# Charmonia production in ALICE

Christophe Suire  for the ALICE Collaboration  Institut de Physique Nucléaire
d’Orsay, CNRS-IN2P3, Université Paris-Sud, France

###### Abstract

Quarkonia states are expected to provide essential information on the
properties of the high-density strongly-interacting system formed in the early
stages of high-energy heavy-ion collisions. ALICE is the LHC experiment
dedicated to the study of nucleus-nucleus collisions and can study charmonia
at forward rapidity ($2.5<y<4$) via the $\mu^{+}\mu^{-}$ decay channel and at
mid rapidity ($|y|<0.9$) via the $e^{+}e^{-}$ decay channel. In both cases
charmonia are measured down to zero transverse momentum. The inclusive
$J/\psi$ production as a function of transverse momentum and rapidity in pp
collisions at $\sqrt{s}=$ 2.76 and 7 TeV are presented. For pp collisions at
$\sqrt{s}=$ 7 TeV, the inclusive $J/\psi$ production as a function of the
charged particle multiplicity, the inclusive $J/\psi$ polarization at forward
rapidity and the $J/\psi$ prompt to non-prompt fraction are discussed.

Finally, the analysis of the inclusive $J/\psi$ production in the Pb-Pb data
collected fall 2011 at a center of mass energy of $\sqrt{s_{NN}}=2.76$ TeV is
presented. Results on the nuclear modification factor are then shown as a
function of centrality, transverse momentum and rapidity and compared to model
predictions. First results on inclusive $J/\psi$ elliptic flow are given.

###### keywords:

Hadron-induced high- and super-high-energy interactions, Relativistic heavy-
ion collisions, Quark-Gluon plasma, $J/\psi$ production and suppression
mechanisms.

##  1 Charmonia in heavy-ion collisions

Charmonia suppression via color-screening of the heavy-quark potential was
originally proposed as a probe of the QCD matter formed in relativistic heavy-
ion collisions in 1986 [1]. $J/\psi$ production was extensively studied at the
Super Proton Synchrotron (SPS) and at the Relativistic Heavy Ion Collider
(RHIC). Indeed, $J/\psi$ suppression in most central heavy-ion collisions was
observed over a large energy range ($\approx$ 20 to 200 GeV/$c$). The LHC has
opened a new energy regime for the study of quarkonium in heavy-ion
collisions. In a Pb-Pb collision at $\sqrt{s_{NN}}=2.76$ TeV, an average of
one $J/\psi$ particle is expected to be produced in every central Pb-Pb
collision, together with about 100 c$\bar{\rm{c}}$ pairs. Several models [2,
3, 4, 5] have included, already at RHIC energy, a $J/\psi$ regeneration
component from deconfined charm quarks in the medium which counteracts the
J/$\psi$ suppression in a QGP. At LHC, this regeneration component may become
important, even dominant.

The in-medium modification of the $J/\psi$ production can be quantified with
the nuclear modification factor $R_{\mathrm{AA}}$, defined as the $J/\psi$
yield measured in nucleus-nucleus collisions divided by the yield measured in
pp collisions and the number of binary nucleon-nucleon collisions occurring in
the nucleus-nucleus collision. To interpret the $R_{\mathrm{AA}}$, one must
keep in mind the following points. First, prompt $J/\psi$ production in
hadronic interactions consists of the sum of direct $J/\psi$ ($\approx$ 65%)
and excited $\rm{c\overline{c}}$ states such as $\chi_{\rm{c}}$ and
$\psi\rm{(2S)}$ ($\approx$ 35%). Since the $\chi_{\rm{c}}$ and $\psi\rm{(2S)}$
have lower dissociation temperatures than the $J/\psi$, a $J/\psi$
$R_{\mathrm{AA}}$ measurement around 0.65 is compatible within errors with the
suppression of excited states only. In addition to these prompt $J/\psi$ one
should also take into account that a non-prompt component from beauty hadron
decays is present at LHC energy. The $R_{\mathrm{AA}}$ includes cold nuclear
matter (CNM) effects, dominated by nuclear absorption and (anti-) shadowing.
These CNM effects can be responsible for $J/\psi$ suppression independently
from the creation of a deconfined medium. To quantify CNM effects, proton-
nucleus collisions are needed. At SPS energy, the observed $J/\psi$
suppression would be compatible with the dissociation of excited states, once
the CNM effects are taken into account. At RHIC, a $J/\psi$ $R_{\mathrm{AA}}$
of $\approx$ 0.25 in most central Au-Au collisions was measured by the PHENIX
experiment [6] with a strong centrality dependence. After estimating the
correction due to the CNM effects, the suppression of direct $J/\psi$ is at
least $\approx$ 40% or more. At the LHC, $J/\psi$ are abundantly produced and
detailed studies of its production are possible in both elementary and heavy-
ion collisions, such as azimuthal asymmetry, polarization,
$R_{\mathrm{AA}}$dependence on rapidity and on transverse momentum, etc. Such
studies may give us some answers about the balance between suppression and
recombination mechanisms of $J/\psi$.

The asymmetry the azimuthal distribution of $J/\psi$ in the plane
perpendicular to the beam direction, the so-called elliptic flow or v2is
indeed a very interesting experimental observable. When heavy-ions collide at
finite impact parameter (non-central collisions), the geometrical overlap
region and therefore the initial matter distribution is anisotropic and is
converted into a momentum anisotropy of the produced particles. The possible
onset of $J/\psi$ production via recombination mechanisms should be, according
to models, accompanied with a non-zero or possibly large v2 value [7] at low
$p_{{\mathrm{T}}}$. Indeed, if charm quarks reach some level of thermalization
in the medium, they may acquire an elliptic flow that can be further
transferred to the $J/\psi$ assuming the $J/\psi$ is formed via recombination.

In the following section, the ALICE experiment and the data samples will be
described. Then, $J/\psi$ production results in pp collisions at $\sqrt{s}=$
2.76 TeV and 7 TeV will be presented. The $J/\psi$ production in Pb-Pb
collisions at$\sqrt{s_{\mathrm{NN}}}=$ 2.76 will be studied through the
$R_{\mathrm{AA}}$ dependence on centrality, $p_{{\mathrm{T}}}$ and $y$.
Finally, $J/\psi$ elliptic flow measurement will be shown.

##  2 Experimental apparatus and data sample

ALICE is a general purpose heavy-ion experiment and is described in [8]. It
consists of a central part covering the pseudo-rapidity $|\eta|<0.9$ and a
muon spectrometer covering $-4<\eta<-2.5$. At forward (mid) rapidity, $J/\psi$
production is measured in the dimuon (dielectron) decay channel; in both cases
the $p_{{\mathrm{T}}}$ coverage extends down to zero. Only detectors that are
relevant to the analysis will be presented.

At mid rapidity, the $J/\psi$ $\longrightarrow e^{+}e^{-}$ analysis makes use
of the high precision tracking and particle identification of the Inner
Tracking System (ITS) and the Time Projection Chamber (TPC). The ITS consists
of six cylindrical layers of silicon detectors; at a radius 3.9 and 7.6 cm,
the first two layers are equipped with silicon pixels (SPD), then two layers
of silicon drifts at radius 15 and 23.9 cm and finally, two layers of silicon
strips at radius 38 and 43 cm. Its main tasks are the primary and secondary
vertex reconstruction; the resolution on the primary vertex ranges from 100
$\mu$m (pp collisions) to 10 $\mu$m (central Pb-Pb collisions). The SPD has
triggering capabilities and can provide a signal at level 0. The large
cylindrical TPC has full azimuthal coverage and extends from z = -2.50 m to z
= 2.50 m 111The z axis is defined here as the beam line axis in the counter
clockwise direction and its origin is at the center of the ALICE detector..
The TPC radial coverage ranges from r = 85 cm to r = 247 cm. This large drift
detector is the main track reconstruction device at central rapidity since it
can provide up to 159 space points per track. Particle identification (PID) is
achieved via the measurement of the specific energy loss (dE/dx) of particles
in the detector gas (Ne/CO2/N2). The excellent dE/dx resolution of $\approx$
5% allows to identify electrons by using inclusion cut around the Bethe-Bloch
fit for electrons and exclusion cuts for protons and pions. ALICE has further
capabilities to improve electron identification and triggering thanks to the
Time-Of-Flight (TOF), the Transition Radiation Detector (TRD) and the
Electromagnetic Calorimeter (EMCAL) detectors. However, these detectors have
not been used in the analysis presented here.

At forward rapidity ($2.5<$ $y$ $<4$) the production of quarkonium states is
measured in the muon spectrometer 222In the ALICE reference frame, the muon
spectrometer covers a negative $\eta$ range and consequently a negative $y$
range. We have chosen to present our results with a positive $y$ notation..
The spectrometer consists of a ten interaction length thick absorber ( -0.9 m
$<$ z $<$ -5.0 m) filtering the muons in front of five tracking stations (MCH)
made of two planes of cathode pad chambers each. The third station is located
inside a dipole magnet with a 3 Tm field integral. The MCH chambers are
positioned between z=-5.2 and z=-14.4 m. The tracking apparatus is completed
by a triggering system (MTR) made of two stations, located at z=-16.1 and
z=-17.1 m, each equipped of two planes of resistive plate chambers. The MTR
chambers are downstream of a 1.2 m thick iron wall, which absorbs secondary
hadrons escaping from the front absorber and low momentum muons coming mainly
from $\pi$ and K decays. Throughout its full length, a conical absorber made
of tungsten, lead and steel protects the muon spectrometer against secondary
particles produced by the interaction of large-$\eta$ primaries in the beam
pipe. The forward VZERO detectors, two arrays of 32 scintillator tiles
covering the range $2.8\leq\eta\leq 5.1$ (VZERO-A) and $-3.7\leq\eta\leq-1.7$
(VZERO-C), are positioned at z=340 and z=-90 cm. And finally, the zero degree
calorimeters (ZDC) placed 116 m down and up-stream ALICE can detect spectator
neutrons and protons.

In proton-proton collisions, the minimum bias (MB) trigger uses information of
the SPD and VZERO detectors. It is defined as the logical OR of the three
following conditions: (i) a signal in two readout chips in the outer layer of
the SPD, (ii) a signal in VZERO-A, (iii) a signal in VZERO-C. This MB trigger
requires the coincidence of the crossing of two proton bunches at the
experiment interaction point (IP). ALICE MB trigger selects about 86% of the
proton-proton inelastic cross section. Specific cross sections were measured
during van der Meer scan in pp collisions at 7 and 2.76 TeV and allowed to
determine the absolute normalization of the inclusive $J/\psi$ cross section.
The muon minimum bias trigger ($\mu$-MB) requires, in addition to the MB
conditions given above, a signal in the MTR system. The MTR can reconstruct a
trigger track, determine its $p_{{\mathrm{T}}}$ and select different
thresholds ($p_{{\mathrm{T}}}$ $\approx$ 0.5, 1 and 4 GeV/$c$). The $\mu$-MB
trigger helps to take advantage of the full luminosity delivered by the LHC in
the muon spectrometer. The MB trigger used in Pb-Pb collisions collected in
2010 requires the logical AND of the conditions (i),(ii) and (iii) given
above. The centrality of the collision is determined from the amplitude of the
VZERO signal fitted with a geometrical-Glauber model [9]. In 2011, the MB
conditions were reduced to the AND of conditions (ii) and (iii) but additional
requirements were added to select rare events. In particular, event
multiplicity and dimuon triggers were added and ZDC were used for rejecting
electromagnetic Pb-Pb interactions and satellite Pb-Pb collisions. Once the
centrality selection cut has been applied, triggers are fully efficient with
negligible contamination.

##  3 $J/\psi$ production in pp collisions

The $J/\psi$ production in pp collisions is extensively studied in ALICE and
only a selection of the available results will be presented in this section.
Further details on the related analysis can be found in [10].

The inclusive $J/\psi$ production was measured in pp collisions at $\sqrt{s}=$
7 TeV in the dimuon and dielectron channels in the rapidity ranges $2.5<y<4$
and $|y|<0.9$ down to $p_{{\mathrm{T}}}$ = 0. The analysis was made with an
integrated luminosity $\mathcal{L}_{\rm int}\approx 16\;(6)\;\mathrm{nb}^{-1}$
in the dimuon (dielectron) channel. The measured cross section values are
$\sigma_{J/\psi}(|y|<0.9)=10.7\pm 1.0(\mathrm{stat.})\pm
1.6(\mathrm{syst.})^{+1.6}_{-2.3}(\mathrm{syst.pol.})\;\mu\mathrm{b}$ and
$\sigma_{J/\psi}(2.5<y<4.)=6.31\pm 0.25(\mathrm{stat.})\pm
0.76(\mathrm{syst.})^{+0.95}_{-1.96}(\mathrm{syst.pol.})\;\mu\mathrm{b}$. At
forward rapidity differential cross section
d${}^{2}\sigma$/d$p_{{\mathrm{T}}}$d$y$ measurement from ALICE fully overlaps
with LHCb and a good agreement is found. At mid rapidity, the situation is
different since ATLAS and CMS cannot measure $J/\psi$ with $p_{{\mathrm{T}}}$
$\lesssim$ 6 GeV/$c$. Thus combining ALICE and CMS/ATLAS data offers a rather
complete inclusive $J/\psi$ production measurement over a large rapidity
range. These comparisons are available in [11].

The same analysis was carried out with pp collisions at $\sqrt{s}=$ 2.76 TeV
collected in March 2011. Since the center of mass energy per nucleon-nucleon
collisions is identical to the one of the Pb-Pb collisions, this analysis
provides an essential reference data to measure the $J/\psi$ nuclear
modification factor. The integrated luminosity for the analysis is
$\mathcal{L}_{\rm int}\approx 20\;(1)\;\mathrm{nb}^{-1}$ in the dimuon
(dielectron) channel. The integrated cross sections are
$\sigma_{J/\psi}(|y|<0.9)=6.71\pm 1.24(\mathrm{stat.})\pm
1.22(\mathrm{syst.})^{+1.01}_{-1.41}(\mathrm{syst.pol.})\;\mu\mathrm{b}$ and
$\sigma_{J/\psi}(2.5<y<4.)=3.34\pm 0.13(\mathrm{stat.})\pm
0.28(\mathrm{syst.})^{+0.53}_{-1.07}(\mathrm{syst.pol.})\;\mu\mathrm{b}$. Note
that the uncertainties quoted here on the pp measurement are one of the main
source of uncertainty of the nuclear modification factor discussed in the next
section. The differential cross sections
d${}^{2}\sigma$/d$p_{{\mathrm{T}}}$d$y$ have been extracted down to
$p_{{\mathrm{T}}}$ = 0 at both rapidities. These results are compared to a
theoretical model, NRQCD calculation that includes Color Singlet and Color
Octet terms at NLO, which describes reasonably well the measurement at
$\sqrt{s}=$ 2.76 TeV and also the one at $\sqrt{s}=$ 7 TeV [12].

In the previous results, one could remark that the $J/\psi$ cross section has
a large uncertainty related to the unknown polarization. ALICE has studied
$J/\psi$ polarization in pp collisions $\sqrt{s}=$ 7 TeV in the dimuon
channel. Measurements of the polar and azimuthal angle distributions of the
decay muons allowed us to extract the $J/\psi$ polarization for 2 $<$
$p_{{\mathrm{T}}}$ $<$ 8 GeV/$c$ and 2.5 $<$ $y$ $<$ 4\. The parameters
describing the $J/\psi$ polarizations are consistent with zero in the
kinematic range under study [13]. This measurement is, at the present date,
the only $J/\psi$ polarization measurement at the LHC. It is crucial in the
near future to extend the polarization measurement down to zero
$p_{{\mathrm{T}}}$ and to high $p_{{\mathrm{T}}}$ in order to provide more
stringent tests to theoretical calculations. In addition, since the pp cross
section enters the nuclear modification factor calculation, the polarization,
if different from zero, may have a strong impact at low transverse momentum.
Such a measurement needs a large statistic and strengthens the requirement to
collect a large amount of data at the same center of mass energy as the Pb-Pb
collisions.

An interesting feature of the $J/\psi$ production in pp collisions at
$\sqrt{s}=$ 7 TeV arises from its dependence on the charged particle
multiplicity. The d$N_{\rm{ch}}$/d$\eta$ is calculated from the number of
tracks reconstructed in $|\eta|<1$ using pairs of hits (tracklets) in the SPD.
These measurements were performed at both rapidities in the dimuon and
dielectron channels. Expressed in terms of the relative $J/\psi$ yield
$\frac{\mathrm{d}N_{J/\psi}/\mathrm{d}\eta}{\langle\mathrm{d}N_{J/\psi}/\mathrm{d}\eta\rangle}$
as a function of the relative charged multiplicity
$\frac{\mathrm{d}N_{\mathrm{ch}}/\mathrm{d}\eta}{\langle\mathrm{d}N_{\mathrm{ch}}/\mathrm{d}\eta\rangle}$,
a linear increase is clearly seen at both rapidities. For
$\mathrm{d}N_{\mathrm{ch}}/\mathrm{d}\eta/\langle\mathrm{d}N_{\mathrm{ch}}/\mathrm{d}\eta\rangle=4$
($\approx 24/6$), the relative $J/\psi$ yield is enhanced by a factor of about
5 at forward rapidity and about 8 at mid rapidity. This trend is not
reproduced by PYTHIA 6.4.25 in the Perugia 2011 tune which exhibits an
opposite tendency, i.e. a decrease of the $J/\psi$ multiplicity with respect
to the event multiplicity [14]. One could infer that the $J/\psi$ production
is strongly connected with the underlying hadronic activity. Whether this
hadronic activity comes from multiple parton interactions remains an open
question. Further investigations are needed to better understand this
measurement that challenges our understanding of the $J/\psi$ production in pp
collisions. In particular, the event multiplicity dependence should be
completed by the $p_{{\mathrm{T}}}$ dependence and extended to the open charm
cross section (e.g. D mesons).

All the results presented up to now refer to inclusive $J/\psi$ production
which sums three distinct contributions: the prompt $J/\psi$ produced directly
in the pp collisions, the prompt $J/\psi$ produced indirectly via the decay of
heavier charmonia states and the non-prompt $J/\psi$ produced in the decay of
beauty hadrons. At central rapidity ($|y|<0.9$), the measurement of the non-
prompt $J/\psi$ was achieved in pp collisions at $\sqrt{s}=$ 7 TeV for $1.3<$
$p_{{\mathrm{T}}}$ $<10$ GeV/$c$. This measurement is only accessible in ALICE
since the other experiments cannot detect $J/\psi$ at mid rapidity below a
$p_{{\mathrm{T}}}$ of 6.5 GeV/$c$ where most of the cross section lies. The
integrated luminosity for the analysis is $\mathcal{L}_{\rm
int}=5.6\mathrm{nb}^{-1}$. This measurement relies on the discrimination of
$J/\psi$ produced detached from the primary vertex of the pp collisions thanks
to the good spatial resolution of the ITS. By fitting simultaneously the
invariant mass spectra and the pseudo-proper decay length of the reconstructed
$J/\psi$, one can can measure the relative abundances of prompt and non-prompt
$J/\psi$, and the background. The fraction of $J/\psi$ from beauty hadrons
($f_{\mathrm{B}}$) in the measured kinematic range is about 15% with a strong
$p_{{\mathrm{T}}}$ dependence. Then, $f_{\mathrm{B}}$ is combined with the the
inclusive $J/\psi$ cross section measured in [11] to extract the prompt
$J/\psi$ cross section
$\sigma_{J/\psi}^{\mathrm{prompt}}(|y|<0.9,p_{\mathrm{T}}>1.3{\mathrm{~{}GeV}/c})=7.2\pm
0.7(\mathrm{stat.})\pm
1.0(\mathrm{syst.})^{+1.3}_{-1.2}(\mathrm{syst.pol.})\;\mu\mathrm{b}$.
Comparisons with models lead to a good description of the prompt $J/\psi$
dependence with $p_{{\mathrm{T}}}$ and of the total beauty cross section [15].

##  4 Nuclear modification factor in Pb-Pb collisions at
$\sqrt{s_{\mathrm{NN}}}=$ 2.76 TeV

Inclusive $J/\psi$ production was studied in Pb-Pb collisions at 2.76 TeV at
mid and forward rapidity in the dielectron and dimuon decay channels using
respectively an integrated luminosity $\mathcal{L}_{\rm int}\approx$ 2.1 and
$\mathcal{L}_{\rm int}\approx 70$ $\mu$b-1. A crucial feature of the ALICE
detector is to measure, in both channels, the $J/\psi$ production down to
$p_{{\mathrm{T}}}$ = 0 GeV/c. The large data sample analyzed in the dimuon
channel allowed us to perform differential analysis of the nuclear
modification factor ($R_{\mathrm{AA}}$) as function of centrality,
$p_{{\mathrm{T}}}$, and $y$. In the dielectron channel, only the centrality
dependence in 3 centrality classes (0–10%, 10–40% and 40–80%) could be
achieved. One should note that the acceptance times efficiency factor in the
dielectron (dimuon) channel is quite high $\approx 8\%$ ($14\%$) and weakly
depends on the collision centrality with a maximum relative loss of 12% (8%)
from peripheral to most central collisions. Details on both analysis are given
in [16].

Figure 1: (Color online) Inclusive $J/\psi$ $R_{\rm{AA}}$ measured in Pb-Pb
collisions at $\sqrt{s_{\mathrm{NN}}}=2.76$ TeV at forward and mid rapidity in
ALICE is shown on the left side. On the right side the $J/\psi$
$R_{\mathrm{AA}}$ at high-$p_{{\mathrm{T}}}$ from ALICE at forward $y$ is
compared to one measured by CMS at central $y$.

On the left side of Fig. 1, the inclusive $J/\psi$ $R_{\mathrm{AA}}$ is shown
as a function of the number of nucleons participating in the collision [9] at
mid and forward rapidity. At forward rapidity, a clear suppression is seen for
$N_{\mathrm{part}}$ $>70$ with almost no centrality dependence. These results
show a good agreement with the ones published in [17] based on
$\mathcal{L}_{\rm int}\approx 2.9$ $\mu$b-1 collected in 2010. At mid
rapidity, a similar pattern could be possible but the coarser centrality
classes and larger uncertainties prevent to draw any firm conclusion. The
centrality integrated $J/\psi$ $R_{\mathrm{AA}}$ at forward and mid rapidity
are $R^{0\%-90\%}_{\rm{AA}}=0.497\pm 0.006\rm{(stat.)}\pm 0.078\rm{(syst.)}$
and $R^{0\%-80\%}_{\rm{AA}}=0.66\pm 0.10\rm{(stat.)}\pm 0.24\rm{(syst.)}$. In
both cases, the systematic uncertainty is dominated by the pp reference. On
the right side of Fig. 1, the centrality dependence of $J/\psi$
$R_{\mathrm{AA}}$ at high-$p_{{\mathrm{T}}}$ is compared between ALICE and CMS
[18]. A larger suppression, $R_{\mathrm{AA}}$ $\approx 0.25-0.30$, is measured
in the most central collisions with a clear centrality dependence. One could
see here an indication that the $J/\psi$ $R_{\mathrm{AA}}$ is
$p_{{\mathrm{T}}}$ dependent at forward rapidity and possibly at mid rapidity;
selecting high-$p_{{\mathrm{T}}}$ $J/\psi$ drives down the $R_{\mathrm{AA}}$.

The $p_{{\mathrm{T}}}$ dependence of the $J/\psi$ $R_{\mathrm{AA}}$ is
confirmed and can be better observed in Fig. 2 (left side). The inclusive
$J/\psi$ $R_{\mathrm{AA}}$ is shown as a function of $p_{{\mathrm{T}}}$ for
the 0%–90% most central Pb-Pb collisions and exhibits a decrease from 0.6 to
0.35 approximately. At high-$p_{{\mathrm{T}}}$ a rather direct comparison with
CMS results [18] is possible; the only difference is that the CMS measurement
covers a more central rapidity range ($1.6<|y|<2.4$). A reasonable agreement
between the two measurements is found for high-$p_{{\mathrm{T}}}$ $J/\psi$
$R_{\mathrm{AA}}$. For $p_{{\mathrm{T}}}$ smaller than 4 GeV/$c$, the
difference with PHENIX measurement [19] is striking. The PHENIX result concern
the 0%–20% most central Au-Au collisions whereas the ALICE result is for a
much wider centrality range (0%–90%). However, the bulk of the $J/\psi$
production ($\approx 60\%$) occurring in 0%–20% most central collisions, the
comparison remains meaningful. In addition, work is ongoing to extract the
$R_{\mathrm{AA}}$ versus $p_{{\mathrm{T}}}$ in smaller centrality classes.

The $J/\psi$ $R_{\mathrm{AA}}$ dependence on rapidity has been measured over a
wide range thanks to the combination of our measurement in the central barrel
and in the muon spectrometer, and is displayed on the right side of Fig. 2. At
forward rapidity, the $J/\psi$ $R_{\mathrm{AA}}$ decreases by $\approx 40\%$
from $y=2.5$ to $y=4$. The measurement at mid rapidity, because of its large
uncertainties, does not allow to draw a clear conclusion but hints towards a
rather flat behavior between $y$ = 2.5 and $y$ = 0. On the same figure, an
estimate of the $J/\psi$ $R_{\mathrm{AA}}$ due only to shadowing effects is
given for two models. Indeed at LHC energies, modification of the gluon
distribution function is dominated by shadowing effects [20]. The first model
is a Next to Leading Order calculation within the Color Evaporation Model [21]
with the EPS09 nuclear PDF (nPDF). The second model is a Leading Order
calculation within the CS Model [22] with the nDSg nPDF. In the first model,
the upper and lower limits correspond to the uncertainty of the EPS09 nPDF,
and in the second model the band covers the uncertainty in the factorization
scale of the for nDSg PDF.

Figure 2: (Color online) ALICE $p_{{\mathrm{T}}}$ dependence of the inclusive
$J/\psi$ $R_{\mathrm{AA}}$ measured in Pb-Pb collisions at
$\sqrt{s_{\mathrm{NN}}}=2.76$ TeV is compared to PHENIX and CMS measurements
on the left side. On the right side, the rapidity dependence of the $J/\psi$
$R_{\mathrm{AA}}$ measured by ALICE is compared with model predictions [21,
22] that implement only shadowing effects.

One could not exclude that shadowing effects are responsible for a large part
of the $J/\psi$ suppression observed in $R_{\mathrm{AA}}$ from $y$ = 0 to $y$
$\approx$ 3, this would imply that the expected color screening $J/\psi$
suppression observed at lower energy (RHIC) or higher $p_{{\mathrm{T}}}$ (CMS)
is either small or compensated by recombination mechanisms. In the rapidity
range from 3 to 4, our results show that the suppression goes beyond the
shadowing-only prediction given by models with our current knowledge of nPDF.

The influence of the contribution of beauty hadron feed-down to the inclusive
$J/\psi$ yield in our $y$ and $p_{{\mathrm{T}}}$ range was estimated. Non-
prompt $J/\psi$ are indeed different since their suppression or production is
insensitive to color screening or recombination phenomena that are expected to
occur in the hot and dense medium created in the Pb-Pb collisions. The beauty
hadron decay mostly occurs outside the fireball, and a measurement of the non-
prompt $J/\psi$ $R_{\mathrm{AA}}$ is connected to the beauty quark in-medium
energy loss. At forward rapidity, the non-prompt $J/\psi$ was measured by the
LHCb collaboration to be about 10% in pp collisions at $\sqrt{s}=7$ TeV [23]
in our $p_{{\mathrm{T}}}$ range. Assuming the scaling of beauty production
with the number of binary nucleon-nucleon collisions and neglecting the
shadowing effects, the prompt $J/\psi$ $R_{\mathrm{AA}}$ would be, and this is
an upper limit, 11% smaller than our inclusive measurement. To estimate the
influence of non-prompt $J/\psi$ as a function of $p_{{\mathrm{T}}}$ and $y$
on our inclusive $R_{\mathrm{AA}}$ measurement, we have extrapolated the LHCb
measurement at $\sqrt{s}=7$ TeV down $\sqrt{s}=2.76$ TeV using an center of
mass energy dependence extracted from CDF and CMS data. Assuming a range of
energy loss for the beauty quarks from $R_{\mathrm{AA}}$(b) = 0.2 to
$R_{\mathrm{AA}}$(b)= 1, we have found that the $J/\psi$ from beauty hadrons
have a negligible influence on our measurement.

In Fig. 3, our $J/\psi$ $R_{\mathrm{AA}}$ measurement is compared with
theoretical models that all include a $J/\psi$ regeneration component from
deconfined charm quarks in the medium.

Figure 3: (Color online) Centrality and $p_{{\mathrm{T}}}$ dependence of the
inclusive $J/\psi$ $R_{\mathrm{AA}}$ measured in Pb-Pb collisions at
$\sqrt{s_{\mathrm{NN}}}=2.76$ TeV at forward rapidity compared to the
predictions by Statistical Hadronization Model [24], Transport Model I [25]
and II [26], see text for details.

The Statistical Hadronization Model [2, 24] assumes deconfinement and a
thermal equilibration of the bulk of the c$\bar{\rm{c}}$ pairs. Then,
charmonium production occurs only at phase boundary by statistical
hadronization of charm quarks. The prediction is given for two values of ${\rm
d}\sigma_{\rm{c}\bar{\rm{c}}}/{\rm d}y$ since no measurements are available at
this rapidity for Pb-Pb collisions. The two transport model results [25, 26]
presented in the same figure differ mostly in the rate equation controlling
the $J/\psi$ dissociation and regeneration. Both are shown as a band that
connects the results obtained with (lower limit) and without (higher limit)
shadowing and can be interpreted as the uncertainty of the prediction. The
model from Zhao & al. implements a simple shadowing estimate leading to a 30%
suppression in most central Pb-Pb collisions. The charm cross-section
d$\sigma_{c\bar{c}}/$d$y$ at $y=3.25$ is $\approx 0.5$ mb and the $J/\psi$
from beauty hadrons is estimated at 10% and no quenching is assumed. The model
from Liu & al. takes the shadowing from EKS98 and uses a smaller charm cross-
section d$\sigma_{c\bar{c}}/$d$y\approx 0.38$ mb. The $J/\psi$ from beauty
hadrons is estimated at 10% and b quenching is fixed at $R_{\mathrm{AA}}$(b)=
0.4 for all the $p_{{\mathrm{T}}}$ range. In both transport models, the amount
of regenerated $J/\psi$ in the most central collisions contributes to about
50% of the production yield, the rest being from initial production. We can
see on the left side of Fig. 3 that all three models reproduce correctly the
centrality dependence of the forward $J/\psi$ $R_{\mathrm{AA}}$ for
$N_{\mathrm{part}}$ $>70$. A similar observation can be made for the mid
rapidity results [16]. The $p_{{\mathrm{T}}}$ dependence of the forward
$J/\psi$ $R_{\mathrm{AA}}$ is also successfully reproduced by the transport
models, as shown on Fig. 3 right side. In addition, both models predict that a
large fraction of $J/\psi$ from regeneration have a $p_{{\mathrm{T}}}$ below
$\approx$ 3.5 GeV/$c$.

##  5 Elliptic flow in Pb-Pb collisions $\sqrt{s_{\mathrm{NN}}}=$ 2.76 TeV

The elliptic flow of inclusive $J/\psi$ has been measured as a function of the
transverse momentum of the $J/\psi$. For this measurement, the reaction plane
has been determined with the VZERO-A detector. The large rapidity gap between
the $J/\psi$ acceptance and the VZEROA detector minimizes the influence of
non-flow effects. One the left side of Fig. 4, an example of the $v_{2}$
signal extraction is given; one can clearly see the cosine shape of the
measured $J/\psi$ signal in the 6 $\Delta\varphi$ bins, where $\Delta\varphi$
is the difference between the azimuthal angle of the $J/\psi$ and the angle of
the reaction plane. Further analysis details can be found in [27]. Figure 4
shows, on the right side, the first measurement of $J/\psi$ elliptic flow at
the LHC. The $J/\psi$ $v_{2}$ is given as a function of $p_{{\mathrm{T}}}$ in
the 20%–60% centrality range. A non-null $J/\psi$ $v_{2}$ seems to be present
at intermediate $p_{{\mathrm{T}}}$ and would tend to vanish at low and high
$p_{{\mathrm{T}}}$. Uncertainties are still too large to draw definitive
conclusions, nevertheless we have a non-zero $v_{2}$ signal for $J/\psi$ with
$p_{{\mathrm{T}}}$ between 2 to 4 GeV/$c$ with 2.2 $\sigma$ significance. At
lower energy, the $J/\psi$ elliptic flow was measured by STAR and appear to be
consistent with zero at $p_{{\mathrm{T}}}$ $<$ 10 GeV/$c$ in 20%–60%
centrality range, whereas charged hadrons and $\phi$ exhibit a rather strong
flow in this same kinematic domain [28]. Model prediction (private
communication) for ALICE was provided by the authors of [26] and is shown on
Fig. 4. The full line assumes a thermalization of the beauty quarks in the
medium, for which there is no evidence so far, and should be considered as an
upper limit. The dashed line, a more realistic prediction in which beauty
quarks are not thermalized, shows indeed a non-zero $J/\psi$ $v_{2}$, which
matches qualitatively our data. It is important to add that this model
reproduces successfully the ALICE $J/\psi$ $R_{\mathrm{AA}}$measurement.

Figure 4: (Color online) Example of the $J/\psi$ $v_{2}$ signal extraction in
$\Delta\varphi$ bins (left). Inclusive $J/\psi$ $v_{2}$ measured in the
20%–60% centrality range for Pb-Pb collisions at $\sqrt{s_{\mathrm{NN}}}=2.76$
TeV (right side) compared to the STAR measurement and to the prediction from a
parton transport model.

##  6 Conclusion

Quarkonia production in ALICE in pp and Pb-Pb collisions at
$\sqrt{s_{\mathrm{NN}}}$ = 2.76 and 7 TeV has been presented. In pp
collisions, the $p_{{\mathrm{T}}}$, $y$ and multiplicity dependence of
$J/\psi$ production, $J/\psi$ polarization and non-prompt $J/\psi$ have been
studied. Results have shown good agreement or complementarity with other LHC
results. All these measurements provide stringent constraints to model
predictions. In Pb-Pb collisions, the $J/\psi$ nuclear modification factor was
studied as a function of centrality, $p_{{\mathrm{T}}}$ and $y$. The $J/\psi$
$R_{\mathrm{AA}}$ dependence on the number of participant nucleons is flat and
centrality integrated values are large at mid and forward rapidity ($\approx
0.7-0.5$). This result is clearly different from the ones seen at lower
energies (e.g. RHIC and SPS). The rapidity dependence of the $J/\psi$
$R_{\mathrm{AA}}$ shows that suppression at large rapidity ( 2.5 $<$ $y$ $<$ 4
) is beyond the one that could be expected from shadowing only predictions.
The $J/\psi$ $R_{\mathrm{AA}}$ is large at low $p_{{\mathrm{T}}}$ and then
decreases with increasing $p_{{\mathrm{T}}}$. The trends observed in the data
as a function of centrality and $p_{{\mathrm{T}}}$ can be reproduced by models
based on deconfinement followed by charm recombination. In these models,
$J/\psi$ from recombination mostly occur at low $p_{{\mathrm{T}}}$ and account
for half of the produced $J/\psi$ in the most central collisions. Finally, we
have presented the $J/\psi$ elliptic flow in semi-central Pb-Pb collisions.
For the first time, a non-zero $J/\psi$ $v_{2}$ is observed in the
intermediate $p_{{\mathrm{T}}}$ range. We have now accumulated hints that the
$J/\psi$ production in Pb-Pb collisions at LHC energy may be governed, for an
important part, by charm quarks recombination processes. In order to confirm
this observation, the shadowing must be measured and constrained since it
remains unknown at LHC energies and this will be addressed by a pPb run
scheduled at the beginning of 2013. One should add here that the uncertainties
in $J/\psi$ $R_{\mathrm{AA}}$ results depend directly on the pp reference
data; thus it is crucial to collect a large amount of pp collisions at the
same collision energy that of Pb-Pb sample, in order to have precise
measurements of $J/\psi$ and charm differential cross section and $J/\psi$
polarization.

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11institutetext: Australian National University, Canberra, Australia
22institutetext: Anyvision, Belfast, United Kingdom
33institutetext: Reconova, Beijing, China 44institutetext: Queens University
of Belfast, Belfast, United Kingdom
44email: Hao.Zhu@anu.edu.au,44email<EMAIL_ADDRESS>
# Imbalance Robust Softmax for Deep Embeeding Learning

Hao Zhu indicates equal contribution.11 Yang Yuan 22 Guosheng Hu 2244 Xiang Wu
33 Neil Robertson 441122

###### Abstract

Deep embedding learning is expected to learn a metric space in which features
have smaller maximal intra-class distance than minimal inter-class distance.
In recent years, one research focus is to solve the open-set problem by
discriminative deep embedding learning in the field of face recognition (FR)
and person re-identification (re-ID). Apart from open-set problem, we find
that imbalanced training data is another main factor causing the performance
degradation of FR and re-ID, and data imbalance widely exists in the real
applications. However, very little research explores why and how data
imbalance influences the performance of FR and re-ID with softmax or its
variants. In this work, we deeply investigate data imbalance in the
perspective of neural network optimisation and feature distribution about
softmax. We find one main reason of performance degradation caused by data
imbalance is that the weights (from the penultimate fully-connected layer) are
far from their class centers in feature space. Based on this investigation, we
propose a unified framework, Imbalance-Robust Softmax (IR-Softmax), which can
simultaneously solve the open-set problem and reduce the influence of data
imbalance. IR-Softmax can generalise to any softmax and its variants (which
are discriminative for open-set problem) by directly setting the weights as
their class centers, naturally solving the data imbalance problem. In this
work, we explicitly re-formulate two discriminative softmax (A-Softmax and AM-
Softmax) under the framework of IR-Softmax. We conduct extensive experiments
on FR databases (LFW, MegaFace) and re-ID database (Market-1501, Duke), and
IR-Softmax outperforms many state-of-the-art methods.

##  1 Introduction

Recently, convolutional neural networks (CNNs) have significantly boosted the
state-of-the-art performance in many computer vision tasks especially in image
classification [1, 2, 3, 4, 5, 6]. Not surprisingly, CNNs have achieved great
success in the field of biometrics, in particular, face recognition (FR) [7,
8, 9, 10] and person re-identification (re-ID) [11, 12, 13]. This success is
derived from the fact that CNNs are able to encode images into rich, semantic
and discriminative representations (features) which can be used to effectively
measure the similarity between two identity-related images. These two tasks
(FR and re-ID) differ from general image classification in terms of two
challenges: open-set setting and data imbalance in the training set.

Open-set setting is much more widely applied than close-set for FR and re-ID.
For open-set setting, the identities of test set are disjoint with those of
training set. In the real world, FR and re-ID system train the CNN (feature
extractor) using images collected from one specific group of people, e.g.
celebrities from IMDb in CASIA WebFace [14] database. During test stage,
however, the FR and re-ID systems work in places, such as one police station,
where the gallery (blacklist) and the probe (people appear in this police
station) are mostly likely disjoint with training set (e.g. those
celebrities). In contrast, classical image classification (e.g. ImageNet
Challenge) uses the close-set setting where training and test sets share the
same classes. Traditionally, both open-set and close-set problems adopt the
softmax function because of its simplicity and probabilistic interpretation.
Together with the cross-entropy loss, they form arguably one of the most
commonly-used components in CNN architectures.

Under open-set setting, however, softmax suffers from one drawback: deep
learning with softmax loss only learns separable features that are not
discriminative enough for ‘unseen’ classes in testing. It results from the
fact that softmax loss does not explicitly optimise the intra- and inter-class
distances. To address this, some methods combine the softmax loss with metric
learning [9, 15, 10] to enhance the discrimination power of features. Metric
learning based methods commonly suffer from the way of building mini-batches
by sampling. Other methods try to add new constraints (e.g. center loss [16],
large-margin term [17, 18], L2 normalization [19, 20]) that make features more
compact and thus more discriminative.

Data imbalance is another challenge for FR and re-ID. Unlike those popular
datasets MNIST [21], CIFAR-10 [22] and ImageNet [23], FR and re-ID datasets
are commonly highly imbalanced. As shown in Fig.1, only a limited number of
identities appear frequently (more than hundreds), while most of the others
appear relatively rarely (fewer than ten times) in the popular face database
CASIA-Webface [14] and re-ID database Market-1501 [24]. Surprisingly, very
little research explores the problem of data imbalance in FR and re-ID. In
this paper, we show that deep embedding learning with the most widely used
softmax (and its variants such as A-Softmax [18]) encounters difficulty in the
presence of imbalanced training data even using either metric learning or
other regularizations. Although some softmax variants such as A-Softmax [18]
can solve the open-set problem by learning compact features, they do not
perform well when the training data is imbalanced. To our knowledge, the only
work exploring the data imbalance problem for FR is the range loss [25].
However, range loss does not deeply investigate the reason why this imbalance
impacts the deep embedding learning.

In this work, we aim to learn deep embeddings which can achieve two targets:
1) being discriminative for open-set and 2) being robust to data imbalance. As
existing works [16, 17, 18, 19, 20], target 1) can be achieved by learning
compact features (i.e. reduce intra-class variance). To achieve target 2), we
have to first investigate why data imbalance influences the performance of
softmax-based deep classification. In this work, we explore the reason.
_During the back-propagation training with imbalanced data, two strengths,
which determine the update of the weights (usually the penultimate fully-
connected layer), are imbalanced (see Eq. 2): the one keeping the weights at
their class centers is much smaller than that pushing them away. This
imbalance causes the weight of the class with minor samples being far away
from its class center, leading to degraded classification performance._ Based
on this analysis, target 2) can be achieved by making the weight from the
class with minor samples close to its class center.

To simultaneously achieve targets 1) and 2) , we propose a uniformed
framework, Imbalance-Robust Softmax (IR-Softmax). First, IR-Softmax solves the
open-set problem by being compatible with the softmax variants ( e.g.
A-Softmax [18], AM-Softmax [26] which can learn discriminative embeddings.
Second, motivated by the aforementioned analysis on data imbalance, IR-Softmax
alleviates the influence of data imbalance by setting the weights as their
class centers in the feature space instead of updating with back-propagation.
In this way, IR-Softmax effectively avoids the shift between the weights and
their centers, which is the main reason of performance degradation caused by
data imbalance detailed in Section 3.1.

Our contributions can be summarised as:

1\. We deeply investigate the reason why data imbalance degrades the
performance of softmax-based classifications in the perspective of neural
network optimisation (Section 3.1) and feature distribution (Section 3.2).

2\. IR-Softmax can learn embeddings which are discriminative under open-set
protocol. In particular, IR-Softmax is a unified framework, e.g. it can
generalise to softmax and its variants (e.g. A-Softmax [18], AM-Softmax [26])
to achieve discriminative feature learning. More importantly, IR-Softmax can
effectively reduce the influence of data imbalance by bridging the gap between
weights (the penultimate fully-connected layer) and their class centers in
feature space.

3\. Our extensive experiments demonstrate the effectiveness and generalisation
of the proposed IR-Softmax, and we achieve state-of-the-art performance on
challenging FR (LFW [27], MegaFace [28]) and re-ID (Market-1501 [24], DUKE-
MTMC [29]) benchmarks. The code will be made publicly available.

(a) WebFace-CASIA (b) Market-1501

Figure 1: Long-tailed distribution on WebFace-CASIA [14] and Market-1501 [24]
database. The number of images per person drops drastically, and only a few
identities have a large number of images.

(a) Softmax, bal.

(b) Softmax, imbal.

(c) A-Softmax, imbal.

(d) IR-Softmax, imbal.

Figure 2: The distribution of deeply learned features. ‘bal.’ (balanced)
setting contains 10 classes, which all has 6,000 images from MNIST [21]. In
contrast, ‘imbal.’ (imbalanced) setting contains 6000 images for all classes
but class ‘3’ with 300 images. A-Softmax in (c) refers to [18]. The label of
each class is plotted on its center. In addition, we also plot the weights
(from the fully-connected penultimate layer) to each class with an arrow in
(b)-(d). Note that our fully-connected layer consists of only 2 neurons to
facilitate visualisation.

##  2 Related Work

In this section, we briefly review the methods of discriminative feature
learning in the field of face recognition (FR) and person re-identification
(re-ID). Recently, two popular ways of deep embedding learning are: (1) metric
learning and (2) discriminative softmax (softmax’s variants which are more
discriminative for open-set problem). Apart from these two strategies, we
discuss the training data imbalanced problem in the field of FR and re-ID.

Metric Learning Metric learning is widely used for FR and re-ID. In practice,
to learn more discriminative features, many works combine softmax loss and
deep metric learning loss (contrastive [9, 30] loss or triplet loss [10]).
Unlike softmax, contrastive and triplet losses accept image pairs or triplets
(3 or a multiple of 3) as input respectively. For contrastive loss, if the
input pair belongs to the same class, their features are required to be as
similar as possible. Otherwise, the contrastive loss would require their
distance larger than a particular margin. Similar to contrastive loss, the
triplet loss [10] encourages a similar distance constraint. Specifically, the
triplet loss minimises the distance between an anchor sample and a positive
sample (of the same identity) and maximises the distance between the anchor
sample and a negative sample (of different identity). Clearly, the contrastive
and triplet losses can encourage intra-class compactness and inter-class
separability, making the learned feature more discriminative. However, both
contrastive and triplet losses require a carefully-designed pair/triplet
selection procedure. For example, using contrastive loss, it is hard to build
training pairs from a mini-batch, especially for the training set with many
classes. Normally the mini-batch size is not more than 256, while the number
of categories is far more than 256 in the application of FR and re-ID. Clearly
the online selection only produces a few positive pairs and much more negative
ones.

Discriminative Softmax Apart from metric learning, some softmax variants are
proposed, aiming to learn more discriminative features to solve the open-set
problem. Wen et al.[16] add a new supervision signal, called center loss, to
softmax loss for face recognition. Specifically, the center loss
simultaneously learns a feature center for each identity and penalises the
distances between the deep features of examples and their corresponding
feature centers. With the joint supervision of softmax loss and center loss,
this method can easily obtain inter-class dispersion and intra-class
compactness. Large-Margin Softmax loss [17] proposes a new perspective to
softmax and optimises the angles between weights and features. However, the
magnitude of weights are also considered, and thus it is also sensitive to
data imbalance just the same as softmax. By contrast, A-Softmax loss [18]
controls the magnitude of weights (i.e. $\|w\|_{2}=1$) and thus make the
weights optimised in an angular space. Although A-Softmax is theoretically
suitable for deep embedding learning, it actually does not work well in the
setting of data imbalance detailed in Section 3. [31, 26, 32] relax the margin
with more efficient and effective ways. Some works [19, 20, 32] try to
optimise the features on a hyper-sphere to make features more discriminative.

Training Data Imbalance The aforementioned methods ignore the problem of
training data imbalance which widely exist in FR and re-ID. In [33],
researchers investigate many factors that influence the performance of fine-
tuning for object detection with long-tailed distributions of samples. Their
analysis and empirical results indicate that classes with more samples will
achieve greater impact on the feature learning, and it is better to make the
sample number more uniform across classes. In the field of FR and re-ID,
unfortunately, the data imbalance problem is much worse than object detection
[33]. Specifically, few identities have more than 1000 images and many
identities have fewer than 10 images. Commonly a large-scale face dataset has
more than 10,000 identities [14]. However, we still cannot simply discard
these identities that only have few images. For face recognition, identities
with few images cannot provide enough intra-class information for the model,
but provide inter-class information which is more useful to open-set protocol.
Many methods [25, 34, 35, 36, 37] have been proposed to solve the data
imbalance in face recognition. However,these works do not deeply investigate
the reason why the imbalance impacts softmax based deep embedding learning.

##  3 Methodology

In this section, we first provide insights into the influence of data
imbalance on CNN performance by training a LeNet [21] on an imbalanced MNIST.
Based on the conclusion drawn from the experiments, we propose a new loss
function, Imbalance Robust Softmax (IR-Softmax), to reduce the influence of
data imbalance while perform discriminative feature learning. Last, we discuss
the relations between the proposed method and metric learning.

###  3.1 Motivation

Softmax regression (or multinomial logistic regression) is a generalisation of
logistic regression to multi-class problem, therefore, softmax can handle
$y_{i}\in\\{1,\ldots,K\\}$ (where $K$ is the number of classes). Given a
training set $\\{(\mathbf{x}_{i},y_{i}),\ldots,(\mathbf{x}_{n},y_{n})\\}$, we
learn an embedding/projection $f(\mathbf{x})$, with which the softmax can be
written as

$J=-\frac{1}{n}\left[\sum_{i=1}^{n}\log\frac{\exp{(f_{y_{i}}(\mathbf{x}_{i}))}}{\sum_{j=1}^{K}\exp(f_{j}(\mathbf{x}_{i}))}\right]$
(1)

where $f_{j}$ denotes the $j$-th dimension of the learned function $f(x)$, and
$n$ is the number of training samples. In CNNs, $f$ is usually the output of a
fully connected layer $\mathbf{W}=[\mathbf{w}_{1},...,\mathbf{w}_{k}]$ , so
$f_{j}=\mathbf{w}_{j}^{T}\mathbf{x}_{i}+b_{j}$ and
$f_{y_{i}}=\mathbf{w}_{y_{i}}^{T}\mathbf{x}_{i}+b_{y_{i}}$.

To analyse the influence of data imbalance, we come to the neural network
optimisation process (we omit the bias term for simplicity):

$\begin{split}\nabla_{\mathbf{w}_{k}}J&=-\frac{1}{n}\sum_{i=1}^{n}{\mathbf{x}_{i}\left(1\\{y_{i}=k\\}-P(k|\mathbf{x}_{i})\right)}\\\
&=\frac{1}{n}(\underbrace{\sum_{i=1}^{n}\mathbf{x}_{i}(P(k|\mathbf{x}_{i})-1)1\\{y_{i}=k\\}}_{term~{}1}+\underbrace{\sum_{i=1}^{n}\mathbf{x}_{i}P(k|\mathbf{x}_{i})1\\{y_{i}\neq
k\\}}_{term~{}2})\end{split}$ (2)

where
$P(k|\mathbf{x}_{i})=\frac{\exp{(f_{k}(\mathbf{x}_{i}))}}{\sum_{j=1}^{K}\exp(f_{j}(\mathbf{x}_{i}))}$,
and $1\\{\cdot\\}$ is the indicator function: $1\\{{true}\\}=1$, and
$1\\{{false}\\}=0$. It can be observed that the gradient of the parameter
$\nabla_{\mathbf{w}_{k}}J$ contains two terms: term 1 (which is activated if
$y_{i}=k$) and term 2 ( if $y_{i}\neq k$). Thus the update of parameter
$\mathbf{w}_{k}$ during optimisation depends on the samples not only from the
$k$-th class but also from the other classes. Term 1 is actually the weighted
center of the observed class; Term 2 can be viewed as the weighted centers of
all the other classes if $n$ is big enough. The update of $\mathbf{w}_{k}$ is
determined by the balance of two strengths: one leads $\mathbf{w}_{k}$ to the
center of class $k$ (term 1), one ‘pushes’ $w_{k}$ away from class $k$
(weighted center of all the other classes). If the training data is
imbalanced, the update of $\mathbf{w}_{k}$ corresponding to class $k$, which
has much fewer samples than other classes, is fully dominated by term 2,
making $\mathbf{w}_{k}$ being far away from center of class $k$.

To further analyse the influence of data imbalance on optimisation, we take
one binary classification with softmax for example. As shown in Fig.3(a) and
(b), there are both five samples for class 1 (blue points) and class 2 (red
points) for balance setting; and nine samples for class 1 and one sample for
class 2 for imbalance setting. Clearly, both $\mathbf{w}_{1}$ and
$\mathbf{w}_{2}$ are influenced by all the samples from class 1 and 2. In
imbalance setting (Fig.3(b)), $\mathbf{w}_{2}$ is determined by term 1 (1
sample from class 2) and term 2 (9 samples of class 1), where term 1 and 2 are
detailed in Eq. (2). Clearly, the update of $\mathbf{w}_{2}$ is dominated by
term 2, which pushes $\mathbf{w}_{2}$ far away from the center of class 2.

Figure 3: One example of binary classification with softmax. (a) balance
setting with five samples of class 1 (blue spots) and five samples of class 2
(red spots). (b) imbalance setting with nine samples of class 1 and one sample
of class 2. Black and green lines indicate term 1 and term 2 in Eq. (2),
influencing the update of $\mathbf{w}_{1}$ and $\mathbf{w}_{2}$ respectively.

To explicitly show the influence of data imbalance on classification
performance, we conduct a toy experiment on MNIST [21]. From Fig.2(b) and
2(c), not surprisingly, the data imbalance degrades the performance of the
models trained with softmax and A-Softmax [18]. We can find the main issue
caused by data imbalance: centers of relevant feature distributions being away
from their weights (from penultimate fully-connected layer). For example, in
Fig. 2(b), the feature center of class ‘3’ (minor training data) and centers
of ‘5’ and ‘7’ (the neighbours of ‘3’) are all distant from their weights.
Thus these biases (feature centers being far from its weights) caused by data
imbalance will induce classification error for the corresponding categories.
Though A-Softmax can learn discriminative features, it suffers from the same
aforementioned bias problem as shown in Fig. 2(c). This observation provides
the cue to solve the data imbalance problem and inspires our solution (Fig.
2(d)) detailed in Section 3.2.

###  3.2 Imbalance Robust Softmax (IR-Softmax)

In this work, we aim to learn features which can (i) improve the
discriminative power of features in open-set protocol, and (ii) alleviate data
imbalance problem. For (i), the desired open-set criterion is that the maximal
intra-class distance is smaller than the minimal inter-class distance.
However, softmax only maximises the the distance between weights rather then
inter-class distance (Fig .4(a)). Derived from softmax, A-Softmax [18],
however, only focuses on minimising intra-class distance, leading to compact
features as shown in Fig. 4(b). For (ii), data imbalance can degrade the
performance of deep CNNs. As analysed in Section 3.1, in particular Eq. (2),
the data imbalance can cause imbalanced gradient updates during optimisation:
the strength of making the weights close to feature center (term 1 in Eq. (2))
is much smaller than the strength of pushing away (term 2). This strength
imbalance causes the weights being far away from their feature centers as
shown in Fig. 4(b) , 2(b) and 2(c).

(a) Softmax, balanced

(b) A-Softmax, imbalanced

(c) IR-Softmax, imbalanced

Figure 4:  Feature distributions in angular space: (a) separable features
under balanced data, feature center being close to its weight (the black
arrow); (b) compact features under imbalanced data, and the feature center
being far from its weight; (c) compact features under imbalanced data, the
feature center being close to its weight.

Based on the above analysis, to simultaneously solve the open-set and data
imbalance problems, two criteria corresponding these two problems are: 1)
minimising the intra-class distance by making the feature distribution with
the same label more compact; 2) maximising the low bound of inter-class
distance by making the center of features of each class being close enough
(ideally equal) to its weights (usually the penultimate fully-connect layer).
Criterion 1) can be achieved by learning compact features e.g. A-Softmax [18]
and AM-Softmax [26]. To our knowledge, we are the first to investigate
Criterion 2). To simultaneously achieve these two targets, we propose a novel
framework, IR-Softmax111code is available in
https://github.com/allenhaozhu/IR-Softmax, which can achieve Criterion 1) by
incorporating itself into discriminative softmaxs e.g. A-Softmax [18] and AM-
Softmax [26].

Now we detail the way of meeting Criterion 2). As the analysis in Section 3.1,
we can find that the imbalanced data causes the weights (from penultimate
fully-connected layer) being away from their class centers after training as
shown in Fig. 2(b), leading to degraded classification performance. Based on
Criterion 2), the key idea of IR-Softmax is _setting the weights as their
corresponding class centers in the feature space_ , naturally avoiding the
shift between the weights and their centers.

IR-Softmax is a unified framework which can be incorporated into softmax and
its variants, leading to different forms of IR-Softmax. For classical softmax
in Eq. (1), $f_{j}=\mathbf{w}_{j}^{T}\mathbf{x}_{i}+b_{j}$ is fed into
softmax. In our IR-Softmax framework,
$f^{{}^{\prime}}_{j}=(\mathbf{c}^{{}^{\prime}}_{j})^{T}\mathbf{x}_{i}+b_{j}$
replaces $f_{j}$, where $\mathbf{c}^{{}^{\prime}}_{j}$, the center of features
from class $j$, is defined as:

$\mathbf{c}_{j}^{\prime}=\frac{1}{\sum_{i}^{n}1\\{y_{i}=j\\}}\sum_{i}^{n}\frac{1\\{y_{i}=j\\}\mathbf{x}_{i}}{\|\mathbf{x}_{i}\|_{2}}$
(3)

Most importantly, in this work, we formulate two discriminative IR-Softmaxs
derived from A-Softmax and AM-Softmax, respectively. To formulate A-Softmax
and AM-Softmax which both normalise the weight $w_{j}$
($||\mathbf{w}_{j}||_{2}=1$), Eq. (1) can be modified as:

$J=-\left[\sum_{i=1}^{m}\log\frac{\exp(\|\mathbf{x}_{i}\|\psi(\theta_{y_{i}}))}{\exp(\|\mathbf{x}_{i}\|\psi(\theta_{\mathbf{y}_{i}}))+\sum_{j\neq
y_{i}}^{K}\exp(\|\mathbf{x}_{i}\|\cos(\theta_{j}))}\right]$ (4)

From $f_{j}=\mathbf{w}_{j}^{T}\mathbf{x}_{i}+b_{j}$ (classical softmax) and
$f^{{}^{\prime}}_{j}=(\mathbf{c}^{{}^{\prime}}_{j})^{T}\mathbf{x}_{i}+b_{j}$
(IR-Softmax version), we use $\mathbf{c}^{{}^{\prime}}_{j}$ to replace
$\mathbf{w}_{j}$. Similarly, we use $\mathbf{c}_{j}^{T}\mathbf{x}_{i}$
($\mathbf{c}_{j}=\frac{\mathbf{c}_{j}^{\prime}}{\|\mathbf{c}_{j}^{\prime}\|_{2}}$)
to replace $\|\mathbf{x}_{i}\|\cos(\theta_{j})$ for both A-Softmax and AM-
Softmax. Thus, our IR-Softmax version of Eq. (4) is:

$J=-\left[\sum_{i=1}^{m}\log\frac{\exp(\|\mathbf{x}_{i}\|\psi(\theta_{y_{i}}))}{\exp(\|\mathbf{x}_{i}\|\psi(\theta_{y_{i}}))+\sum_{j\neq
y_{i}}^{K}\exp(\mathbf{c}_{j}^{T}\mathbf{x}_{i})}\right]$ (5)

For A-Softmax,

$\psi(\theta_{y_{i}})=((-1)^{k}\cos(m\theta_{y_{i}})-2k),\theta_{y_{i}}\in[\frac{k\pi}{m},\frac{(k+1)\pi}{m}]$
(6)

where $k\in[0,m-1]$ and $m\geq 1$ is an integer that controls the size of
angular margin. For original A-Softmax,
$\theta_{y_{i}}(0\leq\theta_{i}\leq\pi)$ is the angle between $\mathbf{w}_{i}$
and $\mathbf{x}_{i}$. Note that, in our IR-Softmax, $\theta_{y_{i}}$ is _the
angle between the $\mathbf{c}_{y_{i}}$ and $\mathbf{x}_{i}$_, where
$\mathbf{c}_{y_{i}}$ is the center of class $i$. For AM-Softmax,

$\psi(\theta_{y_{i}})=\cos(\theta_{y_{i}}+\alpha)$ (7)

$\psi(\theta_{y_{i}})=\cos(m_{1}\theta_{y_{i}}-m_{2})-m_{3}$ (8)

where $\theta_{y_{i}}(0\leq\theta_{y_{i}}\leq\pi)$ of the original AM-Softmax
is the angle between $\mathbf{w}_{i}$ and $\mathbf{x}_{i}$. Note that
$\theta_{y_{i}}$ of IR-Softmax is _the angle between the $\mathbf{c}_{y_{i}}$
and $\mathbf{x}_{i}$_.

Now we can summarise the difference between IR-Softmax and softmax (and its
variants). First, the weight $\mathbf{w}_{i}$ of softmax is updated via back-
propagation, however, $\mathbf{c}_{i}$ of IR-Softmax can be computed directly
from Eq. (3). Second, the update of $\mathbf{w}_{i}$ depends on samples of
class $i$ and samples from other classes as shown in Eq. (2). In contrast, the
update of $\mathbf{c}_{i}$ of IR-Softmax only depends on the samples from
class $i$, effectively avoiding the influence of data imbalance.

In practice, it is impossible to use all samples to calculate the centers as
shown in Eq.(3). We have tried three different updating strategies for feature
centres. i.replacing the weight with an instance feature (which makes the
proposed method like docFace [34]). ii.memory bank: estimate the centre with
last few (in a fixed window) samples in the same class (without BP). iii.a
loss function to estimate centres on the unit sphere for different classes.
The disadvantage of the first solution is that an additional softmax is
necessary in case the convergence is slightly slow and unstable. The second
solution relieves the unstable issue but no improvement in performance. The
third method is equal to adding a new term
$\|\mathbf{c}_{i}-\frac{\mathbf{x}_{i}}{\|\mathbf{x}_{i}\|_{2}}\|_{2}^{2},s.t.\|\mathbf{c}_{i}\|=1$
to Eq.4 and use the corresponding BP process to update feature centres
(weights in Eq.5) rather than Eq.4. After that $\ell 2$ normalization is used
on feature centres to make sure new $c_{i}$ on the unit sphere. We select the
third one in our experiments because it works better than other two
approaches.

###  3.3 Relation to Metric Learning

N-pairs loss [38] enforces softmax cross-entropy loss among the pairwise
similarity in the _mini-batch_.

$E=\frac{-1}{|P|}\sum_{(i,j)\in
P}\log\frac{\exp(S_{i,j})}{\exp(S_{i,j})+\sum_{k:y_{k}\neq
y_{j}}\exp(S_{i,k})}$ (9)

where
$S_{i,j}=f(\mathbf{x}_{i},\mathbf{\Theta})^{T}f(\mathbf{x}_{j},\mathbf{\Theta})$
represents the inner product between two embeddings. and $|P|$ indicates the
number of positive pairs ($i$,$j$). Comparing Eq. (4) and (9), we can see our
method can be viewed as a special form of N-pair loss. However, there are two
main differences: (1) Unlike softmax embedded in N-pairs, we employ A-Softmax
[18] and AM-Softmax [26] to improve the discriminability of features. (2) The
size of _mini-batch_ (where N-pair works) limits the number of negative
samples. In practice, it is usually difficult to make mini-batch more than 256
due to the memory limitation of GPU. In contrast, our method alleviates the
problem by caching historical features. The positive and negative samples are
equal to the number of categories.

##  4 Experiments

In this section, we first describe the experimental settings. We then evaluate
our method on two different tasks, face recognition (FR) and person re-
identification (re-ID), against four different benchmarks. For FR, we use
CASIA-WebFace [14] as training set and evaluate our method on LFW [27] and
MegaFace [28]. For re-ID, we evaluate on the Market-1501 [24] and Duke [29]
datasets.

###  4.1 Face Verification

All the faces and their landmarks are detected by MTCNN [39]. We use the
detected 5 landmarks (two eyes, nose and two mouth corners) to perform
similarity transformation. When the detection fails, we simply discard the
image if it is in the training set, but use the provided landmarks if in the
test set.

We use the publicly available training dataset CASIA-WebFace [14] (excluding
the images of 59 identities appearing in testing sets [26]) to train our CNN
models. CASIA-WebFace has 494,414 face images belonging to 10,575 (in fact,
10,516 after removing) different individuals. As shown in Fig. 1, CASIA-
WebFace is an imbalance dataset. Some identities have very few images (e.g,
only one image), while some have more than 300 images. These face images are
horizontally flipped for data augmentation in the training process. Note that
the number of samples in training set (0.49M) is relatively small compared to
other private datasets used in DeepFace [2] (4M), VGGFace [40] (2M) and
FaceNet [10] (200M). In the testing process, we extract the deep features from
the output of the FC1 layer and do not employ any pre-processing (such as PCA
and flipped features). The cosine distance between two features is applied. A
nearest neighbor classifier and thresholding are used for face identification
and verification, respectively.

To make fair comparison, we use two widely used CNN architectures for face
recognition: 9-layer Light CNN [41] and 20-layer ResNet-20 [18]. Note that the
faces are cropped to two different sizes (128x128 and 112x96) to fit the
setting in [41] and [18] respectively. In the training process, our IR-Softmax
is appended after the feature layer, i.e. the second last inner-product layer.
The networks are trained in an end-to-end way.

For simplicity, we denote IR-Softmax (A) as our IR-Softmax instance derived
from A-Softmax, and IR-Softmax (AM) from AM-Softmax in the whole experiment
section.

####  4.1.1 LFW

The LFW dataset [27] contains 13,233 images from 5,749 identities, with large
variations in pose, expression and illumination. All the images are collected
from the internet. We evaluate our methods on two protocols: (1) official
protocol [27] and (2) BLUFR protocol [42]. For (1), LFW is divided into 10
predefined splits for cross validation. We follow the standard ‘Unrestricted,
Labeled Outside Data’ protocol. Because the performance of face recognition is
almost saturated on this protocol, researchers propose a more challenging
BLUFR protocol [42]. For (2), BLUFR utilises all 13,233 images to evaluate the
performance in the open-set setting. The Verification Rate (VR) at False
Accepted Rate (FAR) 0.1%<EMAIL_ADDRESS>and Detection and Identification rate
(DIR) at FAR 1% (DIR@FAR=1%) are reported under BLUFR. It is noteworthy that
not only three identities exist in both CASIA-Webface [14, 26] and LFW [27].
We removed them according to [26] during training to build a complete open-set
validation.

##### LFW Official Protocol, Light CNN

As shown in Table 2, the performance is evaluated by six methods. The proposed
IR-Softmax(A) and IR-Softmax(AM) greatly outperform their original versions
(A-Softmax and AM-Softmax). Compared with the baseline (i.e. Softmax), IR-
Softmax (A) improves the verification accuracy from 97.15% to 98.38%, and IR-
Softmax (AM) from 97.15% to 98.63%.

##### LFW Official Protocol, ResNet-20

The evaluation results of ResNet-20 are listed in Table 1. Other state-of-the-
art results of A-Softmax and AM-Softmax using ResNet-20 are also presented for
comparison. Compared with the baseline (i.e. Softmax), IR-Softmax (A) loss
improves the verification accuracy from 97.08% to 99.23% on LFW. From the
results, we can see that the proposed methods IR-Softmax(A) and IR-Softmax(AM)
can outperform the corresponding original versions.

##### BLUFR, Light CNN

From Table 2, we can observe that the proposed method significantly
outperforms the other methods [17, 18]. Specifically, IR-Softmax(A) beats the
softmax baseline (which we finetune our model from), and improves the
<EMAIL_ADDRESS>from 83.32% to 94.61%, while DIR@FAR=1% from 60.64% to 75.12%.
Both versions of IR-Softmax are able to outperform their counterparts. It
means that the proposed method can significantly enhance the discriminability
of deeply learned features in the open-set protocol, demonstrating the
effectiveness of the proposed method.

##### BLUFR, ResNet-20

Since ResNet-20 models [18, 19] are also widely used for face recognition, we
make comparisons based on ResNet-20 in Table 1. IR-Softmax with ResNet-20
keeps the similar superiority compared with other models in the BLUFR protocol
of LFW. Note that our approach is better than range loss, which is proposed to
solve the problem of data imbalance in face recognition. Though range loss
uses a larger training set (MS-celeb [43]) and a deeper network (ResNet-50),
our method still outperforms it with<EMAIL_ADDRESS>from 93.72% to 97.08% (IR-
Softmax(A)) or 98.09% (IR-Softmax(AM)) while DIR@FAR=1% from 71.11% to 81.52%
(IR-Softmax(A)) or 85.00% (IR-Softmax(AM)).

Table 1: Performance on ResNet with various loss functions. CenterLoss, NormFace model and sphereface model are provided by authors. NormFace and CenterLoss use ResNet-28 like [16], another two methods use ResNet-20 [18].  loss function | LFW [27] | BLUFR [42] | BLUFR [42] | MegaFace [28] | MegaFace [28]
---|---|---|---|---|---
| 6000 pairs |<EMAIL_ADDRESS>| DIR@FAR=1% | rank1@1e-6 | VR@FAR=1e-6
Softmax | 97.08% | 78.26% | 50.85% | 45.26% | 50.12%
CenterLoss [16] | 99.00% | 94.50% | 65.46% | 63.38% | 75.68%
NormFace [19] | 98.98% | 96.16% | 75.22% | 65.03% | 75.88%
A-Softmax [18] | 99.08% | 96.58% | 79.97% | 67.41% | 78.19%
IR-Softmax(A) | 99.23% | 97.08% | 81.52% | 69.48% | 80.32%
AM-Softmax [26] | 98.98% | 97.69% | 84.82% | 72.47% | 84.44%
IR-Softmax(AM) | 99.21% | 98.09% | 85.00% | 75.28% | 85.67%
Table 2: Performance on Lighten CNN with various loss functions. All Results are derived under the same settings used in [41]. loss function | LFW [27] | BLUFR [42] | BLUFR [42] | MegaFace [28] | MegaFace [28]
---|---|---|---|---|---
| 6000 pairs |<EMAIL_ADDRESS>| DIR@FAR=1% | rank1@1e-6 | VR@FAR=1e-6
Softmax | 97.15% | 83.32% | 60.64% | 47.31% | 54.86%
Large-Margin [17] | 98.35% | 91.62% | 64.76% | 59.03% | 70.57%
A-Softmax [18] | 98.20% | 91.16% | 66.55% | 54.87% | 60.75%
IR-Softmax(A) | 98.38% | 94.61% | 75.12% | 64.71% | 75.94%
AM-Softmax [26] | 98.58% | 94.67% | 72.80% | 65.33% | 78.76%
IR-Softmax(AM) | 98.63% | 95.36% | 79.92% | 66.71% | 78.83%

####  4.1.2 MegaFace

One of the most challenging datasets for face recognition is MegaFace [28].
The MegaFace dataset contains a gallery set and a probe set. The gallery set
contains more than 1 million images from 690K identities; The probe set
consists of two existing datasets: Facescrub [44] and FGNet. MegaFace has
multiple testing scenarios including identification, verification and pose-
invariance under two protocols i.e. large or small training sets. The training
set is considered small if it is less than 0.5M. We evaluate our IR-Softmax
under the small training set protocol.

##### Lighten-CNN

Table 2 shows that our IR-Softmax(A) outperforms A-Softmax result by a margin
(almost 10% for rank-1 identification rate and 15% for VR at 1e-6 FAR) on the
small training dataset protocol while IR-Softmax(AM) outperforms AM-Softmax
result by a margin (7% for rank-1 identification rate and 1.4% for VR at 1e-6
FAR). Compared to the softmax baseline, our method performs significantly
better: 15% from IR-Softmax (A) and 19% IR-Softmax(AM) for identification, 21%
from IR-Softmax (A) and 24% from IR-Softmax (AM) for verification.

##### ResNet-20

Table 1 shows that our IR-Softmax (A) outperforms A-Softmax result by a margin
(almost 2% for rank-1 identification rate and 2% for VR at 1e-6 FAR) on the
small training dataset protocol while IR-Softmax(AM) outperforms AM-Softmax
result by a margin ( almost 3% for rank-1 identification rate and 1.2 % for VR
at 1e-6 FAR). Compared to the softmax baseline, our method performs
significantly better: 24% from IR-Softmax (A) and 30% from IR-Softmax (AM) for
identification, 30% from IR-Softmax (A) and 35% IR-Softmax (AM) for
verification.

Note that the performance of any testing methods on Megaface is intimately
linked to the quality of face alignment. Thus we do not compared with other
methods with different alignments. The results in Table 2 are therefore
computed under the same setting of face alignment and are directly comparable.
These results demonstrate that our IR-Softmax is well designed for open-set
face recognition especially when the training set is imbalanced. One can also
see that, smaller intra-class distance is not the only important issue for
learning features, but larger and evenly inter-class angular margin can
significantly improve face recognition performance.

###  4.2 Person Re-identification

For the evaluation of re-ID, we focus on two well-known re-ID databases:
Market-1501 [24] and DUKE [29] datasets. As shown in Fig. 1, we demonstrate
the distribution of market-1501 database. Although there are no identities
with more than 100 images like WebFace dataset, the number of images per
person ranges from 5 to 80. The DUKE [29] also has the similar imbalance
pattern. We use the standard evaluation metrics for both datasets, namely the
mean average precision score (mAP) and the cumulative matching curve (CMC) at
rank-1. We follow common practice by using random crops and random horizontal
flips during training. Specifically, we resize all images to $256\times 128$,
of which we take random crops of size $224\times 112$. Many methods for re-ID
rely on pre-trained models (e.g. ResNet). Indeed, these models usually lead to
impressive results. However, pre-trained models reduce the flexibility to make
task-specific changes in a network. For example, some application scenarios
need compact models rather than large ones pre-trained on Imagenet. Our method
clearly suggests that it is also possible to learn deep models from scratch
and achieve state-of-the-art performance. We use a Lighten CNN [41] based on
the ResNet Architecture, which is faster than the current ResNet-50 used by
many works [45]. Compared with other methods, we do not use the corresponding
pretrained models in ImageNet for finetuning. Thus we use the softmax to train
a baseline model with the re-ID dataset directly. And other methods (e.g.
large-margin and the proposed method) employ the baseline model as the pre-
trained model and finetune this model further.

####  4.2.1 Market-1501

The Market-1501 dataset contains 1,501 identities, 19,732 gallery images and
12,936 training images captured by 6 cameras. All the bounding boxes are
generated by the DPM detector [46]. The dataset uses both single and multi-
query evaluation, we report the results for both. Table 3 compares our IR-
Softmax (A) to other approaches. For Market-1501, the improvements achieved by
IR-Softmax are significant: (1) Compared with softmax, the Rank-1 accuracy
rises from 81.47% to 91.87%, and the mAP from 57.42% to 76.72% in the setting
of single query; (2) In the setting of multi query, the Rank-1 accuracy rises
from 86.40% to 94.33%, and the mAP from 65.97% to 82.22%. IR-Softmax (A)
significantly outperforms not only the softmax baseline but also other state-
of-the-art methods [13].

####  4.2.2 DukeMTMC-reID

The DukeMTMC-reID dataset is collected via 8 cameras and used for cross-camera
tracking (handover). Table 4 compares our IR-Softmax to other approaches. For
DukeMTMC-reID, IR-Softmax(A) works much better than softmax: the Rank-1
accuracy: 76.84% vs 61.98%, and the mAP 57.47% vs 41.17%. Beyond that, the
imbalance robust softmax also outperforms other state-of-the-art methods[45].

Table 3: Comparison with the state-of-the-art methods on the Market-1501 dataset. The rank-1 accuracy and mAP on single and multiple query are reported respectively. Method | Single Query | Multi. Query
---|---|---
rank-1 | mAP | rank-1 | mAP
BoW +KISSME [24] | 44.42 | 20.76 | - | -
MR CNN [47] | 45.58 | 26.11 | 56.59 | 32.26
DSN [48] | 55.43 | 29.87 | 71.56 | 46.03
Gate Reid [49] | 65.88 | 39.55 | 76.04 | 48.45
SOMAnet [50] | 73.87 | 47.89 | 81.29 | 56.98
DeepTransfer [30] | 83.70 | 65.50 | 89.60 | 73.80
Basel+LSRO [13] | 83.97 | 66.07 | 88.42 | 76.10
SVDNet [45] | 82.30 | 62.10 | - | -
Softmax | 81.47 | 57.42 | 86.40 | 65.97
Large-margin [17] | 90.08 | 72.22 | 92.75 | 78.79
IR-Softmax(A) | 91.87 | 76.72 | 94.33 | 82.88
Table 4: Comparison with state-of-the-art methods on DukeMTMC-reID. Rank-1 accuracy and mAP are reported. Method | Rank-1 (%) | mAP (%)
---|---|---
BoW + KISSME [24] | 25.13 | 12.17
LOMO + XQDA [51] | 30.75 | 17.04
Basel + LSRO [13] | 67.68 | 47.13
ACRN [52] | 72.58 | 51.96
PAN [53] | 71.59 | 51.51
SVDNet [45] | 76.70 | 56.80
Softmax | 61.98 | 41.17
Large-margin [17] | 75.58 | 56.25
IR-Softmax(A) | 76.84 | 57.47

##  5 Conclusion

In this paper, we investigated thoroughly the potential effects of data
imbalance on the deep embedding learning and proposed a new framework,
Imbalance Robust Softmax (IR-Softmax). IR-Softmax can simultaneously solve the
open-set problem and reduce the influence of data imbalance. Extensive
experiments on FR and re-ID are conducted, and the results show the
effectiveness of IR-Softmax. In Future work, we plan to extend this framework
to more softmax based methods and other applications like few-shot learning.

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