# Possibility of generating the ${}^{3}_{\Lambda_{c}}$H in the quark-
delocalization color-screening model

Siyu Wu  Key Laboratory of Atomic and Subatomic Structure and Quantum Control
(MOE), Institute of Quantum Matter, South China Normal University, Guangzhou
Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000,
China  Guangdong Provincial Key Laboratory of Nuclear Science, Institute of
Quantum Matter, South China Normal University, Guangzhou 510006, China
Guangdong-Hong Kong Joint Laboratory of Quantum Matter, Southern Nuclear
Science Computing Center, South China Normal University, Guangzhou 510006,
China  Qian Wu<EMAIL_ADDRESS>(co-first author) Institute of Modern
Physics, Chinese Academy of Sciences, Lanzhou 730000, China  Department of
Physics, Nanjing University, Nanjing 210097, China  Hongxia Huang
<EMAIL_ADDRESS>Department of Physics, Nanjing Normal University, Nanjing
210097, China  Xurong Chen<EMAIL_ADDRESS>Institute of Modern Physics,
Chinese Academy of Sciences, Lanzhou 730000, China  School of Nuclear Science
and Technology, University of Chinese Academy of Sciences, Beijing 100049,
China  Jialun Ping<EMAIL_ADDRESS>Department of Physics, Nanjing Normal
University, Nanjing 210097, China  Qian Wang<EMAIL_ADDRESS>Key
Laboratory of Atomic and Subatomic Structure and Quantum Control (MOE),
Institute of Quantum Matter, South China Normal University, Guangzhou
Guangdong Provincial Key Laboratory of Nuclear Science, Institute of Quantum
Matter, South China Normal University, Guangzhou 510006, China  Guangdong-Hong
Kong Joint Laboratory of Quantum Matter, Southern Nuclear Science Computing
Center, South China Normal University, Guangzhou 510006, China

###### Abstract

We probe the existence of the ${}^{3}_{\Lambda_{c}}$H where the $N\Lambda_{c}$
potentials are derived from the quark-delocalization color-screening model
(QDCSM). The $N\Lambda_{c}$ system is studied and the $N\Lambda_{c}$
scattering length so as the effective range are obtained in the QDCSM. We
construct effective Gaussian-type $N\Lambda_{c}$ potentials which reproduce
the $N\Lambda_{c}$ scattering data given by the QDCSM. By solving the
$NN\Lambda_{c}$ three body Schrödinger equation with the Gaussian expansion
method, we calculate the energies of the ${}^{3}_{\Lambda_{c}}$H with isospin
$I=0$, $J^{P}=1/2^{+}$ and $I=0$, $J^{P}=3/2^{+}$ under different color
screening parameter $\mu$. The $J^{P}=1/2^{+}$ and $J^{P}=3/2^{+}$ states are
both bound when the color screening parameter $\mu$ is set to 1.0 or 1.2,
where the $J^{P}=1/2^{+}$ state is bound by $0.08\sim 0.85$ MeV and the
$J^{P}=3/2^{+}$ state is bound by $0.15\sim 1.31$ MeV with respect to the
deuteron-$\Lambda_{c}$ threshold.

charmed nuclei, light nuclei

##  I Introduction

The low-energy hadronic system and hadron-hadron interactions are difficult to
study due to the non-perturbative nature of Quantum Chromodynamics (QCD). In
order to understand the hadron-hadron interactions and exotic hadronic states,
various sophisticated models have been applied. The nucleon-nucleon (NN),
hyperon-nucleon (YN) and hyperon-hyperon (YY) interactions have been well
constructed by some approaches such as one-boson-exchange (OBE) model
Machleidt1989 ; Rijken2006 ; Rijken2006-2 , chiral effective theory
Epelbaum2008 , chiral quark model Valcarce2005 ; Fujiwara1995 ; Zhang1997  and
quark-delocalization color-screening model (QDCSM) Wang1992 ; Ping1998 ;
Ping2000 . Recently, the hadron-hadron interactions are extended to the heavy
flavor systems. For instance, in Refs. Wu2010 ; Wang2011 , the
$\Sigma_{c}\bar{D}$ bound state are found with chiral quark model. As for the
charmonium-nucleon ($c\bar{c}-N$) interactions, since they have no valence
quarks in common, the single meson exchange is suppressed and the single gluon
exchange is prohibited hiyama2013 . Alternatively, the multi-gluon exchange
interactions could be possible as discussed in Refs. Brodsky1997 ;
Kharzeev1994 ; Luke1992 ; Brodsky1989  and the $c\bar{c}-N$ interaction is
found to be attractive.

The $\Lambda_{c}N$ interaction and the possible $\Lambda_{c}N$ bound state
have been studied in several models. With the chiral constituent quark mode,
the S-wave $\Lambda_{c}$N interaction is investigated in Ref. Gal2014  and
they find a $J^{P}=2^{+}$ candidate of the $\pi\Lambda_{c}$N three body
system. The attractive force between $\Lambda_{c}$ and $N$ is also found in
Ref. Liu2011  within the the one meson exchange model and the quark cluster
model, where a bound $\Lambda_{c}N$ state with 1 MeV binding energy is
obtained. However, the HAL QCD collaboration Miyamoto2017  obtained a weakly
attractive S-wave $\Lambda_{c}N$ interaction based on unphysical quark masses
corresponding to pion masses of $m_{\pi}=410–700$ MeV and no $\Lambda_{c}N$
molecular state is confirmed. Recently, an extrapolation of this HAL QCD
results to the physical pion mass with the chiral effective filed theory (EFT)
Epelbaum2003  was presented Haidenbauer2017LQCD ; Haidenbauer2020LQCD . It
gives that the $\Lambda_{c}N$ interaction at the physical point is slightly
stronger than for that at the large pion masses but still less attractive than
the phenomenological predictions for the $\Lambda_{c}N$ interaction mentioned
above Gal2014 ; Liu2011 . As a summary, the existence of the $\Lambda_{c}N$
bound state is unclear and the theoretical study still remains to be continued
Dover1977 .

Due to the short lifetime of the $\Lambda_{c}$, it is difficult to directly
study the $\Lambda_{c}N$ interaction from the nuclear scattering. Thus,
similar as the hypernuclear physics, people can polish the YN and YY
interactions from the spectra data of the light hypernuclei instead of the
scattering data. For example, in Ref. Hiyama2000 , the two different YN spin-
orbit interactions derived from the quark model and OBE model give different
$5/2^{+}-3/2^{+}$ energy splitting in ${}^{13}_{\Lambda}$C or the
$3/2^{+}-1/2^{+}$ energy splitting in ${}^{9}_{\Lambda}$Be. And the observed
values E929-1 ; E929-2 ; E930  may suggest which model of YN spin-orbit
interaction is favored Hiyama2009 ; Hashimoto2006 . Therefore, in order to
further study the $\Lambda_{c}$N interaction, it’s a natural consideration to
extend our research from the hypernuclei to the charmed hypernuclei.

In experiment, the first charmed hypernuclei is claimed to be found in Ref.
Batusov1981 . The experimental search of the charmed and bottom hypernuclei
was also performed at the ARES facility Bressani1989 . Several factories
including the $\tau$ charm factory Bunyatov1991  and the kaon factory (such as
the Japan Hadron Facility) may have potential to generate the charmed
hypernuclei. In the future, several facilities such as GSI-FAIR, J-PARC
Riedl2007  or HIAF may be available to check whether the charmed hypernuclei
exist or not.

In fact, the theoretical study of the $\Lambda_{c}$ hypernuclei comes back to
the mid 70s Dover1977 ; Bando1981 ; Bando1983 ; Bando1985 ; Gibson1983 ;
Bhamathi1981 ; Tsushima2002a ; Tsushima2002b . Recently, in Ref. Maeda2015 ,
with the one meson exchange model and the quark cluster model, they found a
$NN\Lambda_{c}$ bound state. Soon after that, Garcilazo et al. applied the
$\Lambda_{c}N$ interaction derived from the chiral quark model to the
$J^{P}=3/2^{+}$ charmed hypertriton and obtained a bound state with binding
energy around $0.140\sim 0.715$ MeV. Recently, Lattice QCD calculation
Miyamoto2017  finds that only the $\rm{A}\geq 11$ charmed hypernculei can be
bound with a few MeVs binding energy.

As we know, the forces between atoms (molecular force) are qualitative and
similarly between the nucleons. The quark-delocalization color-screening model
(QDCSM), which models the molecular force, was developed by Wang et al. in
1992 Wang1992  and has been extended to various baryons-baryons interactions
and scattering phase shifts Ping1998 ; Ping2000  in the framework of the
resonating group method (RGM) Kamimura1977 . In QDCSM, quarks confined in one
baryon are allowed to delocalize to another baryon. The delocalization
parameter is determined by the dynamics of the interacting quark system.
Besides, the color-screening confinement interaction is used when the two
quarks are in different baryons. This gives an extra parameter $\mu$ which is
normally determined by the mass of two body hadron-hadron system.

With QDCSM, co-authored by two of the present authors Huang2013 , an
attractive $\Lambda_{c}N$ interaction with a repulsive core was obtained and
no $\Lambda_{c}N$ bound state was found. But no charmed hypertriton was
investigated. Thus in this work, we investigate the possibility of the
existence of the charmed hypertriton. With the effective $\Lambda_{c}N$
interaction derived from the QDCSM, we calculate the $NN\Lambda_{c}$ three
body system. In order to solve the three body Schrödinger equation, we apply
the Gaussian expansion method (GEM) Hiyama2003GEM .

The paper is organized as follows. In Sec. II, we introduce the effective
$\Lambda_{c}N$ interaction obtained from the QDCSM. The detail of GEM is
explained in Sec. III After explaining Method employed, we show the results
and the discussion in Sec. IV. Summary is at the end.

##  II quark-delocalization color-screening model and $N\Lambda_{c}$ potential

In this section, we introduce the QDCSM and its application in the
$\Lambda_{c}N$ system. In order to describe the $\Lambda_{c}N$ system, we use
the chiral constituent quark model and the Hamiltonian of the six quarks is:

$H=\sum_{i=1}^{6}\left(m_{i}+\frac{p_{i}^{2}}{2m_{i}}\right)-T_{c}+\sum_{i<j}\left[V^{G}\left(r_{ij}\right)+V^{\chi}\left(r_{ij}\right)+V^{C}\left(r_{ij}\right)\right],$
(1)

where $i\;(j)=1\sim 6$ is the label of the six quarks composed of the two
baryons. The first term means the sum of the masses and kinetic energies of
all the quarks. $T_{c}$ is the kinetic energy of the center-of-mass. $V^{C}$,
$V^{G}$ and $V^{\chi}$ indicates the confining potential, gluon-exchange
potential and meson-exchange potential where $\chi$ denotes the $\pi$, K or
$\eta$ meson, respectively. The forms of $V^{G}$ and $V^{\chi}$ are given as
follows:

$\displaystyle V^{G}\left(r_{ij}\right)$
$\displaystyle=\frac{\alpha_{s_{ij}}}{4}\lambda_{i}\cdot\lambda_{j}\left[\frac{1}{r_{ij}}-\frac{3}{4m_{i}m_{j}r_{ij}^{3}}S_{ij}\right.$
(2)
$\displaystyle\left.-\frac{\pi}{2}\left(\frac{1}{m_{i}^{2}}+\frac{1}{m_{j}^{2}}+\frac{4\sigma_{i}\cdot\sigma_{j}}{3m_{i}m_{j}}\right)\delta\left(r_{ij}\right)\right],$
$\displaystyle V^{\chi}\left(r_{ij}\right)$
$\displaystyle=\frac{\alpha_{ch}}{3}\frac{\Lambda^{2}}{\Lambda^{2}-m_{\chi}^{2}}m_{\chi}\left\\{\left[Y\left(m_{\chi}r_{ij}\right)-\frac{\Lambda^{3}}{m_{\chi}^{3}}Y\left(\Lambda
r_{ij}\right)\right]\sigma_{i}\cdot\sigma_{j}\right.$
$\displaystyle\left.+\left[H\left(m_{\chi}r_{ij}\right)-\frac{\Lambda^{3}}{m_{\chi}^{3}}H\left(\Lambda
r_{ij}\right)\right]S_{ij}\right\\}\mathbf{F}_{i}\cdot\mathbf{F}_{j},$

where $\lambda_{i}$ and $F_{i}$ are the color Gell-Mann matrices and SU(3)
falvor matrices, respectively. The H(x) and Y(x) are the standard Yukawa
functions Valcarce2005 . The $S_{ij}$ is the tensor operator:

$\displaystyle S_{ij}$
$\displaystyle=\frac{\left(\sigma_{i}\cdot\mathbf{r}_{ij}\right)\left(\sigma_{j}\cdot\mathbf{r}_{ij}\right)}{r_{ij}^{2}}-\frac{1}{3}\sigma_{i}\cdot\sigma_{j}.$
(3)

For the confining potential, $V^{C}$, the color screening confining potential
is used when the two quarks are in different baryons:

$\displaystyle
V^{C}\left(r_{ij}\right)=-a_{c}\lambda_{i}\cdot\lambda_{j}\left[f\left(r_{ij}\right)+V_{0}\right],$
(4) $\displaystyle f\left(r_{ij}\right)=\begin{cases}r_{ij}^{2}&i,j\text{ are
in the same baryon orbit, }\\\ \frac{1-e^{-\mu r_{ij}}}{\mu}&i,j\text{ are in
different baryon orbits. }\end{cases}$

It should be noted that, in order to determine the color screening parameter
$\mu$, we need the total mass of the two baryon system as an input. In the
$NN$ system, the parameter $\mu=1.0$ is used to fit deuteron mass. In this
work, we treat $\mu$ as a relative common parameter since we do not have the
$N\Lambda_{c}$ mass. The other parameters are determined with a global fitting
to the masses of light flavor mesons and charmed baryons $\Lambda_{c}$,
$\Sigma_{c}$ and $\Sigma^{*}_{c}$ Chen2011 ; Huang2013 .

Normally, the usual quark cluster model gives the single quark wave function
as follows:

$\displaystyle\phi_{\alpha}\left(\mathbf{S}_{i}\right)$
$\displaystyle=\left(\frac{1}{\pi
b^{2}}\right)^{3/4}e^{-\frac{1}{2b^{2}}\left(\mathbf{r}_{\alpha}-\mathbf{S}_{i}/2\right)^{2}},$
(5) $\displaystyle\phi_{\beta}\left(-\mathbf{S}_{i}\right)$
$\displaystyle=\left(\frac{1}{\pi
b^{2}}\right)^{3/4}e^{-\frac{1}{2b^{2}}\left(\mathbf{r}_{\beta}+\mathbf{S}_{i}/2\right)^{2}}$

where $r_{\alpha(\beta)}$ is the coordinate of the single quark to the center
of the baryon and $S_{i}$ is the generating coordinate between the two
baryons. In QDCSM, the delocalized quark wave functions are used:

$\displaystyle\psi_{\alpha}\left(\mathbf{S}_{i},\epsilon\right)$
$\displaystyle=\left(\phi_{\alpha}\left(\mathbf{S}_{i}\right)+\epsilon\phi_{\alpha}\left(-\mathbf{S}_{i}\right)\right)/N(\epsilon),$
(6) $\displaystyle\psi_{\beta}\left(-\mathbf{S}_{i},\epsilon\right)$
$\displaystyle=\left(\phi_{\beta}\left(-\mathbf{S}_{i}\right)+\epsilon\phi_{\beta}\left(\mathbf{S}_{i}\right)\right)/N(\epsilon),$
$\displaystyle N(\epsilon)$ $\displaystyle=\sqrt{1+\epsilon^{2}+2\epsilon
e^{-S_{i}^{2}/4b^{2}}},$

where $\epsilon(S_{i})$ is determined by the dynamics of the quark system.

Table 1: The channels coupled to the $N\Lambda_{c}$ states. States |  Channels
---|---
$J^{P}=0^{+}$ |  $N\Lambda_{c}\left({}^{1}S_{0}\right),N\Sigma_{c}\left({}^{1}S_{0}\right),N\Sigma_{c}^{*}\left({}^{5}D_{0}\right)$
$J^{P}=1^{+}$ |  $N\Lambda_{c}\left({}^{3}S_{1}\right),N\Sigma_{c}\left({}^{3}S_{1}\right),N\Sigma_{c}^{*}\left({}^{3}S_{1}\right)$,
|
$N\Lambda_{c}\left({}^{3}D_{1}\right),N\Sigma_{c}\left({}^{3}D_{1}\right),N\Sigma_{c}^{*}\left({}^{3}D_{1}\right),N\Sigma_{c}^{*}\left({}^{5}D_{1}\right)$

In Ref. Huang2013 , the $J^{P}=0^{+}$ and $J^{P}=1^{+}$ $N\Lambda_{c}$
scattering phase shifts and the potentials have been studied in QDCSM with the
well-developed RGM. The coupled channel effects are considered in the
$N\Lambda_{c}$ system. In total, three/seven channels are included in the
$J^{P}=0^{+}$/$J^{P}=1^{+}$ $N\Lambda_{c}$ channel (shown in Tab. 1).

Figure 1: Normalized $N\Lambda_{c}$ potentials with respect to the generating
coordinates S

In this work, we diagonalize the nondiagonal S matrix and normalize the
coupled channel effects into the single channel. We normalize the effects of
all three channels into $N\Lambda_{c}\left({}^{1}S_{0}\right)$ in the
$J^{P}=0^{+}$ case. Similarly in the $J^{P}=1^{+}$ state, the effects of all
seven channels are normalized into $N\Lambda_{c}\left({}^{3}S_{1}\right)$
channel. Thus, the $\Sigma_{c}\;(\Sigma^{*}_{c})$ and the D-wave components
are normalized to the S-wave $N\Lambda_{c}$ potential. In Fig. 1, we depict
the normalized the $J^{P}=0^{+}$ and $J^{P}=1^{+}$ $N\Lambda_{c}$ potentials
where $\mu=1.0$.

We want to stress that the $N\Lambda_{c}$ potentials in Fig. 1 are constructed
with respect to the generating coordinate $S_{i}$, which is not equivalent to
the relative distance $r$ between the center-of-mass of the two baryons. The
generating coordinate $S_{i}$ is constructed based on the reference center of
the three quarks, which is not the center-of-mass of the baryons (N or
$\Lambda_{c}$) especially when the two baryons are close and the quark
delocalization effect is strong. Then, the difference between the V(S) and
V(r) in the short-range could be large. Therefore, it is not a proper way to
directly apply this potential into the $NN\Lambda_{c}$ three body system. On
the other hand, if we study the $NN\Lambda_{c}$ system within framework of the
QDCSM, a nine quarks system is necessary to solve, which is not practical with
our present capacity. Instead, we construct the effective S-wave
$\Lambda_{c}N$ interactions which reproduce the $\Lambda_{c}N$ scattering
length ($a$) and the effective range $r_{0}$ defined as follows:

$\displaystyle k\cot{\delta}=\frac{1}{a}+\frac{1}{2}r_{0}k^{2}+...,$ (7)

where $\delta$ is the scattering phase shift and $k$ is the wave number. The
values are shown in Tab. 2 for three different values, i.e. 0.8, 1.0 and 1.2,
of $\mu$.

Table 2: The scattering length and the effective range of $J^{P}=0^{+}$ and $J^{P}=1^{+}$ $N\Lambda_{c}$ systems. $(J^{P},\mu)$ |  Scattering length (fm) |  Effective range (fm)
---|---|---
$0^{+},0.8$ |  -1.57 |  5.06
$0^{+},1.0$ |  -4.88 |  3.62
$0^{+},1.2$ |  -2.58 |  3.65
$1^{+},0.8$ |  -1.92 |  3.70
$1^{+},1.0$ |  -7.11 |  3.15
$1^{+},1.2$ |  -2.88 |  3.50

The functional form of the effective $N\Lambda_{c}$ interactions are given by:

$V_{\mathrm{eff}}(r)=V_{1}e^{-r^{2}/b_{1}^{2}}+V_{2}e^{-r^{2}/b_{2}^{2}}.$ (8)

The fitting parameters are listed in Tab. 3. The obtained effective
$\Lambda_{c}N$ interactions are shown in Fig. 2 and Fig. 3, respectively for
the $J^{P}=0^{+}$ and $J^{P}=1^{+}$ channel. As we can see, the effective
potential V(r) and the potential V(S) shown in Fig. 1 have a significant
difference in the short range of the r (or S), which consists with our
conjectures that the quark delocalization effect is strong when the two
baryons are close.

Table 3: Fitting parameters of $V_{\mathrm{eff}}(r)$ defined in Eq. 8 for the $N\Lambda_{c}$ effective potentials. The $V_{1}$ and $V_{2}$ are in unit of MeV. $b_{1}$ and $b_{2}$ are in unit of fm. | ${}^{1}S_{0}$ | ${}^{3}S_{1}$
---|---|---
$\mu$ |  0.8 |  1.0 |  1.2 |  0.8 |  1.0 |  1.2
$V_{1}$ |  -115.5 |  -42.0 |  -223.5 |  255.0 |  -216.4 |  -226.5
$b_{1}$ |  1.29 |  1.58 |  1.20 |  1.12 |  1.30 |  1.20
$V_{2}$ |  160.6 |  81.3 |  261.2 |  292.8 |  253.3 |  262.6
$b_{2}$ |  1.05 |  0.91 |  1.05 |  1.01 |  1.12 |  1.05
Figure 2: Effective $N\Lambda_{c}$ interaction of ${}^{1}S_{0}$ case. Figure
3: Effective $N\Lambda_{c}$ interaction of the ${}^{3}S_{1}$ case.

##  III Gaussian Expansion method

In order to solve the $NN\Lambda_{c}$ three-body Schrödinger equation, we
apply the Gaussian expansion method and set the three body wave function as:

$\displaystyle\Psi_{JMTT_{z}}\left({}_{\Lambda_{c}}^{3}\rm{H}\right)$
$\displaystyle=\sum_{c=1}^{2}\sum_{s,S,L}\sum_{n_{1},l_{1}}\sum_{n_{2},l_{2}}C_{\gamma}^{(c)}\mathcal{A}$
(9)
$\displaystyle\times\left\\{\left\\{\left(\phi_{n_{1}l_{1}}^{(c)}\left(\bm{r}_{c}\right)\psi_{n_{2}l_{2}}^{(\mathrm{c})}\left(\bm{R}_{c}\right)\right)_{L}\right.\right.$
$\displaystyle\left.\times\left[\left(\chi^{1}_{1/2}\chi^{2}_{1/2}\right)_{s}\chi^{3}_{1/2}\right]_{S}\right\\}_{JM}\left.\left(\tau^{1}_{1/2}\tau^{2}_{1/2}\right)_{TT_{z}}\right\\},$

where the two sets of Jacobian coordinates (labeled as 1 and 2) are shown in
Fig. 4. Here $\gamma$ denotes $\\{L,s,S,n_{1},l_{1},n_{2},l_{2}\\}$. $\chi$
and $\tau$ represent the spin and isospin wave function of the nucleon or
$\Lambda_{c}$, respectively. Note that we omit the isospin wave function of
the $\Lambda_{c}$ since it is isospin singlet. The $\mathcal{A}$ is the anti-
symmetric operator between the two nucleons. The relative wave functions
between the baryons, corresponding to the two Jacobi coordinates,
$\phi_{n_{1}l_{1}}(\bm{r})$, $\psi_{n_{2}l_{2}}(\bm{R})$ are expanded by using
the following Gaussian basis functions, applying the GEM:

$\displaystyle\phi_{n_{1}l_{1}}(\bm{r})=r^{\ell_{1}}e^{-(r/r_{n_{1}})^{2}}Y_{\ell_{1}m_{1}}(\hat{\mbox{\boldmath$r$}}),$
$\displaystyle\psi_{n_{2}l_{2}}(\bm{R})=R^{\ell_{2}}e^{-(R/R_{n_{2}})^{2}}Y_{\ell_{2}m_{2}}(\hat{\mbox{\boldmath$R$}}).$
(10)

The Gaussian variational parameters are chosen to have geometric progression:

$\displaystyle r_{n_{1}}=r_{\rm min}A_{1}^{n_{1}-1},\quad(n_{1}=1\sim
n_{1}^{\rm max}),$ $\displaystyle R_{n_{2}}=R_{\rm
min}A_{2}^{n_{2}-1},\quad(n_{2}=1\sim n_{2}^{\rm max}).$ (11)

Then, the eigen energies and the coefficients $C_{\gamma}$ are obtained with
applying the Rayleigh-Ritz variational method.

Figure 4: Jacobi coordinates of NN$\Lambda_{c}$ three body system.

##  IV Results and Discussions

The energy levels of ${}^{3}_{\Lambda_{c}}$H with $I=0$, $J^{P}=1/2^{+}$ and
$J^{P}=3/2^{+}$ states are shown in Fig. 5. The binding energies of
${}^{3}_{\Lambda_{c}}$H are given with respect to the $N+N+\Lambda_{c}$ three
body break-up threshold. We calculate the energies of ${}^{3}_{\Lambda_{c}}$H
without or with the coulomb force between the $\Lambda_{c}^{+}$ and the
nucleon, which are shown in the left and right column of Fig. 5. As shown in
Fig. 5, it is obvious that the binding energies are smaller when we include
the repulsive coulomb force.

Figure 5: The calculated energy levels of ${}^{3}_{\Lambda_{c}}$H. The levels
without and with considering the coulomb force are shown in the left figure
and the right figure, respectively, for the color screening parameter
$\mu=0.8,1.0,1.2$. The energies are given with respect to the $NN\Lambda_{c}$
three body breakup threshold. The $d+\Lambda_{c}$ threshold is also shown at
$E=-2.24$ MeV.

The binding energy for the $\mu=1.0$ case is larger than the $\mu=1.2$ and
$\mu=0.8$ cases. This can be expected from the $N\Lambda_{c}$ scattering
length in Table. 2, in which the $\mu=1.0$ case has the largest scattering
length for both ${}^{1}S_{0}$ and ${}^{3}S_{1}$. In the $\mu=0.8$ case, no
bound state is obtained with the coulomb force. And when we exclude coulomb
force, a very weakly bound state for the $1/2^{+}$ state is obtained with the
$\Lambda_{c}$ separation energy $B_{\Lambda_{c}}=0.01$ MeV. As for the
$\mu=1.2$ case, very weakly bound states with $B_{\Lambda_{c}}=0.08$ MeV and
0.15 MeV are obtained, respectively for the $1/2^{+}$ and $3/2^{+}$ states.

On the other hand, the energy splitting between the $1/2^{+}$ and $3/2^{+}$
states does not change significantly when we include the coulomb force since
the coulomb force does not have a spin-dependent term. The $\mu=1.2$ case has
the smallest energy splitting while the $\mu=0.8$ and 1.0 cases are similar.

We find that the energy levels of $J^{P}=3/2^{+}$ state are lower than the
$J^{P}=1/2^{+}$ state in all of the three cases of $\mu$. In order to study
this, we calculate the fraction of the S-wave wave function (L=0 and S=3/2) in
the $J^{P}=3/2^{+}$ state which is around $94\%$ and the S-wave component (L=0
and S=1/2) is around $97\%$ for the $J^{P}=1/2^{+}$ state. Thus, the
${}^{1}S_{0}$ $N\Lambda_{c}$ component is nearly excluded in the
$J^{P}=3/2^{+}$ state while it has a dominant contribution in the
$J^{P}=1/2^{+}$ state. Besides, the ${}^{3}S_{1}$ $N\Lambda_{c}$ interaction
is more attractive than the ${}^{1}S_{0}$ case. Therefore, the binding energy
of the $J^{P}=3/2^{+}$ state are larger than the $J^{P}=1/2^{+}$ state.

It’s interesting that the similar conclusions were made in Refs. Garcilazo2015
; Maeda2015  for the three body ${}^{3}_{\Lambda_{c}}$H calculations. As shown
in Fig. 9 of Ref. Maeda2015 , the $J^{P}=1^{+}$ interaction is also stronger
than the $J^{P}=0^{+}$ case though their calculated $N\Lambda_{c}$ interaction
is much stronger than ours. For the three body ${}^{3}_{\Lambda_{c}}$H
calculation, the binding energy of the $J^{P}=3/2^{+}$ state is larger than
the $J^{P}=1/2^{+}$ state. In Ref. Garcilazo2015 , they also found that the
binding energy for the $J^{P}=3/2^{+}$ state in ${}^{3}_{\lambda_{c}}$H is
larger than the $J^{P}=1/2^{+}$ state.

Table 4: The $\Lambda_{c}$ separation ($B_{\Lambda_{c}}$) energy of ${}^{3}_{\Lambda_{c}}$H under different models. The units are in MeV. |  $\mu=1.0$ |  $\mu=1.2$ |  Ref. Maeda2015  |  Ref. Garcilazo2015
---|---|---|---|---
$B_{\Lambda_{c}}(1/2^{+})$ |  0.85 |  0.08 |  18.23 |  -
$B_{\Lambda_{c}}(3/2^{+})$ |  1.31 |  0.15 |  18.89 |  $0.14\sim 0.75$
$1/2^{+}-3/2^{+}$ |  0.46 |  0.07 |  0.66 |  -

The comparison with those two works Garcilazo2015 ; Maeda2015  are collected
in Tab. 4, where we show the $\Lambda_{c}$ separation energies for the
$J^{P}=1/2^{+}$ and $J^{P}=3/2^{+}$ states in the $\mu=1.0$ and $\mu=1.2$
cases, as well as the energy difference between the $1/2^{+}$ and $3/2^{+}$.
In our calculation, the $\Lambda_{c}$ separation energies in $\mu=1.2$ case
and the energy of $1/2^{+}-3/2^{+}$ are smaller than the $\mu=1.0$ case.
Despite the extremely large binding energies in Ref. Maeda2015 , the
$1/2^{+}-3/2^{+}$ energy is close to our $\mu=1.0$ result. As for the
$\Lambda_{c}$ separation energies of the $J^{P}=3/2^{+}$ state, our results
are consistent with the values in Ref. Garcilazo2015 .

As people know exp-hypernuclei , the lightest hypernucleus,
${}^{3}_{\Lambda}$H ($1/2^{+}$), is a weakly bound state and the
$B_{\Lambda}=0.13\pm 0.05$ MeV. And according to our model, a charmed
hypertrion may also be bound with the $B_{\Lambda_{c}}=0.08\sim 0.85$ MeV for
the $1/2^{+}$ state. Though both the hypertriton and the
${}^{3}_{\Lambda}$H($1/2^{+}$) might have small binding energies, the
dynamical behaviors between the $\Lambda$ and $\Lambda_{c}$ could be different
significantly in the nucleus due to the much heavier mass of the $\Lambda_{c}$
baryon. In order to investigate this issue, we study the three body
${}^{3}_{\Lambda}$H wave functions with calculating the following form:

$\displaystyle\rho(R_{2})=\int|\Psi(^{3}_{\Lambda_{c}}{\rm
H})|^{2}d^{3}{\bm{r}}_{2}d^{2}\hat{\bm{R}}_{2},$ (12)
$\displaystyle\rho(r_{2})=\int|\Psi(^{3}_{\Lambda_{c}}{\rm
H})|^{2}d^{3}{\bm{R}}_{2}d^{2}\hat{\bm{r}}_{2},$

where $\emph{R}_{2}$ and $\emph{r}_{2}$ are the Jacobian coordinates with
channel c=2, defined in Fig. 4 and $\Psi(^{3}_{\Lambda_{c}}{\rm H})$ is the
wave function of the ${}^{3}_{\Lambda_{c}}$H.

In Fig. 6, we depict the $\rho(r_{2})$ and $\rho(R_{2})$ of the
${}^{3}_{\Lambda_{c}}$H( $1/2^{+}$) state in the $\mu=1.0$ case. It’s obvious
that $\rho(r_{2})$ reflects the dynamical behavior between the two nucleons
and $\rho(R_{2})$ reflects the $\Lambda_{c}$’s behavior. For comparison, we
also show the $\rho(r_{2})$ and $\rho(R_{2})$ of the hypertriton. The
calculated $B_{\Lambda}$ is 0.13 MeV which agrees with the experimental data.
The NN interactions used in the hypertriton and
${}^{3}_{\Lambda_{c}}$H($1/2^{+}$) are the same and we also apply the GEM to
solve the three body $NN\Lambda$ problem. The $\Lambda$N interaction used here
is same as the one used in Ref. wu2019  which simulates the G-matric (YN)
interaction derived from the Nijmegen model F (NF). Then, as shwon in Fig. 6,
we find that the wave function of $\Lambda_{c}$ is more compact in the
${}^{3}_{\Lambda_{c}}$H( $1/2^{+}$) state than the $\Lambda$ in the
hypertriton though they have similar binding energies. On the other hand, the
nucleon-nucleon wave functions are similar with each other in the
${}^{3}_{\Lambda_{c}}$H($1/2^{+}$) and the hypertriton. It’s well known that
in the hypertrion, the $\Lambda$ wave function is very dilute and is around 10
fm away from the deuteron core. But such behavior may not happen for the
$\Lambda_{c}$ in the charmed hypertrion according to our calculation.

Figure 6: $\rho(r_{2})$ and $\rho(R_{2})$ defined in Eq. 12. The black ones
are the ${}^{3}_{\Lambda_{c}}$H system and the red ones are the hypertriton.

##  V Summary

In this work, we investigate the three body $NN\Lambda_{c}$ system within the
framework of the quark-delocalization color-screening model and probe the
possibility of a bound ${}^{3}_{\Lambda_{c}}$H ($T=0$).

First, with QDCSM, we normalize the $\Sigma\;(\Sigma^{*})$ and the $l=2$
components into the S-wave $N\Lambda_{c}$ constituents. Considering the
differences between the generating coordinates $S$ and the relative coordinate
$r$ between the baryons, we construct effective Gaussian type $\Lambda_{c}N$
potentials with reproducing the $\Lambda_{c}N$ scattering lengths and the
effective ranges for the ${}^{1}S_{0}$ and ${}^{3}S_{1}$ cases. As for the
$NN$ interaction, we apply the AV8’ realistic $NN$ interaction.

With the above potentials, we solve the three body problem with GEM and obtain
the energy levels of ${}^{3}_{\Lambda_{c}}$H $J^{P}=1/2^{+}$ and
$J^{P}=3/2^{+}$ states with different color screening parameters $\mu$. We
find that both $J^{P}=1/2^{+}$ and $J^{P}=3/2^{+}$ states are bound under the
$\mu=1.0$ and $\mu=1.2$ cases while it’s unbound when $\mu=0.8$. Our
calculated binding energy of $J^{P}=3/2^{+}$ state consists with that of Ref.
Garcilazo2015 . We also find that the energy difference of $1/2^{+}-3/2^{+}$
is $70\sim 460$ keV and close to another calculation in Ref. Maeda2015 , i.e.
660 keV. At the end, we also compare the dynamical behavior of the
$\Lambda_{c}$ in the ${}^{3}_{\Lambda_{c}}$H and the $\Lambda$ in the
hypertrion. We find that the $\Lambda_{c}$ is closer to the nucleon core than
the $\Lambda$ in hypertrion due to the $\Lambda_{c}$’s heavier mass.frr The
experimental search of the charmed hypertriton is urgently needed in order to
further study the charmed hypernuclei.

###### Acknowledgements.

This work is supported by the Strategic Priority Research Program of Chinese
Academy of Sciences under the Grant NO. XDB34030301, the National Natural
Science Foundation of China under the Grant NOs. 12005266, 12075288, 11735003,
11961141012, 12035007, Guangdong Provincial funding with Grant No.
2019QN01X172, and Guangdong Major Project of Basic and Applied Basic Research
No. 2020B0301030008. It is also supported by the Youth Innovation Promotion
Association CAS. Q.W. is also supported by the NSFC and the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) through the funds
provided to the Sino-German Collaborative Research Center TRR110 “Symmetries
and the Emergence of Structure in QCD” (NSFC Grant No. 12070131001, DFG
Project-ID 196253076-TRR 110).

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# Consistent Explanations by Contrastive Learning

Vipin Pillai
University of Maryland, Baltimore County
<EMAIL_ADDRESS>Soroush Abbasi Koohpayegani
University of Maryland, Baltimore County
<EMAIL_ADDRESS>Ashley Ouligian
Northrop Grumman
<EMAIL_ADDRESS>Dennis Fong
Northrop Grumman
<EMAIL_ADDRESS>Hamed Pirsiavash
University of California, Davis
<EMAIL_ADDRESS>
###### Abstract

Post-hoc explanation methods, e.g., Grad-CAM, enable humans to inspect the
spatial regions responsible for a particular network decision. However, it is
shown that such explanations are not always consistent with human priors, such
as consistency across image transformations. Given an interpretation
algorithm, e.g., Grad-CAM, we introduce a novel training method to train the
model to produce more consistent explanations. Since obtaining the ground
truth for a desired model interpretation is not a well-defined task, we adopt
ideas from contrastive self-supervised learning, and apply them to the
interpretations of the model rather than its embeddings. We show that our
method, Contrastive Grad-CAM Consistency (CGC), results in Grad-CAM
interpretation heatmaps that are more consistent with human annotations while
still achieving comparable classification accuracy. Moreover, our method acts
as a regularizer and improves the accuracy on limited-data, fine-grained
classification settings. In addition, because our method does not rely on
annotations, it allows for the incorporation of unlabeled data into training,
which enables better generalization of the model. Our code is available here:
https://github.com/UCDvision/CGC

##  1 Introduction

Deep neural networks have become ubiquitous in many applications owing to
their performance on several computer vision tasks. Although they have been
instrumental in achieving state-of-the-art accuracy, deep neural networks are
widely considered to be black box systems, which is not desirable. For
example, if an AI system is deployed to identify a malignant tumor from CT
scans, it is important for medical experts to understand the reasoning behind
the decision-making process [39]. This not only enables building trust, but
also helps identify any spurious correlations that the network may have
inadvertently learned to use to make its decision [34]. In recent years, there
have been attempts to open this black box by designing frameworks to explain
the network’s decision-making process. Post-hoc explanation methods such as
CAM [46], Grad-CAM [32], and Full-Grad [36] generate a heatmap in the size of
the image with higher values corresponding to the regions that contributed
most to the network’s decision.

Original | Augmented |  |  |  |  |  |  |  |  | Original | Augmented
---|---|---|---|---|---|---|---|---|---|---|---
|  |  |  |  |  |  |  |  |  |  |
Baseline |  |  |  |  |  |  |  |  |  |  |
Ours |  |  |  |  |  |  |  |  |  |  |
R.V | Puma
Figure 1: Our method significantly improves the consistency of Grad-CAM
explanation heatmaps under data augmentation. For both the RV and the Puma,
our method highlights the same portions of the image in both the original and
augmented versions. Figure 2: The block diagram of our method. Our method
consists of both cross-entropy loss ($L_{CE}$) and contrastive Grad-CAM
consistency loss ($L_{CGC}$). We load a batch of random images, and consider
one to be the query image. We feed the query image to the network and
calculate $L_{CE}$. We calculate Grad-CAM for this image on the top predicted
category and then augment the heatmap. We then augment all the images in the
batch, feed them to our model, and calculate the Grad-CAM heatmap for the top
predicted category. The top category is chosen using the original image and
not the augmented one. (Note that all random images are also augmented with
independently sampled parameters. We do not show this to reduce the clutter.)
The heatmap from the augmented query image is considered to be the positive
example. The heatmaps from the other random images in the batch are the
negative examples. The augmented query heatmap, the positive heatmap, and the
negative heatmaps are all fed into our contrastive comparison function to
produce our $L_{CGC}$ term.

Unlike image labels, there can be multiple valid explanations for a given
image category. Furthermore, a valid explanation might not involve the entire
object-segmentation area. For example, in order to correctly classify an image
of a dog, the network might rely on just the facial features of the dog, or
the texture of the fur and the tail, or a combination of both. Each of these
are valid explanations, and hence, generating ground truth annotations for
explanations is a not well defined task. This makes it difficult to directly
supervise the network’s explanation during training.

It is shown that most interpretation methods are not consistent with spatial
transformation of the images. For instance, shifting an image does not shift
the interpretation heatmap in the same way [11, 18]. In addition, in fine-
grained visual categorization, it is important to learn the subtle yet
discriminative features across classes (e.g., wing color, beak and eyes for a
bird) [42] and hence the network interpretation should focus on the most
salient features that discriminate the correct class from other classes.
Assuming Grad-CAM is a truthful interpretation method, we are interested in
improving the training process of the deep network so that its Grad-CAM
interpretation is more consistent with respect to spatial transformations,
thus making the model more interpretable. We use Grad-CAM as the key
interpretation method for the rest of the paper since it passes the sanity
check introduced in [1] and is end-to-end differentiable.

Inspired by self-supervised learning, we argue that a spatial affine
transformation on an image should correspond to such transformation in the
interpretation. For instance, given image of a “dog”, the heatmap for “dog”
category should highlight the dog and it should shift or zoom if we shift or
zoom into the image. Figure 1 shows an example where the change in Grad-CAM
heatmap for the baseline network computed for the “RV” and “Puma” categories
is not consistent with the spatial affine transformation applied to the
images. We adopt ideas from recently developed contrastive self-supervised
learning literature [12, 24, 14] and design a loss function that encourages
the Grad-CAM of an image to be close to the Grad-CAM of an augmented version
of the same image while being far from the Grad-CAM of other random images.

We evaluate our CGC method for both classification accuracy and quality of
explanations measured by the “Content Heatmap” (CH) metric introduced in [28].
A similar metric was also introduced in [37] to measure the energy of
explanation heatmap inside an adversarial patch. CH measures the cumulative
heatmap contained within an annotated object mask and is a proxy to measure
the consistency of an explanation heatmap with respect to human annotations.
Moreover, since our method guides the model to focus mainly on the most
discriminative features of the images, it improves the accuracy in fine-
grained classification settings while improving the consistency of
interpretations. The effect is even more prominent in learning with fewer
labels on fine-grained tasks. We believe this is due to the regularization
effect that our method adds to the training process. In addition, since our
loss function does not need labeled data, it can benefit from unlabeled data
during training. Our experiments show that our method improves the consistency
of interpretation with respect to human annotations as well as the
classification accuracy in limited labels and fine-grained settings.

##  2 Related Work

Explanation Methods: Early methods used to explain the decision-making process
of a deep neural network have relied on developing post-hoc explanation
frameworks that generate explanation heatmaps for a given network, input
image, and the corresponding predicted class. [33] introduced a method that
measured the gradient of the predicted class with respect to the input image
and used spatial locations with large gradient magnitudes to obtain a saliency
map. This was improved upon by [35, 38] to obtain sharper saliency maps. Such
gradient-based saliency methods have been shown to match the state-of-the-art
example-based explanations on several datasets [30]. Class Activation Mapping
(CAM) [46] was introduced to generate a coarse localization heatmap by using a
global average pooling (GAP) layer to compute the gradients flowing into the
final convolution layer. Gradient-weighted Class Activation Mapping (Grad-CAM)
[32] generalized CAM by eliminating the need for a GAP layer to compute coarse
localization heatmaps. Another class of explanation methods perturb the input
to the model and observe the resulting changes to the output [43, 5, 9, 26,
8].

Self-Supervised Learning: Self-supervised learning methods rely on introducing
pretext tasks based on structural priors of the data without using image
labels to learn meaningful representations. Spatial structure in the visual
domain [7, 23, 24], color information [45, 20], and spatial orientation [10]
have been used to design pretext tasks. Recently, such hand-crafted pretext
tasks have been superseded by methods that instead learn representations by
contrasting the feature vectors of positive pairs with those of negative pairs
[14, 3]. We build upon these ideas by leveraging contrastive learning on the
network’s explanation heatmaps instead of the representations. This aligns
with our prior that the network’s explanation for a given image should be
consistent under augmentations of that image.

Consistency for Explanations: [29] use domain knowledge to align explanations
with prior knowledge in the form of importance scores. [13] use adversarial
perturbation on the input images for explanation consistency with the original
image. [40] use causal masking to remove salient regions of the input image
and generate positive and negative contrast images to improve model
interpretability. [16, 17] propose contrastive learning to improve
interpretability for NLP models. [11] introduced the idea of imposing a
perceptual consistency prior on the attention heatmaps while training the
network for multi-label image classification. The key idea in [11] is that the
CAM [46] attention heatmap of an image should follow the same transformation
if the image is transformed. A similar idea to enforce consistency
regularization on the CAM attention heatmaps using the concept of attention
separability and cross-layer attention consistency [41] was introduced for the
task of weakly-supervised semantic segmentation.

Our work differs from these in that we use a set of negative examples along
with the standard image transformations in a contrastive setting. Negative
examples play an important role in ensuring that the interpretation relies on
image-specific regions rather than being biased towards a blob in the middle
of the image or spread around the image uniformly. Moreover, most of these
works evaluate the interpretation by using it as a semantic segmentation tool.
However, we believe that an interpretation heatmap should not necessarily
highlight the whole object mask. Instead, it should highlight the “most
discriminative regions” of an image, which is a strict subset of the semantic
segmentation mask. We emphasize that we are not introducing a new explanation
method, but rather learning a model that is tuned to be explainable for a
given explanation method.

Probably, [28] is the closest to our work as it uses self-supervised learning
ideas to remove spurious correlations in the interpretation heatmap. [28] uses
a different self-supervised pseudo task for reducing the contextual reasoning
by feeding synthetic composite images during training which is out-of-
distribution data. Our work is focused on contrastive learning which has
driven the recent progress in self-supervised learning community and uses in-
distribution images only during training. Unlike [28], our method does not
require labels to encourage explanation consistency and hence we are able to
leverage additional unlabeled data for our contrastive explanation consistency
loss together with the labeled data used for the standard cross-entropy loss.

##  3 Method

Figure 2 shows a block diagram of our method. Our training process consists of
both categorical cross-entropy loss ($L_{CE}$) and contrastive Grad-CAM
consistency loss ($L_{CGC}$). In this section, we first present a brief
overview of the Grad-CAM interpretation algorithm and then describe the
contrastive Grad-CAM consistency loss term.

Background on Grad-CAM [32]: Consider an input image $x$ and a deep neural
network $f$. Let $y$ be the vector of output logits when we feed $x$ to the
model $f$ where, $y^{t}$ corresponds to the output for category $t$. The Grad-
CAM of model $f$ for image $x$ and a given category $t$ is a heatmap that
highlights the regions of image $x$ responsible for the model’s classification
of the image as category $t$. We calculate this heatmap by choosing an
intermediate convolutional layer and then linearizing the rest of the network
to be interpretable. More specifically, we calculate the derivative of the
predicted output with respect to each channel of the convolutional layer
averaged over all spatial locations. This results in a scalar for each channel
that captures the importance of that channel in making the current prediction.
Then, we calculate a weighted average of all activations of the convolutional
layer with the above importance weights for each channel to get a 2D matrix
over spatial locations. Finally, we keep only positive numbers and resize it
to the size of input image to get the interpretation heatmap.

###  3.1 Contrastive Grad-CAM Consistency Loss

We are interested in training the image classification model so that its Grad-
CAM heatmaps are consistent with spatial transformations. For instance, when
we shift the image, the interpretation heatmap should also shift in the same
way. Also, the heatmap should be specific to the image, as in not always
focused on a blob in the middle of the image or be spread around the whole
image.

Inspired by contrastive learning methods in self-supervised learning, we
design a contrastive loss function for the interpretation heatmaps that acts
as a regularizer when added to the standard cross entropy loss for supervised
learning. We want the transformed interpretation of a query image to be close
to the interpretation of the transformed query image while being far from
interpretations of other random images.

More formally, we assume $g(.)$ is the Grad-CAM operator that calculates the
interpretation heatmap for the top predicted category of an input image. Given
a set of $n$ random images $\\{x_{j}\\}_{j=1}^{n}$, we augment them with
independent random spatial transformations $T_{j}(.)$ which involves a
combination of random scaling, cropping, and flipping. This is similar to the
standard augmentation usually done in deep learning. Then, we feed the
augmented images through the model and calculate their Grad-CAM heatmaps to
get $\\{g_{j}(T_{j}(x_{j})\\}_{j=1}^{n}$. We assume one of the images $x_{i}$
where $i\in{1..n}$ is the query image and calculate its Grad-CAM heatmap
without any transformation. We then apply the same transformation we had
applied to $x_{i}$ to the Grad-CAM heatmap, instead of the image, to get
$T_{i}(g_{i}(x_{i}))$.

Our main idea is that if we transform an image, the interpretation should also
be transformed in the same way. In addition, the interpretation should be
specific to each image. We want $T_{i}(g_{i}(x_{i}))$ to be close to
$g_{i}(T_{i}(x_{i})))$ and far from $\\{g_{j}(T_{j}(x_{j})\\}_{j\neq i}$.
Hence, we define the following loss function:

$L_{i}=-\text{log}\frac{\text{exp}\big{(}\text{sim}\big{(}T_{i}(g_{i}(x_{i})),g_{i}(T_{i}(x_{i}))\big{)}/\tau\big{)}}{\sum_{j=1}^{n}\text{exp}\big{(}\text{sim}\big{(}T_{i}(g_{i}(x_{i})),g_{j}(T_{j}(x_{j})\big{)}/\tau\big{)}}$

where $\tau$ is the temperature hyperparameter and $\text{sim}(a,b)$ measures
a similarity between two heatmaps. In our experiments, we use cosine
similarity. Note that cosine similarity is equivalent to L2 distance metric on
normalized features. $g_{i}(.)$ always calculates the Grad-CAM of the top
prediction of the original image regardless of the transformation since it is
important to keep the category that the Grad-CAM heatmap is calculated on
consistent for the positive pair involving the query image $x_{i}$.

We call our loss term Contrastive Grad-CAM Consistency Loss ($L_{CGC}$). This
loss is similar to the standard contrastive self-supervised learning loss [14]
with two main differences: (1) Our loss is defined on the interpretation of
the network output instead of the image features; (2) The interpretation of
the original query image is also augmented with the same parameters to match
the interpretation of the augmented image. This compensates for the
transformation to make the interpretations aligned.

In practice, since we run the optimization on mini-batches, we assume each
image is the query once and sum over all losses optimizing
$L_{CGC}=\sum_{i}L_{i}$. This can be implemented efficiently for the whole
mini-batch by augmenting each image once and calculating Grad-CAM for each
image twice. Some contrastive self-supervised learning algorithms like MoCo
[14] use a memory bank to increase the number of negative pairs, but for
simplicity, we do not use a memory bank and use mini-batches of size $256$.
Thus, our method is more similar to SimCLR [4] than MoCo [14].

Our final loss is the combination of the standard cross-entropy loss
($L_{CE}$) and our contrastive Grad-CAM consistency loss ($L_{CGC}$). Hence,
we minimize the following loss function:

$L=L_{CE}+\lambda L_{CGC}$

where, $\lambda$ is a hyper-parameter that controls the trade-off between the
two loss terms. Note that our $L_{CGC}$ loss term does not use image labels as
it uses pseudo labels for Grad-CAM. This enables us to use additional
unlabeled data to improve both the accuracy and explainability of the
resulting model in Section 4.3.

##  4 Experiments

In this section, we perform a variety of experiments using our CGC method. For
each of the experiments, ‘baseline’ refers to a model trained from scratch
using the standard cross-entropy loss unless noted otherwise. We will report
the classification accuracy along with the following metrics used for
evaluating the explanation heatmaps:

Content Heatmap (CH): Introduced in [28], this metric is a measure of the
summation of $\ell_{1}$-normalized heatmap contained within the annotated
bounding box of the object. If the model interpretation is consistent with
human annotations of the object location, we can assume that the percentage of
the heatmap that lies inside the object annotation mask should be close to
100%. Hence, we expect this metric to be high.

CGC loss: We also evaluate the explanation heatmaps using the same $L_{CGC}$
loss that we use for training our models. Although this loss is already used
in our optimization, we believe it is important to show that the loss is small
on the unseen test data as well, i.e., the method generalizes from the
training to test set. We use ImageNet [31] validation set to report this
metric. We use a batch size of 32 to compute this loss term. For every query
image in the batch, we pair it with an augmentation of the corresponding query
image as the positive pair and consider each of the remaining 31 images in the
batch to be the negative pairs.

Insertion AUC score (IAUC): This metric [26] successively inserts pixels from
the highest to lowest attribution scores and makes predictions. The area under
the curve defined by the prediction scores is then defined as the IAUC score.
We expect the IAUC score to be higher for a better interpretation.

Implementation Details: We use PyTorch [25] to train and evaluate our models
for all experiments. We use SGD (weight decay=$1e-4$, learning rate=$0.1$,
momentum=$0.9$, and batch size=$256$) to optimize both ResNet18 and ResNet50
[15] models. We train ImageNet [31] models for 90 epochs and decay the
learning rate by 0.1 every $30$ epochs. For transformation $T_{i}$ in our loss
term, we use standard data augmentations (scaling, flipping, and translation).
The baseline models also use these same augmentations. We use ResNet50
architecture for all our experiments unless otherwise noted. Training our
ResNet50 on 2 Titan-RTX GPUs takes approximately 100 hours, whereas training
the baseline takes approximately 70 hours on the same setting. We use
$\tau=0.5$ for all experiments using our method. For ImageNet dataset, we use
$\lambda=1.0$ for our method using ResNet18 and $\lambda=0.5$ for ResNet50.

We first report our results on ImageNet and UnRel [27] datasets. We then show
results on fine-grained classification tasks including limited data settings.
Finally, we discuss results using unlabeled data with our loss term.

###  4.1 Contrastive Grad-CAM Consistency (CGC)

We train a network from scratch using the CGC method (cross-entropy loss and
contrastive loss) and compare it to a baseline network trained from scratch
using cross-entropy loss only. We report the Top-1 classification accuracy as
well as heatmap evaluation metrics. We show results for both the ImageNet and
UnRel [27] datasets in order to show that our approach generalizes across
multiple datasets. For both datasets, on the ResNet50 architecture, we show
that our model has a slight (less than 2% points) decrease in classification
accuracy but shows a significant (greater than 15% points) improvement on the
explanation metrics.

ImageNet: Quantitative results on ImageNet validation set are reported in
Table 1. For the ResNet50 model, we have a marginal drop in classification
accuracy (1.5 points), whereas CH increases by 17 points. Since, ImageNet is a
large dataset, we do not expect our regularizer to improve the classification
accuracy on ImageNet. Moreover, our main focus is on improving the consistency
of the explanations and we are willing to accept a marginal drop in
classification accuracy at the cost of improved explanation consistency.

Table 2 reports the evaluation results using the IAUC metric [26] on the
ImageNet validation set and shows that our method quantitatively improves the
Grad-CAM explanation heatmaps of the underlying model.

Arch | Method | Top-1 | CH (%) | CGC
---|---|---|---|---
|  | Acc (%) |  | Loss
ResNet18 | Baseline | 69.76 | 54.47 | 3.19
GCC [28] | 67.74 | 57.73 | 3.14
Ours (CGC) | 66.37 | 65.83 | 2.59
ResNet50 | Baseline | 76.13 | 54.77 | 3.15
GCC [28] | 74.40 | 59.42 | 3.09
Ours (CGC) | 74.60 | 71.75 | 2.64
Table 1: Classification accuracy along with the Content Heatmap (CH) and CGC Loss explanation metrics on ImageNet validation set. Note that lower is better for CGC Loss. Method | Insertion AUC
---|---
Baseline | 0.4860
Ours (CGC) | 0.5216
Table 2: The Grad-CAM explanation maps generated by our ResNet50 model
outperforms the baseline ResNet50 model on the IAUC metric [26] using the
ImageNet validation set.

UnRel: The UnRel dataset [27] consists of images capturing unusual relations
between objects. These images were collected from the web using triplet
queries such as ‘person ride giraffe’. This dataset thus captures objects
occurring in unusual spatial configurations and contexts. Both the baseline
and CGC models are trained on ImageNet and evaluated on the 28 object
categories of the UnRel dataset that overlap with ImageNet. We report the
corresponding quantitative results in Table 3. We observe that although our
CGC model was trained on ImageNet, which is an object centric dataset, the
improvement in our explanation heatmap generalizes to objects occurring in
unusual backgrounds and spatial configurations.

Model | Top-1 Acc (%) | CH (%)
---|---|---
Baseline | 40.66 | 51.66
Ours (CGC) | 38.25 | 74.20
Table 3: Classification accuracy and Content Heatmap (CH) evaluation on the 28
UnRel categories [27] that overlap with ImageNet.

Generalization to another interpretation method: We also show that the
improved explanations of our model trained using Grad-CAM generalize to
Contrastive Top-down Attention (cMWP) [44], which is another explanation
method. We report CH evaluation results using cMWP in Table 4.

Method | CH (%) using cMWP [44]
---|---
Baseline | 74.78
GCC [28] | 75.08
Ours (CGC) | 75.50
Table 4: Our method generalizes to another explanation method, Contrastive
Top-down Attention (cMWP) [44] although our model was trained using Grad-CAM.
We report the CH metric computed using cMWP on ImageNet validation set. All
models use ResNet50 architecture.

Role of negative heatmaps in $L_{CGC}$: As part of our $L_{CGC}$ loss,
together with an augmented image as the positive pair, we use the rest of the
images in the batch to compute the negative heatmaps. The presence of negative
heatmaps is expected to encourage the model to learn explanation heatmaps
specific to each image. This would result in the model learning features
corresponding to the most discriminative regions of the object. If the
negative heatmaps are not used as part of the contrastive loss, the model can
cheat by learning a trivial solution for the explanation consistency and would
result in the explanation heatmaps being spread uniformly across the image. We
verify this be training a model where the $L_{CGC}$ uses a simple $\ell_{2}$
loss on the normalized heatmap of the positive pair and does not use negative
heatmaps. We observe that the resulting model indeed results in explanation
heatmaps which are diffused across the image as verified by the low CH in
Table 5.

Model | ResNet18 | ResNet50
---|---|---
Top-1 Acc (%) | CH (%) | Top-1 Acc (%) | CH (%)
CGC | 66.37 | 65.83 | 74.60 | 71.75
CGC w/o neg | 67.00 | 44.08 | 74.80 | 39.84

Table 5: Comparison of ImageNet evaluation against model trained without using negative heatmaps as part of $L_{CGC}$. The model trained without the negative heatmaps as part of $L_{CGC}$ loss results in a very low CH, thus confirming our hypothesis that the lack of negative heatmaps would result in a model learning a trivial solution of generating heatmaps diffused throughout the image. Method | CUB-200 | FGVC-Aircraft | Cars-196 | VGG Flowers-102
---|---|---|---|---
Baseline | 80.09 $\pm$ 0.89 | 83.65 $\pm$ 0.15 | 89.71 $\pm$ 0.14 | 96.09 $\pm$ 0.23
Ours (CGC) | 81.49 $\pm$ 0.09 | 85.72 $\pm$ 0.20 | 90.28 $\pm$ 0.08 | 96.18 $\pm$ 0.09
Table 6: Evaluation of classification accuracy of the baseline against our CGC method for fine-grained datasets. We run 3 trials and report the mean and standard deviation. We observe consistent improvements in the classification accuracy across all four datasets with the largest gain on the FGVC-Aircraft dataset. | CUB200 | Cars196
---|---|---
Method | 1-shot | 5-shot | 10-shot | 1-shot | 5-shot | 10-shot
Baseline | 13.7 $\pm$ 0.3 | 51.7 $\pm$ 0.3 | 66.4 $\pm$ 0.2 | 6.1 $\pm$ 0.2 | 34.3 $\pm$ 0.4 | 61.1 $\pm$ 0.4
Ours (CGC) | 15.8 $\pm$ 0.3 | 55.2 $\pm$ 0.3 | 68.4 $\pm$ 0.3 | 6.5 $\pm$ 0.2 | 36.9 $\pm$ 0.4 | 63.0 $\pm$ 0.4
| Aircrafts | Flowers
| 1-shot | 5-shot | 10-shot | 1-shot | 5-shot | 10-shot
Baseline | 7.7 $\pm$ 0.3 | 25.7 $\pm$ 0.4 | 41.4 $\pm$ 0.3 | 52.1 $\pm$ 0.5 | 85.6 $\pm$ 0.4 | 93.2 $\pm$ 0.2
Ours (CGC) | 8.0 $\pm$ 0.3 | 26.9 $\pm$ 0.4 | 42.9 $\pm$ 0.3 | 53.3 $\pm$ 0.5 | 85.8 $\pm$ 0.4 | 93.4 $\pm$ 0.2
Table 7: Classification accuracy for the limited-data setting on the fine-grained datasets. We run 20 trials and report the mean and standard deviation. We observe consistent improvement across all datasets with the largest improvements observed on the CUB-200 and Cars-196 datasets. | CUB-200 | Cars-196
---|---|---
Method | 1-shot | 5-shot | 10-shot | Full training set | 1-shot | 5-shot | 10-shot | Full training set
Baseline | 55.54 | 57.60 | 61.39 | 63.71 | 60.51 | 64.08 | 64.31 | 65.58
Ours (CGC) | 61.63 | 63.55 | 73.86 | 71.08 | 60.48 | 62.13 | 64.63 | 69.04
Table 8: Content Heatmap (CH) evaluation results for limited and full training
data settings on CUB-200 and Cars-196 datasets. CH results are better for all
CUB-200 sets but are marginally worse for 1-shot and 5-shot settings on
Cars-196. We believe the lower accuracy of the model for few-shot setting
results in noisy interpretations, so the CH metric becomes less reliable.

###  4.2 Fine-Grained Classification

We now report the results for fine-grained classification on CUB-200 [42],
FGVC-Aircraft [21], Stanford Cars-196 [19], and VGG Flowers [22] datasets. All
models used for fine-grained classification are pretrained on ImageNet and all
layers are fine-tuned for the new dataset.

We find that our method can be used as a form of regularization, which is
particularly helpful in these fine-grained classification scenarios. While our
method trains to produce better interpretation, using a contrastive loss on
the interpretation will allow the model to learn unique attention for
individual classes. Note that the negative samples in the contrastive loss
encourages the interpretation to be different from other samples. As a result,
the trained model limits the attention to the most discriminative part(s) of
the object and hence, our method acts like a regularizer when adopted for
fine-grained classification tasks.

For our CGC method, we use $\lambda=0.25$ for the CUB-200 and Cars-196
datasets, and $\lambda=0.5$ for the FGVC-Aircraft and VGG Flowers datasets.

Results for classification accuracy are shown in Table 6, and results for
Content Heatmap evaluation are shown in Table 8. For all fine-grained
datasets, the model trained with our CGC method achieves both improved
classification accuracy and improved explanation scores. For most datasets,
improvements to classification accuracy is marginal, but the FGVC-Aircraft
dataset shows an improvement of over 2% points. This experiment demonstrates
that our method not only improves explainability, but also classification
accuracy for datasets that require the network to focus on the “most
discriminative” features of an object.

Limited Training Data: Using our method for regularization can be particularly
helpful when the amount of training data is limited. We evaluate our method
with the same fine-grained classification settings, but limit the amount of
training data for each class. We evaluate 1, 5, and 10-shot training settings.
We repeat our experiment for 20 episodes and report mean and standard
deviation over all episodes. In each episode, we randomly select
$n\in\\{1,5,10\\}$ samples from each class as training set and use the rest of
the samples as a validation set. We initialize the model from a ResNet50 model
pretrained on ImageNet with categorical cross-entropy loss only, then finetune
all layers on the limited training set. We use the same random seed for our
method as well as the baseline. For our method in this limited-data setting,
we use $\lambda=0.8$ for the CUB-200, Cars-196, and VGG Flowers datasets and
$\lambda=0.6$ for the FGVC-Aircraft datasets. For all datasets we use
$\tau=0.5$ as the value for the temperature hyperparameter.

Results for classification accuracy in this limited-data setting are found in
Table 7. Both baseline and CGC models perform better with more samples per
class, but our method consistently outperforms the baseline. The magnitude of
improvement varies from 1-4% points. Notably, our method outperforms the
baseline by 4% points (5-shot and 10-shot) on CUB-200 and by 2% points (5-shot
and 10-shot) on Cars-196. Since only CUB-200 and Cars-196 contain bounding box
annotations, we report the CH evaluation metric for these two datasets in
Table 8. CH results are better for all CUB-200 sets but are marginally worse
for 1-shot and 5-shot settings on Cars-196. We believe the lower accuracy of
the model for the few-shot setting results in noisy interpretations, so the CH
metric becomes less reliable.

Polaroid | Boathouse | Great White | Cowboy | Snowmobile | Volleyball | Dogsled
---|---|---|---|---|---|---
Camera |  | Shark | Hat |  |  |
Original |  |  |  |  |  |
Baseline G-CAM |  |  |  |  |  |
Ours G-CAM |  |  |  |  |  |
Figure 3: Grad-CAM visualization results for images from ImageNet validation
set using ResNet50. We observe that the model trained using our method focuses
on the most discriminative regions of the object instead of the background
pixels. In the 6th column above, we compute the Grad-CAM explanation for the
category “volleyball” and we see that our model no longer focuses on the
players and correctly highlights the volleyball as the cause of the target
category. The last column contains a failure example for the category
“Dogsled”. The baseline model correctly highlights the sled whereas our model
incorrectly highlights the dog.

###  4.3 Using Unlabeled Data for CGC Loss

Models trained with categorical cross-entropy loss only have no way to
incorporate information from unlabeled data. Since our method does not rely on
labels for computing the explanation heatmaps, we extend our method to use
mostly unlabeled data with just a small fraction of labeled data for training.
We use $1\%$ of ImageNet training images as labeled data and leverage the rest
of ImageNet training images as the unlabeled data for the CGC loss term. We
initialize both the baseline and our method from a ResNet50 model trained
using SwAV [2].

Table 9 shows that our method outperforms the baseline by 1% point on both the
Top-1 and Top-5 accuracy metrics. Since we are using only 1% labeled data, the
resulting Grad-CAM heatmap will be less accurate and hence the 1% point
improvement is not insignificant. Even though our model is only able to
leverage the additional unlabeled data for the CGC loss term, and not the
categorical accuracy, we can see that the CGC loss acts as a regularizer to
improve the generalization of the model.

Model | Top-1 (%) | Top-5 (%) | CH (%)
---|---|---|---
Baseline | 54.00 | 78.69 | 46.08
Ours (CGC) | 55.18 | 79.12 | 46.76
Table 9: On the 1% ImageNet limited-label setting, our CGC method leverages
unlabeled data for the $L_{CGC}$ loss term and is able to improve the
classification accuracy and explanations when evaluated on ImageNet validation
data. Both models are initialized from a ResNet50 model trained in an
unsupervised manner using SwaV [2].

###  4.4 Ablation on $\lambda$

We perform an ablation experiment to study the sensitivity of our method to
the $\lambda$ hyperparameter. If $\lambda$ is too low, the cross-entropy term
will dominate the optimization and thus the resulting improvement in the
explanation consistency will be marginal, whereas if $\lambda$ is too high,
the CGC loss will be applied on noisy heatmaps resulting in lower accuracy and
lower explanation consistency. Table 10 shows the Top-1 accuracy and CH scores
for different values of $\lambda$ for ResNet18 on ImageNet dataset. We choose
$\lambda=1.0$ for ResNet18 as the best trade-off between accuracy and CH.

$\lambda$ | 0 | 0.01 | 0.1 | 0.5 | 1.0 | 2.0
---|---|---|---|---|---|---
Top-1 Acc | 69.76 | 66.97 | 66.90 | 65.14 | 66.37 | 65.89
CH | 54.47 | 56.18 | 61.97 | 66.19 | 65.83 | 52.60
Table 10: Ablation to study the sensitivity of our method to the $\lambda$
hyperparameter for ResNet18 on ImageNet dataset.

###  4.5 Qualitative Results

Figure 3 compares the Grad-CAM heatmaps generated by the baseline model
against our CGC model on ImageNet. Our model consistently focuses on the
discriminatory parts of the object and does not highlight background pixels.
Additional results for CUB-200, Cars-196 and Aircrafts datasets are included
in the appendix.

##  5 Conclusion

We introduce a contrastive learning method for improving the explanations
generated by a deep neural network, training them to be consistent with
spatial transformations. We emphasize the importance of evaluating the network
based on its quality of explanation, and not only classification accuracy. Our
CGC method significantly improves the explanation heatmaps while obtaining
comparable classification accuracy on ImageNet and UnRel datasets.
Furthermore, our method is able to boost the classification accuracy on fine-
grained classification datasets such as CUB-200, Cars-196, VGG Flowers-102,
and FGVC-Aircraft while improving the consistency of explanation heatmaps with
human annotations. This demonstrates that our method acts as a regularizer
that focuses more attention on the discriminating aspects of the image. We
also show that our method is able to leverage unlabeled data to improve the
classification accuracy in limited-label data settings.

Limitations: Our method uses Grad-CAM [32] algorithm to compute the
explanation heatmaps for the original set of images as well as the augmented
images, which are then used to compute the contrastive loss term $L_{CGC}$. In
comparison to standard training with cross-entropy loss, our method requires
additional compute to account for storing the additional gradient graph in
memory during backpropagation. While this is an overhead during the training
stage, we believe the incurred compute cost is offset by the improved
explainability of the resulting model.

Ethics Statement: Our method improves the explainability of image
classification models and thereby increases trust and transparency of the
underlying decision making process. However, our method is a data-driven
approach and hence could reflect potential negative biases present in the
training dataset. Moreover, explanation methods such as Grad-CAM [32] can be
an unreliable estimate of model interpretability (i.e., evidence for an
incorrect prediction looking identical to evidence for a correct prediction).

Acknowledgment: This material is based upon work partially supported by the
U.S. Air Force under Contract No. FA8750‐19‐C‐0098, U.S. Department of
Commerce, National Institute of Standards and Technology under award number
60NANB18D279, NSF grant numbers 1845216 and 1920079, and funding from Northrop
Grumman and SAP SE. Any opinions, findings, and conclusions or recommendations
expressed in this material are those of the authors and do not necessarily
reflect the views of the U.S. Air Force, DARPA, or other funding agencies. We
would also like to thank the reviewers for their valuable feedback.

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##  6 Appendix

Seaside Sparrow | Anna Hummingbird | Eared Grebe | Pileated | Red Cockaded | Red headed
---|---|---|---|---|---
|  |  | Woodpecker | Woodpecker | Woodpecker
Original |  |  |  |  |
Baseline G-CAM |  |  |  |  |
Ours G-CAM |  |  |  |  |
Figure 4: Grad-CAM visualization results for images from the CUB-200 validation set using ResNet50. Interestingly, on the right column, the baseline is focusing on the whole bird while our method focuses on the head of the bird only. Given the name of the category “Red headed Woodpecker”, it makes sense that the head should be the most discriminative region. BMW ActiveHybrid 5 | Lamborghini | Porsche | Jeep | Mercedes-Benz
---|---|---|---|---
Sedan | Reventon Coupe | Panamera Sedan | Patriot SUV | E-Class Sedan 2012
Original |  |  |  |
Baseline G-CAM |  |  |  |
Ours G-CAM |  |  |  |
Figure 5: Grad-CAM visualization results for images from the Cars-196
validation set using ResNet50. In the second column above, we see that our
model is able to correctly focus on the object regions of “Lamborghini”
whereas the baseline incorrectly highlights the fighter jet as well. The last
column shows a failure example where our model incorrectly focuses on the
“Mercedes” logo on the building instead of the car itself. This is an
interesting failure case since the logo is a valid discriminatory attribute
for a fine-grained car classification.

DHC-6 | DC-8 | DHC-6 | A380 | A340-500 | Hawk T1
---|---|---|---|---|---
Original |  |  |  |  |
Baseline G-CAM |  |  |  |  |
Ours G-CAM |  |  |  |  |
Figure 6: Additional Grad-CAM visualization results on the FGVC Aircraft validation set using ResNet50. While our method correctly highlights most discriminative part of the aircraft in the first and the third column, the baseline incorrectly highlights the water along with the aircraft. Note that the last column shows a failure case for our model which incorrectly highlights the smoke trajectory of the aircraft.  Shoe Store | Knot | Tench | Water Hen | Albatross | Airship | Minivan
---|---|---|---|---|---|---
Original |  |  |  |  |  |
Baseline G-CAM |  |  |  |  |  |
Ours G-CAM |  |  |  |  |  |
Figure 7: Additional Grad-CAM visualization results on the ImageNet validation set using ResNet50. Our model is able to improve upon the baseline by not relying on background pixels and instead focusing on the most discriminative regions of the object. In the 3rd column, Grad-CAM is computed for the category “Tench”and we see that the baseline incorrectly highlights the person along with the fish whereas our model correctly highlights the fish. The last column shows a failure case where our model incorrectly highlights the license plate along with the other regions of the minivan. Original | Baseline | CGC w/o neg | CGC
---|---|---|---
Parking meter |  |  |
Volley ball |  |  |
Figure 8: Qualitative results for Table 5 in the main paper. The Grad-CAM
heatmap for the model trained with CGC loss, but without the negative examples
results in a uniform heatmap spread across the image (column 3). In the second
row, the baseline model relies on the arms of the player for classifying the
image as ’Volleyball’, whereas our method is able to reduce this spurious
correlation.

Model | ResNet50 ImageNet-100
---|---
Top-1 Acc (%) | CH (%)
Baseline | 86.40 | 53.60
CGC | 84.04 | 72.32
CGC w/o neg | 81.94 | 38.46

Table 11: Results similar to Table 5 of the main paper with ResNet50 on
ImageNet-100 (subset of ImageNet with 100 classes). The low CH for CGC w/o
negatives shows that this method could result in heatmaps diffused across the
image. The model trained without the negative heatmaps as part of $L_{CGC}$
loss results in a very low CH, thus confirming our hypothesis that the lack of
negative heatmaps would result in a model learning a trivial solution of
generating heatmaps diffused throughout the image.

##  7 License for assets

We list the license for each of the dataset and code assets used for our
experiments.

ImageNet: We have been granted access to the ImageNet [6] dataset for non-
commercial research/educational purposes and we abide by the terms of the
license of this dataset.

CUB-200: This dataset was introduced in [42] and we use the images and the
accompanying annotations for non-commercial/education research purposes only.

FGVC-Aircraft: The images in the FGVC-Aircraft [21] dataset has been made
available exclusively for non-commercial research/educational purposes and as
such we only use this dataset for non-commercial research/educational
purposes.

Stanford Cars-196: This dataset [19] has been made available for non-
commercial research purposes only and we abide by the terms of the license of
this dataset.

VGG Flowers-102: The images and annotations in the VGG Flowers-102 [22]
dataset are released under the GNU General Public License, version 2.

TorchRay: This framework was introduced in [8] and is licensed under CC-BY-NC.
We use this framework for evaluating the explanation heatmaps using the
Content Heatmap (CH) metric.

Insertion AUC metric: This metric was introduced in [26] and the accompanying
code is released under the MIT license.