spin-dimer compound, comprising magnetic dimers of
Mn5+ ions arranged on triangular planesWeller_1999 . The Mn ions occupy
equivalent sites with a distorted tetrahedral coordination, resulting in a
quenched orbital angular momentum and a total spin $\mathbf{S}=1$ Uchida_2002
. Antiferromagnetic intradimer exchange leads to a singlet ground state with
excited triplet and quintuplet states. Weaker interdimer exchange leads to
dispersion of these excitations, as previously revealed by inelastic neutron
scattering Stone_2008 . The rhombohedral R$\bar{3}$m crystal structure of
Ba3Mn2O8 comprises staggered hexagonal planes (Fig. 1(a)), leading to
geometric frustration both within individual planes, and also between adjacent
planes. Magnetic fields can be used to close the spin gap, and the competition
between interdimer interaction on this highly frustrated lattice, and the
uniaxial single ion anisotropy associated with the $\mathbf{S}=1$ ions, leads
to a very complex phase diagram with at least three distinct ordered states
Samulon_2008 ; Samulon_2009 .

Despite the complexity of the phase diagram of Ba3Mn2O8, we have previously
shown that for fields oriented along the principal crystalline axes the high-
field behavior of this material can be described by a remarkably simple spin
Hamiltonian, including terms representing superexchange between nearest and
next-nearest Mn ions within and between planes, the Zeeman energy and uniaxial
single ion anisotropy Samulon_2008 . These terms have been determined through
a combination of inelastic neutron scattering (INS) and electron paramagnetic
resonance (EPR). INS studies revealed the dominant exchange within a dimer as
$J_{0}$ = 1.642(3) meV; the nearest and next nearest out-of-plane exchanges,
$J_{1}$ = 0.118(2) meV and $J_{4}$ = 0.037(2) meV; and the dominant in-plane
exchanges $J_{2}-J_{3}$ = 0.1136(7) meV (Fig. 1) Stone_2008 . EPR measurements
in the diluted compound Ba3(V1-xMnx)2O8 revealed a nearly isotropic
$g$-tensor, with $g_{aa}$ = 1.96 and $g_{cc}$ = 1.97, and an easy axis single
ion anisotropy $D$ = -0.024 meV Whitmore_1993 . Similar measurements in the
pure Ba3Mn2O8 compound revealed a zero field splitting of the triplet states
characterized by $D$ =- 0.032 meV Hill_2007 , the difference reflecting the
additional effect of intradimer dipolar coupling in the undiluted compound.

Figure 1: (Color online) (a) Schematic diagram illustrating the exchange
interactions between Mn ions (red spheres) in Ba3Mn2O8 (Ba and O ions are not
shown). Values for the exchange constants are given in the main text (b, c)
Phase diagram of singlet-triplet ordered states for fields aligned along the
$c$ and $a$ axes, respectively, taken from our previous workSamulon_2008 . I,
II and PM mark phase I, phase II and the paramagnetic phase as described in
the main text. Solid and open symbols represent data points obtained via heat
capacity and MCE, respectively.

At low temperatures, the field dependence of the magnetization of Ba3Mn2O8
reveals two regions of linearly increasing moment as first the $S^{z}=1$
triplet (between $H_{c1}=8.7$ T and $H_{c2}=26.5$ T) and then $S^{z}=2$
quintuplet (between $H_{c3}=32.5$ T and $H_{c4}=47.9$ T) states are polarized,
separated by a plateau at half the saturation magnetization Uchida_2002 ;
Samulon_2009 . Here we concentrate solely on the first region of increasing
magnetization, which can be accessed by moderate laboratory fields. In this
field range the ground state can be approximated as a coherent mixture of
singlet and $S^{z}$=1 triplet states. Previous measurements revealed a
distinct anisotropy in the phase diagram depending on the field direction
Samulon_2008 . For fields applied along the $c$ axis, a single ordered state
was found in this regime (fig. 1(b)). Analysis of the minimal spin Hamiltonian
suggests that this state is an incommensurate XY antiferromagnet with
wavevector shifted slightly away from the 120∘ structure favored for a 2D
triangular layerSamulon_2008 ; Diep_2005 . NMR measurements appear to confirm
this conclusionSuh_2009 , but to date the magnetic structure has not been
explicitly solved. Alternatively, fields applied perpendicular to the $c$ axis
revealed two ordered states in this field regime (Fig. 1(c)). Analysis of the
minimal spin Hamiltonian for this field direction yields two distinct
incommensurate modulated states - an Ising phase stabilized closed to $H_{c1}$
and $H_{c2}$, as well as a canted XY state in between. The modulation in these
two phases is stabilized by the uniaxial anisotropy associated with the zero
field splitting of the triplets. Similar to the situation for fields oriented
along the $c$ direction, the exact form of the magnetic structure has not been
explicitly solved, and it remains to be seen whether the minimal spin
Hamiltonian that has been used to describe this system so far is complete.

To this end, careful measurements of the phase diagram of Ba3Mn2O8 for fields
at intermediate angles between the $a$ and $c$ axes have the potential to
determine the presence or absence of additional terms. In this paper we
present results of heat capacity and torque magnetometry measurements
revealing how the two distinct ordered states for fields perpendicular to the
$c$ axis evolve into a single phase for fields along the $c$ axis. Through
analysis of the previously established minimal spin Hamiltonian we can
quantitatively account for the angular dependence of $H_{c1}$ solely via
consideration of the triplet dispersion. However, the same analysis,
incorporating the predicted magnetic structures, fails to quantitatively
account for the angular dependence of the transition between the two ordered
states. We discuss the implications of this observation.

##  II Experimental Methods

Single crystals of Ba3Mn2O8 were grown from a NaOH flux according to our
previously published procedureSamulon_2008 . Heat capacity ($C_{p}$) studies
were performed on a Quantum Design physical properties measurement system
(PPMS) using standard thermal relaxation-time calorimetry. These measurements
were performed in fields up to 14T and temperatures down to 0.35 K. The sample
was mounted on angled brackets made from oxygen-free high conductivity copper
and the field was oriented in the [100]-[001] plane.

Cantilever torque magnetometry experiments were performed at the National High
Magnetic Field Laboratory (NHMFL) in a superconducting magnet for fields up to
18T in a dilution refrigerator. A crystal was mounted on one face of a
capacitance cantilever which was attached to a rigid plate rotatable about an
axis parallel to the torque axis and perpendicular to the magnetic field. The
magnetic field was aligned away from the principal crystalline axes yielding a
finite torque.

##  III Results

Figure 2: (Color Online) Representative heat capacity data taken at 12T for
for fields in the $a$-$c$ plane. Labels indicate the angle between the field
and the $c$ axis. Successive data sets are offset vertically by 1.6 J/molK for
clarity.

Representative heat capacity measurements, taken at 12 T for several angles,
are shown in Figure 2. These data show a single peak for fields aligned along
the $c$ axis, a peak and a shoulder for fields 15∘ from the $c$ axis and two
peaks for larger angles. Significantly, comparison of the data at 75∘ and 90∘
degrees shows that the 75∘ data has both a slightly higher critical
temperature between the paramagnetic phase and phase II ($T_{c_{II}}$) and
also a substantially lower critical temperature between phase II and phase I
($T_{c_{I}}$) than the 90∘ data.

The phase diagram derived from the complete set of $C_{p}$ measurements, shown
in Figure 3(a), reveals the evolution as a function of angle of the two
distinct singlet-triplet ordered states for fields in the [100]-[001] plane.
The data show a single transition for all fields for $H\|c$ and two
transitions for $H\|a$. The extent in temperature of phase II,
$\Delta_{T}=T_{c_{II}}-T_{c_{I}}$, is shown in Figure 3(b). $\Delta_{T}$
increases as a function of angle as the field is rotated away from the $c$
axis, reaches a maximum at 75∘, and decreases at the $a$ axis (90∘). For
example, for a field of 11T, $\Delta_{T}$ is $\sim$ 0.07 K larger at 75∘ than
at 90∘.

Figure 3: (Color online) (a) Phase diagram showing the transitions between the
paramagnetic state and phase II ($T_{II}$), and between phase II and phase I
($T_{I}$), as a function of temperature and angle in the [100]-[001] plane for
various fields, where $\theta$ indicates the angle between the field and the
$c$ axis. (b) Width of phase II, $\Delta_{T}=T_{c_{II}}-T_{c_{I}}$, as a
function of angle for 11T, 12T, 13T and 14T (black circles, red up triangles,
green down triangles, and blue squares, respectively.)

Torque magnetometry measurements, taken at 25 mK, are shown in Figure 4 for
three representative angles. The sample was inclined slightly so that the
field did not exactly rotate within the $a$-$c$ plane such that a finite
torque was generated for all angles studied. Angles are quoted in terms of the
angular position with respect to the closest approach to the $c$ axis, but it
is important to note that the field was never less than $\sim 10^{\circ}$ from
the $c$ axis. Consequently two phase transitions are observed for all angles
studied. Critical fields, marking the transition between the paramagnetic
phase and phase II ($H_{c1}$), and between phase II and phase I ($H_{II-I}$),
were determined from maxima and minima in the second field derivative of the
torque divided by field Samulon_2008  and are marked with dashed vertical
lines. The sign of the peak in the second derivative changed with the
evolution of angle, reflecting a change in anisotropy for the two different
phases. This leads to minor discontinuities in the determination of the phase
boundary (dashed lines in Fig. 5).

Figure 4: Torque scaled by field, and its first two derivatives with respect
to field, plotted versus field for representative angles at 5.5∘, 55.0∘ and
88.0∘ with respect to the nearest approach to the $c$ axis (see main text).
The critical fields were estimated from peaks in the 2nd derivative as shown
by the dashed lines.

Similar to the phase diagram obtained from heat capacity measurements (Fig.
3), the phase diagram obtained from torque measurements at 25mK reveals a non-
monotonic angle dependence (Fig. 5). The maximum value of $H_{II-I}$ occurs
between 65-75∘ from the closest approach to the $c$ axis. This is in agreement
with the heat capacity data, for which the smallest $T_{c_{I}}$ occurs at 75∘
from the $c$ axis. Additionally, the field extent of phase II,
$\Delta_{H}=H_{II-I}-H_{c1}$, is largest at 75∘, and decreases by $\sim$0.05 T
from the maximum at the highest angles.

##  IV Discussion

The previously established minimal spin Hamiltonian for arbitrarily oriented
field direction in Ba3Mn2O8 is:

$\displaystyle\mathcal{H}$ $\displaystyle=$
$\displaystyle\sum_{i,j,\mu,\nu}\frac{J_{i\mu
j\nu}}{2}\textbf{S}_{i\mu}\cdot\textbf{S}_{j\nu}+D\sum_{i,\mu}\left(S^{z}_{i\mu}\cos{\theta}-S^{x}_{i\mu}\sin{\theta}\right)^{2}$
(1)
$\displaystyle-\mu_{B}H\sum_{i\mu\alpha\beta}\left({\tilde{g}}_{zz}S^{z}_{i\mu}+{\tilde{g}}_{xz}S^{x}_{i\mu}\right),$

where ${\tilde{g}}_{zz}=g_{aa}\sin^{2}{\theta}+g_{cc}\cos^{2}{\theta}$,
${\tilde{g}}_{xz}=(g_{cc}-g_{aa})\sin{\theta}\cos{\theta}$, $g_{\alpha\beta}$
is the diagonal gyromagnetic tensor with components $g_{cc}$, $g_{aa}=g_{bb}$,
and $\theta$ is the angle between the applied field and the $c$-axis. The
quantization $z$ axis is set along the field direction. Here $i$, $j$
designate the dimer coordinates, $\alpha,\beta=\\{x,y,z\\}$,
$\mu,\nu=\\{1,2\\}$ denote each of the two S=1 spins in each dimer. The
various exchange constants are shown in Fig. 1(a) and are defined as follows:
the exchange within a dimer is $J_{0}=J_{i,1,i,2}$; the dominant out-of-plane
exchange is $J_{1}=J_{i,2,j,1}$ for $i,j$ nearest neighbor dimers between
planes; the dominant in-plane exchanges between dimers is
$J_{2}=J_{i,\mu,j,\mu}$ and $J_{3}=J_{i,\mu,j,\nu}$ for $i,j$ in plane nearest
neighbor dimers and $\mu\neq\nu$; and finally the second largest out-of-plane
exchange is $J_{4}=J_{i,2,j,1}$ for $i,j$ next nearest neighbor dimers between
planes.

Figure 5: (Color online) Phase diagram at 25 mK determined from torque
magnetometry measurements. Circles (triangles) mark transitions between the
disordered phase and phase II (phase II and phase I). Open (closed) symbols
signify that the transition was determined from a peak (trough) in the second
derivative. Angles are measured relative to the closest approach to the $c$
axis as described in the main text. Red dotted line shows the calculated
transition between the paramagnetic and ordered phases as described in the
main text.

Using this spin Hamiltonian and the measured values of $J_{0}$-$J_{4}$ and $D$
given in the introduction, we have previously been able to quantitatively
account for the observed critical fields and magnetization of the ordered
states of Ba3Mn2O8 for fields oriented along the principal axes Samulon_2008 ;
Samulon_2009 ; Suh_2009 . The calculation is based on a generalized spin-wave
approach in which we only keep the singlet and the three triplet states of
each dimer Khaled2010 . The critical field $H_{c1}(\theta)$ corresponds to the
value for which the energy of lowest energy triplet mode becomes equal to
zero. The softening of this triplet mode signals the onset of the magnetic
instability towards an ordered state (phase II). For field directions along
the principal axes, it is possible to obtain simple analytical expressions for
the critical field. The expression for $H\|c$ is:

$(g_{cc}\mu_{B}H_{c1})^{2}=\left(J_{0}-\frac{D}{3}\right)^{2}+\frac{8}{3}\left(J_{0}-\frac{D}{3}\right)\mathcal{J}_{min},$
(2)

while for $H\perp c$ the expression is:

$\displaystyle(g_{aa}\mu_{B}H_{c1})^{2}$ $\displaystyle=$
$\displaystyle\left(J_{0}+\frac{D}{6}\right)^{2}+\frac{8}{3}\left(J_{0}+\frac{D}{6}\right)\mathcal{J}_{min}$
(3) $\displaystyle-\frac{D^{2}}{4}-\frac{4}{3}|D||\mathcal{J}_{min}|$

where $\mathcal{J}_{min}$ is the minimum of the interdimer exchange portion of
the dispersion and is fully described in equation (3) of reference
[Stone_2008, ]. The difference in the first two terms between these
expressions stems from a change in the zero field splitting of an isolated
dimer depending on quantization direction expressed in the reduced basis of
dimer states. For the quantization axis along the $c$ axis, the zero field gap
of an isolated dimer between the $S^{z}=1$ triplet and the singlet is
$J_{0}-D/3$, while for the quantization axis along the $a$ axis the equivalent
zero field gap is $J_{0}+D/6$. The two additional terms of $H_{c1}$ for fields
perpendicular to the $c$ axis arise from a second order process mixing
singlets and triplets described in our previous work Samulon_2008 . This state
mixing causes the gap between the $S^{z}=1$ triplet and singlet states to
close as $\sqrt{H-H_{c1}}$, as expected for an Ising-like QCP. Evaluating
these expressions using the values of the exchanges and single ion anisotropy
described earlier yield values for the critical fields which are in good
accord with the measured data. Numerical calculation of $H_{c1}$ for arbitrary
field orientations in the [001]-[100] plane yields the red dotted curve shown
in Fig. 5. The calculation was performed using the values of $D$ and the
interdimer couplings given in the introduction while $J_{0}$ was allowed to
vary leading to a fit value of 1.567 meV. The calculated values agree well
with the measured data up to the inherent uncertainty associated with the
misalignment of the sample in the torque measurements described above.

The single-ion anisotropy term of ${\cal{H}}$ is:

$\displaystyle
D\bigg{[}\left(S^{z}\right)^{2}\cos^{2}\left(\theta\right)+\left(S^{x}\right)^{2}\sin^{2}\left(\theta\right)$
$\displaystyle-\left(S^{z}S^{x}+S^{x}S^{z}\right)\cos\left(\theta\right)\sin\left(\theta\right)\bigg{]}.$
(4)

The last term of eq. (4) is zero for fields along the $a$ and $c$ axes, but
adds a small contribution for intermediate angles. In particular, this term
grows linearly in small deviations of $\theta$ from $\pi/2$,
$\delta\theta=\pi/2-\theta$, while the other two terms vary quadratically in
$\delta\theta$. Thus, the last term of Eq. (4) determines the shape of the
boundary between phase I and phase II slightly away from $\theta=\pi/2$, and
could be responsible for the striking non-monotonic behavior observed for
$H_{II-I}$ in Fig. 5 as we describe in greater detail below.

To understand the effect that the last term of eq. (4) has on the ground
state, it is convenient to analyze the effective low-energy Hamiltonian,
${\cal H}_{\rm eff}$, introduced in ref. [Samulon_2008, ]. The low-energy
Hamiltonian results from projecting the original Hamiltonian ${\cal H}$ onto
the low-energy subspace generated by the singlet and the $S^{z}=1$ triplet
states of each dimer. This two-level Hilbert space is described by a local
pseudo-spin $\frac{1}{2}$ in each dimer, ${\bf s}_{i}$, such that
$s^{z}_{i}=\frac{1}{2}$ if the dimer $i$ is in the $S^{z}=1$ triplet state and
$s^{z}_{i}=-\frac{1}{2}$ if it is in the singlet state. In our earlier work we
provided the expression of ${\cal H}_{\rm eff}$ for $\theta=0$ and
$\theta=\pi/2$ Samulon_2008 . In particular, for $\theta=\pi/2$, we showed
that the second term of Eq. (4) generates an effective exchange anisotropy
that is responsible for the emergence of phase II. According to our analysis,
this exchange anisotropy favors an Ising-like phase in which the transverse
spin components (perpendicular to the applied field) are aligned along the
easy $c$-axis. In contrast, phase I is an elliptical spiral phase in which the
transverse spin components of adjacent dimers rotate around the field axis.

The effect the last term of Eq. (4) has on ${\cal H}_{\rm eff}$ at
intermediate angles can be determined using second order degenerate
perturbation theory. Such an analysis yields an effective Dzyaloshinskii-
Moriya (DM) interaction between dimers on adjacent bilayers connected by the
$J_{1}$ and $J_{4}$ exchange constants:

$\sum_{\langle\langle i\rightarrow
j\rangle\rangle}{\bf{\tilde{D}}}_{1}\cdot{\bf s}_{i}\times{\bf
s}_{j}+\sum_{\langle\langle i\rightarrow
j\rangle\rangle^{\prime}}{\bf{\tilde{D}}}_{4}\cdot{\bf s}_{i}\times{\bf
s}_{j},$ (5)

with ${\bf{\tilde{D}}}_{l}={\tilde{D}}_{l}{\bf{\hat{y}}}$,
${\tilde{D}}_{l}={\cal O}(DJ_{l}/J_{0})$ and $l=1,4$. The arrow indicates how
the bonds $\langle\langle i\rightarrow j\rangle\rangle$ are oriented ($i$
always denotes the dimer in the lower bilayer). Microscopically, this process
turns one singlet into an $S^{z}=1$ triplet or vice versa.

This effective DM coupling between pseudo-spins results from two important
symmetry considerations. First, the singlet and the triplet states of a given
dimer have opposite parity under exchange of the two sites of the dimer:
$1\leftrightarrow 2$. Second, the $J_{1}$ and $J_{4}$ terms of ${\cal H}$ are
invariant under the inversion symmetry transformation around the center of the
corresponding bonds: $1\leftrightarrow 2$ and $i\leftrightarrow j$ (see Fig.
6(a)). This implies that a DM term is allowed between pseudo-spins connected
by the $J_{1}$ and $J_{4}$ exchanges. In contrast, the effective DM
interaction cannot occur for pairs of dimers within a plane because the
$J_{2}$ and $J_{3}$ terms of ${\cal H}$ are invariant under the symmetry
transformation $i\leftrightarrow j$, while the effective DM interaction
changes sign under such transformation. These symmetries are fundamentally
equivalent to the selection rules governing the DM vector in the effective
lattice, where each dimer constitutes a single, unique site. In such a
lattice, there is a center of inversion symmetry at the midpoint of the
effective $J_{2}$ and $J_{3}$ exchange, precluding a DM term by the normal
selection rules, while no such inversion symmetry exists at the midpoints of
the effective $J_{1}$ and $J_{4}$ exchanges. This is contrast to the
symmetries of the real lattice, where there are inversion symmetries at the
midpoint of the $J_{1}$ and $J_{4}$ exchanges but none at the midpoint of the
$J_{2}$ and $J_{3}$ exchanges.

Figure 6: (a) Schematic diagram illustrating the two dimers involved in
processes of order $J_{l}D/J_{0}$ that leads to Eq. (5). The $c$ axis is
vertical, and the two dimers belong to adjacent planes. $J_{l}$ represents
either $J_{1}$ or $J_{4}$. (b) (i), (ii) Schematic diagram for two pseudo-
spins on adjacent dimers connected by the $J_{l}$ interaction for arbitrary
field direction in the $a$-$c$ plane. Red arrows show uniform moment, while
green arrows show ordered moment. Field direction along $z$ axis,
${\bf{\tilde{D}}}_{l}$ along $y$ axis, and $c$ axis vertical.

This effective DM interaction between dimers on adjacent layers is frustrated
in both ordered states at a mean field level (the mean value of Eq. (5) is
zero for the semi-classical states associated with phases I and II).
Therefore, the small contribution of the effective DM interaction to the
ground state energy must be produced by quantum fluctuations. Because the DM
vectors point along the $y$ direction (perpendicular to the applied field and
to the easy $c$ axis) this contribution term will favor phase II, for which
the pseudo-spins only have $x$ and $z$ components (Fig. 6(b)(ii)), as opposed
to phase I for which the pseudo-spins have an additional third component along
the $y$ direction (Fig. 6(b)(i)) Samulon_2008 . Thus this contribution
strengthens phase II relative to phase I near $\theta=\pi/2$ and leads to a
small non-monotonic behavior of the $H_{II-I}$ curve (see Fig. 5).

Although this simple analysis captures the qualitative non-monotonic behavior
of the $H_{II-I}$ curve, it cannot account for the magnitude of the observed
effect. The amplitude ${\cal D}(\theta)=D\sin{\theta}\cos{\theta}$ of the
effective DM interaction is of order 100 mK for ${\theta}\simeq 75^{\circ}$.
Because the interaction mixes the singlet and $S^{z}=1$ triplet dimer states,
the mean value of the DM term is less than $\sqrt{m}{\cal D}(\theta)$ for any
state with magnetization $m$ (the mean density of $S^{z}=1$ triplets). Noting
that $m\simeq 0.06$ at $H_{II-I}$, the upper bound on the DM term of $\simeq
25$ mK is the order of magnitude of the observed non-monotonic effect of 5-10
mK in the $H_{II-I}$ curve. Given that ${\cal D}(\theta)$ is much weaker than
the dominant terms of ${\cal H}$, it is clear that the effective DM term can
only explain the magnitude of the non-monotonic effect if it gives a first
order contribution to the energy of Phase II. However, as established above
for the proposed ordered states, the effective DM interaction contributes via
a second order correction and must therefore be considerably smaller. This
leaves us with two possibilities: a) The magnetic structure of Phase II is
different from the simple Ising phase proposed in Ref. [Samulon_2008, ] in
such a way that the mean value of the effective DM term is non-zero, or b) The
non-monotonic effect is caused by a term that has not been included in ${\cal
H}$. At present it is impossible to distinguish between these possibilities,
but ongoing efforts to experimentally determine the magnetic structure have
the potential to directly address option (a), while EPR experiments should, at
least in principle, be able to determine the energy scale of additional
interactions not considered in the minimal spin Hamiltonian (eq. (1)).

##  V Conclusion

In summary, via heat capacity and torque magnetometry measurements we have
established the angular dependence of the phase boundary for singlet-triplet
ordered states of Ba3Mn2O8. The data reveal a striking non-monotonicity of the
phase boundary as the field is rotated between the principal axes. The angle-
dependence of $H_{c1}$ can be quantitatively understood in terms of the
original minimal spin Hamiltonian that we had proposed for this material. This
quantity does not depend on details of the magnetically ordered states but
only on the minimum of the triplet dispersion. However, the observed non-
monotonicity in $H_{I-II}$ is at least an order of magnitude larger than
anticipated based on this model and assuming the magnetic structures
previously proposed. This indicates that a complete theoretical description of
this material requires either subtle changes in the proposed magnetic ordered
structures or an additional low-energy term in the Hamiltonian.

##  VI Acknowledgements

Work at Stanford University is supported by the Division of Materials
Research, National Science Foundation under Grant No. DMR-0705087. LB is
supported by DOE-BES and the NHMFL-UCGP program. A portion of this work was
performed at the National High Magnetic Field Laboratory, which is supported
by the National Science Foundation under Cooperative Agreement No.
DMR-0084173, by the state of Florida, and the Department of Energy.

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  * (10) While the $g$ factor will also play a role in the angle dependence of phase II, it is nearly isotropic and will vary monotonically as a function of angle.
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# Feature-aware Diversified Re-ranking with Disentangled Representations for
Relevant Recommendation

Zihan Lin†<EMAIL_ADDRESS>School of Information, Renmin University of China
,  Hui Wang†<EMAIL_ADDRESS>School of Information, Renmin University of
China ,  Jingshu Mao<EMAIL_ADDRESS>Kuaishou Inc.BeijingChina ,
Wayne Xin Zhao♣ ♠🖂<EMAIL_ADDRESS>Gaoling School of Artificial
Intelligence, Renmin University of China ,  Cheng Wang
<EMAIL_ADDRESS>Kuaishou Inc.BeijingChina ,  Peng Jiang
<EMAIL_ADDRESS>Kuaishou Inc.BeijingChina and  Ji-Rong Wen♣ ♠
<EMAIL_ADDRESS>Gaoling School of Artificial Intelligence, Renmin University
of China

(2022)

###### Abstract.

Relevant recommendation is a special recommendation scenario which provides
relevant items when users express interests on one target item (_e.g.,_ click,
like and purchase). Besides considering the relevance between recommendations
and trigger item, the recommendations should also be diversified to avoid
information cocoons. However, existing diversified recommendation methods
mainly focus on item-level diversity which is insufficient when the
recommended items are all relevant to the target item. Moreover, redundant or
noisy item features might affect the performance of simple feature-aware
recommendation approaches.

Faced with these issues, we propose a Feature Disentanglement Self-Balancing
Re-ranking framework (FDSB) to capture feature-aware diversity. The framework
consists of two major modules, namely disentangled attention encoder (DAE) and
self-balanced multi-aspect ranker. In DAE, we use multi-head attention to
learn disentangled aspects from rich item features. In the ranker, we develop
an aspect-specific ranking mechanism that is able to adaptively balance the
relevance and diversity for each aspect. In experiments, we conduct offline
evaluation on the collected dataset and deploy FDSB on KuaiShou app for online
$A$/$B$ test on the function of relevant recommendation. The significant
improvements on both recommendation quality and user experience verify the
effectiveness of our approach.

Relevant Recommendation; Diversity; Disentangled Representation Learning

$\dagger$ Equal contribution. Work done during internship at KuaiShou.

$\spadesuit$ Beijing Key Laboratory of Big Data Management and Analysis
Methods

$\clubsuit$ Beijing Academy of Artificial Intelligence, Beijing, 100084, China

🖂 Corresponding author.

††copyright: acmcopyright††journalyear: 2022††doi:
10.1145/3534678.3539130††conference: Proceedings of the 28th ACM SIGKDD
Conference on Knowledge Discovery and Data Mining; August 14–18, 2022;
Washington, DC, USA.††booktitle: Proceedings of the 28th ACM SIGKDD Conference
on Knowledge Discovery and Data Mining (KDD ’22), August 14–18, 2022,
Washington, DC, USA††price: 15.00††isbn: 978-1-4503-9385-0/22/08††ccs:
Information systems Recommender systems

##  1\. Introduction

Nowadays, recommender systems are playing an increasingly important role in
delivering suitable content to users and boosting the exposure of high-quality
resources (Ricci et al., 2011). From classic collaborative filtering (Su and
Khoshgoftaar, 2009) to recent neural recommenders (Gupta et al., 2020), the
underlying recommendation algorithms (Zhao et al., 2021) have been evolving
during the past two decades, for better satisfying user preferences. Besides
the algorithmic aspect, various auxiliary functionalities have been introduced
in recommender systems to further improve user experiences. Among these new
functionalities, _relevant recommendation_ has been widely deployed in
e-commerce platforms and video platforms (Xie et al., 2021), both recommend
relevant items (products or videos) of a specific item (called _trigger
item_). For example, Youtube uses “Up next” section to show a suggestion of
what to watch next when you are watching a video. Amazon would present
relevant products (“ _Products related to this item_ ”) on product description
page. With relevant recommendations, Amazon is able to keep consumers engaged
and offer products of interest to them that they might not even think about
before.

Figure 1. A snapshot of the relevant recommendations in KuaiShou. Left: The
video which contains three feature tags (_e.g.,_ _sunset_ , _train_ and _sea_
) is liked by a user. Right: A small number of relevant videos are displayed
below.

In this paper, we study the task of _relevant recommendation_ in a special
scenario of micro-video recommendation on _KuaiShou_
111https://www.kuaishou.com/, which is a popular Chinese micro-video app
(similar to _TikTok_). Compared with other types of apps, it is easier to
activate a consecutive watching of videos, since it takes less time to watch
micro-videos than ordinary videos. We present an illustrative snapshot of this
function on Kuaishou app in Figure 1. When a user clicks on the ‘ _like_ ’
button for a micro-video (the left part of Figure 1), the system will present
a small number of micro-videos for recommendation (the right part of Figure
1). Furthermore, these recommended micro-videos can be directly clicked for
watching. Generally speaking, we refer to the micro-video being liked _trigger
item_ and the recommended micro-videos as _relevant items_.

The main purpose of relevant recommendation is to recommend both _relevant_
and _diverse_ items. For relevance, the videos recommended should be relevant
to the trigger item, since the user expresses positive preference on it. For
diversity, the recommendations should be diverse to include new features for
avoiding information cocoons, and explore more interests of users. In fact,
such a task setting is similar to _diversified recommendation_ (Wu et al.,
2019a). A significant difference is that relevant recommendation is conducted
based on both the trigger item and user preference, while diversified
recommendation only considers the user preference. To incorporate the trigger
item for diversified recommendation, we need to explicitly model fine-grained
item characteristics in order to better characterize item-to-item relations.
For example, in Figure 1, the liked micro-video is attached with a set of
semantic labels {  _sunset_ , _train_ and _sea_ }, and it is meaningful to
conduct the relevant recommendations according to these fine-grained semantic
labels (_i.e.,_ features). However, existing diversified recommendation
methods mainly focus on item-level diversity (Huang et al., 2021), which is
not able to capture feature-level diversity effectively. In addition, these
side information can be rather redundant or noisy, and a direct feature-level
recommendation approach (_e.g.,_ simple finding and merging relevant items for
each feature) may not work well in practice (Cheng et al., 2022).

Considering the above issues, we design a general re-ranking framework,
Feature Disentanglement Self-Balanced Re-ranking (FDSB), to capture _feature-
aware diversity_ for relevant recommendation. Our framework is developed based
on a multi-factor re-ranking paradigm, consisting of two major modules, namely
disentangled attention encoder and self-balanced multi-aspect re-ranker. For
the disentangled attention encoder, we learn disentangled dimensions from rich
item features, called _aspects_ , based on multi-head attention. The learned
aspects provide a more compact representation to extract item features, which
will be used in subsequent re-ranking. For the self-balanced multi-aspect re-
ranker, we develop an aspect-specific re-ranking mechanism that is able to
adapt balance relevance and diversity. We set aspect-specific coefficients for
each dimension to tune the importance of both targets, and associate these
coefficients of relevance and diversity for balancing the two factors.

To verify the effectiveness of proposed framework, we conduct both offline
evaluation experiments on a large user logged dataset and online A/B testing
experiments on the Kuaishou app. Experimental results show that our proposed
framework outperforms the comparison baselines in recommendation quality and
is more suitable to promote both relevance and diversity for relevant
recommendation. Our contributions can be summarized as follows:

  * •

We formulate the task of relevant recommendation in a multi-factor ranking
paradigm, which jointly considers the user preference, relevance and diversity
for re-ranking.

  * •

We propose a feature disentanglement self-balanced re-ranking framework for
this task, which learns more compact aspects from rich item features and
conducts multi-aspect adaptive ranking to balance different factors.

  * •

We demonstrate the effectiveness of proposed framework through both offline
experiments and online $A/B$ test.

##  2\. Methodology

In this section, we present the proposed FDSB framework for relevant
recommendation. We first define the notations and formulate the task in
Section 2.1. Then we introduce the disentangled attention encoder in Section
2.2, which encodes multiple features into different representations. After
that, a feature-aware diversified recommendation algorithm is proposed in
Section 2.3. Finally, we discuss the proposed framework in Section 2.4.

###  2.1. Task Formulation

The task of _relevant recommendation_ aims to return a set of suitable items
to a target user given a trigger item. Since such a task is seldom considered
by previous work (Xie et al., 2021), we first formulate the task and present
an approximate greedy approach.

####  2.1.1. A Multi-Factor Ranking Formulation

We formulate the task of relevant recommendation as a multi-factor ranking
problem, where the preference score of users, the relevance to the trigger
item and the diversity of the returned recommended items are considered for a
balanced ranking. Formally, let $\mathcal{I}$ denote the full item candidate
set for recommendation, $i_{t}$ denote the trigger item and $u$ denote a
target user. This task can be described as an conditional subset selection
problem (Daszykowski et al., 2002) from the item set $\mathcal{I}$ as follows:

(1)
$\mathcal{R}=\arg\max_{\mathcal{I}{{}^{\prime}}\subset\mathcal{I}}\underbrace{\text{Pref}(u,\mathcal{I}{{}^{\prime}})}_{Preference}+\lambda\cdot\underbrace{\text{Rel}(\mathcal{I}{{}^{\prime}},i_{t})}_{Relevance}+\underbrace{\gamma\cdot\text{Div}(\mathcal{I}{{}^{\prime}}|i_{t})}_{Diversity},$

where $\mathcal{R}$ is the returned set of relevant recommended items to user
$u$ given the trigger item $i_{t}$ and $\mathcal{I}{{}^{\prime}}$ denotes the
selected items. Here, $\text{Pref}(\cdot)$ represents the preference score of
user $u$ to returned items, $\text{Rel}(\cdot)$ measures the relevance between
trigger item $i_{t}$ and $\mathcal{I}{{}^{\prime}}$, and $\text{Div}(\cdot)$
calculates the diversity of returned items in term of trigger item. We set two
hyper-parameters $\lambda$ and $\gamma$ to balance the weights of the three
factors.

####  2.1.2. Feature-aware Relevance and Diversity

In the above formulation, we can utilize existing recommendation methods to
compute the preference term $\text{Perf}(u,\mathcal{I}{{}^{\prime}})$. While,
the major difficulty lies in how to derive the latter two terms, _i.e.,_
_relevance_ and _diversity_ , which is the focus of this paper. To model these
two terms, we consider a feature-aware way, where we assume the side
information of items (_e.g.,_ brand, color and avatar) is available for
recommendation, denoted as a feature set $\mathcal{F}_{i}$ for item $i$.
Furthermore, we assume that a pre-trained item embedding, $\bm{e}_{i}$, can
also be used for re-ranking. In this setting, the terms of relevance and
diversity can be defined as:

(2) $\displaystyle\text{Rel}(\mathcal{I}^{\prime},i_{t})$ $\displaystyle=$
$\displaystyle\sum_{i^{\prime}\in\mathcal{I}^{\prime}}\text{sim}^{I}(\bm{e}_{i^{\prime}},\bm{e}_{i_{t}})+\text{sim}^{F}(\mathcal{F}_{i^{\prime}},\mathcal{F}_{i_{t}}),$
$\displaystyle\text{Div}(\mathcal{I}{{}^{\prime}}|i_{t})$ $\displaystyle=$
$\displaystyle\text{Div}^{I}(\\{\bm{e}_{i{{}^{\prime}}}|i{{}^{\prime}}\in\mathcal{I}{{}^{\prime}}\\})+\text{Div}^{F}(\\{\mathcal{F}_{i{{}^{\prime}}}\cap\mathcal{F}_{i_{t}}|i{{}^{\prime}}\in\mathcal{I}{{}^{\prime}}\\}),$

where $i{{}^{\prime}}\in\mathcal{I}{{}^{\prime}}$ is a candidate item to be
recommended, and the two factors are modeled at both item level (with the
superscript of $I$) and feature level (with the superscript of $F$). For
relevance, it is calculated according to the item embedding similarity
($\text{sim}^{I}(\cdot)$) and the feature similarly ($\text{sim}^{F}(\cdot)$).
Similarly, the diversity is computed according to embedding diversity
($\text{Div}^{I}(\cdot)$) and feature diversity ($\text{Div}^{F}(\cdot)$).
Most of existing works mainly focus on item-level relevance or diversity (Xie
et al., 2021; Huang et al., 2021). Here, we consider more fine-grained
characterization of the two factors at the feature level.

####  2.1.3. Greedy Approximation

Above, we have formulated the diversified re-ranking problem of relevant
recommendation as a combinatorial optimization problem based on item
embeddings and features in Eq. (1) and Eq. (2). However, this problem has been
proved to be NP-hard (Toth, 2000) for exact solutions. So, we turn to a greedy
selection approach (Nemhauser et al., 1978) that decomposes the set based
objective into a single-item objective according to the marginal gain as
follow:

(3)
$i_{l}=\arg\max_{i\in\mathcal{I}\backslash\mathcal{R}_{1:l-1}}\text{Pref}(u,i)+\lambda\cdot\text{Rel}(i,i_{t})+\gamma\cdot\text{Div}(\mathcal{R}_{1:l-1}\cup
i|i_{t}),$

where $l$ is the current selection step and we make the selection based on the
results of previous $l-1$ steps. Through such a greedy approximation, we can
easily compute the second term of relevance score, and the key is how to model
the third term of diversity, considering both item-level and feature-level
diversity. This is a major difference compared with previous works (Carbonell
and Goldstein, 1998; Chen et al., 2018; Huang et al., 2021) which mainly focus
on item-level diversity.

To address this difficulty, we design a novel relevant recommendation
framework, consisting of two major modules, namely _disentangled attention
encoder_ that disentangles the features into different aspects and _self-
balanced multi-aspect re-ranker_ that balances the diversity of multiple
feature aspects. In what follows, we describe the two parts in detail.

###  2.2. Disentangled Attention Encoder

In recommender systems, the item resources are usually attached with rich
feature information, which correspond to different latent aspects describing
the item characteristics from different perspectives. However, these features
are usually correlating or redundant. Considering this issue, we propose a
disentangled attention encoder (DAE) to disentangle these features into fine-
grained feature-aware representations corresponding to different aspects. This
module is the base to model the terms of relevance and diversity in our
framework. The overall architecture of DAE is depicted in Figure 2.

####  2.2.1. Attention-based Feature Disentanglement

For an item $i$, an embedding $\bm{e}_{i}\in\mathbb{R}^{d}$ (either pre-
trained or to be learned) is associated. For simplicity, we omit the index of
item $i$ in the following notations. Besides, it corresponds to a feature set
of side information, denoted as $\mathcal{F}=\\{f^{1},f^{2},\ldots,f^{F}\\}$.
Following the idea of disentangled representation learning (Tran et al.,
2017), we aim to learn multi-aspect disentangled representations over the
feature set, denoted as
$\bm{V}=\\{\bm{v}^{1},\bm{v}^{2},\ldots,\bm{v}^{A}\\}$, where $A$ is the
number of disentangled dimensions (_i.e.,_ aspects).

To learn the disentangled representations, instead of following existing
approaches (Ma et al., 2019; Mu et al., 2021), we adopt a simpler approach
suited to our setting based on the multi-head attention (Vaswani et al., 2017)
to conduct disentangled representation learning. The input query vectors and
key vectors are projected while the value vectors remain fixed across
different heads. Specifically, input features are firstly mapped to learnable
vectors through an embedding layer:

(4) $\bm{F}=\text{Emb}(\mathcal{F}),$

where $\bm{F}\in\mathbb{R}^{F\times d}$ is the initial embedding matrix of $F$
features. Then, item embeddings and feature embeddings are projected $A$ times
in parallel corresponding to $A$ aspect representations where the $a$-th
projected vector is produced by multi-layer perceptrons (MLP):

(5) $\displaystyle\bm{q}^{a}$ $\displaystyle=\text{MLP}_{Q}^{a}(\bm{e}_{i}),$
$\displaystyle\bm{P}^{a}$ $\displaystyle=\text{MLP}_{K}^{a}(\bm{F}).$

After that, we use the original feature embeddings as value and projected
vectors as query and key to calculate attention scores. The $a$-th aspect
representation is computed by scaled attention:

(6) $\bm{v}^{a}=\text{Attention}(\bm{q}^{a},\bm{P}^{a},\bm{F}).$

The weights of each feature in the attention block are calculated by the
scaled dot product of projected query vector and key matrix in the form of
$\text{Softmax}(\frac{\bm{q}^{a}\cdot\bm{P}^{a}}{\sqrt{d}})$.

####  2.2.2. Alignment with Item Representations

Since these aspects capture part of characteristics of items, their
representations should be closer to the overall item embedding in order to
reflect item-specific characteristics. Based on this idea, we design two
semantic alignment methods for distilling the information of item embedding
into aspect representations:

(7) $\displaystyle\mathcal{L}^{A}_{MSE}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\parallel\bm{e}_{i}-\widetilde{\bm{v}}\parallel_{2}^{2},$
(8) $\displaystyle\mathcal{L}^{A}_{InfoNCE}$ $\displaystyle=$
$\displaystyle-\log\frac{\exp(\bm{e}_{i}\cdot\widetilde{\bm{v}}/\tau)}{\sum_{i{{}^{\prime}}\in\mathcal{I}\backslash
i}\exp(\bm{e}_{i{{}^{\prime}}}\cdot\widetilde{\bm{v}}/\tau)},$

where $\widetilde{\bm{v}}=\frac{1}{A}\sum_{a=1}^{A}\bm{v}^{a}$ is the average
of the aspect representations. The first alignment loss is based on point-wise
Mean Squared Error (MSE) loss while the second alignment loss conducts the
InfoNCE loss (Oord et al., 2018) by treating other items in the same batch as
negative examples. Besides the alignment loss, we further require that the
disentangled aspects capture different semantic characteristics. Therefore, we
introduce the orthogonalization loss $\mathcal{L}^{O}$ which is conducted in a
similar contrastive manner:

(9)
$\mathcal{L}^{O}=\sum_{a=1}^{A}-\log\frac{\exp(\bm{v}^{a}\cdot\bm{v}^{a}/\tau)}{\sum_{a{{}^{\prime}}=1}^{A}\exp(\bm{v}^{a{{}^{\prime}}}\cdot\bm{v}^{a}/\tau)},$

where $\tau$ is a temperature parameter and $A$ is the number of disentangled
aspects representations. By optimizing the above two parts jointly, we can
efficiently disentangle rich feature information into compact feature-aware
representations.

Figure 2. Architecture of our proposed disentangled attention encoder (DAE),
which takes pre-trained item embedding and feature set as input and produces
multiple disentangled representations (_i.e.,_ aspects).

###  2.3. Self-balanced Multi-aspect Re-Ranker

A fundamental approach to diversity ranking is the _Maximal Marginal
Relevance_ (MMR) (Carbonell and Goldstein, 1998), and a number of studies have
been proposed to improve and generalize the MMR-based algorithm (Di Noia et
al., 2014; Ashkan et al., 2015). In our setting, we aim to capture the diverse
characteristics corresponding to different aspects (_i.e.,_ the disentangled
representations based on item features). For this purpose, we extend the
classic MMR framework and design a self-balanced multi-aspect re-ranker that
is able to adaptively tune the aspect weight for diversity re-ranking,
summarized in Algorithm 1.

####  2.3.1. Aspect-specific Adaptive Balance

In order to model relevance and diversity over different aspects, we introduce
learnable coefficient vectors $\bm{w}^{R}\in\mathbb{R}^{A}$ and
$\bm{w}^{D}\in\mathbb{R}^{A}$, to adaptively integrate different aspects. To
guide the update of combination vectors at each selection step, we calculate
the accumulated relevance (denoted as $\bm{r}\in\mathbb{R}^{A}$) before each
selection. Furthermore, let $r_{a}$ denote the similarity between the
proceeding selections $\mathcal{R}_{1:l-1}$ and the trigger item $i_{t}$ for
the $a$-th aspect. For multi-factor ranking, the major difficulty lies in the
balance between relevance and diversity, since the two factors are essentially
contradictive: the more relevant to the trigger item, the less diverse the
relevant recommendations are, and vice versa.

Intuitively, we should enhance the weight of relevance when diversity is well
captured, and reduce the weight of relevance when diversity is less well
captured. There should be a trade-off between the two factors. Since we
consider the balance at the aspect level, we design a relevance-diversity
relevant weighting mechanism that adaptively captures the balance as follows:

(10) $\displaystyle\bm{w}^{D}$ $\displaystyle=$
$\displaystyle\text{softmax}(\bm{r}),$ (11) $\displaystyle\bm{w}^{R}$
$\displaystyle=$ $\displaystyle\text{softmax}(\frac{1}{\bm{r}}),$

where $\frac{1}{\bm{r}}$ denotes the vector by performing the element-wise
reciprocal computation for $\bm{r}$, and each entry in $\bm{r}$ is computed by
$r_{a}=\sum_{i\in\mathcal{R}}\bm{v}_{i}^{a}\cdot\bm{v}_{i_{t}}^{a}$ where
$\bm{v}_{i}^{a}$ is the disentangled representation of item $i$ in $a$-th
aspect which is calculated by Eq. (6). By adaptively updating the coefficients
of multiple aspects, the selection process can be conducted in an aspect-
specific adaptive way.

####  2.3.2. Greedy Selection

We firstly consider the selection of the first item $i_{1}$. In this simple
case, since there are no selected items (_i.e.,_
$\mathcal{R}_{1:l-1}=\emptyset$), it degenerates into the single-item
relevance score as:

(12)
$i_{1}=\arg\max_{i\in\mathcal{I}}\text{Pref}(u,i)+\lambda\big{(}\bm{e}_{i}\cdot\bm{e}_{i_{t}}+\sum_{a=1}^{A}\bm{v}^{a}_{i}\cdot\bm{v}^{a}_{i_{t}}\big{)},$

where we compute the relevance score in terms of item embeddings and
representations for $A$ disentangled aspects. When $l>1$, the overall
objective of relevance and diversity are determined by the selected items, and
the self-balanced coefficients are utilized to control the balance in multiple
aspects. The relevance scores of each aspect between candidate item $i$ and
trigger item $i_{t}$ are weighted by the aspect-specific coefficients as:

(13)
$\text{Rel}(i,i_{t})=\bm{e}_{i}\cdot\bm{e}_{i_{t}}+\sum_{a=1}^{A}w^{R}_{a}\big{(}\bm{v}^{a}_{i}\cdot\bm{v}^{a}_{i_{t}}\big{)}.$

Correspondingly, the diversity scores from multiple aspects can be also
weighed as follows:

(14) $\text{Div}(\mathcal{R}_{1:l-1}\cup
i|i_{t})=1-\max_{i{{}^{\prime}}\in\mathcal{R}_{1:l-1}}\bigg{(}\bm{e}_{i{{}^{\prime}}}\cdot\bm{e}_{i}+\sum_{a=1}^{A}w^{D}_{a}\big{(}\bm{v}_{i^{\prime}}^{a}\cdot\bm{v}_{i}^{a}\big{)}\bigg{)}.$

In this diversity function, we subtract the maximal similarity between
selected items and candidate item as the diversity score which follows the MMR
approach (Carbonell and Goldstein, 1998). While, other choices of diversity
functions can be easily adapted, which will be discussed in the following.
After defining the relevance and diversity scores, the $l$-th item can be
selected according to Eq. (3).

Input: preference scores $\\{\text{Pref}(u,i)\\}_{i=1}^{N}$ of user $u$ on $N$
items, original item representations $\\{\bm{c}_{i}\\}_{i=1}^{N}$,
disentangled aspects representations
$\\{\\{\bm{v}_{i}^{a}\\}_{a=1}^{A}\\}_{i=1}^{N}$, trigger item $i_{t}$, trade-
off weights$\\{\lambda,\gamma\\}$, output sequence length $L$;

Output: recommended item sequence
$\mathcal{R}=\\{i_{1},i_{2},\ldots,i_{L}\\}$;

1 Initialize: $l=1$, accumulated relevance $\bm{r}=\bm{0}$ ;

2 Select first item $i_{1}$ by Eq. (12);

3

4while  _$l <L$_ do

5 for  _$a=1\rightarrow A$_ do

6 $\bm{r}_{a}=\bm{r}_{a}+\Braket{\bm{v}_{i_{l}}^{a},\bm{v}_{i_{t}}^{a}}$ ; //
accumulated relevance

7

8 end for

9 $l++$;

10 Update diversity weights $\bm{w}^{D}$ by Eq. (10);

11 Update relevance weights $\bm{w}^{R}$ by Eq. (11);

12 for  _$i\in I\backslash\mathcal{R}_{1:l}$_ do

13 Calculate relevance scores $\text{Rel}(i,i_{t})$ by Eq. (13);

14 Calculate diversity scores $\text{Div}(\mathcal{R}_{1:l-1}\cup i|i_{t})$ by
Eq. (14);

15

16 end for

17 Select $i_{l}$ by Eq. (3);

18

19 end while

20return $\mathcal{R}=\\{i_{1},i_{2},\ldots,i_{L}\\}$

Algorithm 1 The algorithm for self-balanced multi-aspect re-ranking

###  2.4. Discussion

In this part, we make some discussions about the framework flexibility and
complexity analysis.

Framework Flexibility. Although we design the framework for relevant
recommendation, it is easy to adapt it to other recommendation scenarios.
Actually, our framework presents a general way to balance multiple factors
over the disentangled aspects. By zooming into our definition of relevance and
diversity scores (see Eq. (13) and Eq. (14)), we can see that it only involves
the inner product operation between item embeddings. Such a design is
essentially model-agnostic, and any embedding-based recommendation models can
be easily fit into our framework. As we do not introduce additional constraint
on user/item embeddings, our framework can be also implemented with different
diversity algorithms. In this work, we adopt the classic MMR approach
(Carbonell and Goldstein, 1998) as the backbone algorithm for diversity re-
ranking, which can be replaced by Determinantal Point Process (DPP) (Chen et
al., 2018) and Gram-Schmidt Process (GSP) (Huang et al., 2021). In our study,
we test the performance of our framework with other diversity ranking methods
(_e.g.,_ Fast-DPP (Chen et al., 2018) (a DPP variant) and SSD (Huang et al.,
2021) (a GSP variant), but observe no significant improvement over MMR. Due to
the simplicity, we adopt MMR as the diversity ranking approach.

Complexity Analysis. The proposed self-balanced multi-aspect re-ranking
algorithm iteratively selects items from candidates according to the
similarity with selected items. Formally, it has comparable complexity with
vanilla MMR (Carbonell and Goldstein, 1998) where the time complexity can be
roughly estimated as $\mathcal{O}\big{(}L^{2}NAd\big{)}$ and space complexity
is $\mathcal{O}\big{(}NAd\big{)}$ where $N$ is the number of candidate items
and $L$ is the number of recommended items. The additional costs are mainly
attributed to the involving of the disentangled representations. To further
reduce the costs, we can offline compute the embedding similarity (_e.g.,_
using an efficient vector recall algorithm MIPS (Shrivastava and Li, 2014))
and store the top similar embeddings via ID-based index. In this way, the time
and space complexity can be reduced to $\mathcal{O}\big{(}LNAd\big{)}$ and
$\mathcal{O}\big{(}NAd+LN\big{)}$ respectively. To sum up, our framework has
comparable complexity with those lightweight diversity re-ranking algorithms
(_e.g.,_ Fast-DPP (Chen et al., 2018) and SSD (Huang et al., 2021)) when $M\ll
L\ll N$.

##  3\. Experiments

In this section, we first introduce the dataset which is collected from real
relevant micro-video recommendation scenario in Section 3.1. Then, the
implementation details and experimental settings are presented in Section
3.2.1 for reproducibility. Afterwards, the evaluation results of the proposed
FDSB on both collected dataset and online $A/B$ test are shown in Section 3.2
and 3.4 respectively. Finally, we conduct detailed analysis and case studies
in Section 3.3 to further verify the effectiveness of proposed components.

###  3.1. Datasets

To the best of our knowledge, there is no available dataset customized for the
relevant recommendation task. Therefore, we construct a new dataset, from
KuaiShou app. We randomly sample 0.3 million active users and trace their logs
in the _Relevant Videos_ page for 24 hours. We group the interaction data by
user and trigger item to form different sessions, which consist of clicked
items that both satisfy the user preferences and keep relevant to the trigger
item. After filtering the items that are clicked fewer than 5 times, we
collect nearly 4M interactions with 0.25M items and 0.6M sessions. In order to
capture feature-aware characteristics of items, we collect the tags of each
item as features. Besides, we initialize the embeddings of users and items
with pre-trained parameters which are produced by the traditional
collaborative filtering (CF) model with large-scale training data to alleviate
data sparsity problem. The detailed statistics of our collected dataset are
shown in Table 1.

Table 1. Statistics of our dataset. #user | #item | #interaction | #session | #tag | $\frac{\\#(\text{item},\text{tag})}{\\#\text{item}}$
---|---|---|---|---|---
369,329 | 241,143 | 4,016,182 | 648,777 | 97,840 | 9.1

###  3.2. Offline Evaluation

To verify the effectiveness of the proposed FDSB in re-ranking, we first
conduct offline experiments on the collected dataset.

####  3.2.1. Experimental Details

In the experiments, the number of embedding dimension is set to 128, which is
the same as the hidden dimension of disentangled aspect representations. We
employ a two-layer fully connected network with ReLU activation as our
projection layer in DAE, where the hidden size is set to 128 and 64
respectively. The hyper-parameter, $\tau$, is set to 0.1 and we train the DAE
module for 100 epochs with learning rate of 1e-4. The number of disentangled
aspects is set to 5 and we conduct a grid search for all the trade-off weights
in [0.001, 0.01, 0.1, 1], then report the best overall performance for all the
compared methods. We use the dot product between the pre-trained user
embeddings and item embedding as preference scores and relevance scores are
computed from item embeddings. Then, the initial ranking is determined by the
weighted sum of these two scores, and the weights are carefully tuned
according to the recommendation performance.

####  3.2.2. Comparison Methods

We compare our proposed framework with several baselines. The first one is a
Relevance-aware Ranking (RR) algorithm (Pang et al., 2017), which jointly
considers the user-item preference scores and item-item similarity scores for
ranking. As we implement the diversity scoring function following MMR
(Carbonell and Goldstein, 1998), we select it as the main competitor which
incorporates coarse-grained item-level diversity through re-ranking.
Furthermore, we also select an efficient DPP-based algorithm, FastDPP (Chen et
al., 2018), and a more recent diversified ranking algorithm, SSD (Huang et
al., 2021), as our compared methods. To explicitly verify the effectiveness of
re-ranking and avoid sampling bias (Krichene and Rendle, 2020), we treat the
whole item set as candidates for re-ranking (Zhao et al., 2020). Owing to the
unaffordable time and space costs, we omit the comparison with other DPP-based
algorithms (Wilhelm et al., 2018; Gartrell et al., 2017).

####  3.2.3. Evaluation Metrics

In addition to recommendation accuracy, our work also focuses on relevance and
diversity. Specifically, we employ six metrics to comprehensively evaluate
recommendation performance from different facets. Firstly, we exploit two
commonly used metrics, Recall and Mean Reciprocal Rank (MRR) to measure the
accuracy. Then, we introduce a feature-aware relevance metric, Mean Feature
Hit Ratio (MFHR), to represent the relevance between recommended items and the
trigger item:

(15)
$MFHR=\frac{1}{|\mathcal{R}|}\sum_{i\in\mathcal{R}}\frac{|\mathcal{F}_{i}\cap\mathcal{F}_{i_{t}}|}{|\mathcal{F}_{i_{t}}|},$

where $\mathcal{R}$ denotes the recommended item set and $\mathcal{F}_{i}$ is
the feature set of item $i$. Feature set of the trigger item is indicated as
$\mathcal{F}_{i_{t}}$. This metric calculates the average hit ratio of
features which can be seen as an indicator of relevance. To evaluate the
feature-aware diversity of recommended items, we modify two diversity metrics,
Feature Coverage Ratio (FCR) and Intra List Average Distance (ILAD) to fit the
task of relevant recommendation (similar metrics have been widely used by
previous works (Chen et al., 2018; Wu et al., 2019b)) as follows:

(16) $\displaystyle FCR=$
$\displaystyle\frac{|\bigcup\limits_{i\in\mathcal{R}}\mathcal{F}_{i}\cap\mathcal{F}_{i_{t}}|}{|\mathcal{F}_{i_{t}}|},$
(17) $\displaystyle ILAD=$ $\displaystyle\frac{\sum_{i,j\in\mathcal{R},i\neq
j}1-s(i,j)}{L(L-1)},$

where
$s(i,j)=\frac{|\mathcal{F}_{i}\cap\mathcal{F}_{j}\cap\mathcal{F}_{i_{t}}|}{|\mathcal{F}_{i}\cup\mathcal{F}_{j}\cap\mathcal{F}_{i_{t}}|}$
denotes the overlap ratio of features between item $i$ and $j$. Note that we
only consider the overlapped features between recommended items and trigger
item which shows the particularity of our task compared with conventional
diversified recommendation task. Finally, we calculate the F-score over Recall
and MFHR to find the balance between accuracy and relevance. For computational
efficiency, we calculate metrics over the top-20 items after re-ranking.

####  3.2.4. Experimental Results

Table 2. Results of offline experiments Methods | Accuracy | Relevance | Diversity | F-score
---|---|---|---|---
Recall | MRR | MFHR | FCR | ILAD
RR | 0.0894 | 0.0624 | 0.3511 | 0.9507 | 0.4222 | 0.1425
FastDPP | 0.0856 | 0.0601 | 0.3497 | 0.9501 | 0.4269 | 0.1375
SSD | 0.0856 | 0.0586 | 0.3460 | 0.9493 | 0.4317 | 0.1373
MMR | 0.1436 | 0.0818 | 0.3781 | 0.9968 | 0.4496 | 0.2082
FDSB | 0.1798 | 0.0968 | 0.3936 | 0.9989 | 0.4927 | 0.2468
Improv. | +25.2% | +18.3% | +4.1% | +0.32% | +9.6% | +18.5%

The offline evaluation results of different methods on our industrial dataset
are shown in Table 2. Based on the results, we can observe that:

(1) For those baselines that consider diversity, the performance of DPP and
SSD is close to RR. A potential reason is they both assume that more
orthogonality of the recommended items embeddings would result in larger
diversity in the recommendation results. When the recommended items are highly
relevant to the target item, the diversity score of these item embeddings will
become relatively small compared to the relevance score. Therefore, it is
difficult for those algorithms to effectively balance relevance and diversity
in recommendation results. In contrast, MMR outperforms the simple method RR
in both accuracy and diversity, which indicates that using the dot product of
embeddings as diversity score is more suitable than using orthogonality when
item embeddings are similar in relevant recommendation. (2) By comparing our
framework with all the baselines, it is clear to see that FDSB outperforms all
baseline methods on all metrics. Different from baselines, our framework
focuses on capturing fine-grained feature-aware relationships among items. As
for the diversity metrics, the proposed FDSB performs better than baselines by
a large margin on ILAD metric. These findings are consistent with our
assumption that it is insufficient to only consider item-level diversity when
candidate items are very similar in relevant recommendation. Finally, the
F-score is also significantly improved, which further verifies that our
framework can capture fine-grained preference characteristics of users for
making personalized recommendations.

(3) Moreover, the improvement on both relevance and diversity indicates that
our self-balanced multi-aspect re-ranker is able to adaptively adjust the
weights of two targets in each aspect. These results also show that two
proposed modules (_i.e.,_ DAE and re-ranker) can coordinate well for improving
the overall recommendation.

###  3.3. Further Analysis of FDSB

In this section, we conduct more detailed analysis experiments to demonstrate
the effectiveness of FDSB .

####  3.3.1. Ablation Study

To improve the performance of the relevant recommendation task, our proposed
FDSB has incorporated several technical components. Next, we examine how each
of them affects the final performance. Specifically, we consider the following
variants of our framework for comparison through offline experiments:

  * •

$\neg Disen$: the variant replaces the disentangled component with average
pooling operation over feature embeddings, which produces a composite feature
representation. In this variant, we only utilize this composite representation
to promote feature-aware relevance and diversity.

  * •

$\neg SB$: the variant removes the self-balancing (SB) strategy from the
multi-aspect re-ranking algorithm.

  * •

$\neg Div$: the variant removes the fine-grained diversity target and only
considers the relevance factor for re-ranking.

  * •

$\neg Rel$: this variant is opposite to the third case which ignores the
relevance target and only maximizes the diversity score.

Table 3. Performance comparison of different variants. Variants | Recall | MFHR | ILAD
---|---|---|---
FDSB | 0.1798 | 0.3936 | 0.4927
$\neg Disen$ | 0.1581 | 0.4035 | 0.4221
$\neg SB$ | 0.1534 | 0.4209 | 0.3938
$\neg Div$ | 0.1823 | 0.4219 | 0.4279
$\neg Rel$ | 0.1388 | 0.3395 | 0.5365

From the results in Table 3, we have the following observations. Firstly,
replacing disentanglement aspect representations yields a large performance
drop on both recall and diversity metrics. One possible reason is that simply
introducing features into diversified re-ranking is unable to capture fine-
grained feature interactions, thus resulting in sub-optimal performance.
Secondly, removing the self-balancing strategy would achieve a high relevance
score but lead to a decrease in diversity score. It indicates that manual
weights are difficult to effectively balance the diversity of multiple
aspects, which further verifies the necessity of our proposed self-balancing
strategy. Finally, relevance and diversity are two vital factors that should
be considered jointly. The two variants (_i.e.,_ $\neg Div$ and $\neg Rel$)
that only focus on one of the two factors achieve worse performance on the
other factor. These results indicate that our proposed framework which jointly
consider both relevance and diversity is effective to improve the overall
performance.

Figure 3. Performance on relevance and diversity w.r.t. different hyper-
parameters. The left shows the results when varying $\lambda$ and the right
shows the results when varying $\gamma$.

####  3.3.2. Performance Tuning

Since our framework utilizes two hyper-parameters $\lambda$ and $\gamma$
(parameters in Eq. (3)) to balance the relevance and diversity, we continue to
examine the effect of these two parameters. As we can see from Figure 3, as
$\lambda$ becomes larger, MFHR and Recall increase, and ILAD decreases in the
meantime. While, the increasing tuning of $\gamma$ lead to opposite results.
This observation demonstrates the importance of the relevance factor in
relevant recommendation which directly determines the final recommendation
performance (_i.e.,_ Recall). Moreover, compared to relevance, diversity is
more sensitive to the choice of hyper-parameters. A possible explanation is
that our proposed FDSB is better at depicting fine-grained diversity, so that
it is more capable in obtaining high diversity with less loss on accuracy.

####  3.3.3. Case Study

In this part, we present one case for illustrating how our framework
disentangles item features into different aspects. We randomly sample a video
and visualize the attention weights of each aspect in Figure 4. As we can see,
features are disentangled into different aspects according to their semantics.
The first aspect focuses on abstract features like “ _Animal_ ” and “ _Wild
Animal_ ”. The second aspect captures fine-grained features like “ _Panda_ ”
and “ _Giant Panda_ ”. The third aspect extracts uncommon features where “
_Metal Eater_ ” is a nickname of panda. Note that the tag “ _Pets_ ” that is
somehow redundant to this video has not been attended by all aspects. This
example qualitatively illustrates the effectiveness of our proposed DAE in
learning disentangled feature representations.

Figure 4. An example of the feature disentanglement. We plot six features and three disentangled aspects. Two primary features of each aspect are shown with attention weights. Table 4. Results of online A/B tests on KuaiShou app. | Watch Time | Video Play | Watched Tags | Very Relevant
---|---|---|---|---
FDSB | +0.293% | +1.169% | +4.656% | +0.189%
Figure 5. The workflow of our algorithm in relevant recommendation system.

###  3.4. Online Deployment and $A/B$ Test

To further verify the effectiveness of FDSB, we deploy it on the function of “
_more relevant videos_ ” on KuaiShou app for online $A/B$ test. The entire
workflow is shown in Figure 5.

We utilize the tags of videos as the item features, which are consistent with
the offline experiments. To demonstrate the superiority of our framework in
feature-aware relevance and diversity, we compare our framework with a
baseline, MMR, to show the relative improvements.

We evaluate the performance of the proposed framework in real application
scenario of Kuaishou, with three metrics considering both user engagement and
recommendation diversity: (1) _Watch Time_ , (2) _Video Play_ , and (3)
_Watched Video Tags_. The first two metrics reflect the users’ satisfaction,
and the third metric is commonly used for diversity. Furthermore, to examine
the relevance between recommended items and trigger item, we randomly deliver
a questionnaire to a small population of users during the recommendation
service. The users are required to rating on relevance according to a 3-point
Likert scale 222https://en.wikipedia.org/wiki/Likert_scale, _i.e.,_ the fourth
metric. We report the proportion that users assign a 3-point rating, which
means “Very Relevant”. The $A/B$ test is conducted for seven consecutive days
and we report the average performance on these four metrics.

From Table 4, we can have the following observations. Firstly, FDSB achieves
significant improvements in both _Watch Time_ and _Video Play_ , indicating
that our framework is able to increase user loyalty (more per-video watching
time and more video plays, _i.e.,_ watching videos). Secondly, the number of
tags for watched videos also largely increases, which suggests we promote the
diversity in the recommendation results. Thirdly, more users respond with “
_Very Relevant_ ” in our questionnaires. This result shows that FDSB can
recommend more relevant videos and further explains why FDSB has longer _Watch
Time_ in the relevant recommendation. Overall, by incorporating item features
and self-balanced strategy, both the diversity in recommendation results and
the relevance between recommended items and trigger item are enhanced, showing
that our framework has made a good trade-off between relevance and diversity
in the relevant recommendation task.

##  4\. Related work

In this section, we review related works on relevant recommendation, aspect-
aware recommendation and diversified recommendation.

Relevant Recommendation. To the best of our knowledge, there are few works
that focus on the relevant recommendation (Xie et al., 2021). In (Xie et al.,
2021), the authors proposed a framework named R3S to jointly rank relevant
items and decide whether to display the items. They designed a multi-critic
multi-gate mixture-of-experts strategy to jointly model the information of
user, trigger and context. While this method can not effectively distinguish
features of different aspects, which is important to promote the diversity for
relevant recommendation. Moreover, this work has not explicitly model the
recommendation diversity. Besides, R3S is an end-to-end model that requires
more labeled data to optimize, while our FDSB is a lightweight re-ranking
algorithm which is easy to optimize and deploy. Furthermore, it is flexible to
integrate our method with existing methods such as R3S for relevant
recommendation. For example, our approach can take the predictions of R3S as
the preference scores for making more accurate recommendations.

Aspect-aware Recommendation. Several studies have been proposed to extract
aspects from side information to guide the recommendation (Guan et al., 2019;
Cheng et al., 2018). Most of existing works design aspect-aware architecture
to understand text (_e.g.,_ reviews and item description), which is able to
model users and items from different aspects for fine-grained matching. For
example, Guan et al. (2019) proposed a well-designed aspect-level attention
module to select related description from user and item reviews. As a recently
proposed technique, disentangled representation learning (Tran et al., 2017;
Yang et al., 2021) has been widely applied to extract latent semantic
dimensions from data in a variety of fields, _e.g.,_ recommender systems
(Zheng et al., 2021b; Fan et al., 2021; Wu et al., 2021). Most of the previous
works learn disentangled representations with graph-based model (Chen et al.,
2021; Zheng et al., 2021b; Mu et al., 2021) (either based on interaction graph
or knowledge graph) or self-attention model (Fan et al., 2021). Furthermore,
to alleviate the data bias in recommendation, an additive causal model was
proposed to learn disentangled representations of interest and conformity
(Zheng et al., 2021b). In the task of sequential recommendation, LightSANs
(Fan et al., 2021) introduced decomposed self-attention and decoupled position
encoding for context-aware representations. Different from existing works, we
consider learning feature-level aspects for modeling relevance and diversity
in a specific setting of relevant recommendation.

Diversified Recommendation. Recently, the diversity of recommendations has
received much attention in the research community (Wu et al., 2019a), which is
proved to be a NP-hard problem (Toth, 2000). Several works conducted the
diversified recommendation as an optimal selection problem and utilized post-
processing (Carbonell and Goldstein, 1998; Chen et al., 2018; Antikacioglu and
Ravi, 2017) or adversarial sampling (Sun et al., 2020; Zheng et al., 2021a; Ye
et al., 2021) for enhancing the diversity. Overall, there are two mainstream
categories of diversified recommendation, namely aggregation diversity (Ge et
al., 2010; Zhang et al., 2021) and individual diversity (Cheng et al., 2017;
Sha et al., 2016). In this work, we focus on the individual diversity of
relevant recommendation. Recently, determinantal point process (DPP)
(Gillenwater et al., 2014) has been widely used as a principled approach to
conducting diversified recommendation. To tackle the problem of high
complexity in these methods, a sampling algorithm was proposed based on eigen-
decomposition (Gillenwater, 2014) for approximation. Furthermore, SSD (Huang
et al., 2021) leveraged the Gram-Schmidt process to reduce the space
complexity. Actually, diversity was considered in the field of information
retrieval at a much earlier time, and a classical work MMR (Carbonell and
Goldstein, 1998) proposed to greedily select docs based on the similarity with
both the query and the currently selected docs, inspiring a number of
extension works (Qin and Zhu, 2013; Sha et al., 2016). Different from the
above approaches, our work is the first to consider both diversity and
relevance in re-ranking stage for relevant recommendation. Although we
currently adopt the MMR approach for modeling the diversity, FDSB is a model-
agnostic framework and can be instantiated with other diversity methods (Chen
et al., 2018; Huang et al., 2021).

##  5\. Conclusion and Future work

In this work, we study the task of relevant recommendation from a real
application scenario, where relevance and diversity are two important factors
that need to be considered by recommenders. In order to better characterize
item-to-item relations and capture feature-aware diversity, we propose a re-
ranking framework, FDSB, based on a multi-factor ranking approach. There are
two modules, namely disentangled attention encoder (DAE) and self-balanced
multi-aspect re-ranker in FDSB. In the DAE, we learn disentangled
representations from rich item features. Then we adaptively balance the
relevance and diversity with a multi-aspect re-ranker. Both offline evaluation
and online $A/B$ test have demonstrated the effectiveness of the proposed
framework.

For future work, we will study how to capture both relevance and diversity in
an end-to-end ranking model for relevant recommendation. Besides, we will also
consider modeling feature-aware fine-grained diversity in other recommendation
tasks or scenarios.

###### Acknowledgements.

This work was partially supported by National Natural Science Foundation of
China under Grant No. 61872369, Beijing Natural Science Foundation under Grant
No. 4222027, and Beijing Outstanding Young Scientist Program under Grant No.
BJJWZYJH012019100020098. This work was also partially supported by Beijing
Academy of Artificial Intelligence (BAAI). Xin Zhao is the corresponding
author.

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