# Exploring Origins for Correlations between Flow Harmonics and Transverse
Momentum in Small Collision Systems
(Unambiguous Ambiguity)

S.H. Lim  Pusan National University, Busan, 46241, South Korea  J.L. Nagle
University of Colorado, Boulder, Colorado 80309, USA

###### Abstract

High statistics data sets from experiments at the Relativistic Heavy Ion
Collider (RHIC) and the Large Hadron Collider (LHC) with small and large
collision species have enabled a wealth of new flow measurements, including
the event-by-event correlation between observables. One exciting such
observable $\rho(v^{2}_{n},[p_{T}])$ gauges the correlation between the mean
transverse momentum of particles in an event and the various flow coefficients
($v_{n}$) in the same event Bożek (2016). Recently it has been proposed that
very low multiplicity events may be sensitive to initial-state glasma
correlations Giacalone  _et al._ (2020a) rather than flow-related dynamics. We
find utilizing the ip-jazma framework that the color domain explanation for
the glasma results are incomplete. We then explore predictions from pythia8,
and the version for including nuclear collisions called pythia-angantyr, which
have only non-flow correlations and the ampt model which has both non-flow and
flow-type correlations. We find that pythia-angantyr has non-flow
contributions to $\rho(v^{2}_{n},[p_{T}])$ in $p$$+$O, $p$$+$Pb, O$+$O
collisions that are positive at low multiplicity and comparable to the glasma
correlations. It is striking that in pythia8 in $p$$+$$p$ collisions there is
actually a sign-change from positive to negative $\rho(v^{2}_{n},[p_{T}])$ as
a function of multiplicity. The ampt results match the experimental data
general trends in Pb+Pb collisions at the LHC, except at low multiplicity
where ampt has the opposite sign. In $p$$+$Pb collisions, ampt has the
opposite sign from experimental data and we explore this within the context of
parton geometry. Predictions for $p$$+$O, O$+$O, and Xe$+$Xe are also
presented.

###### pacs:

25.75.Dw

##  I Introduction

The standard time-evolution model for relativistic heavy ion collisions at the
Relativistic Heavy Ion Collider (RHIC) and the Large Hadron Collider (LHC)
includes an “epoch” of quark-gluon plasma (QGP) formation and hydrodynamic
expansion Busza  _et al._ (2018). The QGP expands as a strongly coupled fluid
with very small specific shear viscosity $\eta/s$ Romatschke and Romatschke
(2019); Heinz and Snellings (2013). Detailed computer modeling of said time
evolution requires inputs including the initial energy deposit geometry and
any pre-hydrodynamic evolution. Higher level “flow” correlation observables,
such as the event-by-event correlation of anisotropies ($v_{n}^{2}$) and mean
transverse momentum $([p_{T}])$, can provide additional insights and
constraints on the interplay of geometry and subsequent transport Bożek
(2016).

The ATLAS experiment has carried out measurements of the
$\rho(v^{2}_{n},[p_{T}])$ defined as

$\displaystyle\rho(v^{2}_{n},[p_{T}])=\frac{{\rm
cov}(v_{n}^{2},[p_{T}])}{\sqrt{{\rm Var}(v_{n}^{2})}\sqrt{{\rm
Var}([p_{T}])}}$ (1)

in $p$$+$Pb and Pb$+$Pb collisions at $\sqrt{s_{{}_{NN}}}$ = 5.02 TeV at the
LHC Aad  _et al._ (2019). The ${\rm cov}(v_{n}^{2},[p_{T}])$ is the covariance
between $v_{n}^{2}$ and $[p_{T}]$, and ${\rm Var}(v_{n}^{2})$ and ${\rm
Var}([p_{T}])$ are the variances of $v_{n}^{2}$ and $[p_{T}]$. Focusing on the
correlation of elliptic flow $v_{2}^{2}$ and $[p_{T}]$, in Pb$+$Pb collisions
the correlator is negative at low event multiplicity, then turning positive
and increasing with multiplicity, and finally slightly decreasing for the very
highest multiplicity events. These features are semi-quantitatively reproduce
in a simple calculation of initial Glauber geometry combined with hydrodynamic
flow Bozek (2016); Giacalone  _et al._ (2020b); Schenke  _et al._ (2020),
though the results depend sensitively on the geometry model in small collision
systems Bozek and Mehrabpour (2020).

The basic idea is that there are event-to-event fluctuations in the geometric
size of the overlap region in the transverse plane at fixed multiplicity.
Smaller overlap area with the same multiplicity leads to larger pressure
gradients and thus larger radial flow and hence larger mean transverse
momentum $\left<p_{T}\right>$. If there is a positive correlation in geometry
for smaller overlap area events to have a larger eccentricity, and vice versa,
then one gets a positive $\rho$ correlator. If there is the opposite
correlation such that for smaller overlap area events, they have a smaller
eccentricity, and vice versa, then one gets a negative $\rho$ correlator. It
has been shown that alternative variables to the overlap area present a
stronger correlation Giacalone  _et al._ (2020b); Schenke  _et al._ (2020);
however, the basic picture above still holds.

However, the detailed behaviour in the lowest multiplicity events warrants
further examination. The question of whether small collision systems, for
example $p$$+$$p$ and $p$$+$Pb at the LHC and $p$$+$Au, $d$$+$Au, 3He$+$Au at
RHIC, have correlations induced via hydrodynamic expansion of small QGP
droplets or via a mimic from initial-state correlations, for example glasma
correlations, has been studied in detail Nagle and Zajc (2018); Dusling  _et
al._ (2016). Extensive data from the LHC and the geometry scan from RHIC
Aidala  _et al._ (2019), combined with theoretical clarification for glasma
correlations Mace  _et al._ (2018, 2019); Nagle and Zajc (2019), have given a
clear answer for the higher multiplicity events in small collision systems –
there is overwhelming evidence for correlations via initial geometry coupled
with strong final-state interactions (e.g. hydrodynamics).

Even more recently, it has been proposed that initial-state correlations may
yet dominate over final-state collectivity in the very lowest multiplicity
collisions, i.e. for $\mbox{$dN_{\rm ch}/d\eta$}<10$ Schenke  _et al._ (2019).
Current there is no clear experimental evidence for such glasma correlations
at these multiplicities. A proposal has been put forth that the correlation
between anisotropies $v_{n}^{2}$ and event average $p_{T}$ can provide
definitive evidence: Ref. Giacalone  _et al._ (2020a) states that
“experimental observation of these clean qualitative signatures in peripheral
heavy ion and small system collisions will be the first evidence for the
presence and importance of initial state momentum anisotropies predicted by an
effective theory of QCD.”

The calculations, in a hybrid glasma and hydrodynamic model, indicate that the
$\rho$ correlator should change sign from negative at low multiplicity back to
positive at the very lowest multiplicities $\mbox{$dN_{\rm ch}/d\eta$}<10$
where glasma correlations dominate. The basic argument is that in the glasma
color domain picture, events with a smaller transverse size will have fewer
color domains and hence a stronger glasma type correlation. Ref. Schenke
(2021) notes that at fixed multiplicity, the saturation scale $Q_{s}$ of the
projectile, which drives the $\langle\mbox{$p_{T}$}\rangle$, decreases with
increasing transverse area and hence the positive $\rho$ correlator from
initial-state glasma correlations alone. We highlight that the connection
between the calculation in momentum-space and the spatial domain picture (see
Ref. Lappi  _et al._ (2016) for details) makes the cause-effect relation for
the positive $\rho$ correlation via glasma diagrams to some degree
speculative. In Appendix IX, we utilize the ip-jazma framework to show that
the scaling of projectile saturation scale and eccentricity with overlap area
do not follow the ordering postulated above and thus require further
validation.

Due diligence requires the checking of whether non-exotic (i.e. not glasma
correlations) could also mimic said sign change and hence positive $\rho$
correlations at the lowest multiplicities. It is notable that within the
calculation of Ref. Giacalone  _et al._ (2020a), there is no inclusion of what
are often referred to as non-flow correlation effects (e.g. from dijet
correlations, overall momentum conservation, etc.).

In this paper we explore the expectations for the $\rho$ correlator in the
pythia8 and pythia-angantyr models Sjostrand  _et al._ (2008); Bierlich  _et
al._ (2018), which have neither initial-state glasma correlations nor final-
state interactions. pythia-angantyr is an extension of the publicly available
pythia8 code that allows the modeling of $p$$+$$A$ and $A$$+$$A$ collisions,
and is named for Angantyr the Berserker in Norse mythology saga og Heioreks
(1618-1682). We highlight that there is a version of pythia-angantyr with so-
called “string shoving” Bierlich  _et al._ (2016) that may effectively model
final-state interactions; however, we are running the code without this option
turned on.

We also explore calculations from the A-Multi-Phase Transport (ampt) model Lin
_et al._ (2005), which again has no initial-state glasma correlations but has
both initial state non-flow (e.g. dijets) and final-state parton and hadron
scattering. In the process of completing this manuscript, a complementary
study using the pythia8 and hijing Gyulassy and Wang (1994) models has been
submitted Zhang  _et al._ (2021).

##  II Methodology

There are different methods for constructing the $\rho$ correlator, in part in
an effort to reduce non-flow contributions. We follow the three-subevent
method Aad  _et al._ (2019) using three separate $\eta$ regions labeled $a$,
$b$, and $c$: $-2.5<\eta_{a}<-0.75$, $|\eta_{b}|<0.5$, and
$0.75<\eta_{c}<2.5$. Charged particles in subevents $a$ and $c$ are used to
calculate $v_{n}^{2}$ to suppress the non-flow contribution, and the $[p_{T}]$
is obtained with charged particles from subevent $b$. The
$\rho(v^{2}_{n},[p_{T}])$ is calculated using particles in
$0.3<p_{T}<2.0~{}\mathrm{GeV}$, and particles in $|\eta|<2.5$ and
$0.5<p_{T}<5.0~{}\mathrm{GeV}$ are used to calculate the event multiplicity
$N_{\rm ch}$.

The ${\rm cov}(v_{n}^{2},[p_{T}])$ term in the $\rho$ correlator is defined as

$\displaystyle{\rm cov}(v_{n}^{2},[p_{T}])={\rm
Re}\left(\left<\sum_{a,c}e^{in(\phi_{a}-\phi_{c})}\left([p_{T}]-\langle[p_{T}]\rangle\right)\right>\right),$
(2)

where $\phi_{a(c)}$ is the azimuthal angle of particles in the region $a$
($c$), and $\langle[p_{T}]\rangle$ is the average $[p_{T}]$ in the entire
events at a certain multiplicity range. The variance of $v_{n}^{2}$ is
calculated with two-particle and four-particle correlations,

$\displaystyle{\rm Var}(v_{n}^{2})$
$\displaystyle=v_{n}\\{2\\}^{4}-v_{n}\\{4\\}^{4}$ $\displaystyle=\langle{\rm
corr}_{n}\\{4\\}\rangle-\langle{\rm corr}_{n}\\{2\\}\rangle^{2},$ (3)

where $v_{n}\\{2\\}$ and $v_{n}\\{4\\}$ are the flow coefficients obtained
from two- and four-particle correlations, respectively. The ${\rm
corr}_{n}\\{2\\}$ and ${\rm corr}_{n}\\{4\\}$ are the two-particle and four-
particle correlations with the subevent method Jia  _et al._ (2017) with
particles in the region $a$ and $c$. Events required to have at least two
charged particles in each region ($a$ and $c$) for the four-particle
correlation. The variance of $[p_{T}]$ is estimated by the dynamical $p_{T}$
fluctuation magnitude Abelev  _et al._ (2014) $c_{k}$ defined as

$\displaystyle c_{k}=\left<\frac{1}{N_{\rm pair}}\sum_{b}\sum_{b^{\prime}\neq
b}(p_{T,b}-\langle[p_{T}]\rangle)(p_{T,b^{\prime}}-\langle[p_{T}]\rangle)\right>,$
(4)

and particles in region $b$ are used. The $\rho(v^{2}_{n},[p_{T}])$ is
calculated with ${\rm cov}(v_{n}^{2},[p_{T}])$, ${\rm Var}(v_{n}^{2})$, and
$c_{k}$, and the ${\rm Var}(v_{n}^{2})$ should be positive for valid
$\rho(v^{2}_{n},[p_{T}])$ value.

##  III pythia8 Results

We have run millions of events for various collision systems within the
pythia8 and pythia-angantyr frameworks. The output particle truth list is then
processed through the full three-subevent calculation detailed above. Here we
utilize the ATLAS define regions of $-2.5<\eta_{a}<-0.75$, $|\eta_{b}|<0.5$,
and $0.75<\eta_{c}<2.5$ and consider charged hadrons with
$0.3<p_{T}<2.0~{}\mathrm{GeV}$. For the event category, the $N_{\rm ch}$ is
determined within the range $-2.5<\eta<2.5$ and for charged hadrons with
$0.5<p_{T}<5.0~{}\mathrm{GeV}$. Figure 1 shows the resulting $\rho$ correlator
from pythia8 for $p$$+$$p$ at 5 TeV and 13 TeV and from pythia-angantyr for
$p$$+$O at 5 TeV, $p$$+$Pb at 8 TeV, and O$+$O at 5 TeV. Note that future
running of $p$$+$O and O$+$O at the LHC may end up being run at a different
collision energy. The full eight-panel set of calculated ingredients that go
into the $\rho$ correlator are shown in the Appendix VIII in Figure 11.

Figure 1: Results shown from the pythia8 model for $p$$+$$p$ at 5 and 13 TeV
and the pythia-angantyr model for $p$$+$O, O$+$O at 5 TeV, $p$$+$Pb at 8 TeV.
In all cases the event categories on the $x$-axis are defined by $N_{\rm ch}$
for charged hadrons with $|\eta|<2.5$ and
$0.5<\mbox{$p_{T}$}<5.0~{}\mathrm{GeV}$. The particles entering the $\rho$
correlations are from $0.3<\mbox{$p_{T}$}<2.0~{}\mathrm{GeV}$ and are utilized
using the three-subevent method.

The correlations in $p$$+$O, $p$$+$Pb, and O$+$O collisions are positive for
all $N_{\rm ch}$ and are around 10%. In contrast, the results in $p$$+$$p$ at
5 and 13 TeV (consistent with each other) are positive for $\mbox{$N_{\rm
ch}$}<30$ and then change sign to negative for $\mbox{$N_{\rm ch}$}>30$. These
non-zero correlations cannot be from spatial geometry coupled to final-state
interactions or flow, since pythia-angantyr has neither. Instead they result
from non-trivial correlations due to jets, correlation between jets and
underlying event, and multiplicity category correlations.

Figure 2 shows a comparison in $p$$+$Pb and O$+$O between the pythia-angantyr
results and glasma initial-state calculations Giacalone  _et al._ (2020a). In
this case, the glasma correlator is a proxy determined in momentum space from
the correlations encoded in the $T_{\mu\nu}$ prior to any hydrodynamic
calculation in their hybrid framework. What is striking is that both
calculations are positive and of a similar order of magnitude. It decidedly
means that a full accounting of non-flow effects, as coded into pythia8 and
pythia-angantyr, are needed to make any statement about more exotic glasma
correlations.

It should be be pointed out that such contributions as seen in the $\rho$
correlation may be additive to leading order if the sources of flow and non-
flow particles are independent, as is often assumed in non-flow subtraction
techniques. However, in low multiplicity events this independence must be
violated due to momentum conservation, overall color neutrality, as well as
the possible direct scattering of particles from each source. Thus, one can
postulate additive contributions Zhang  _et al._ (2021), though at the lowest
multiplicities it is unknown the level of violation.

Figure 2: Results shown from the pythia-angantyr model for $p$$+$Pb at 8 TeV
and O$+$O at 5 TeV. In all cases the event categories on the $x$-axis are
defined by $N_{\rm ch}$ for charged hadrons with $|\eta|<2.5$ and
$0.5<\mbox{$p_{T}$}<5.0~{}\mathrm{GeV}$. The particles entering the $\rho$
correlations are from $0.3<\mbox{$p_{T}$}<2.0~{}\mathrm{GeV}$ and are utilized
using the three-subevent method. In comparison are results from initial-state
glasma correlations Giacalone  _et al._ (2020a).

##  IV ampt Results

We now compare with calculations from the ampt model (version
v1.26t9b-v2.26t9b), which is publicly available code Lin  _et al._ (2005). The
ampt model has been run it the default mode, with a parton screening mass of
3.22 fm-1 (or equivalently a parton-parton cross section of 3 mb) and with
Lund symmetric splitting function parameters ($a=0.4$ and $b=0.8$), set to
optimize matching with particle multiplicities and $\left<p_{T}\right>$ at the
LHC Zhang  _et al._ (2019). ampt has been successful at describing a number of
global and flow features of heavy ion data, including in small collision
systems – see Refs. Zhang  _et al._ (2019); He  _et al._ (2016); Orjuela Koop
_et al._ (2015); Bzdak and Ma (2014) for examples.

Again, millions of ampt events have been run for $p$$+$$p$ at 5 and 13 TeV,
$p$$+$O and O$+$O at 5 TeV, $p$$+$Pb at 5 and 8 TeV, and Xe$+$Xe and Pb$+$Pb
at 5 TeV. The results for large $A$$+$$A$ systems are shown in Figure 3 and
compared with ATLAS results in Pb$+$Pb collisions Aad  _et al._ (2019). The
AMPT results are calculated with the three-subevent method in
$-2.5<\eta_{a}<-0.75$, $|\eta_{b}|<0.5$, and $0.75<\eta_{c}<2.5$ and consider
charged hadrons with $0.5<p_{T}<2.0~{}\mathrm{GeV}$, matching the higher
low-$p_{T}$ selection used by ATLAS in Ref. Aad  _et al._ (2019). The ATLAS
data $x$-axis $N_{\rm ch}$ values are re-scaled since ATLAS does not
correction for reconstruction efficiency in the published results.

The ampt Pb$+$Pb results show a qualitatively similar trend to the ATLAS data,
but are approximately a factor of two higher. It is also notable that neither
the ampt Pb$+$Pb nor Xe$+$Xe go negative for the lowest $N_{\rm ch}$ as seen
in the data. ATLAS has shown a modest dependence in Pb$+$Pb collisions of
$\rho$ on the $p_{T}$ selection of the charged hadrons used in their analysis
Aad  _et al._ (2019). They observe a modest 15% increase in $\rho$ when
switching to charged hadrons with $0.5<\mbox{$p_{T}$}<5.0~{}\mathrm{GeV}$,
while in ampt the has a roughly 15% decrease in $\rho$ with the same momentum
selection change.

Figure 3: Results shown from the ampt model for Pb$+$Pb and Xe$+$Xe at 5 TeV
compared with published ATLAS results for Pb$+$Pb collisions at the same
energy. In all cases the event categories on the $x$-axis are defined by
$N_{\rm ch}$ for charged hadrons with $|\eta|<2.5$ and
$0.5<\mbox{$p_{T}$}<5.0~{}\mathrm{GeV}$. The particles entering the $\rho$
correlations are from $0.5<\mbox{$p_{T}$}<2.0~{}\mathrm{GeV}$ and are utilized
using the three-subevent method.

The full eight panel comparison of ingredients for Pb$+$Pb that go into the
$\rho$ calculation are shown in Appendix VIII in Figure 13. Again one observes
qualitatively similar trends between ampt and data, but significant
quantitative differences. One other striking observation from ampt is that the
average $p_{T}$ as a function of $N_{\rm ch}$ actually decreases with
increasing $N_{\rm ch}$ over a significant range of $N_{\rm ch}$– see Figure
13 in Appendix VIII. This is contrary to results in experimental data Abelev
_et al._ (2013) where average transverse momentum always increases with
increasing $N_{\rm ch}$ in $p$$+$$p$, $p$$+$Pb and Pb$+$Pb collisions. One key
question regarding finite parton-parton scattering pictures is how they build
up radial flow and thus increase the $\left<p_{T}\right>$. This ampt result
indicates that this effect is not modeled fully in ampt, even for large
collision systems.

ampt predictions for small collision systems, $p$$+$$p$ at 5 and 13 TeV,
$p$$+$O and O$+$O at 5 TeV, and $p$$+$Pb at 8 TeV are shown in Figure 4. In
this case the results vary from a large negative $\rho$ correlation in
$p$$+$$p$ collisions to positive correlations in $p$$+$Pb and O$+$O. It is
striking that the positive correlations in $p$$+$Pb and O$+$O are quite
comparable to those from pythia-angantyr in Figure 1, even though ampt has
both flow and non-flow contributions.

ATLAS has measured the correlator $\rho$ in $p$$+$Pb collisions, but from
earlier data taken at 5 TeV. ampt calculations are compared with ATLAS data
for $p$$+$Pb collisions at 5 TeV for the selection $0.5<\mbox{$p_{T}$}<2.0$ in
Figure 9. The ampt result is positive, though decreasing with $N_{\rm ch}$,
and is opposite to the consistently negative $\rho$ result in data. The full
eight panel comparison of ingredients for $p$$+$Pb that go into the $\rho$
calculation are shown in Appendix VIII in Figure 14.

Figure 4: Results shown from the ampt model for $p$$+$$p$, $p$$+$O, O$+$O at 5
TeV, $p$$+$Pb at 8 TeV, and $p$$+$$p$ at 13 TeV. In all cases the event
categories on the $x$-axis are defined by $N_{\rm ch}$ for charged hadrons
with $|\eta|<2.5$ and $0.5<\mbox{$p_{T}$}<5.0~{}\mathrm{GeV}$. The particles
entering the $\rho$ correlations are from
$0.3<\mbox{$p_{T}$}<2.0~{}\mathrm{GeV}$ and are utilized using the three-
subevent method. Figure 5: Results shown from the ampt model for $p$$+$Pb at 5
TeV compared with published ATLAS results for the same system and energy. In
all cases the event categories on the $x$-axis are defined by $N_{\rm ch}$ for
charged hadrons with $|\eta|<2.5$ and $0.5<\mbox{$p_{T}$}<5.0~{}\mathrm{GeV}$.
The particles entering the $\rho$ correlations are from
$0.5<\mbox{$p_{T}$}<2.0~{}\mathrm{GeV}$ and are utilized using the three-
subevent method.

Since there may be O$+$O data taking at RHIC by the STAR experiment and/or the
sPHENIX experiment at 200 GeV, as well as at the LHC in Run-3, we present
predictions for the $\rho$ correlator from those systems in Figure 6. We
highlight that for the results at 200 GeV, we have modeled the acceptance
selections for a detector with $|\eta|<1.0$ and as such have regions
$-1.0<\eta_{a}<-0.35$, $|\eta_{b}|<0.3$, and $0.35<\eta_{c}<1.0$ and consider
charged hadrons with $0.2<p_{T}<2.0~{}\mathrm{GeV}$. Also, the $x$-axis values
of $N_{\rm ch}$ are defined by $|\eta|<0.5$ and
$0.2<p_{T}<2.0~{}\mathrm{GeV}$. The results indicate a positive $\rho$
correlator with a very similar shape and magnitude at the two energies if one
considers a rescaling of the $x$-axis between collision energies.

Figure 6: Predictions for O$+$O at 200 GeV (RHIC) and 5 TeV (LHC) from the
ampt model. See text for details on the different charged hadron kinematic
selections.

##  V Discussion

Figure 7: ampt results for multiple collision systems. The panels from left to
right show (i) the multiplicity distribution of events normalized to unity,
(ii) the average eccentricity squared $\varepsilon_{2}^{2}$ as a function of
$N_{\rm ch}$, (iii) the average overlap area $S_{T}$ as a function of $N_{\rm
ch}$, and the correlation between $\varepsilon_{2}^{2}$ and $S_{T}$ normalized
by their individual average values. The $N_{\rm ch}$ axis is zoomed in to
focus on the lower multiplicity range.

Within the ampt model, one can examine the geometry as defined by the partons
that are “string-melted” out of the color strings – as has been done
previously Nagle  _et al._ (2018); Orjuela Koop  _et al._ (2016). The partons
after appearing have a formation time set in ampt corresponding to the
uncertainty principle before they can start scattering with other partons. It
is their location after the formation time utilized here for calculating the
eccentricity of the event $\varepsilon_{2}$ and the overlap area $S_{T}$ Alver
_et al._ (2008). Here we define the overlap area
$S_{T}=\pi\sqrt{\sigma_{x}^{2}\sigma_{y}^{2}-\sigma_{xy}^{2}}$, noting there
are different prefactors used in the literature.

Figure 7 shows results from $p$$+$$p$ at 5 and 13 TeV, $p$$+$O and O$+$O at 5
TeV, $p$$+$Pb at 8 TeV, and Xe$+$Xe at 5 TeV. The panels from left to right
show (i) the multiplicity distribution of events normalized to unity, (ii) the
average eccentricity squared $\varepsilon_{2}^{2}$ as a function of $N_{\rm
ch}$, (iii) the average overlap area $S_{T}$ as a function of $N_{\rm ch}$,
and the (iv) correlation between $\varepsilon_{2}^{2}$ and $S_{T}$ normalized
by their individual average values. Thus, the correlation is $>1$ if there is
a correlation between eccentricity and area and $<1$ if there is an anti-
correlation.

The correlation (or anti-correlation) in geometry is a good predictor for ampt
for whether the $\rho$ correlation will be positive or negative. The positive
correlation of eccentricity and area in $p$$+$$p$ collisions at 5 and 13 TeV
is mirrored in the negative $\rho$ correlation shown in Figure 4, though not
at larger $N_{\rm ch}$. The negative geometry correlation for $p$$+$$A$ and
O$+$O is mirrored in the positive $\rho$ correlation again shown in Figure 4.

For comparison, we calculate the geometry correlation for Monte Carlo Glauber
type models Loizides (2016). The Glauber calculation is carried out with
standard nuclear geometry parameters and including three constituents per
nucleon. The results for the (i) multiplicity, (ii) eccentricity squared,
(iii) the overlap area, and (iv) the correlation of eccentricity and area are
shown in Figure 8 in the panels from left to right.

Figure 8: Monte Carlo Glauber geometries including three constituents for each
nucleon. Note that the $x$-axis is simply an arbitrary scaling times the
number of constituent quark-quark collisions. For $p$$+$$p$ collisions in
particular this does not model the multiplicity distribution properly and
additional fluctuations are necessary to describe data.

In this zoomed in view for $\mbox{$N_{\rm ch}$}<300$, the geometry
correlations are often opposite between ampt and Monte Carlo Glauber with
three constituents. We know that in the case of $p$$+$$p$ collisions, the ampt
geometry is almost exclusively two strings and thus the partons have a
geometry that appears as two radial distributions of partons around each
string with the distance in the transverse plane based on the formation time –
see Ref. Nagle  _et al._ (2018) for details. In this case, two strings that
are farther apart will lead to a much more elliptical geometry and a larger
area; hence the positive correlation seen in the ampt results in Figure 7. In
contrast, in the Monte Carlo Glauber with three constituents, the larger area
typically leads to a more circular geometry and thus a negative correlation.
However, in ampt as soon as one has a $p$$+$$A$ collision, there are multiple
strings and they have a geometry correlation that is negative and once again
flipped from the Monte Carlo Glauber case. The disagreement of ampt with
experimental $p$$+$Pb data shown in Figure 5 may lead one to conclude that
ampt string geometry is not a good descriptor, or that a substantial over-
prediction of non-flow in ampt changes the sign. Additional comparisons with
measurements in $p$$+$$p$ data should prove useful.

One question is whether the formation time implemented in ampt might be
altering the initial geometry as it mimics free streaming. Thus, for $p$$+$Pb
collisions, we have re-run ampt with the formation time reduced by a factor of
10. Those results, labeled short $\tau_{f}$, are also shown in Figure 7 and
show a modestly smaller overlap area and a modestly larger eccentricity. These
effects do impact the $\rho$ correlator as shown in Figure 9. These changes do
modify the $\rho$ value by almost a factor of two, though do not change the
sign.

Figure 9: Results shown from the ampt model for $p$$+$Pb at 8 TeV in different
running modes. In all cases the event categories on the $x$-axis are defined
by $N_{\rm ch}$ for charged hadrons with $|\eta|<2.5$ and
$0.3<\mbox{$p_{T}$}<5.0~{}\mathrm{GeV}$, except in the comparison case with
$0.3<\mbox{$p_{T}$}<5.0~{}\mathrm{GeV}$. Also shown are results with a shorter
formation time in ampt. Lastly, results with no interactions for ampt are
shown, though they have low statistical significance due to the very small
$v_{2}^{2}$ values overall.

Finally, for the Pb$+$Pb case, the geometry correlation is opposite in the
very low $N_{\rm ch}$ region between ampt and Monte Carlo Glauber, where again
ampt may have the wrong geometry as indicated by the lack of sign change that
is seen in data – see Figure 3. However, for larger $N_{\rm ch}$ as shown in
Figure 10, the correlation in both ampt and Monte Carlo Glauber is negative
and thus would predict a positive $\rho$ correlator – exactly as seen in
Figure 3 at $\mbox{$N_{\rm ch}$}>250$. It is notable that the correlation is
weaker, i.e. closer to one, in ampt, though it over-predicts $\rho$ by almost
a factor of two.

Figure 10: Comparison of the correlation between overlap area and eccentricity
in Pb$+$Pb collisions at 5 TeV from ampt and Monte Carlo Glauber. The $x$-axis
highlights the larger $N_{\rm ch}$ range.

##  VI Summary

In summary, we have carried out a study of elliptic flow – transverse momentum
correlations within the context of pure non-flow pythia8 and pythia-angantyr
models and the combination of flow and non-flow ampt model. The results
indicate that even with the three-subevent method to reduce non-flow,
substantial non-zero $\rho$ correlations persist for small collision systems
and peripheral event classes in large collision systems. The pythia8 and
pythia-angantyr correlations are comparable in size to predicted glasma
correlations. It is striking that ampt calculations have the opposite sign to
measured ATLAS data in $p$$+$Pb and peripheral Pb$+$Pb collisions, which may
indicate that the string geometry, combined with string melting, has an
incorrect modeling of the initial geometry. Finally, the pure glasma results
for the $\rho$ correlator are interesting, and the simple color domain
explanation appears incorrect, and thus more study is warranted as
explanations can be as important as the result itself.

##  VII Acknowledgements

We gratefully acknowledge useful discussions with Giuliano Giacalone,
Jiangyong Jia, Blair Seidlitz, Bjoern Schenke, Christian Bierlich, and Ross
Snyder. We also want to remember Jack Sandweiss, who was an inspiration to so
many of us and would want us to continue to “take nature to the mat” and
uncover it’s secrets. JLN acknowledges support from the U.S. Department of
Energy, Office of Science, Office of Nuclear Physics under Contract No. DE-
FG02-00ER41152. SHL acknowledges support from the National Research Foundation
of Korea (NRF) grant funded by the Korea government (MSIT) under Contract No.
2020R1C1C1004985.

##  VIII Appendix A

For completeness we include the full set of calculated quantities in the model
calculations. Shown in Figure 11 are the pythia8 and pythia-angantyr results
from upper left to upper right and then lower left to lower right: (i) the
minimum bias $N_{\rm ch}$ distribution with the integral normalize to one,
(ii) the event $\langle p_{T}\rangle$ as a function of $N_{\rm ch}$, (iii) the
coefficient $c_{k}$, (iv) the variance of $v_{2}^{2}$, (v) the average two-
particle cumulant, (vi) the average four-particle cumulant, (vii) the
covariance, and finally (viii) the $\rho$ correlation all as a function of
$N_{\rm ch}$. We highlight that the range of $N_{\rm ch}$ shown here is meant
to highlight small systems, and thus the O$+$O distribution extends beyond the
range of the figure. Also, at some large $N_{\rm ch}$ at the very tail of the
distribution for each collision systems, the statistical uncertainties get
large. In the $\rho$ distributions, we have eliminated points with
uncertainties larger than a set value for clarity of comparison between
systems. Note that results from pythia8 and pythia-angantyr for $p$$+$$p$ at
13 TeV are compared, and no significant difference is observed.

Shown in Figure 12 are the same quantities for small systems from ampt.
Lastly, in Figures 13 and 14 are the eight panels for Pb$+$Pb and $p$$+$Pb at
5 TeV compared with ATLAS results Aad  _et al._ (2019).

Figure 15 shows results of the coefficient $c_{k}$, the variance of
$v_{2}^{2}$, the covariance $\mathrm{cov}(v_{2}^{2},[p_{T}])$, and the $\rho$
correlation for two $p_{T}$ ranges in $p$$+$$p$ at 13 TeV and $p$$+$Pb at 8
TeV. The $\rho$ values in $0.5<p_{T}<2.0~{}\mathrm{GeV}$ are lower than those
in $0.3<p_{T}<2.0~{}\mathrm{GeV}$ where the non-flow contribution is expected
to be smaller. The $\rho$ correlations at low multiplicity are positive in
both $p_{T}$ ranges and collision systems, but the sign change has been
reported in $p$$+$Pb from hijing Zhang  _et al._ (2021).

Figure 11: Results from the pythia8 model for $p$$+$$p$ at 5 and 13 TeV, and
the pythia-angantyr model for $p$$+$O and O$+$O at 5 TeV and $p$$+$Pb at 8 TeV
are shown. The quantities in the subpanels are defined in the text. Figure 12:
Results from the ampt model for $p$$+$$p$ at 5 and 13 TeV, $p$$+$O and O$+$O
at 5 TeV, and $p$$+$Pb at 8 TeV are shown. The quantities in the subpanels are
defined in the text. Figure 13: Results from the ampt model for Pb$+$Pb at 5
TeV are shown as circular markers. The quantities in the subpanels are defined
in the text. Also shown are the ATLAS results for the quantities in the four
right subpanels plotted as lines. The colored lines are for different $p_{T}$
selections – $0.5<p_{T}<2.0$ (blue), $0.5<p_{T}<5.0$ GeV (black), and
$1.0<p_{T}<2.0$ GeV (red). Figure 14: Results from the ampt model for $p$$+$Pb
at 5 TeV are shown as circular markers. The quantities in the subpanels are
defined in the text. Also shown are the ATLAS results for the quantities in
the four right subpanels plotted as lines. The colored lines are for different
$p_{T}$ selections – $0.3<p_{T}<5.0$ (blue), $0.3<p_{T}<2.0$ GeV (black), and
$0.5<p_{T}<2.0$ GeV (red). Figure 15: Results from the pythia8 model for
$p$$+$$p$ at 13 TeV and the pythia-angantyr model for $p$$+$Pb at 8 TeV are
shown. The quantities in the subpanels are defined in the text. Results for
two $p_{T}$ selections, $0.3<p_{T}<2.0~{}\mathrm{GeV}$ and
$0.5<p_{T}<2.0~{}\mathrm{GeV}$, are presented.

##  IX Appendix B

Here we briefly explore the relationship between area, eccentricity, and
saturation scale within the ip-jazma framework Nagle and Zajc (2019). The ip-
jazma calculation includes the IP-Sat features of treating each nucleon, or in
this case each constituent quark, as a two-dimensional Gaussian distribution
for the saturation scale $Q_{s}^{2}$. The saturation scale is allowed to
fluctuate via a Gaussian parameter $\sigma=0.5$ on a log-normal distribution –
thus the fluctuations are quite large and have a high-side tail. These
features mimic the ip-glasma calculations and then the energy deposit is
calculated on a grid in each cell as the product of the summed
$Q_{s}^{2}(x,y)$ in the projectile and the summed $Q_{s}^{2}(x,y)$ in the
target. This calculation mimics the dense-dense limit of saturation
calculations and reproduces the ip-glasma geometry with good accuracy Snyder
_et al._ (2021). We can also use ip-jazma to mimic the dilute-dense limit
where the target contribution is treated logarithmically – see Ref. Nagle and
Zajc (2019) for details.

We simulate $p$$+$Pb collisions at 5 TeV and categorize each event via the
overlap area, the eccentricity, and the saturation scale of the projectile
over the overlap region. For the overlap region, to avoid over-counting the
area in cases where there may be so-called “islands” of energy deposit (i.e.
where different regions are disconnected) we simply count cells above a
minimum energy rather than use the pocket equation
$S_{T}=\pi\sqrt{\sigma_{x}^{2}\sigma_{y}^{2}-\sigma_{xy}^{2}}$. Similarly for
the saturation scale of the projectile, we simply average the $Q_{s}^{2}$
value over the cells included in the area calculation. The absolute values
will thus have a sensitivity to this minimum energy, but for the correlations
the relative values should be mostly insensitive.

Figure 16 shows the dilute-dense case for $p$$+$Pb collisions. The upper left
panel shows the energy deposit distribution normalized to the mean for minimum
bias collisions. The lower panels from left to right show the overlap area in
the transverse plane, the average projectile, and separately target,
$Q_{s}^{2}$ over the overlap area, and the eccentricity as a function of the
energy deposit normalized to the mean. The points are the mean values while
the vertical lines represent the rms of the distribution about the mean. We
then select a fixed energy deposit bin, as shown by the vertical dashed lines.
The upper middle panel shows the variation in $Q_{s}^{2}$ and
$\varepsilon_{2}$ with transverse area for this fixed energy selection.
Lastly, in the upper right, again in this fixed energy bin, the correlation of
the projectile $Q_{s}^{2}$ and $\varepsilon_{2}$ is shown. The same results
but in the dense-dense case are shown in Figure 17.

One simple remark is that the projectile saturation scale does not increase
with decreasing area as conjectured in Ref. Schenke (2021), and rather has a
more complicated dependence. The very large fluctuations in $Q_{s}^{2}$ added
by hand for each constituent quark play a key role here. At fixed energy
deposit, to have a small area, it is most likely that one quark has fluctuated
to a very large $Q_{s}^{2}$, which would then explain the anti-correlation of
$Q_{s}^{2}$ projectile and area, for cases where the area is less than 0.4
fm2. It also then explains the very large drop off in eccentricity, since the
single quark distribution is circularly symmetric. The behaviour for area
greater than 0.4 fm2 has an increasing $Q_{s}^{2}$ projectile, and since it is
at fixed energy deposit, can only be explained by having a decrease in the
$Q_{s}^{2}$ target in the region of overlap – also shown in the upper middle
panel of Figure 16. It may seem surprising that the $Q_{s}^{2}$ in the proton
projectile and Pb target are so similar, but this is a consequence of higher
multiplicity events arising from large $Q_{s}^{2}$ fluctuations in the
projectile – see Ref. Nagle and Zajc (2019) for example. Arguments about the
expectations ignoring the very large spatial and $Q_{s}^{2}$ fluctuations in
the target need to be revisited.

In both the dilute-dense and dense-dense cases, the relationship between
$Q_{s}^{2}$ projectile and eccentricity is a modest anti-correlation, almost
uncorrelated in the dense-dense case. Thus, from this simple viewpoint, one
might expect a negative $\rho$ correlator. In contrast, the ip-glasma yields a
positive $\rho$ Giacalone  _et al._ (2020a). Thus, further work is warranted
to understand the source of this positive correlation.

Figure 16: Results from the ip-jazma model for $p$$+$Pb collision geometry in
the dilute-dense case. Details on the various panels is given in the main
text. Figure 17: Results from the ip-jazma model for $p$$+$Pb collision
geometry in the dense-dense case. Details on the various panels is given in
the main text.

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# Object Instance Identification in Dynamic Environments

Takuma Yagi, Md. Tasnimul Hasan, and Yoichi Sato
Industrial Institute of Science, The University of Tokyo
<EMAIL_ADDRESS>
###### Abstract

We study the problem of identifying object instances in a dynamic environment
where people interact with the objects. In such an environment, objects’
appearance changes dynamically by interaction with other entities, occlusion
by hands, background change, etc. This leads to a larger intra-instance
variation of appearance than in static environments. To discover the
challenges in this setting, we newly built a benchmark of more than 1,500
instances built on the EPIC-KITCHENS dataset which includes natural activities
and conducted an extensive analysis of it. Experimental results suggest that
(i) robustness against instance-specific appearance change (ii) integration of
low-level (_e.g_., color, texture) and high-level (_e.g_., object category)
features (iii) foreground feature selection on overlapping objects are
required for further improvement.

##  1 Introduction

The ability to identify whether an object appearing in an image is the same
instance as previously seen is one of the essential abilities to understand
physical environments over time. For example, when a robot is instructed by
the user to “bring my phone”, the robot needs to identify the phone that has
observed in the past from its appearance. This object instance identification
task offers several applications such as product search, robot control, and
assistive technology.

Previously, recognition of object instances has been primarily studied in a
static environment where the appearance of the object does not change across
time. However, in the real-world, objects are used for their intended use, and
their appearance change through interactions. For example, a mug may be filled
with liquids or washed by hands during its use, causing significant foreground
and background clutters (Figure 1, top row). Although identifying instances in
such a dynamic environment is demanded, challenges in this situation has not
been seriously discussed yet.

We analyze the problem of object instance identification under dynamic
environment where people interact with objects. To this end, we contribute a
new annotation built on the large-scale EPIC-KITCHENS dataset [3] that
contains natural cooking activities. We choose first-person videos as a
subject because it contains realistic hand-tool use and habitual activities
across times. The new EK-Instance dataset (Figure 1) consists of more than
1,500 object instances, and contains various clutter in the foreground
(_e.g_., pouring drinks) and background (_e.g_., overlap with other objects).

To discover the challenges in this setting, we introduce baseline methods
using the off-the-shelf metric learning and clustering, and conduct (1)
performance evaluation in EK-Instance dataset (2) cross-dataset transfer for
analysis. The results suggest that (i) robustness against instance-specific
appearance change (ii) integration of low-level and high-level features (iii)
foreground feature selection are required for further improvement.

Figure 1: Examples of EK-Instance dataset. Each row denotes images from same
instance.

##  2 Related Work

Instance recognition/detection in a monotonous background [8] and re-
identification in a static environment [2] have been studied, aiming for
applications such as product recognition and robot control. Numbers of
instance-level datasets have been proposed in parallel. Their source is
roughly divided into (i) recording under controlled environments and (ii)
Internet images. In the first group, at most, a few hundred instances were
collected on a turntable or by hand-held. Robustness against pose change and
lighting conditions is their primary interests [18]. Except for a few datasets
that intentionally collected varied backgrounds [17, 12], most datasets are
captured in a controlled environment where objects are captured without
interaction with monotonous background. The second group collects data from
E-Commerce sites for product recognition. Compared to the former, it is easier
to scale up the number of instances to tens of thousands of instances (_e.g_.,
[13]). However, for their original purpose, objects were shot with a simple
background without real-world clutters. Different from the above, we collect a
large number of object instances that appear in natural environments with
interaction.

##  3 EK-Instance Dataset

Figure 2: Criteria for determining an instance.

###  3.1 Definition of Instance

We first clarify what an instance refers to. Compared to the previous works
that captured objects apart from clutters, real-world environments include
corner cases caused by interaction and it is not trivial how to determine the
same instance. We propose four rules for consistent annotation.

(a) Identification by appearance: We define an instance as an object which has
a unique appearance that is distinguishable from other objects. Visually
identical objects (_e.g_., set of tableware) are counted as a single instance.

(b) Single object per bounding box: We use a bounding box that surrounds the
object just enough on determining the target object. When part of an object is
occluded, we just cover the visible region. If a bounding box surrounds
multiple objects (_e.g_., stacked dishes), it is considered invalid and
excluded from the evaluation (see Figure 2 (b)).

(c) Exception on unshared view: Ideally, a model should identify an object as
the same even if it is viewed from different directions. However, as shown in
Figure 2 (c), in some cases, it will be difficult to do so when there is no
common part across views, such as the colors of the front and back of the lid
being completely different. We exceptionally count as different instances in
such a situation.

(d) Priority on foreground: Some objects can be separated into multiple parts.
For example, a frying pan can be separated into a body and a lid. If we look
at it from a top-down view, it will be impossible to distinguish between (i) a
set of body and lid and (ii) the lid itself. While multiple interpretations
are possible, we give priority to the part on the foreground and count as the
same if objects share the foreground. In Figure 2 (d), we consider (i) and
(ii) as the same while counting the body alone as a different instance.

###  3.2 Annotation Procedure

Based on the aforementioned criteria, annotations were made on the EPIC-
KITCHENS dataset [3] which includes continuous cooking activities across
dates. Because it is too labor-intensive to manually annotate instances across
long videos, we adopt a semi-automatic procedure for annotation. Specifically,
a category-agnostic object detector [9] and an object tracker are used to
automatically extract and track the objects. Then we manually inspect whether
the obtained trajectories contain a valid object and the same instance as
other tracks. Finally, we sample detections by 1 fps and corrected the
bounding boxes.

As a result, we collected 1,554 object instances, 38,479 tracks, and 92,376
frames from the 20 hours of video from 23 camera wearers in the EPIC-KITCHENS
dataset. It consisted of containers (_e.g_., bottle), cooking tools (_e.g_.,
frying pan and paper roll), tableware (_e.g_., knife), food (_e.g_., apple),
and electronics (_e.g_., toaster). It also included objects which are
difficult to define in a clear category (_e.g_., fridge magnet, portafilter,
and product package).

##  4 Experiments

Using the EK-Instance dataset, we evaluated the instance identification task
in dynamic environments by building baseline models and comparing them across
datasets.

###  4.1 Problem Statement

The problem is formulated as clustering of image tracks to a group of
instances. Given a set of image tracks $\\{X_{1},\dots,X_{N}\\}$, we assign
cluster labels $\\{y_{1},\dots,y_{N}\\}(y_{i}\in\\{1,\dots,K\\})$, which is
grouped by instance. In a novel environment, the true number of instances will
not given in advance. Therefore, we formulated the task as clustering.

###  4.2 Image Track Encoder

We use ResNet-34 [7] pre-trained with the ImageNet dataset [4] as a backbone
network to extract frame-level features. We pass the images to the backbone
layers before the final average pooling layer and obtain a 512-dimensional
feature vector for each image. Next, they are averaged among images to
aggregate image-level features into a track-level feature vector. Finally, we
pass it to a single fully-connected layer to obtain a 256-dimensional track
embedding. We used two loss functions for comparison:

Additive angular margin (ArcFace) [5]: Loss based on classification.
Dicriminative features are learned by giving a penalty that the feature
vectors between different classes are separated from each other by a certain
amount or more.

N-pair [15]: Loss based on sample-wise comparison. Increase the similarity of
pairs of feature vectors in the same class while decreasing the similarity of
different classes.

In addition, we evaluated the ImageNet baseline which extracts 512-dimensional
feature vectors from ImageNet pre-trained model as a strong baseline model.

###  4.3 Clustering and Evaluation

We performed clustering using the embeddings from the trained models. We
compared Spherical K-means [6] and Hierarchical Agglomerative Clustering (HAC)
[11]. Cosine similarity was used as a similarity measure in both algorithms.
In Spherical K-means, the true number of clusters was given while in HAC the
threshold was adjusted by the validation set. We used Adjusted Mutual
Information (AMI) [16], unsupervised accuracy rate (ACC), Paired ($F_{P}$) and
BCubed F-score ($F_{B}$) [1] as metrics.

###  4.4 Evaluation on EK-Instance Dataset

Model | Clustering Method | AMI | ACC | $F_{P}$ | $F_{B}$
---|---|---|---|---|---
ImageNet | K-means | 0.755 | 0.590 | 0.547 | 0.628
HAC | 0.849 | 0.735 | 0.724 | 0.770
ArcFace | K-Means | 0.794 | 0.631 | 0.606 | 0.670
HAC | 0.892 | 0.796 | 0.784 | 0.827
N-pair | K-Means | 0.834 | 0.713 | 0.646 | 0.724
HAC | 0.924 | 0.854 | 0.847 | 0.874

Table 1: Performance on EK-Instance Dataset.

Table 1 shows the performance in the EK-Instance dataset. While ImageNet
without fine-tuning showed moderate performance, the fine-tuned model in the
training set showed better performance. In particular, the combination of
N-pair loss and HAC showed the best performance. The reason why HAC exceeded
K-means was that the K-means method tended to generate relatively even
clusters, while the actual frequency distribution was biased.

Figure 3: Example of obtained clusters.

Figure 4: Failed HAC merging attepnts. First and second row of each item show
tracks that are wrongly merged during HAC.

Figure 3 shows an example of clusters extracted by the combination of N-pair
and HAC. The model was not only robust to changes in the pose of the object
and occlusion by hand (upper) but was also able to absorb foreground clutter
such as putting a plastic bag on the trash can (lower).

However, some failure cases were seen as shown in Figure 4. First, there was
intra/inter-category confusion with similar colors and textures. In
particular, there were cases where completely different types of objects were
confused by superficial features. As a unique failure in dynamic environments,
confusion due to severe foreground clutter was seen when the same meal was
served in another container as in the second line. Moreover, when multiple
objects overlap, the model misunderstood the target object specified by a
bounding box (_e.g_., a cutting board and a knife on it).

Dataset | EK-Instance | Online Products | INSTRE | CORe50
---|---|---|---|---
| AMI | $F_{P}$ | $F_{B}$ | AMI | $F_{P}$ | $F_{B}$ | AMI | $F_{P}$ | $F_{B}$ | AMI | $F_{P}$ | $F_{B}$
ImageNet | 0.849 | 0.724 | 0.770 | 0.332 | 0.023 | 0.434 | 0.944 | 0.836 | 0.890 | 0.497 | 0.311 | 0.446
EK-Instance | 0.924 | 0.847 | 0.874 | 0.358 | 0.360 | 0.448 | 0.930 | 0.837 | 0.884 | 0.578 | 0.415 | 0.532
Online Products | 0.800 | 0.620 | 0.694 | 0.649 | 0.321 | 0.633 | 0.870 | 0.756 | 0.773 | 0.090 | 0.063 | 0.216
INSTRE | 0.795 | 0.634 | 0.703 | 0.304 | 0.047 | 0.445 | 0.984 | 0.963 | 0.972 | 0.667 | 0.440 | 0.599

Table 2: Results of cross-dataset transfer. Left column shows source dataset
and top row shows target dataset. Numbers in bold and blue show best and
second best results across source datasets, respectively.

###  4.5 Cross-Dataset Transfer

To investigate the properties of the learned feature representations in each
dataset, we evaluated the performance when a model trained in one dataset was
transferred to another dataset. We used Stanford Online Products (SOP) [13] as
a dataset using internet images, INSTRE [17] and CORe50 [12] as a dataset
containing instances on various backgrounds. CORe50 was used only for
evaluation.

As shown in Table 2, the models trained by SOP and INSTRE did not generalize
to EK-Instance (worse than ImageNet), while the model trained by EK-Instance
showed improvement in SOP and CORe50, showing that it can respond to both
foreground and background clutter.

##  5 Discussion

We draw the following suggestions: (i) Robustness against instance-specific
appearance change is needed: The newly constructed EK-Instance dataset showed
that the appearance of an object changes across interactions in a dynamic
environment. Towards practical object instance identification, it is necessary
to acquire not only viewpoint invariance but also robustness to changes in
appearance due to interaction. (ii) Both low-level/high-level features are
important: There was confusion between instances with similar low-level
features such as color and texture but from completely different categories.
We suspect that the category-level information was lost during training. A
more robust approach is needed that uses both low-level and high-level
features as clues for identification. (iii) Selective foreground feature
extraction is demanded: In this work, bounding boxes were used to indicate the
target object. However, when instances intersect with each other, confusion in
determining the target object was observed. An explicit method to selectively
filter features of the target object based on contextual information will be
demanded.

##  6 Conclusion

In this study, we tackled the problem of object instance identification in a
dynamic environment with human-object interaction. We proposed the EK-Instance
dataset to analyze its difficulty. Specifically, we formulated the problem as
clustering of image tracks and conducted an extensive analysis by
within/cross-dataset evaluation. The analyses have revealed the existing
biases in the trained model, and have discovered challenges for further
development such as robustness against appearance change, feature fusion, and
selective foreground feature extraction.

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##  Appendix A Details on EK-Instance Dataset

###  A.1 Statistics

As mentioned in the main text, we collected 1,554 instances, 38,479 tracks,
and 92,376 frames from 23 participants in total. Following the split used in
the EPIC-KITCHENS dataset [3], we split the dataset into training, validation,
and test set. Concretely, we used instances from P33, P34, and P36 for
validation and P09, P11, P18, and P32 for testing. Table 3 shows the
statistics of each split. Figure 5 and 6 show the images of the instances
included in validation and test set, respectively. By collecting more than
1,500 instances, we were able to train a metric learning-based neural network
model. In addition, the subject-based split allowed us to evaluate the
instance identification performance in novel environments.

The frequency of the instances is highly biased as it reflects the occurrence
rate of objects in real-world videos. Figure 7 and 8 shows the frequency of
the number of images per track and the number of tracks per instance,
respectively. It can be seen that most instances appear a small number of
times in a very short time, whereas a very small number of instances appear a
long time and a large number of times.

Split | #participants | #instances | #tracks | #frames
---|---|---|---|---
Training | 16 | 1051 | 27234 | 64300
Validation | 3 | 193 | 4647 | 11851
Test | 4 | 310 | 6598 | 16225
Total | 23 | 1554 | 38479 | 92376
Table 3: Statistics of each split.

###  A.2 Additional Examples

Figure 9 shows additional examples on the instances included in the EK-
Instance dataset. Not only changes in the posture of the object and the
lighting environment, but also various changes such as disturbances caused by
hands and other objects, changes in appearance due to operation (_e.g_., using
the contents, peeling off the packaging), changes in the surrounding
background, etc. are within the same instance. It is characteristic that it
occurs in.

##  Appendix B Details on Baseline Methods

###  B.1 Normalized Softmax (N-Softmax) Loss [19]

Normalized softmax loss is one of the simplest loss function for metric
learning, extending the cross-entropy loss for classification task. It is
almost the same as the cross-entropy loss except the embedded vector and the
weight vector of the final layer are L2 normalized.

$L_{\mathrm{nsoftmax}}=-\frac{1}{N}\sum_{i=1}^{N}\log{\frac{e^{W_{y_{i}}^{\mathsf{T}}{\bf
z}^{\prime}_{i}/\tau}}{\sum_{k=1}^{K}e^{W_{k}^{\mathsf{T}}{\bf
z}^{\prime}_{i}/\tau}}},$

where ${\bf z}^{\prime}_{i}=\frac{{\bf z}_{i}}{\|{\bf z}_{i}\|}$ is a vector
by applying L2-normalization to ${\bf z}_{i}$ and $\tau$ denotes a temperature
parameter that adjusts the scale of the numerator and denominator. We set
$\tau=0.2$.

###  B.2 Additive Angular Margin (ArcFace) Loss [5]

ArcFace loss is described as follows:

$L_{\mathrm{arcface}}=-\frac{1}{N}\sum_{i=1}^{N}\log{\frac{e^{\cos{(\theta_{y_{i}}+m)}/\tau}}{e^{\cos{(\theta_{y_{i}}+m)}/\tau}+\sum_{k=1,k\neq
y_{i}}^{K}e^{\cos{\theta_{j}}/\tau}}},$

where $\theta_{k}$ denotes the angle between L2-normalized weight $W_{k}$ and
a feature vector ${\bf z}^{\prime}_{i}$ which belongs to class $y_{i}$. $m$
denotes a margin term to increase the inter-class distance. This loss function
tries to maintain a feature vector to have a margin of $m$ against all the
feature vectors that belongs to a different class. As a result, feature
vectors of a same class will be distributed in a narrower space. We set
$m=0.5$ and $s=1/30$.

###  B.3 N-pair Loss [15]

Different from the classification-based losses above, the n-pair loss tries to
increase the similarity between feature vectors from the same class while
decrease the similarity between feature vectors from different classes.
Especially, the n-pair loss enables efficient training by taking all the
possible pairs within the batch, without using classification weight $W$.

$L_{\mathrm{npair}}=-\frac{1}{N}\sum_{i=1}^{N}\log{\frac{\sum_{j=1,y_{j}=y_{i}}^{N}e^{{\bf
z}_{i}^{\mathsf{T}}{\bf z}_{j}/\tau}}{\sum_{j=1}^{N}e^{{\bf
z}_{i}^{\mathsf{T}}{\bf z}_{j}/\tau}}}.$

We set $\tau=0.07$.

##  Appendix C Details on Evaluation

###  C.1 Data Split

We performed clustering within the images from the same participant and
reported the average score among them.

###  C.2 Evaluation Metrics

We describe the four evaluation metrics in detail:

Adjusted Mutual Information (AMI) [16]: A metric based on mutual information.
Given a set of ground truth clusters $C=\\{c_{i}\\}_{i=1}^{N}$ and predicted
clusters $K=\\{k_{i}\\}_{i=1}^{N}$, we calculate AMI as follows:

$AMI=\frac{I(C,K)-\mathbb{E}[I(C,K)]}{\frac{1}{2}[H(C)+H(K)]-\mathbb{E}[I(C,K)]},$

$I$ and $H$ denote mutual information and entropy, respectively. AMI will be
1.0 if the ground truth clusters and the predicted clusters are identical.

Unsupervised Accuracy (ACC): Given a set of ground truth clusters and a set of
predicted clusters, the maximum accuracy can be achieved when a one-to-one
correspondence between the two sets of clusters is made. Hungarian algorithm
[10] is used to calculate the optimal assignment $m$.

$ACC=\max_{m}\frac{\sum_{i=1}^{N}\mathbf{1}\\{c_{i}=m(k_{i})\\}}{N}.$

Paired F-score ($F_{P}$): Given $N$ samples, takes all the $\frac{1}{2}N(N-1)$
combinations and counts them as correct if the relationship between the two
samples is the same in predicted clusters and ground truth clusters.

BCubed F-measure ($F_{B}$) [1]: This metric calculates cluster-wise precision
and recalls while $F_{P}$ takes all the pairs regardless of the predicted
clusters. Specifically, the correctness between samples $i$ and $j$ will be
defined as follows:

$Correctness\>(i,j)=\begin{cases}1&\mathrm{if}\>L(i)\>=\>L(j)\>\mathrm{and}\>C(i)\>=\>C(j)\\\
0&\mathrm{otherwise},\end{cases}$

where $L(i)$ and $C(i)$ refers to the $i$-th predicted clusters and ground
truth clusters. Finally, BCubed Precision and BCubed Recall will be calculated
as follows:

$BCubed\>Precision=\frac{1}{N}\sum_{i}^{N}\sum_{j\in
C(i)}\frac{Correctness\>(i,j)}{|C(i)|},$
$BCubed\>Recall=\frac{1}{N}\sum_{i}^{N}\sum_{j\in
L(i)}\frac{Correctness\>(i,j)}{|L(i)|}.$

##  Appendix D Details on Datasets

#### Stanford Online Products [13]:

This dataset is composed of 120,053 images from 22,634 online product
categories collected from eBay. Following the original split, we used 11,318
categories for training and 11,316 categories for testing, while the
hyperparameter was tuned by further partitioning the training set for
validation.

#### INSTRE [17]:

This dataset consists of distinctive objects such as architecture, planar
objects, and toys. The Internet images and manually collected images are mixed
and divided into three subsets based on filming conditions. Notably, manually
collected images were intentionally shot in 25 different backgrounds. For the
evaluation, we used 100 instances from the INSTRE-S1 subset which contains
11,011 images, and 50 instances each are used for training and testing.

#### CORe50 [12]:

This dataset contains 50 hand-held objects belonging to 10 categories such as
scissors. Although the number of instances is as small as 50, we used as a
target data set because it was intentionally photographed in various
backgrounds with hand occlusion, which shares the difficulties in our EK-
Instance dataset. Since this data set consists of videos for a total of 550
trials, images were extracted at 1 fps to compose a image track. Since the
bounding box of the object attached to the data set was cut out looser than
the target object, we crop 28 pixels at each of the four corners of the
original 128-pixel square bounding box to form an input.

##  Appendix E Visualization of Learned Features

To understand which region the model is focusing on, we applied Grad-CAM [14]
and visualized the activation heatmap of each predicted instance (class). For
simplicity, we used the model trained with N-softmax loss. Figure 10 shows the
results of discriminative regions taken from the last layer of the ResNet-34
backbone.

First, we found that the model focused on the peripheral region of specific
objects such as dishes and mugs (1-3rd row). Such unchanged regions may be
discovered during the training since such objects typically have foreground
clutters on their centers. Similarly, the model focused on regions other than
hands that are not discriminative for instance identification (4th row). On
the other hand, unreasonably sharp regions are generated in some instances
even though the entire region contains unique patterns (5th and 6th row).
Furthermore, as mentioned in the failure examples, the model focused on
backgrounds in some instances, leaving room for improvement.

Figure 5: Images used in validation set (193 instances).

Figure 6: Images used in test set (310 instances).

Figure 7: Distribution of number of images per track.

Figure 8: Distribution of number of tracks per instance.

Figure 9: Additional examples of EK-Instance dataset. Each row refers to same
instance.

Figure 10: Grad-CAM visualization on training set. First and second column are
input image and its support for ground-truth category. Other columns are top-5
“hard” classes which showed high instance classification probability and their
corresponding supports, respectively.