Telangana Social Welfare Residential Degree College for Women (TSWRDC-W),
Nizamabad, 503002, Telangana, India. Department of Physics, National Institute
of Technology-Warangal, Telangana, India

# Lattice instability, anharmonicity and Raman spectra of BaO under high
pressure: A first principles study

K. Lavanya  Jawaharlal Nehru Technological University (JNTU), Hyderabad,
500085, Telangana, India.  N. Yedukondalu<EMAIL_ADDRESS>Department of
Geosciences, State University of New York, Stony Brook, New York 11794-2100,
USA  S. C. Rakesh Roshan  Rajiv Gandhi University of Knowledge Technologies,
Basar, Telangana-504107, India  Shweta D. Dabhi  Department of Physical
Sciences, P D Patel Institute of Applied Sciences, Charotar University of
Science and Technology, CHARUSAT Campus, Changa, 388421, Gujarat, India
Suresh Sripada  Department of Physics, JNTUHCEJ, 505501 Karimnagar, Telangana,
India  M.Sainath<EMAIL_ADDRESS>Department of Physics, IcfaiTech, IFHE,
Hyderabad 501203, Telangana, India.  Lars Ehm  Department of Geosciences,
State University of New York, Stony Brook, New York 11794-2100, USA  John B.
Parise  Department of Geosciences, State University of New York, Stony Brook,
New York 11794-2100, USA

###### Abstract

Alkaline-earth metal oxides, in particular MgO and CaO dominate Earth’s lower
mantle, therefore, exploring high pressure behavior of this class of compounds
is of significant geophysical research interest. Among the alkaline-earth
metal oxides, BaO exhibits rich polymorphism in the pressure range of 0-1.5
Mbar. Static enthalpy calculations revealed that BaO undergoes a pressure
induced structural phase transition from NaCl-type (B1) $\rightarrow$ NiAs-
type (B8) $\rightarrow$ distorted CsCl-type (d-B2) $\rightarrow$ CsCl-type
(B2) at 5.1, 19.5, 120 GPa respectively. B1 $\rightarrow$ B8 $\&$ B8
$\rightarrow$ d-B2 transitions are found to be first order in nature whereas
d-B2$\rightarrow$ B2 is a second order or weak first order phase transition
with displacive nature. Interestingly, d-B2 phase shows stability over a wide
pressure range of $\sim$19.5-113 GPa. Mechanical and dynamical stabilities of
ambient and high pressure phases are demonstrated through the computed second
order elastic constants and phonon dispersion curves, respectively. Under high
pressure, significant phonon softening and soft phonon mode along M-direction
are observed for B8, d-B2 and B2 phases, respectively. Pressure dependent
Raman spectra suggests a phase transition from d-B2 to Raman inactive phase
under high pressure. Overall, the present study provides a comprehensive
understanding of lattice dynamics and underlying mechanisms behind pressure
induced structural phase transitions in BaO.

Keywords:- High pressure, Phase transitions, Elastic constants, Lattice
dynamics, Phonon softening, Lattice instability, Anharmonicity, Earth’s
interior

##  1 Introduction

Alkaline-earth metal oxides, MO (M = Mg, Ca, Sr, Ba) have gained a
considerable attention due to their technological applications in diverse
fields,1, 2, 3, 4, 5 ranging from catalysis 6, 7, 8 to electrode coating 9 and
interfaces 10. These metal oxides are suitable candidates for CO2 sorbent
applications due to their high absorption capacity at moderate temperatures.3
Moreover, these mono-oxides are prime candidates for the Earth’s interior,
especially light metal oxides such as MgO and CaO are abundant in the Earth’s
lower mantle, in which the pressure varies from 24 to 136 GPa from top of the
upper mantle to the lower mantle.11, 12, 13, 14, 15 Therefore, exploring high
pressure and/or temperature behavior of these metal oxides provides an insight
on the deep interior of the Earth. These metal oxides crystallize in the
rocksalt NaCl-type (B1) structure with space group $Fm\bar{3}m$ at ambient
conditions. Upon compression, these metal oxides usually undergo a body
centred cubic CsCl-type (B2) structure and it is well established for MgO, CaO
and SrO. Amongst all these compounds, BaO has interesting structural
properties and it exhibits a rich polymorphism in the pressure range of 0-1.5
Mbar over the remaining alkaline earth oxides, CaO,16, 17, 18, 19, 20, 21, 22,
23, 24 SrO.16, 17, 18, 19, 25, 26, 27, 28, 29 Therefore, understanding the
high pressure structural and lattice dynamical behavior of BaO is crucial as
this can be a possible candidate of the Earth’s mantle12. Several high
pressure X-ray diffraction (HP-XRD) measurements30, 29, 23, 31, 32, 33, 34, 35
and theoretical studies using first principles calculations were focused on
exploring the structural phase transitions36, 37, 38, 27, 18, 21, 25, 39,
metallization 40, 41, 42, 43, 44, 20, 45, 46, elastic properties 47, 48, 49,
50, thermodynamic stability 25, 51, 52 of MO (M = Mg, Ca, Sr, Ba) compounds
under high pressure. Similarly, efforts have been put forward by the
researchers to explore the phase diagram of BaO under high pressure from both
experimental and theoretical perspective. A HP-XRD study30 revealed that BaO
undergoes a structural transition from B1 to distorted B2 (d-B2) at 9.2 GPa
and remains in the same structure till 29 GPa. Energy dispersive X-ray
diffraction study 31 disclosed that BaO transforms from B1 to an intermediate
hexagonal NiAs-type (B8) structure at around 10 GPa, later to a tetragonal
PH4I-type (d-B2) structure with pseudo 8-fold coordination at around 15 GPa
and this phase approaches to B2 structure with increasing pressure above 1
Mbar. First principles studies53, 17, 54 reported that BaO transforms from B1
to B8 phase at 11.3 GPa,53 5 GPa,17 4.3 GPa54 then to d-B2 from B8 at 21.5
GPa,53 13 GPa,17 23.2 GPa,54 upon further compression, the d-B2 phase
transforms to B2 phase at 62 GPa,53 50 GPa,17 108 GPa.54 Pandey et al 49 and
Jog et al 55 suggested a direct B1 to B2 phase transition at 100 and 86 GPa,
respectively without any intermediate high pressure phases.

On the other hand, lattice dynamics of B1 phase is well studied at ambient as
well as at high pressure using first principles calculations56, 16, 57, 51 and
inelastic neutron scattering technique.58 However, very limited studies52 are
available exploring lattice dynamics of high pressure phases, underlying
mechanisms behind the observed structural phase transitions and spectroscopic
properties under high pressure in BaO. Therefore, in the present study, we
have systematically investigated the structural phase transitions, lattice
dynamics of different polymorphs and Raman spectra of B8 and d-B2 phases of
BaO under high pressure.

##  2 Computational details

All the first principles calculations were carried out using projected
augmented wave (PAW) method as implemented in Vienna ab-initio simulation
package (VASP)59. Electron-electron interactions are treated within
generalized gradient approximation (GGA) with Perdew–Burke-Ernzerhof (PBE)
functional while PAW pseudopotentials are used to capture the electron-ion
interactions. The plane wave basis orbitals 5s25p66s2 for Ba
(Ba$\\_$sv$\\_$GW) and 2s22p4 for O (O$\\_$GW) are used as valence electrons.
Plane wave cutoff energy and k-spacing are tested against the total energy to
ensure the convergence. The plane wave cutoff energy was set to 600 eV and a
k-spacing of 2$\pi$ $\times$ 0.024 $\AA^{-1}$ with $\Gamma$-centered
Monkhorst-Pack scheme to compute structural and mechanical properties at
ambient as well as at high pressure. To compute phonon dispersion curves,
forces are calculated using finite displacement method for ambient and high
pressure phases and then force constants are extracted and then post-processed
by using phonopy package. 60 Pressure dependent Raman spectra is calculated
using norm conserving pseudo potentials within linear response approach as
implemented in CASTEP package61. Crystal structures are visualized and
analyzed using the VESTA software62. To include anharmonic effects, we also
have calculated the finite temperature lattice dynamics for B2 phase of BaO
using ab-initio molecular dynamics (AIMD) simulations and then post-processed
using temperature dependent effective potential method (TDEP).63

##  3 Results and Discussion

###  3.1 Crystal structure and pressure-induced structural phase transitions

As shown in Figure 1a, BaO crystallizes in the face centred cubic (SG:
$Fm\bar{3}m$) rocksalt NaCl-type (B1) structure with 6-fold coordination and
having Z = 4 formula units (f.u.) per unit cell at ambient conditions.31 High
pressure X-ray diffraction measurements64, 30, 31 revealed that BaO exhibits
rich polymorphism under high pressure. BaO undergoes to a high pressure NiAs-
type (B8) structure (SG: $P6_{3}/mmc$) from B1 phase at 9.2 GPa without
changing the coordination number (CN) by having Z = 2 f.u. per primitive cell.
The crystal structure of B8 phase is an hexagonal analog of the B1 structure.
Later, the B8 phase transforms to PH4-type distorted CsCl-type (d-B2) at 14.0
GPa,30 18 GPa30 with pseudo 8-fold coordination (SG: $P4/nmm$) with Z = 2 f.u.
per primitive cell, and then to a 8-fold coordinated CsCl-type (B2) structure
(SG: $Pm\bar{3}m$) with Z = 1 f.u. per primitive cell upon further compression
$i.e.,$ above 1 Mbar pressure (see Figure 1 and Table 1).64, 30, 31 The
crystal structure of d-B2 phase is analogous to PbO-type structure without a
lone-pair at Ba atom having two in-equivalent bond lengths as shown in Figure
1c. To get further insights on the observed structural phase transitions and
their sequence, as a first step, we have tested with various DFT functionals
such as LDA, PBE, PW91 and SCAN to reproduce the B1 to B8 transition pressure
(see Figure S1). The transition pressure obtained using PBE, PW91 and SCAN
functionals are consistent with each other while LDA poorly describe the phase
stability between B1 and B8 phases. Later, we carried out static enthalpy
calculations with PBE functional for the polymorphic phases of BaO and plotted
enthalpy difference with respect to B2 phase as a function of pressure as
shown in Figure 2a. The obtained phase transition sequence is B1 $\rightarrow$
B8 $\rightarrow$ d-B2 $\rightarrow$ B2 and the corresponding transition
pressures 5.1, 19.5, 120 GPa are in good agreement with the HP-XRD
measurements30, 31 as well as with previous first principles calculations53,
52. Moreover, TlI-type (B33) phase is found to be meta-stable for CaO65 and we
also have included this B33 phase in the phase diagram of BaO to verify it’s
relative phase stability w.r.t high pressure phases of BaO and then we found
that the B33 phase is also meta-stable for BaO over the maximum studied
pressure range (see Figure 2a). As presented in Figure 2b, the enthalphy
difference between d-B2 and B2 phases is continuously decreasing with
increasing pressure and they become iso-enthalpic at around 120 GPa. As shown
in Figure 2c, the calculated lattice constants for the ambient and high
pressure phases are monotonically decreasing with pressure. The lattice
constant ’c’ of d-B2 phase becomes lattice constant ’a’ for B2 phase (cd-B2 =
aB2). The detailed transition mechanism for d-B2 to B2 transition is explained
through elastic constants in the following section. As shown in Figure 2d, the
calculated volume decreases monotonically with pressure. We observe a volume
collapse during the phase transition from B1 $\rightarrow$ B8 is $\sim$ 3.65
$\%$ at 5.1 GPa and it is $\sim$ 6.4 $\%$ at 19.5 GPa for the transition B8
$\rightarrow$ d-B2. The obtained volume reduction is consistent with high
pressure X-diffraction measurements and previous first principles calculations
as presented in Table 2. This clearly demonstrates that B1 to B8 $\&$ B8 to
d-B2 transitions are first order in nature, while d-B2 to B2 transition is a
second order or weak first order in nature due to continuous change in volume
during the phase transition (see Figure 2d). We then calculated the
equilibrium bulk modulus (B0) and its pressure derivative (B0’) by fitting the
obtained pressure-volume (P-V) data to 3rd order Birch-Murnaghan equation of
state,66 B0 measures the ability of a material to resist changes in volume
under hydrostatic expansion or compression. The calculated B0 value is found
to be 71.4 GPa for the ambient B1 phase, which is in good agreement with
previous first principles calculations.67 The obtained relatively low B0
values 68.8 GPa for B8 and 51.8 for d-B2 intermediate phases suggest the soft
nature of these phases over ambient B1 (71.4 GPa) and a high B0 value 92.4 GPa
for high pressure B2 phase disclose that B2 phase is harder than B1 phase (see
Table 3).

In addition, the calculated in-equivalent bond lengths are plotted as a
function of pressure and the same are presented in Figure 2e. The in-
equivalent bond lengths of d-B2 phase are found to converge to became a single
bond length for B2 phase above 120 GPa. In addition, in the d-B2 phase, the Ba
atoms are located at (0.5, 0.0, 0.5+$\Delta$) and (0.0, 0.5, 0.5-$\Delta$),
where $\Delta$ is the atomic position parameter, $\Delta$ in combination axial
ratio (c/a) provides a useful information on d-B2 $\rightarrow$ B2 transition.
If $\Delta$ = 0 $\&$ c/a = 1/$\sqrt{2}$ = 0.7071, then the structure is of
undistorted B2 phase.31 To confirm this, we have calculated c/a ratio and
$\Delta$ as a function of pressure for d-B2 phase. As shown in Figure 2f, both
c/a ratio and $\Delta$ are decreasing with pressure and then approaches to
$\Delta$ = 0 $\&$ c/a = 0.7071 around 120 GPa. The continuous volume change at
the transition pressure, variation in the fractional coordinate of Ba atom
along c-direction (see Figure S2), c/a = 0.7071 and $\Delta$ = 0 values
clearly demonstrate that d-B2 to B2 transition to be a displacive transition
under pressure. Displacive nature of $z_{Ba}$ with pressure suggests phonon
softening in d-B2 phase as function of pressure. Further to explore the
microscopic origin of the continuous phase transition in BaO, pressure
dependent elastic constants, lattice dynamics and Raman spectra are calculated
and analyzed them in detail in the following sections.

###  3.2 Elastic constants under high pressure

To get deeper insight on the structural phase transition(s) and verify
mechanical stability of the BaO polymorphs, we have calculated second order
elastic constants (Cij) as a function of pressure for all the four B1, B8,
d-B2 and B2 phases. The cubic ($Fm\bar{3}m$, $Pm\bar{3}m$), hexagonal
($P6_{3}/mmc$) and tetragonal ($P4/nmm$) crystal symmetries possess 3, 5 and 6
independent elastic constants.

When a finite hydrostatic pressure is applied to a crystal system, the Born
stability criteria68 at ambient conditions must be revised with modified
elastic constants at a given hydrostatic pressure conditions. According to
Sinko and Smirnov the modified elastic constants at a given pressure (P) are
$\widetilde{C}_{ii}$ = $C_{ii}$ \- P, (for $i$ = 1-6) and $\widetilde{C}_{1j}$
= $C_{1j}$ \+ P (for $j$ = 2-3) for cubic and hexagonal/tetragonal systems.
Therefore, the modified Born stability criteria under hydrostatic pressure for
cubic (equation 1) and hexagonal/tetragonal (equation 2) are given as follows:

$C_{11}-C_{12}-2P>0,\hskip 14.45377ptC_{11}+2C_{12}+P>0,\hskip
14.45377ptC_{44}-P>0$ (1) $\begin{split}C_{11}-C_{12}-2P>0,\\\
P^{2}+P(C_{11}+C_{12}+4C_{13})+2C^{2}_{13}-C_{33}(C_{11}+C_{12})>0,\\\
C_{44}-P>0,\hskip 14.45377ptC_{66}-P>0\end{split}$ (2)

The calculated pressure dependent Cij for B1, B8, d-B2 and B2 phases satisfy
the necessary and sufficient conditions for the revised Born stability
criteria under hydrostatic pressure indicating that the polymoprhs are
mechanically stable at the corresponding pressures. We shed more light on
understanding lattice transformation from d-B2 to B2 phase through the
computed pressure dependent elastic constants of d-B2 and B2 phases. The
relationships between lattice constants of d-B2 and B2 phases at the phase
transition are given as follows: (a=b)d-B2 = $\sqrt{2}c_{d-B2}$ and cd-B2 =
aB2 (see Figure 3a), thus, results in merging of the following elastic
constants of d-B2 to B2 phase under pressure are as depicted in Figure 3 as
given below: C${}^{d-B2}_{33}$ $\rightarrow$ C${}^{B2}_{11}$,
C${}^{d-B2}_{44}$ $\rightarrow$ C${}^{B2}_{44}$ and C${}^{d-B2}_{13}$
$\rightarrow$ C${}^{B2}_{12}$, which corroborates with pressure dependent
structural and bond parameters at the d-B2 $\rightarrow$ B2 transition.

##  4 Lattice dynamics and Raman spectra under high pressure

To compliment the structural stability obtained from total energy calculations
and verify dynamical stability of the ambient and high pressure phases of BaO,
we calculated phonon dispersion curves including LO-TO splitting at 0 and 9
GPa for B1 phase (see Figure 4a $\&$ b) at 9 GPa and 20 GPa for B8 phase (see
Figure 4c $\&$ d), at 30, 60, 90 and 120 GPa for d-B2 phase (see Figure 5) and
finally at 100, 110, 120 and 140 GPa for B2 phase (see Figure 6). As shown in
Figure 4a $\&$ b, B1 and B8 phases are dynamically stable at 0, 9 GPa and 9,
20 GPa respectively. The phonon dispersion curves of B1 phase show large LO-TO
splitting ($\sim$ 312.3 cm-1) along $\Gamma$-direction with two degenerate TO
and one LO modes and this large LO-TO splitting causes a fall of low lying
optical (LLO) phonon modes into acoustic mode region, which can increase
overlap between acoustic and LLO phonon modes there by enhancing the
scattering phase space and eventually this leads to lower the lattice thermal
conductivity in B1 phase at ambient conditions.69 However, the LO-TO splitting
decreases under pressure (see Figure 4a $\&$ b) for B1 phase. Moreover, the
optical phonon modes are hardening with pressure while phonon softening is
observed for acoustic phonon modes along X-direction in B1 phase and along M
and K-$\Gamma$ directions in B8 phase. Phonon softening is observed along
X-direction for B1 phase previously,52 which actually drives the transition
from B1 $\rightarrow$ B8 phase in BaO under pressure. In addition, the B1 and
B8 phases are dynamically stable above the transition pressures, which are in
consistent with the first order nature of the structural phase transitions, B1
$\rightarrow$ B8 $\&$ B8 $\rightarrow$ d-B2. The d-B2 phase is found to be
dynamically stable over a wide pressure range of $\sim$ 20-120 GPa (see Figure
5) for BaO in contrast to the light alkaline-earth metal oxides, MgO, CaO and
SrO. The calculated phonon dispersion curves show a significant phonon
softening at 120 GPa along Z-$\Gamma$ and ’A’ high symmetry directions as
depicted in Figure 5d. On the other hand, the calculated phonon dispersion
curves for B2 phase at 100 and 110 GPa show soft phonon mode along M high
symmetry point, which clearly indicates that B2 is unstable below 110 GPa
pressure. In addition, we could also observe a significant phonon softening of
TO and LO modes of B2 phase at 140 GPa (see Figure 6d) compared to 120 GPa
(see Figure 6b), which indicates that B2 phase becomes dynamically unstable
and it may undergo a structural phase transition upon further compression
above 150 GPa.

The cubic ambient B1 and high pressure B2 phases of BaO are Raman inactive,
and they have only one IR active (T1u) mode. However, the intermediate high
pressure B8 and d-B2 phases are Raman active, and are thoroughly analyzed (see
Figure 7). This result is in very good agreement with the Raman spectroscopic
measurements,70 which show that BaO is Raman inactive until 10 GPa (B1 phase)
and above 11 GPa it becomes Raman active (B8), where one Raman active E2g mode
is observed and this Raman mode shows blue-shift with increasing pressure.
Later, significant changes are observed in the Raman spectra at around 14 GPa
and it indicates a transition from B8 to to d-B2 phase. The similar trends are
observed in the present work with deviations in the transition pressures which
might be due to Raman measurements were carried out at 300 K, while the
calculations are performed at 0 K. Since, B8 and d-B2 phases crystallize in
$P6_{3}/mmc$ and $P4/nmm$ space groups, respectively and they consist of four
atoms per primitive cell, the symmetry decomposition of group theory predicts
12 vibrational modes along $\Gamma$ for both of these phases, which are given
as follows:

$\Gamma_{12}^{B8}$ = 2E2g $\oplus$ B2g $\oplus$ 4E1u $\oplus$ 2E2u $\oplus$
2A2u $\oplus$ B1u

Among the 12 vibrational modes of B8 phase, only E2g is Raman active mode
which is consistent with high pressure Raman measurements,70 E1u, E2u, A2u and
B1u are IR active modes but B2g is neither Raman nor IR active (silent) mode;
where E1u, E2u and E2g are doubly degenerate modes. Variation of E2g mode as a
function of pressure is shown in Figure 7a, which is mainly due to in-plane
vibrational motion of Ba atom and its intensity decreases with pressure.

$\Gamma_{12}^{d-B2}$ = 4Eg $\oplus$ B2g $\oplus$ A1g $\oplus$ 4Eu $\oplus$
2A2u

On the other hand, for Litharge PbO-type d-B2 phase, the zone-center Raman
active modes B2g, A1g and Eg are polarized along c- and a/b-axis,
respectively. Eu and A2u are IR active modes; where Eu and Eg modes are doubly
degenerate. Pressure dependence of vibrational frequency modes are presented
in Figure 7d. For each vibrational mode, frequency increases with pressure
except for A1g mode. When the pressure is applied, the interatomic distance
becomes smaller and hence interaction between neighboring atoms becomes
stronger. Therefore, usually, the vibrational modes increase with pressure as
shown in Figure 7b $\&$ d except for A1g mode. As Eg mode is in plane
vibrational mode, the vibrational frequency/wavenumber increases as the
interaction of atoms becomes stronger. As shown in Figure 2f; c/a ratio
decreases with increasing pressure. Due to the out-of-plane vibrations of
heavier Ba atoms in A1g mode (Figure 7c), the decrease in the wavenumber of
Raman-shift is observed with pressure. The red-shift of the A1g mode with
pressure shows a negative pressure coefficient and it might be responsible for
introducing dynamical instability at higher pressure in the d-B2 phase. In
contrast, this mode was tentatively assigned as Eg mode due to in-plane
vibration of Ba atoms in the HP-Raman study.70. However, we thoroughly
analyzed and assigned it as A1g mode rather than Eg mode, which actually shows
phonon softening with pressure which is associated with displacive nature of
Ba atoms along c-axis. The Intensity of four Raman active modes (Eg, B2g and
A1g) are gradually diminishing with pressure (Figure 7b) for d-B2 phase, this
strongly suggests that high pressure phase might be Raman inactive. Finally,
the high pressure B2 phase can be stabilized above 113 GPa (see Figure 7e)
which is consistent with the observations from the HP-XRD measurements.31.

As shown in Figure S3 $\&$ 6, the calculated room temperature phonon
dispersion curves in combination with AIMD and TDEP disclose that B2 phase is
dynamically unstable at 80 GPa and 300 K whereas it becomes dynamically stable
at 100 GPa and 300 K, while harmonic lattice dynamical calculations show that
B2 phase is dynamically unstable below $\sim$ 113 GPa. This clearly shows the
importance of anharmonicity to determine the stability of B2 phase above 85
GPa and this is in good accord with the HP-XRD diffraction measurements.31

##  5 Conclusions

To summarize, we have systematically investigated the polymorphism of BaO
under hydrostatic compression using first principles calculations based on
density functional theory. We found that BaO undergoes series of pressure-
induced structural phase transitions from B1 $\rightarrow$ B8 $\rightarrow$
d-B2 $\rightarrow$ B2 and the corresponding transition pressures are 5.1 ,
19.5, 120 GPa respectively. The obtained structural phase transition sequence
and transition pressures are consistent with HP-XRD measurements and previous
first principles calculations. TlI-type (B33) phase is found to be metastable
for BaO over the studied pressure range. The d-B2 phase exhibits stability
over a wide pressure range $\sim$ 19.5-113 GPa in contrast to light alkaline-
earth metal oxides. The transitions from B1 $\rightarrow$ B8 and B8
$\rightarrow$ d-B2 are first order in nature with a volume collapse of $\sim$
3.65 and 6.4 $\%$, respectively while d-B2$\rightarrow$ B2 is a second order
or weak first order phase transition with displacive nature. Mechanical
stability has been established for ambient and high pressure phases of BaO
through computed elastic constants. The pressure dependent elastic constants
provide an insight on lattice transformation from d-B2 to B2 phase. The
calculated phonon dispersion curves indicate the dynamical stability of BaO
polymorphs and a significant phonon softening is observed for B8 and d-B2
phases at high pressures and also a soft phonon mode (along M-direction) in B2
phase explained its dynamical instability below 110 GPa. Finally, gradual
diminishing of Raman active modes of d-B2 phase under pressure suggests that
high pressure phase could be a Raman inactive. Overall, the present study
provides an in depth understanding of structural phase transitions in BaO
under high pressure and further exploration of high pressure behavior of light
alkaline-earth oxides (MgO, CaO and SrO) have important implications on
composition of the Earth’s lower mantle.

##  6 Acknowledgments

LV and NYK contributed equally to this manuscript. NYK would like to thank
Science and Engineering Research Board and Indo-US Scientific Technology Forum
for providing financial support through SERB Indo-US postdoctoral fellowship
and Institute for Advanced Computational Science, Stony Brook University for
providing computational resources (Seawulf cluster). SCRR would like thank
RGUKT Basar for providing computational facilities.

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Figure 1: Crystal structure of (a) ambient rocksalt NaCl-type (B1) and high
pressure (b) NiAs-type (B8), (c) PH4-type distorted CsCl-type (d-B2) and (d)
CsCl-type (B2) phases of BaO. Where SG and CN denote space group and
coordination number, respectively. Green and red color balls represent Ba and
O atoms, respectively. Figure 2: Calculated (a,b) relative enthalpy difference
of B1, B8, B33, d-B2 phases w.r.t B2 phase, (c) lattice constants, (d) volume,
(e) bond lengths for B1, B8, DB2 and B2 phases as a function of pressure and
(f) variation of c/a ratio and $\Delta$ with pressure for d-B2 phase. Figure
3: (a) Lattice transformation from d-B2 to B2 and (b) calculated elastic
constants of d-B2 and B2 phases of BaO as a function of pressure.

Figure 4: Calculated phonon dispersion curves of (a, b) B1 phase at (a) 0, (b)
9 GPa and for (c, d) B8 phase at (c) 9 GPa, (d) 20 GPa. Phonon softening is
observed along X-direction for B1 phase and for B8 phase at high pressure (20
GPa) along M and K-$\Gamma$ directions, which destabilize the B1 and B8 phases
upon further compression.

Figure 5: Calculated phonon dispersion curves for d-B2 phase at (a) 30, (b)
60, (c) 90 and (d) 120 GPa. Optical phonon softening along Z-$\Gamma$
direction and acoustic phonon softening along A-direction are observed with
pressure and it is predominant above 90 GPa.

Figure 6: Calculated phonon dispersion curves for B2 phase at high pressures
(a) 100 GPa, (b) 110 GPa (c) 120 GPa and (d) 140 GPa at 0K as well as at 300
K. B2 phase is dynamically stable including anharmonic effects at 100 and 110
GPa at 300 K but it is dynamically unstable within the harmonic approximation
at 0 K. Figure 7: Calculated Raman spectra of (a) B8 and (b) d-B2 phase as a
function of pressure. Eigen vectors of Raman active modes are presented for
(a) B8 and (c) d-B2 phases of BaO. Pressure evolution of (d) Raman and IR
active modes of d-B2 phase and (e) Transverse acoustic mode of B2 phase along
M-high symmetry direction without and with inclusion of anharmonic effects.

Table 1: Calculated equilibrium lattice constants (a, c in Å) of BaO at
ambient and high pressure and are compared with the available X-ray
diffraction data and other first principles calculations.

Phase | Pressure | This work | Expt. | Others
---|---|---|---|---
B1 | 0 | a=5.533 | 5.520a | 5.562b, 5.454b ,5.614c,
|  | - | - | 5.604d, 5.552e,5.462f
B8 | 13.9 | a= 3.612,c= 6.365 | a= 3.617,c= 6.349 g | -
d-B2 | 60.5 | a= 4.187,c= 3.007 | a= 4.1,c= 2.998 g | -
B2 | 130 | a=2.8 | - | -

aRef.71 bRef. 72 cRef.51 dRef.67 eRef. 47 f Ref.19 g Ref.31

Table 2: Calculated phase transition pressure (Pt, in GPa) and volume collapse
(VC in $\%$) of BaO and are compared with the available experimental data and
other first principles calculations.

Parameter | Transition | This Work | Expt. | Others
---|---|---|---|---
Pt | B1 $\rightarrow$ B8 | 5.1 | 9.2a,10 b | 5 c,4.3 d
| B8 $\rightarrow$ d-B2 | 19.5 | 18a,18.8 b | 13 c,23.2d
VC | B1 $\rightarrow$ B8 | 3.65 | 5a | 3.6e,1.5f
| B8 $\rightarrow$ d-B2 | 6.4 | 7a | 8.7e

a Ref. 30 b Ref.31 c Ref.17 d Ref.54 eRef.53 fRef.52

Table 3: Calculated equilibrium bulk modulus (B0) and its pressure derivative
(B0’) of ambient and high pressure phases of BaO and are compared with the
available experimental data and previous first principles calculations.

Phase | Pressure range | B0 (B0’) | Expt. | Others
---|---|---|---|---
B1 | 0-9 | 71.4 (4.35) | 69 a ,66.2 b , 61c | 68.912 d (4.09d), 73e (4.3e),
|  | - | 74.1b, 74.06h (5.07h) | 71.22 f (4.52f),75.7 g,
|  |  |  | 62.75 i, 71.72 j ,73 k
B8 | 6-30 | 68.6(4.32) | - | 68.04f (4.19f),72.7g
d-B2 | 20-120 | 51.3 (4.67) | 33.2 b (6.02b), | 27.13 f ,41.8g
B2 | 70-150 | 90.6 (4.15) | - | 74.69f (4.07f), 72.57 d (3.7d),
|  |  |  | 86 l,59.1g

a Ref.30 b Ref.31 c Ref.73 d Ref.19 eRef. 47 fRef.53 g Ref.54 hRef. 74 iRef.
49 jRef. 75 k Ref.55 l Ref. 76


# Exploiting Data Hierarchy as a New Modality for Contrastive Learning

Arjun Bhalla
M.Eng. Computer Science
<EMAIL_ADDRESS>
&Daniel Levenson11footnotemark: 1
M.S. Health Tech
<EMAIL_ADDRESS>
&Jan Bernhard11footnotemark: 1
M.Eng. Computer Science
<EMAIL_ADDRESS>
Anton Abilov
M.S. Connective Media
<EMAIL_ADDRESS>
These authors equally contributed to this work.Dropped Deep Learning course
mid-semester. Contributed to the brainstorming and data preprocessing.

###### Abstract

This work investigates how hierarchically structured data can help neural
networks learn conceptual representations of cathedrals. The underlying
WikiScenes dataset111We would like to thank Elor et al. [6] for granting us
access to the WikiScenes dataset pre-release. provides a spatially organized
hierarchical structure of cathedral components. We propose a novel
hierarchical contrastive training approach that leverages a triplet margin
loss to represent the data’s spatial hierarchy in the encoder’s latent space.
As such, the proposed approach investigates if the dataset structure provides
valuable information for self-supervised learning. We apply t-SNE to visualize
the resultant latent space and evaluate the proposed approach by comparing it
with other dataset-specific contrastive learning methods using a common
downstream classification task. The proposed method outperforms the comparable
weakly-supervised and baseline methods. Our findings suggest that dataset
structure is a valuable modality for weakly-supervised learning.

Figure 1: Images of the Berliner Dom used to demonstrate the hierarchical
structure of the WikiScenes dataset.

##  1 Introduction

An active open-question in Machine Learning research is how to learn useful
representations from unstructured or loosely structured data. Data is often
loosely structured in hierarchies: these data relate to each other, but don’t
have in a consistent structure. We investigate if there is value in including
hierarchical structure as input into a weakly-supervised learning problem.

In order to perform this investigation, we leverage the novel WikiScenes
dataset[6] which contains hierarchical representations of cathedrals, as
illustrated by Figure 1. While cathedrals alone are a narrow concept, the
dataset is expected to extend to other conceptual categories included on
WikiMedia222https://www.wikimedia.org. Moreover, we expect our contributions
to apply to other conceptual categories, independently of WikiMedia. Our
approach builds upon recent breakthroughs of learning multi-modal neural
networks using contrastive learning [16, 20]. However, instead of using text
as an additional modality for a contrastive loss, we relate images by using
their relative location in the hierarchy. We introduce a training procedure
that trains a neural network ordered by the data’s hierarchical structure and
demonstrate the utility of the learned latent space.

##  2 Related Work

In 2013, Bengio et al. [2] presented a list of strengths of representation
learning. One characteristic of good representations is that they embed the
hierarchical structure of the data, even from a weakly-supervised task.
Additionally, good representations naturally form clusters in the learned
latent space [2]. Given our objective to learn characteristic representations
of cathedrals from a hierarchical dataset, applying a representation learning
approach is an obvious fit for the learning task. Yet, learning these
representations is not trivial.

Contrastive learning, which is part of the representation learning landscape,
has shown to learn powerful representations in a series of recent works [11,
4, 3, 1, 12, 16, 8]. These approaches typically learn a latent representation
of unlabeled data in which similar samples cluster together [7]. Many of them
have only relied on a single modality [11, 4, 3, 1, 12, 8] by augmenting
samples to generate similar pairs for the learning process. Dropping the data
augmentation component, Radford et al. [16]’s multimodal approach outperformed
the previous start-of-the-art self-supervised contrastive model [4], and even
matched the performance of supervised methods on ImageNet. Recent work by
Salesforce Research [14] has focused on using hierarchical structural as input
into a contrastive learning process by using an Expectation-Maximization
objective to discover structure. For our approach, the hierarchy of the
dataset provides an additional modality to inform the otherwise purely image-
based self-supervised learning process. We apply an approach similar to the
contrastive losses of weakly-supervised approaches and use the signal from the
hierarchy as additional modality instead of relying on data augmentations.

Due to the incremental nature of our proposed hierarchical training procedure,
we need to consider issue of catastrophic forgetting [15]. To address this
issue, we experiment with revisiting the data of all layers repeatedly. This
approach is similar to memory replay, which has been beneficial when
considered to address catastrophic forgetting [9].

##  3 Dataset

###  3.1 WikiScenes Dataset

The WikiScenes dataset consists of data from 99 cathedrals. Images can be
associated with a leaf component of the hierarchy of the cathedral (e.g
“Stained-glass windows in the Cathedral of St. John the Divine”) or with a
parent component that itself is broken down into sub-components (e.g.
“Interior of the Cathedral of St. John the Divine”). Figures 1 and 2(a)
illustrate the hierarchical structure of the dataset. The hierarchy of
information for each cathedral corresponds with the hierarchy of "concepts"
encoded in Wikimedia333Example Wikimedia concept: ”Views from Notre-Dame de
Paris” - https://commons.wikimedia.org/wiki/Category:Views_from_Notre-
Dame_de_Paris. Notably, images are not mutually exclusive to a particular
concept in the hierarchy: an image for the altar concept might appear in the
images for the interior concept. The WikiScenes dataset includes three data
modalities: image, image description (text) and 3D image correspondence. Our
work focuses on the image modality and the hierarchical modeling of the
dataset.

###  3.2 Contrastive Training Dataset Construction

Elor et al.’s method focused on contrastive training using only examples that
are labeled with a small set of "distilled concepts" [6]. Evaluating the
impact of hierarchical training required a larger dataset that spans multiple
hierarchy levels. To explore the ability of our approach to generalize to
other cathedrals, we consider datasets of multiple sizes: (1) Small: Only
images of Notre-Dame ($\sim$23K images), (2) Medium: Images of Notre-Damn +
six other cathedrals ($\sim$40K images), (3) Large: Images of all cathedrals
($\sim$104K images). In order to maintain a rigorous evaluation approach, we
construct datasets for this hierarchical contrastive pre-training which do not
include any examples from the pre-defined image classification validation set.

(a) Dataset structure and contrastive sampling

(b) Images of descendent nodes contribute to their parent node’s image pool.

Figure 2: Visualizations of node structure underlying the hierarchical
learning procedure.

##  4 Method

###  4.1 Hierarchical Training Algorithm

####  4.1.1 Training Data Sampling

We sample training data triplets by leveraging WikiScene’s tree structure. A
node represents a directory in the WikiScenes dataset, e.g. Exterior, which
holds a set of images directly associated to the concept represented by the
dataset folder. The edges between nodes represent folder and sub-folder
relationships. A node’s image pool could be confined to only the images of its
corresponding dataset directory and not any of its sub-directories. However,
preliminary experiments showed that training on these image pools results in
latent space collapse. We mitigate this problem by creating a node’s image
pool from all images in its corresponding dataset directory and sub-
directories, as shown in Figure 2(b), and revisiting a larger subset of images
on consecutive hierarchy levels.

####  4.1.2 Contrastive Loss

Our data sampling technique allows us to sample triplets of anchor, positive
and negative samples given a node, its position in the dataset, and its image
pool. Given this triplet, we can train a neural network using a triplet margin
loss [19]. We sample images first by node and then by image to balance against
nodes with an unbalanced distribution of images. As presented in Equation 1,
the triplet margin loss operates on three image samples: an anchor sample
$(\mathbf{x}_{a})$, a positive example, $(\mathbf{x}_{p})$, from the same
image pool as the anchor, and a negative example $(\mathbf{x}_{n})$ from a
node that shares the same parent as the anchor node.

$\mathcal{L}(\mathbf{x}_{a},\mathbf{x}_{p},\mathbf{x}_{n};f)=max(0,\|f(\mathbf{x}_{a})-f(\mathbf{x}_{p})\|^{2}_{2}-\|f(\mathbf{x}_{a})-f(\mathbf{x}_{n})\|^{2}_{2}+\alpha)$
(1)

To compute the feature difference between the samples, the model, $f$, maps
the input images onto three corresponding feature vectors. The margin
hyperparameter $\alpha$ is tied to the current hierarchy level $h$, as shown
in Equation 2.

$\alpha(h)=(h_{max}-h)^{2}+\alpha_{min}$ (2)

####  4.1.3 Level-Specific Triplet Margin

The calculation of $\alpha$ requires the specification of $h_{max}$ and
$\alpha_{min}$, which represent the deepest hierarchy level to consider and
the minimum triplet margin at that deepest hierarchy level, respectively. By
defining a level-specific triplet margin, each hierarchy level is evaluated
with its corresponding unique loss function.

This approach aims to enforce large separation of high level concepts in their
latent space representations and separate leaf concepts to a lesser extent:
when $h$ is small compared to $h_{max}$, the difference is large and therefore
the margin hyperparameter $\alpha$ is large, increasing loss and latent space
separation on convergence. The goal of this approach is to learn strong
representations by mapping the relationships between parent and child nodes
onto a model’s latent space. For the top level of the hierarchy, the model
segments the latent space into three distinct classes: interior, exterior, and
views. After stepping down the hierarchy once, the model segments each of
these three concepts further into sub-classes, depending on the level and
number of classes at that level. This pattern repeats until the training
process reaches the predefined $h_{max}$ or concepts leaf nodes which do not
have sub-classes. By reducing the triplet margin when moving from super- to
sub-class, the sub-classes of a given super-class can separate while also
maintaining a larger margin to the sub-classes of another super-class.

To keep the margin between super-classes after stepping to lower, sub-class
hierarchy levels, it is important to occasionally revisit (replay) previous
hierarchy layers. For that purpose, we introduced a replay regularization
hyper-parameter $r_{p}\in[0.0,1.0]$, which specifies the probability that a
current batch is drawn from the current hierarchy level or any of the previous
hierarchy levels.

###  4.2 Model Selection

When exploring model architectures for the hierarchical training procedure, we
focused on common image encoder models pretrained on ImageNet [5], such as
ResNet50 [10] and VGG-16 [17]. Preliminary experiments indicated that the
fine-tuning of the convolutional layers resulted in poor performance.
Consequently, we focus on the VGG-16, which allows us to freeze the
convolutional layers and fine-tune the first and second fully connected layers
of its classification module.

###  4.3 Evaluation

We evaluate the effectiveness of our novel hierarchical training approach in
two ways: quantitatively via an image classification task and qualitatively
via visualization of the latent space produced using the t-SNE [18] method.

####  4.3.1 Quantitative Evaluation

We evaluate the learned representations’ quality by performing a supervised
classification task on images from the WikiScenes dataset. To evaluate the
learned representations, we freeze the pre-trained encoder networks, and train
a single layer softmax classifier for the 10-class supervised single-class
classification problem. We train the softmax layer with the same hyper-
parameters for all models using the Adam optimizer [13] and cross-entropy
loss, batch size of 64, four epochs, and a learning rate of 0.001. Our
baseline models for this evaluation are an ImageNet-pretrained VGG-16 model
with no contrastive pre-training and a VGG-16 model with contrastive pre-
training over just Wikiscenes leaf nodes. We directly compare our approach
with Elor et al. [6]’s "Towers of Babel" encoder, which is a multi-modal
contrastively trained ResNet-50 model [10]. We then compare these baseline and
comparison models against our best performing hierarchical-constrastively
pretrained model.

For consistency with Elor et al. [6], we report on the precision of the
classification against the validation set averaged over classes, averaged over
all examples, and broken out by each individual concept. The precision
averaged over all classes, $mAP$, gives weight to concepts less represented in
the validation set, and the precision averaged over all examples, $mAP^{*}$,
gives every sample equal weight independent of its class.

####  4.3.2 Qualitative Evaluation

In order to investigate our hypothesis surrounding segmentation of the latent
space, we perform a qualitative evaluation by using t-SNE [18] to reduce the
feature maps from the encoder models to two dimensions and visualize them.
This is done by first using Principal Component Analysis to reduce the
dimensionality from 4096 to 50, and then further reduce this to two dimensions
using t-SNE. We do this in order to provide a more numerically stable baseline
on which to run t-SNE (as it gave worse results if reducing directly) and
reduce noise. We then observe how different conceptual clusters form in the
latent space. Thus, we are able to both qualitatively compare different models
to explain the quantitative results, and use these visualisations to further
understand what is happening during the training process.

##  5 Results and Discussion

This section presents the results of the various experiments. First, we
present the downstream classification performance of our hierarchically
trained, baseline, and comparison models. Second, we visualize t-SNE
projections of the models’ encoded features. Third, we compare the effect of
hierarchy depth, dataset size, and replay frequency on the training outcome in
an ablation study.

###  5.1 Downstream Classification Task

Model | mAP | mAP∗ | Facade | Window | Chapel | Organ | Nave | Tower | Choir | Portal | Altar | Statue
---|---|---|---|---|---|---|---|---|---|---|---|---
ImageNet VGG-16 |  $6$plus1.5minus4plus23plus2minus2 |  $6$plus1.5minus4plus23plus2minus2 |  $6$plus1.5minus4plus23plus2minus2 |  $6$plus1.5minus4plus23plus2minus2 |  $6$plus1.5minus4plus23plus2minus2 |  $6$plus1.5minus4plus23plus2minus256.7 |  $6$plus1.5minus4plus23plus2minus262.2 |  $6$plus1.5minus4plus23plus2minus2 |  $6$plus1.5minus4plus23plus2minus256.4 |  $6$plus1.5minus4plus23plus2minus2 |  $6$plus1.5minus4plus23plus2minus2 |  $6$plus1.5minus4plus23plus2minus2
Contrastive VGG-16 |  $6$plus1.5minus4plus23plus2minus2 |  $6$plus1.5minus4plus23plus2minus2 |  $6$plus1.5minus4plus23plus2minus2 |  $6$plus1.5minus4plus23plus2minus294.7 |  $6$plus1.5minus4plus23plus2minus220.7 |  $6$plus1.5minus4plus23plus2minus2 |  $6$plus1.5minus4plus23plus2minus2 |  $6$plus1.5minus4plus23plus2minus2 |  $6$plus1.5minus4plus23plus2minus2 |  $6$plus1.5minus4plus23plus2minus2 |  $6$plus1.5minus4plus23plus2minus2 |  $6$plus1.5minus4plus23plus2minus230.5
Towers Of Babel [6] |  $6$plus1.5minus4plus23plus2minus2 |  $6$plus1.5minus4plus23plus2minus2 |  $6$plus1.5minus4plus23plus2minus291.6 |  $6$plus1.5minus4plus23plus2minus2 |  $6$plus1.5minus4plus23plus2minus2 |  $6$plus1.5minus4plus23plus2minus2 |  $6$plus1.5minus4plus23plus2minus2 |  $6$plus1.5minus4plus23plus2minus2 |  $6$plus1.5minus4plus23plus2minus2 |  $6$plus1.5minus4plus23plus2minus2 |  $6$plus1.5minus4plus23plus2minus228.2 |  $6$plus1.5minus4plus23plus2minus2
Ours |  $6$plus1.5minus4plus23plus2minus241.6 |  $6$plus1.5minus4plus23plus2minus251.9 |  $6$plus1.5minus4plus23plus2minus2 |  $6$plus1.5minus4plus23plus2minus2 |  $6$plus1.5minus4plus23plus2minus2 |  $6$plus1.5minus4plus23plus2minus2 |  $6$plus1.5minus4plus23plus2minus2 |  $6$plus1.5minus4plus23plus2minus211.9 |  $6$plus1.5minus4plus23plus2minus2 |  $6$plus1.5minus4plus23plus2minus215.6 |  $6$plus1.5minus4plus23plus2minus2 |  $6$plus1.5minus4plus23plus2minus2
Table 1: Classification Performance. The table above shows mean average
precision (mAP), and per distilled concept average precision. mAP* is averaged
over all images, not per concept.

These results are comparable to WS-U results in Elor et al. [6]. Our best
performing model is contrastively pretrained with a small set of cathedrals
that are not within the validation set. While optimal models vary for each
concept, our hierarchical model outperforms others on average. This implies a
more consistent performance across classes.

Our optimal model is contrastively trained on the medium dataset described in
Section 3.2 and leverages only the first level of the hierarchy for each
cathedral. Our best model is trained with a replay regularization
hyperparameter of 0.5, learning rate of 0.0001, and contrastive triplet batch
size of 16.

We likely perform best overall because we benefit from using all the images
associated with the provided cathedral instances during contrastive pre-
training. The comparison model only exploits the data of the leaf nodes.

###  5.2 t-SNE Latent Space Visualisations

(a) Our Method

(b) Contrastive VGG-16 Baseline

(c) Elor et al. [6]’s Method

Figure 3: t-SNE plots of model feature maps for an unseen cathedral’s first
level

As seen in Figure 3(a), our method provides clear feature-space separation
between each of the classes provided by the hierarchy. The contrastive
baseline (Figure 3(b)) manages to separate the “interior" from the other two
classes fairly effectively, but mixes the “exterior” and “views” classes
together. However, this is to be expected to a degree as they are similar in
that they both include outdoor environments. Similarly, Figure 3(c) shows that
Elor et al. [6]’s model generates significant separation between “interior”
and other concepts, but maps “exterior” and “views” close together. Figure 3
shows that our method can clearly and effectively distinguish between semantic
classes. This implies that this hierarchical data structure modality can be
useful for various applications and that our model is a good approach to
exploiting this modality.

(a) Our Method

(b) Naive Contrastive Baseline

(c) Elor et al. [6]’s method

Figure 4: t-SNE plots of the latent space with respect to the downstream
classification labels for an unseen cathedral.

As seen in Figure 4(a), our method does not clearly have distinct clusters for
each classified cathedral concept. However, upon further inspection, we see
that some of the more prominent classes, such as facade, window, tower, and
nave, are strongly concentrated in distinct regions in feature space. There
are also weaker clusters for other concepts. We see a similar occurrence in
Figure 4(b), for the contrastive baseline, but with weaker clusters for
concepts where our model is strong, such as window and nave. Elor et al. [6]’s
method clearly has the strongest segmentation, but even this suffers from a
fair amount of class overlap and ambiguity.

Furthermore, this implies that despite not explicitly training on these
classes, our method can still learn conceptual relationships effectively. In
turn, this suggests that learning from the data’s structural hierarchy
modality using our method has strong conceptual representational ability. It
also further supports our hypothesis that gradually segmenting the latent
space with our method groups spatially similar concepts despite being given no
information about their semantic relationship (supervised classification
labels). However, when comparing these plots, it is important to note that
Elor et al. [6]’s model is trained with the downstream classification labels
as part of the contrastive loss function, whereas our model has no knowledge
of these classes, which we believe is a major reason that they have more
robust latent space separation.

###  5.3 Ablation Study

Figure 5 presents the impact of hierarchical information, dataset size, and
replay frequency on the downstream classification performance. The star
markers in each plot represent one full training run including contrastive
pre-training and linear probing to assess downstream classification
performance. Within a given plot, only the specified parameter changed between
training runs. The y-axes of Figure 5 specify classification performance
relative to the top performing model in the set presented a given plot. Figure
5(b) shows the impact of hierachical training on the downstream task.
Hierarchy level zero corresponds to no hierarchical pre-training, and, thus,
represents the vanilla ImageNet VGG-16. A significant finding is that training
a model on just the first and a model on all hierarchical layers result in
significant performance improvements, whereas training on layers two through
five result in comparable performances to the vanilla model. This indicates
that the concepts of the intermediate hierarchy levels provide less strong of
a signal. This observation likely relates to the inconsistency in the number
of hierarchy levels per cathedral instance, which is discussed in the next
paragraph.

The impact study of dataset size suggests that our model is performing best
for the medium sized dataset, as illustrated by Figure 5(a). This regression
in performance can be explained with the varying number of hierarchy levels
between top level concepts, such as “interior", and leaf node concepts, such
as “altar.” The large dataset includes many cathedral instances for which the
hierarchical structure is flat which means leaf node concepts may occur as
early as the second hierarchy level. Consequently, some leaf node concepts are
evaluated with a large margin loss, which likely causes deterioration of
performance. To address this shortcoming of our method, we would need to
introduce an adaptive hierarchy logic that sets $h_{max}$ programmatically
based on the depth of the currently considered cathedral instance. The medium
sized dataset is likely the best performing because it comprises cathedral
instances of similar hierarchy depth. Consequently, the performance gains due
to the additional data outweigh the performance regressions due to
inconsistent hierarchy depths.

Including replay into the training procedure resulted in significant
improvements on downstream classification performance. Figure 5(c) shows that
a replay probability above 0.4 improves training outcomes significantly. A
replay probability of 0.0 indicates that previous hierarchy levels are never
revisited. The impact of the replay regularization parameter indicates that
the current approach is prone to catastrophic forgetting. The given replay
approach seems to mitigate this failure mode.

(a)

(b)

(c)

Figure 5: Each star marker represents a trained encoder model evaluated on the
classification task. Relative $mAP*$ reflects the relative regression compared
to the best classification performance of all models in the plot. For example,
a Relative $mAP*$ of 0.10 is 90% worse than the best performing model.

##  6 Limitations

Our proposed hierarchical training method suffers from multiple limitations.
First, its downstream task performance doesn’t scale with dataset size, as
shown in the ablation study. Second, we only explore this training approach on
image data. It is unlikely that this exact method can be successfully applied
to other modalities, such as language. Second, our method is specific to the
spatial hierarchy of the WikiScenes dataset. To evaluate our method’s full
merit, it is necessary to look at other hierarchical datasets, e.g., the
ImageNet database [5]. Using ImageNet, one could investigate if hierarchical
training would work for semantic hierarchical structures of animal breeds,
such as dogs. Another limitation of this work is that hierarchical training
could not successfully fine-tune or train from scratch convolutional layers
and should be explored further.

##  7 Future Work

Given these promising initial results, there are many avenues for future work.
The next step in our research would be to apply this to other structured
datasets as more inherently structured WikiMedia datasets with different
content is released. Along this same vein, it would be interesting to both run
ImageNet classification on our method as it stands, and apply this method to
ImageNet’s purely semantic hierarchy, to see if our hypothesis for spatial
hierarchy reasoning applies to other types of hierarchies. To improve latent
feature representations, it is important to attempt more computationally
intensive work on this matter - for example, training a model purely from
scratch instead of using an ImageNet pretrained initialisation, and conducting
more thorough hyperparameter sweeps to explore the effect of a wider spectrum
of hyperparameters on the latent space. For example, we anticipate that
different margin distance metrics and hierarchy scheduling functions might
have a positive impact on the overall model performance.

##  8 Conclusion

We proposed a new training method that uses the hierarchical structure of data
scraped from WikiMedia as a modality to inform weak supervision. The
experiment’s results suggest that considering hierarchical training can
improve performance on downstream tasks. Future work is required to assess the
full merit of this approach and address some issues around downstream task
performance regressions for hierarchical training on large datasets.
Nevertheless, the proposed method performs slightly better than comparable
contrastive learning methods. Consequently, if given a structured dataset,
hierarchical training is a competitive alternative to other self-supervised
pre-training methods.

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