$a_{1}$ meson-nucleon coupling constant at finite temperature from the soft-
wall AdS/QCD model

Shahin Mamedov a,b,c111corresponding author<EMAIL_ADDRESS>and
Shahnaz Taghiyeva b222<EMAIL_ADDRESS>

${}^{a}\,$ Institute for Physical Problems, Baku State University,

Z. Khalilov street 23, Baku, AZ-1148, Azerbaijan,

bTheoretical Physics Department, Physics Faculty, Baku State University,

Z.Khalilov street 23, Baku, AZ-1148, Azerbaijan,
cInstitute of Physics, Azerbaijan National Academy of Sciences,

H. Cavid avenue. 33, Baku, AZ-1143 Azerbaijan,

Abstract

We continue to study the temperature dependence of the meson-nucleon coupling
constants and consider the axial-vector $a_{1}$ meson case. Profile functions
for the axial-vector and fermion fields in the AdS-Schwarzschild metric are
presented in the framework of the soft-wall AdS/QCD model with thermal dilaton
field. Bulk interaction Lagrangian are constructed for the interacting
fermion-axial-vector-thermal dilaton fields system. For the $g_{a_{1}NN}$
coupling constant is obtained an integral representation and the temperature
dependence of this coupling is analyzed numerically.

##  I Introduction

The AdS/CFT correspondence is one of the most important discoveries in the
modern theoretical physics. This principle states the equivalence of two
different physical theories, a gravitational theory in the bulk of AdS space-
time and a quantum field theory on the boundary of this space-time. Moreover,
the duality between the two theories is the strong-weak duality, which means
that it relates the strong coupling sector of one theory to the weak coupling
sector of the other one 1 ; 2 ; 3 ; 4 . Further this duality was adapted to
describe the low-energy dynamics of QCD and called as AdS/QCD 5 ; 6 ; 7 ; 8 ;
9 . There are two approaches in AdS/QCD: top-down and bottom-up ones. The
bottom-up approach is based on direct application of holographic duality,
while the top-down approach is based on the duality between the open and
closed string amplitudes. The models of the bottom-up and top-down approaches
are widely applied to predict the phenomenological quantities and to study
other strong interaction problems in particle physics. Holographic soft-wall
model 7 ; 10 ; 11 ; 12 , which is one of two main models in the bottom-up
approach, also is useful for the studies of hadron interactions in the nuclear
medium333more precisely, in the nucleon medium, which is formed in the heavy
ion or proton collisions until the energies, when the confinement-
deconfinement phase transition occur. Temperature dependencies of the
quantities describing the hadrons in the nuclear medium are investigated in
the framework of different approaches and models. The holographic soft-wall
model is one of good tools for the studies in the medium and this model was
applied for solving different kinds of problems: determining of the
temperature influence on in-medium quantities such as screening mass 13 ,
hadron form factors 14 ; 15 ; 16 ; 17 , quarkonium 18 , decay constants 19 ,
transport coefficients 20 , anisotropy in such medium 21  and so on. In the
holographic models the temperature of the medium is taken into account by
means of the black hole temperature in the dual AdS-Schwarzschild background
22 ; 23 . Different modifications of original holographic thermal models were
applied in order to solve different phenomenological problems. In the approach
in Refs. 24 ; 14 ; 15 ; 16  the sot-wall model was modified by considering the
dilaton field, which is responsible for the chiral and conformal symmetries
breaking in the model, as a thermal one, since this field is in the thermal
medium. This field is related by the chiral condensate and so, the
thermalization of dilaton field physically means that, the authors in fact
take into account the temperature dependence of the chiral quark condensate in
the medium. As a check of this idea’s predictions, the authors investigate the
hadron form factors, transition and electromagnetic form factors of nucleons
in the framework of this model and find reasonable results. Continuing these
investigations in the Ref. 25  authors studied the temperature dependence of
the vector meson - nucleon coupling constant as well. It was observed that the
value of this constant decreases on temperature increasing and vanishes at the
temperature close to the confinement-deconfinement phase transition
temperature. It is interesting to check, whether this situation takes place
for other meson sectors of the model as well or not. Here we aim to study
temperature dependence in the axial-vector sector and consider the lightest
axial-vector meson, namely the $a_{1}$ meson-nucleon coupling constant
dependence on temperature.

The paper is organized as follows: In section II we briefly review the basic
definitions in the soft-wall model having thermal dilaton field and present
profile function for the thermal $a_{1}$ meson in this model. In Section III
for the nucleon and in section IV for the scalar field the temperature
dependent profile functions are briefly derived. In section V an integral
expressions for the temperature- dependent coupling constants are obtained. In
section VI the coupling constant terms are numerically analyzed and in
conclusion section we discuss our results for the temperature dependencies.

##  II Soft-Wall Model at Finite Temperature

According to AdS/CFT correspondence the quantum field theory at finite
temperature on the boundary of the space-time is described by the AdS-
Schwarzschild background in the bulk, which has following metric:

$ds^{2}=g_{MN}dx^{M}dx^{N}=e^{2A(z)}\left(f(z)dt^{2}-dx_{i}dx^{i}-\frac{dz^{2}}{f(z)}\right),$
(1)

where the $A(z)$ and $f(z)$ functions have explicit form:

$\displaystyle A(z)=log(R/z),$ (2) $f(z)=1-\frac{z^{4}}{z_{h}^{4}}.$ (3)

$R$ is the AdS radius and $x=(t,\overrightarrow{x})$ are the set of Minkowski
coordinates. The $z$ coordinate is chosen in the $0\leq z\leq z_{h}$ interval
in order to describe the confinement phase of medium. The black hole Hawking
temperature $T$ is related by the position of the black hole horizon $z_{h}$:

$T=\frac{1}{4\pi}|\frac{df}{dz}|_{z=z_{h}}=\frac{1}{\pi z_{h}}.$ (4)

This temperature in the dual QCD theory corresponds to the temperature of
hadronic matter.

Main idea of this model is to consider the dilaton field $\varphi(x)$ as a one
depending on the temperature of the medium 14 ; 15 ; 16 . To this end the $z$
coordinate was replaced by the Regge-Wheeler tortoise coordinate $r$, which
was introduced in Ref. 26 . Relation between these coordinates in the finite
temperature limit is following one:

$r=z[1+\frac{t^{4}}{5}+\frac{t^{8}}{9}+O(t^{12})],$ (5)

where $t=z/z_{h}$. An explicit form of the dilaton field finds from the idea
of sameness of the temperature dependencies of dilaton field and quark
condensate $\Sigma(T)$. Last one is known from chiral perturbation theory and
we have the dilaton field of form below444More explanation will be done in the
IV section.:

$\varphi(r,T)=K^{2}(T)r^{2}=(1+\rho(T))k^{2}r^{2}.$ (6)

The thermal addition $\rho_{T}$ term has a form:

$\rho(T)=\delta_{T_{1}}\frac{T^{2}}{12F^{2}}+\delta_{T_{2}}\left(\frac{T^{2}}{12F^{2}}\right)^{2}+O(T^{6}),$
(7)

with the denotations of constant parameters $\delta_{T_{1,2}}$

$\delta_{T_{1}}=-\frac{N_{f}^{2}-1}{N_{f}},$ (8)
$\delta_{T_{2}}=-\frac{N_{f}^{2}-1}{2N_{f}^{2}}.$ (9)

Here $N_{f}$ is the number of quark flavors. $F$ is the decay constant in the
chiral limit in zero temperature. In Ref. 15 ; 16  authors pointed out that in
the soft-wall AdS/QCD model the dilaton parameter $k$ and the pion decay
constant $F$ in the chiral limit have a relation

$F=k\frac{\sqrt{3}}{8},$ (10)

which is right at zero and finite temperatures.

###  II.1 Profile function for $a_{1}$ meson at finite temperature

Let us at first briefly present the $a_{1}$ meson at zero temperature
following to Ref. 7 . The gauge field sector of the model consists of the
$A_{L,R}$ gauge fields, which are coming from the $SU(2)_{L,R}$ symmetries of
the $SU(2)_{L}\times SU(2)_{R}$ flavor symmetry group of the model.
$V=\frac{1}{2}(A_{L}+A_{R})$ vector and $A=\frac{1}{2}(A_{L}-A_{R})$ axial-
vector fields are composed of these gauge fields and bulk action for the gauge
sector will be written in terms of the composed fields:

$\displaystyle
S=-\frac{1}{4g_{5}^{2}}\int_{0}^{\infty}d^{5}xe^{-\varphi(z)}\sqrt{g}Tr[F_{L}^{2}+F^{2}_{R})]=-\frac{1}{2g_{5}^{2}}\int_{0}^{\infty}d^{5}xe^{-\varphi(z)}\sqrt{g}Tr[F_{V}^{2}+F^{2}_{A}].$
(11)

Here $g$ is $g=detg_{MN}$, $\varphi(z)=-k^{2}z^{2}$ is the dilaton field, $X$
is scalar field. $F_{V,A}$ are the field strength tensors of the vector and
axial-vector fields. Here we shall deal with the only axial-vector field, the
strength tensor of which has a form:

$F_{A}^{MN}=\partial^{M}A^{N}-\partial^{N}A^{M}-\frac{i}{\sqrt{2}}[A^{M},A^{N}].$
(12)

For further calculations the axial gauge $A_{5}=0$ is chosen. Fourier
components $A_{\mu}(q,z)$ of the axial-vector field satisfies the boundary
conditions $A(q,\epsilon)=1$, $\partial_{{}_{z}}A(q,z=z_{m})=0$ at ultraviolet
and infrared boundaries, respectively. Transverse part
$(\partial^{\mu}A_{\mu}=0)$ of the axial-vector field will be decomposed into
the Kaluza-Klein modes $A_{\mu}(q,z)=\sum_{n=0}A_{n\mu}(q)A_{n}(z)$ and the
equation for this part has normalizable solutions for the discrete values of
the $4D$ momentum $q^{2}=m_{n}^{2}$. Equation of motion leads to the following
equation for the $A_{n}(z)$ mode profile function 7 :

$\partial_{z}\left(e^{-B(z)}\partial_{z}A_{n}(z)\right)+\left(m_{n}^{2}-g_{5}^{2}e^{2A(z)}v^{2}(z)\right)e^{-B(z)}A_{n}(z)=0.$
(13)

This equation cannot be solved analytically. However, its solution near the UV
boundary $(z\rightarrow 0)$ can be found for the $m_{q}=0$ chiral limit case.
In these limits the equation (13) gets a form, which coincide with the one for
the vector field

$\partial_{z}\left(e^{-B(z)}\partial_{z}A_{n}\right)+m_{n}^{2}e^{-B(z)}A_{n}=0$
(14)

and is solved similarly to the vector field case. Making the
$A_{n}(z)=e^{B(z)/2}\psi_{n}(z)$ substitution in the equation of motion in
(14), this equation will get the Schrödinger-type equation form and has a
solution expressed in the terms of Laguerre polynomials $L^{n}_{m}$ 12 :

$\psi_{n}(z)=e^{-k^{2}z^{2}/2}(kz)^{m+\frac{1}{2}}\sqrt{\frac{2n!}{(m+n)!}}L_{n}^{m}(k^{2}z^{2}).$
(15)

Since the UV boundary value of the $A_{\mu}(q,z)$ function corresponds to the
wave function of the axial-vector meson on this boundary, we may accept that
the UV asymptotic solution (15) is the wave function of the $a_{1}$ meson. For
the $a_{1}$ meson, which is the lightest axial-vector meson, we take $m=1$ in
this solution:

$A_{n}(z)=k^{2}z^{2}\sqrt{\frac{2}{n+1}}L_{n}^{1}(k^{2}z^{2}).$ (16)

Since the zero-temperature equation (14) for the axial-vector meson in the
aforementioned limits is in the same form as one for the vector field, the
thermal dilaton field and the thermalization procedure, which was applied in
Ref. 14  for the vector field is available for the axial-vector field case as
well. Now let us briefly present formulas from this thermalization of vector
field applied in Ref. 14 . In the profile function for the vector field
$\Phi_{n}(r,T)$ a substitution
$\phi_{n}(r,T)=e^{-\frac{B_{T}(r)}{2}}\Phi_{n}(r,T)$ with
$B_{T}(r)=\varphi(r,T)-A(r)$ is useful and in the rest frame the equation for
$\phi_{n}(r,T)$ obtains a form of Schrödinger equation:

$\left[-\frac{d^{2}}{dr^{2}}+U(r,T)\right]\phi_{n}(r,T)=M_{n}^{2}(T)\phi_{n}(r,T).$
(17)

Here $U(r,T)$ is the effective potential, which is written in the sum of the
temperature-dependent and zero-temperature terms:

$U(r,T)=U(r)+\Delta U(r,T).$ (18)

The $U(r)$ and $\Delta U(r,T)$ potentials were found in the following:

$\displaystyle U(r)=k^{4}r^{2}+\frac{(4m^{2}-1)}{4r^{2}},$
$\displaystyle\Delta U(r,T)=2\rho(T)k^{4}r^{2}.$ (19)

Here $m=N+L-2$. $N$ is number of partons in the meson and $N=2$ for our case.
$L$ is the angular momentum and $L=1$ for our case. The meson mass spectrum
$M_{n}^{2}$ in (17) is written in the sum of discrete zero-temperature part
$M_{n}^{2}(0)$ and continuous finite-temperature part $\Delta M_{n}^{2}(T)$:

$\displaystyle M_{n}^{2}(T)=\ M_{n}^{2}(0)+\Delta M_{n}^{2}(T),$
$\displaystyle M_{n}^{2}(0)=4k^{2}\left(n+\frac{m+1}{2}\right),$
$\displaystyle\Delta
M_{n}^{2}(T)=\rho(T)M_{n}^{2}(0)+\frac{R\pi^{4}T^{4}}{k^{2}},$ $\displaystyle
R=(6n-1)(m+1).$ (20)

Finally, the solution of equation (17) for the bulk profile $\phi_{n}(r,T)$
was found in the following form 14 :

$\phi_{n}(r,T)=\sqrt{\frac{2\Gamma(n+1)}{\Gamma(n+m+1)}}K^{m+1}r^{m+\frac{1}{2}}e^{-\frac{K^{2}r^{2}}{2}}L_{n}^{m}(K^{2}r^{2}),$
(21)

which coincides with the zero-temperature solution found in Ref. 12  on making
replacements $r\rightarrow z$, $K(T)\rightarrow k$ in it.

Thus, in the chiral limit and near the UV boundary the finite-temperature
profile function of the axial-vector field can be taken the (21) solution and
the thermal $A_{n}(z,T)$ in $r$ coordinate will have form below:

$A_{n}(r,T)=K^{2}z^{2}\sqrt{\frac{2}{n+1}}L_{n}^{1}(K^{2}r^{2}).$ (22)

##  III NUCLEON PROFILE FUNCTION AT FINITE TEMPERATURE

In the bottom-up approach of the holographic QCD in order to describe the left
and the right handed components of the nucleons two fermion fields
$\left(N_{1},N_{2}\right)$ are introduced in the bulk of AdS space, which are
not interrelated 27 ; 28 . Following to Ref. 15  here we present briefly the
solution of equation of motion for the fermion fields interacting with the
thermal dilaton field. The action for such fermion field $N(x,r,T)$ in the
background (1) and in $r$ coordinate is written as below:

$S=\int d^{4}xdre^{-\varphi(r,T)}\sqrt{g}{\bar{N}}(x,r,T)D_{\pm}(r)N(x,r,T),$
(23)

where the $D_{\pm}(r)$ covariant derivative contains a temperature-dependent
part as well:

$D_{\pm}(r)=\frac{i}{2}\Gamma^{M}\left[\partial_{M}-\frac{1}{4}\omega_{M}^{ab}\
\left[\Gamma_{a}\Gamma_{b}\right]\right]\mp\left[\mu_{F}(r,T)+U_{F}(r,T)\right].$
(24)

The ”mass” $\mu_{F}(r,T)$ of the $N(x,r,T)$ thermal fermion field is related
with the $\mu_{F}$ mass in zero-temperature case:

$\mu_{F}(r,T)=\mu_{F}\ f^{\frac{3}{10}}(r,T)$ (25)

and $\mu_{F}$ is determined by the following equation

$\mu_{F}=\emph{N}_{B}+\emph{L}-\frac{3}{2}.$ (26)

Here $\emph{N}_{B}=3$ is the number of partons in the composite fermion (in
the nucleon) and L is the orbital angular momentum. The temperature-dependent
potential $U_{F}(r,T)$ for the fermion field is related with the dilaton field
and the $f(r,T)$ blackening function:

$U_{F}(r,T)=\varphi(r,T)/f^{\frac{3}{10}}(r,T).$ (27)

Non-zero components of the spin connection $\omega_{M}^{ab}$ are given by
equation:

$\omega_{M}^{ab}=(\delta_{M}^{a}\delta_{r}^{b}-\delta_{M}^{b}\delta_{r}^{a})\
rf^{\frac{1}{5}}(r,T).$ (28)

The $\sigma^{MN}=\left[\Gamma^{M},\Gamma^{N}\right]$ in the (24) derivative is
the commutator of the Dirac $\Gamma^{M}$ matrices in the curved space-time,
which are related with the reference frame $\Gamma^{a}$ matrices by the
$\Gamma^{M}=e_{a}^{M}\Gamma^{a}$ relation. Inverse vielbeins are
$e_{a}^{M}=r\times diag\\{\frac{1}{f(r)},1,1,1,-f(r)\\}$. Reference frame
$\Gamma^{a}$ matrices are chosen as $\Gamma^{a}=(\gamma^{\mu},\
-i\gamma^{5})$. For the fifth component of the fermion field we choose an
axial gauge $N_{5}(x,r,T)=0$ and decompose $N(x,r,T)$ into the left- and
right-chirality components:

$N(x,r,T)=N^{R}(x,r,T)+N^{L}(x,r,T),$ (29)

where the chiral components are defined as
$N^{R}(x,r,T)=\frac{1-\gamma^{5}}{2}N(x,r,T)$,
$N^{L}(x,r,T)=\frac{1+\gamma^{5}}{2}N(x,r,T)$.

The Kaluza-Klein decomposition for these components is written in terms of the
profile functions $\Phi_{n}^{L,R}(r,T)$, which are temperature-dependent as
well:

$N^{L,R}(x,r,T)=\sum_{n}N_{n}^{L,R}(x)\Phi_{n}^{L,R}(r,T).$ (30)

For the nucleons we consider the $L=0$ case and the total angular momentum
will be $\emph{J}=\frac{1}{2}$. It is useful to write the
$\Phi_{n}^{L,R}(r,T)$ profiles with the prefactors $e^{-\frac{3}{2}A(r)}$:

$\Phi_{n}^{L,R}(r,T)=e^{-\frac{3}{2}A(r)}F_{n}^{L,R}(r,T).$ (31)

After the substitution of these profile functions into the equations of motion
in the rest frame of nucleon $(\vec{p}=0)$, the following equations for the
$F_{n}^{L,R}(r,T)$ profile functions were obtained in the Ref. 15 :

$\left[\partial_{r}^{2}+U_{L,R}(r,T)\right]F_{n}^{L,R}(r,T)=M_{n}^{2}(T)F_{n}^{L,R}(r,T).$
(32)

The spectrum $M_{n}^{2}(T)$ is divided into the temperature-dependent part,
which is continuous, and ”cold” part, which is quantized. It is obvious, the
quantized part

$M_{n}^{2}(T)=4K^{2}(T)\left(n+m+\frac{1}{2}\right)=4k^{2}\left(1+\rho(T)\right)\left(n+m+\frac{1}{2}\right)$
(33)

at $T=0$ will coincide with the known zero-temperature spectrum 7 . The
effective potentials $U_{L,R}(r,T)$ in Eq. (32) are written in the sum of
zero- and finite-temperature parts:

$\displaystyle U_{L,R}(r,T)=U_{L,R}(r)+\Delta U_{L,R}(r,T),$
$\displaystyle\Delta
U_{L,R}\left(r,T\right)=2\rho(T)k^{2}\left(k^{2}r^{2}+m\mp\frac{1}{2}\right).$
(34)

Here

$m=N_{B}+L-\frac{3}{2}.$ (35)

Finally, solutions to the equations (32), which are profile functions for the
boundary thermal nucleons were found in the form 15 :

$\displaystyle
F_{n}^{L}(r,T)=\sqrt{\frac{2\Gamma(n+1)}{\Gamma(n+m_{L}+1)}}K^{m_{L}+1}r^{m_{L}+\frac{1}{2}}e^{-\frac{K^{2}r^{2}}{2}}L_{n}^{m_{L}}\left(K^{2}r^{2}\right),$
$\displaystyle
F_{n}^{R}(r,T)=\sqrt{\frac{2\Gamma(n+1)}{\Gamma(n+m_{R}+1)}}K^{m_{R}+1}r^{m_{R}+\frac{1}{2}}e^{-\frac{K^{2}r^{2}}{2}}L_{n}^{m_{R}}\left(K^{2}r^{2}\right),$
(36)

where $m_{L,R}=m\pm\frac{1}{2}$. The $\Phi_{n}(r,T)$ and $F_{n}(r,T)$ profile
functions obey normalization conditions:

$\int_{0}^{\infty}dre^{-\frac{3}{2}A(r)}\Phi_{m}^{L,R}(r,T)\Phi_{n}^{L,R}(r,T))=\int_{0}^{\infty}drF_{m}^{L,R}(r,T)F_{n}^{L,R}(r,T)=\delta_{mn}.$
(37)

With the replacements $r\rightarrow z$ and $K(T)\rightarrow k$ the (36)
functions coincide with the profiles at zero temperature case in the Ref. 12 .

##  IV Bulk Vacuum expectation value of scalar meson field

The action for the pseudo-scalar $X$ field in AdS/QCD has a form:

$S=\int_{0}^{\infty}d^{5}x\sqrt{g}e^{-\varphi(z)}Tr\\{|DX|^{2}-m_{5}^{2}|X|^{2}\\}.$
(38)

In terms of the tortoise coordinate $r$ and with the thermal dilaton this
action will be slightly changed:

$S_{X}=\int
d^{4}xdr\sqrt{g}e^{-\varphi(r,T)}Tr\left[|DX|^{2}+3|X|^{2}\right],$ (39)

where $DX$ is the covariant derivative including the interaction with the
gauge fields $A_{L,R}$. In terms of the vector $M_{M}$ and the axial-vector
$A_{M}$ fields terms this derivative looks like:
$D^{M}X=\partial^{M}X-iA_{L}^{M}X+iXA_{R}^{M}=\partial^{M}X-i\left[M_{M},X\right]-i\\{A_{M},X\\}$,
$A_{L,R}^{M}=A_{L,R}^{M}t^{a}$ and $F_{L,R}^{MN}$ are the field strengths of
these fields. The $X$ field transforms under the bifundamental representation
of the flavor symmetry group $SU(2)_{L}\times SU(2)_{R}$ of the model and
performs the breaking of this symmetry by Higgs mechanism 27 ; 29 ; 30 . E. o.
m. for this field, which is obtained from the action (38), has a solution
below:

$\langle X\rangle=\frac{1}{2}v(z),$ (40)

where

$v(z)=\frac{1}{2}M_{q}az+\frac{1}{2a}\Sigma z^{3},$ (41)

where $a=\sqrt{N_{c}}/(2\pi)$ $(N_{c}=3)$ is the normalization parameter 27 .
According to dictionary of the bulk/boundary correspondence, the parameters
$M_{q}$ and $\Sigma$ are identified with the $u,d$ quark mass matrix and with
the chiral condensate $\Sigma=<0|\bar{q}q|0>$ correspondingly. For the finite
temperature case in the solution (41) the quark condensate depends on
temperature and the $z$ coordinate should be replaced by the tortoise one.
Then in the solution (41) a replacement $\Sigma\rightarrow\Sigma(T)$ will be
done and it accepts a form:

$v(r,T)=\frac{1}{2}M_{q}ar+\frac{1}{2a}\Sigma(T)r^{3}.$ (42)

Temperature dependence of the quark condensate was defined in 31  using two-
loop chiral perturbation theory at finite temperature in the following:

$\Sigma(T)=\Sigma[1-\frac{N_{f}^{2}-1}{N_{f}}\frac{T^{2}}{12F^{2}}-\frac{N_{f}^{2}-1}{2N_{f}^{2}}(\frac{T^{2}}{12F^{2}})^{2}+O(T^{6})]=\Sigma[1+\Delta_{T}+O(T^{6})],$
(43)

where $N_{f}$ is the number of quark flavor and $F$ is the decay constant. The
dilaton field $\phi(r)$ is responsible for the dynamical broken of the chiral
symmetry in AdS/QCD. The chiral condensate formed as a result of this symmetry
break in the boundary QCD. Both constants have dependence only on temperature,
so, in Refs. 14 ; 15 ; 16  authors suppose that the temperature dependence of
the $\Sigma(T)$ quark condensate should be the same as the temperature
dependence of the dilaton parameter $K^{2}(T)$:

$K^{2}(T)=k^{2}\frac{\Sigma(T)}{\Sigma}.$ (44)

In addition, it was conjectured that known zero-temperature relation between
the quantities quark condensate $\Sigma$, number of flavors $N_{f}$,
condensate parameter $B$ and the pseudo-scalar meson decay constant $F$ in the
chiral limit

$\Sigma=-N_{f}BF^{2}$ (45)

holds for the finite temperature case as well:

$\Sigma(T)=-N_{f}B(T)F^{2}(T).$ (46)

Then, according to the (6) and (44) relations the $\Sigma(T)$ dependence will
be written in the following 30 :

$\Sigma(T)=\Sigma\left[1+\rho(T)\right].$ (47)

Let us note, that the relation (47) is correct up to $T^{6}$ degree of the
temperature. The $F(T)$ and $B(T)$ dependencies have been studied in the Ref.
31 .

##  V $g_{a_{1}NN}$ coupling constant

The action for the interaction between axial-vector, fermion and scalar fields
in the soft-wall model with the thermal dilaton and in the background (1) will
be written as below:

$S=\int_{0}^{\infty}d^{4}xdz\sqrt{g}e^{-\varphi(z,T)}L(x,z,T).$ (48)

Here $L(x,z,T)$ is the interaction Lagrangian. In terms of the $r$ tortoise
coordinate and fields defined above in this coordinate the action will be
written without the exponent factor:

$S=\int_{0}^{\infty}d^{4}xdr\sqrt{g}L(x,r,T).$ (49)

The meson-nucleon coupling constant, which we want investigate here, can be
derived from the bulk interaction action involving the gauge, bulk scalar and
fermion fields. Let us list here possible interactions at the lowest order of
the fields. Corresponding Lagrangian terms should be Hermitian scalars under
the existing symmetries in the model, namely gauge, chiral and 5D ”Lorentz”
symmetries. The minimal gauge coupling term is usual 5D Lagrangian 27 ; 12 ;
33 ; 34 :

$L^{(0)}=\frac{1}{2}[\overline{\Psi}_{1}\Gamma^{M}A_{M}\Psi_{1}-\overline{\Psi}_{2}\Gamma^{M}A_{M}\Psi_{2}].$
(50)

Another well-known interaction term, which obeys the symmetries mentioned
above, is a magnetic gauge couplings in the bulk 27 ; 28 ; 12 ; 25 ; 33 ; 34 :

$L^{(1)}=\frac{i}{2}k_{1}[\overline{\Psi}_{1}\Gamma^{MN}F_{MN}\Psi_{1}+\overline{\Psi}_{2}\Gamma^{MN}F_{MN}\Psi_{2}].$
(51)

Tripartite interaction between the bulk spinors, gauge and scalar fields was
constructed in 35 ; 33  and is similar to Yukawa coupling term introduced in
27 :

$L^{(2)}=g_{Y}[\overline{\Psi}_{1}X\Gamma^{M}A_{M}\Psi_{2}+\overline{\Psi}_{2}X^{\dagger}\Gamma^{M}A_{M}\Psi_{1}].$
(52)

According to AdS/CFT correspondence, the generating functional in the gauge
field theory is equivalent to the exponential of an on-shell action in the
gravity theory (is called GKP-W relation) 18 ; 19 ; 20 :

$Z_{AdS}[\phi_{0}]=\langle e^{i\int
dx\phi_{0}(x)O(x)}\rangle_{gauge}=e^{iS_{gravity}[\phi_{0}]}$ (53)

where $Z_{AdS}[\phi_{0}]$ is the generating functional with the source
$\phi_{0}$ coupled with an operator $O(x)$ and $S_{gravity}$ is an on-shell
action with boundary condition $\phi\rightarrow\phi_{0}$ at the UV boundary.
In our case the axial-vector current $(J_{\mu})$ is an operator and its
holographic dual is the axial-vector field $A_{\mu}$. The holographic
principle will give us following relation between these quantities:

$\langle J_{\mu}\rangle=-i\frac{\delta Z_{QCD}}{\delta
A_{\mu}^{0}}|_{A_{\mu}^{0}}=0.$ (54)

Here $Z_{QCD}=e^{iS_{int}}$ is the generating functional for the boundary QCD,
$A_{\mu}^{0}=A_{\mu}(q,z=0)=A_{\mu}(q)$ is a boundary value of the axial-
vector field. In the boundary QCD theory the four dimensional axial-vector
current of nucleons is defined as follows:

$J_{\mu}(p^{\prime},p)=g\overline{u}(p^{\prime})\gamma^{5}\gamma_{\mu}u(p).$
(55)

For the nucleon-$a_{1}$ meson interaction the $g$ constant is the
$g_{a_{1}NN}$ coupling constant and it can be determined from the equivalence
of the right-hand side of (54) and (55) formulas. After integrating out space-
time coordinates from the 5D action integral we obtain integral expressions
for the $g_{i}(r,T)$ constants corresponding to the $L^{(i)}$ Lagrangian
terms:

$g_{0}(r,T)=\frac{1}{2}\int^{\infty}_{0}\frac{dr}{r^{4}}A_{0}(r,T)(|\phi_{1R}(r,T)|^{2}-|\phi_{1L}(r,T)|^{2}),$
(56)
$g_{1}(r,T)=\frac{k_{1}}{2}\int^{\infty}_{0}\frac{dr}{r^{3}}\partial_{z}A_{0}(r,T)(|\phi_{1R}(r,T)|^{2}+|\phi_{1L}(r,T)|^{2}),$
(57)
$g_{2}(r,T)=2g_{Y}\int^{\infty}_{0}\frac{dr}{r^{4}}A_{0}(r,T)v(r,T)(\phi_{1R}(r,T)\phi_{1L}(r,T)).$
(58)

Temperature dependence of each $g_{i}(r,T)$ constant can be studied
numerically. In addition, we can investigate the temperature dependence of the
$a_{1}$ meson decay constant, which was defined in the Ref. 7  for the ’cold’
case by the holographic formula:

$F_{a_{1}}^{2}=\frac{1}{g_{5}^{2}}\left[v^{\prime\prime}(0)\right]^{2}.$ (59)

For the finite-temperature case this formula will be calculated with the
thermal profile $A_{n}(r,T)$ in Eq. (8):

$F_{a_{1}}^{2}\left(T\right)=\frac{1}{g_{5}^{2}}\left[A^{\prime\prime}_{n}(r,T)|_{r=0}\right]^{2}$
(60)

and the temperature dependence of this constant also can be investigated
numerically.

##  VI Numerical analysis

In order to visualize the temperature dependence of the terms in the Eqs.
(56), (57), (58) and the constant in the Eq. (60) we have to perform
numerically an integration over the $r$ variable of these temperature-
dependent integrals. All values of parameters are chosen in GeV units. We use
the values of the parameters $k_{1}$, $m_{q}$, $\Sigma$ and $g_{Y}$, which
were fixed in the earlier works. The $k_{1}=-0.98$ value of the parameter was
obtained from the fitting of the couplings $g_{\pi NN}=13.5$ and $g_{\rho
NN}=-8.6$ of the ground state nucleons in hard-wall AdS/QCD model in Ref. 27 .
The numerical values of the quark condensate and quark mass were taken from
Ref. 29  and are equal respectively, $\Sigma=(0.213$ MeV$)^{3}$, $m_{q}=8.3$
MeV. The constant $g_{Y}=9.182$ was fixed to get nucleon mass $m_{N}=0.94$ GeV
in the framework of hard-wall 27 . The numerical values of the number of quark
flavors $N_{f}$ and the pseudo-scalar meson decay constant $F$ were taken from
the Refs. 14 ; 31 . We consider ground state of the $a_{1}$ meson and take
$n=0$ in the profile function (22).

In Figs. 1-4 we present the numerical results for the temperature dependencies
of the $g_{0}$, $g_{1}$, $g_{2}$ terms and for the $g_{a_{1}NN}$ coupling
constant. As we seen in the section IV, the dilaton parameter $K^{2}(T)$ is
defined through the quark condensate $\Sigma(T)$, which is determined by the
$N_{f}$ and $F$ parameters. So, in order to answer to question how the $g_{i}$
terms and the $g_{a_{1}NN}$ constant depend on values of the $N_{f}$ and $F$
parameters, in these figures we have applied following values of these
parameters: $N_{f}=2$, $F=87$ MeV; $N_{f}=3$, $F=100$ MeV and $N_{f}=5$,
$F=140$ Mev, which were considered in Refs. 14 ; 15 ; 16 . In Fig.5 we have
plotted the temperature dependence of the squared decay constant
$F_{a_{1}}^{2}$ of the $a_{1}$ meson. As is seen from the graphs all
$g(T),F_{a_{1}}^{2}(T)$ dependencies decrease and become zero around the
temperature value $T=200$ MeV, which is confinement-deconfinement phase
transition temperature.

##  VII Conclusion

Numerical analysis here for the $g_{a_{1}NN}$ coupling constant shows the
temperature dependence of this constant has similar shape of behavior as was
for the $g_{\rho NN}$ constant, i.e. the $g_{a_{1}NN}$ constant goes down on
increasing temperature and around the confinement-deconfinement phase
transition temperature the $g_{a_{1}NN}$ constant becomes zero. Shape of
dependence almost does not depends on the values of the $N_{f}$ and $F$
parameters. Also, from a comparison of the graphics we observe, the minimal
coupling is not leading term in the $a_{1}$ meson-nucleon interaction.It
should be noted a quantity vanishing at confinement-deconfinement phase
transition temperature result was obtained in works 22 ; 14 ; 15 ; 16  as
well, where was applied AdS/Schwarzchild metric.

As was noted in Ref. 25 , the study of couplings at finite temperature may
have a use for the early Universe physics studies. For a completeness of these
investigations it is reasonable to make a similar analysis in the scalar
(pseudo-scalar) meson sector of this model and to check the temperature
dependency results, which were obtained here and in the Ref. 25 , for the
pion-nucleon coupling constant. This question under consideration.

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Figure 1:  $g_{0}(T)$ term at $N_{f}=2$, $F=87$ MeV (orange line); $N_{f}=3$,
$F=100$ MeV (green line); $N_{f}=5$, $F=140$ MeV (blue line)  Figure 2:
$g_{1}(T)$ term at $N_{f}=2$, $F=87$ MeV (green line); $N_{f}=3$, $F=100$ MeV
(blue line); $N_{f}=5$, $F=140$ MeV (orange line) Figure 3: $g_{2}(T)$ term at
$N_{f}=2$, $F=87$ MeV (green line); $N_{f}=3$, $F=100$ MeV (blue line);
$N_{f}=5$, $F=140$ MeV (orange line)  Figure 4: $g_{a_{1}NN}(T)$ constant at
$N_{f}=2$, $F=87$ MeV (orange line); $N_{f}=3$, $F=100$ MeV (green line);
$N_{f}=5$, $F=140$ MeV (blue line)  Figure 5: Temperature dependence of the
$F^{2}_{a_{1}}(T)$ decay constant at $N_{f}=2$, $F=87$ MeV (orange line);
$N_{f}=3$, $F=100$ MeV (green line); $N_{f}=5$, $F=140$ MeV (blue line).


11institutetext: Department of Computer Science, University of Oxford, Oxford,
UK 22institutetext: UiT The Arctic University of Norway, Tromsø, Norway
22email<EMAIL_ADDRESS>
# Learning Underrepresented Classes from Decentralized Partially Labeled
Medical Images

Nanqing Dong 11 0000-0001-5014-1993 Michael Kampffmeyer 22 0000-0002-7699-0405
Irina Voiculescu 11 0000-0002-9104-8012

###### Abstract

Using decentralized data for federated training is one promising emerging
research direction for alleviating data scarcity in the medical domain.
However, in contrast to large-scale fully labeled data commonly seen in
general object recognition tasks, the local medical datasets are more likely
to only have images annotated for a subset of classes of interest due to high
annotation costs. In this paper, we consider a practical yet under-explored
problem, where underrepresented classes only have few labeled instances
available and only exist in a few clients of the federated system. We show
that standard federated learning approaches fail to learn robust multi-label
classifiers with extreme class imbalance and address it by proposing a novel
federated learning framework, FedFew. FedFew consists of three stages, where
the first stage leverages federated self-supervised learning to learn _class-
agnostic_ representations. In the second stage, the decentralized partially
labeled data are exploited to learn an energy-based multi-label classifier for
the common classes. Finally, the underrepresented classes are detected based
on the energy and a _prototype_ -based nearest-neighbor model is proposed for
few-shot matching. We evaluate FedFew on multi-label thoracic disease
classification tasks and demonstrate that it outperforms the federated
baselines by a large margin.

###### Keywords:

Federated Learning Partially Supervised Learning Multi-Label Classification.

##  1 Introduction

Learning from partially labeled data, or partially supervised learning (PSL),
has become an emerging research direction in label-efficient learning on
medical images [27, 8, 20, 4, 6]. Due to high data collection and annotation
costs, PSL utilizes multiple available partially labeled datasets when fully
labeled data are difficult to acquire. Here, a partially labeled dataset
refers to a dataset with only a specific _true subset_ of classes of interest
annotated. For example, considering a multi-label thoracic disease task on
chest X-ray (CXR) images (_i.e. missing_ a CXR could contain several diseases
at the same time), a pneumonia dataset may only have labels for pneumonia and
not the labels for the other diseases of interest.

Figure 1: An example problem setup with multi-label thoracic disease
classification. In this example, each client in the red dashed box has only
one underrepresented class, while the clients in the green dashed box share a
set of common classes.

In this work, we extend the discussion of PSL to an unexplored federated
setup, where the partially labeled datasets are stored separately in different
clients (_e.g. missing_ hospitals and research institutes). In this work, we
denote the common classes (CCs) as the classes with enough labels to learn a
multi-label classifier. Meanwhile, in a _open-world_ scenario, there are also
newly-found or under-examined classes, which tend to have much fewer labeled
instances than the CCs. This extreme class imbalance makes learning these
underrepresented classes (UCs) a difficult task. Furthermore, the main
assumption of this work is that the CCs and UCs are annotated at disjoint
clients, which makes this practical problem even more challenging. While a
formal problem setup will be described in Sec. 2, an intuitive illustration of
the problem is presented in Fig. 1. Due to decentralization and extreme class
imbalance, improving the performance of UCs can inevitably decrease the
performance of CCs, while maintaining the performance of CCs might lead to UCs
being completely ignored. To the best of our knowledge, this is the first
study on decentralized partially labeled rare classes.

Contributions We formalize this under-explored problem and present FedFew, the
first solution to it. FedFew is a three-stage federated learning (FL)
framework. Firstly, we utilize federated self-supervised learning (FSSL) to
learn _class-agnostic_ transferable representations in a pre-training stage.
Secondly, in the fine-tuning stage, we propose an energy-based multi-label
classifier that aims to utilize the knowledge from the CCs to learn
representations for the UCs. Finally, we utilize a _prototype_ -based nearest-
neighbor classifier for few-shot matching given only few partially labeled
examples for the UCs. We evaluate FedFew on a set of multi-label thoracic
disease classification tasks on the Chest-Xray14 dataset [25] in a simulated
federated environment. The empirical results show that the proposed framework
outperforms existing methods by a large margin.

##  2 Problem Setup

Task and Data Setup The task of interest is multi-label classification (MLC)
with $C$ non-mutually exclusive classes of interest, where the decentralized
data are stored in a federated system with $K>1$ clients. Let
$\mathcal{D}_{k}$ denote the data stored in client $k\leq K$, we have
$\mathcal{D}_{k}\cap\mathcal{D}_{l}=\emptyset$ for $k\neq l$ and
$\\{\mathcal{D}_{k}\\}_{k=1}^{K}$ are non-IID data. For convenience, we define
$\mathcal{D}_{k}=\mathcal{P}_{k}\cup\mathcal{U}_{k}$, where $\mathcal{P}_{k}$
is a _partially labeled_ dataset and $\mathcal{U}_{k}$ is an _unlabeled_
dataset. We define $n_{k}^{p}=|\mathcal{P}_{k}|$ and
$n_{k}^{u}=|\mathcal{U}_{k}|$.111$|\cdot|$ is the cardinality of a set. We
assume that the classes of interest, denoted as $\mathcal{C}$, can be split
into two mutually exclusive subsets, namely a set of UCs (denoted as
$\mathcal{C}_{u}\subset\mathcal{C}$), which is also the primary target of this
work, and a set of CCs (denoted as
$\mathcal{C}_{c}=\mathcal{C}\setminus\mathcal{C}_{u}$). For simplicity, we
consider a representative case that there are $|\mathcal{C}_{u}|<K$ clients
and each of these clients is annotated for only one UC222This is the most
fundamental case. As a trivial extension, each client could have labels for
more than one class and multiple clients could have labels for the same set of
classes. The proposed method could be easily adapted to these extensions.. For
the remaining $K-|\mathcal{C}_{u}|$ clients, we assume that each client has
_partially labeled_ 333Here, the images with CCs are partially labeled with
respect to the missing labels of the UCs, _i.e. missing_ they are fully
labeled if we only consider CCs. The assumption here is that the CCs are
diseases with high prevalence, which can be easily collected and diagnosed;
but the UCs are rare and only spotted in certain clients. data for all CCs
($\mathcal{C}_{c}$). We additionally require $n_{i}^{p}\ll
n_{j}^{p}~{}\forall~{}i\in\mathcal{C}_{u},j~{}\in\mathcal{C}_{c}$ to enforce
the assumption of UCs. The learning outcome is to leverage the decentralized
training data to train an MLC model for $\mathcal{C}$.

Federated Environment Setup In addition to $K$ clients, there is a parameter
server (PS) [16] for model aggregation. Let $f_{\theta}$ be the model of
interest. In the PS, the parameter set $\theta_{0}$ is randomly initialized
and sent out to $K$ clients as $K$ copies $\\{\theta_{k}\\}_{k=1}^{K}$ for
full synchronization. During the federated optimization phase, the client $k$
updates $\theta_{k}$ by training on $\mathcal{D}_{k}$ independently for a
number of local epochs. Then, the PS aggregates $\\{\theta_{k}\\}_{k=1}^{K}$
collected from $K$ clients to update $\theta_{0}$. Under the data regulations
in the medical domain [22, 7], we assume that the patients’ data (either raw
data or encoded data) in a client can not be uploaded to the PS or other
clients, _i.e. missing_ only parameters $\\{\theta_{k}\\}_{k=0}^{K}$ and
_metadata_ (_e.g. missing_ the statistics of data) can be exchanged between
the PS and the clients.

##  3 Method

In this section, we first provide the preliminaries that FedFew builds on in
Sec. 3.1. The first training stage of FSSL is briefly described in Sec. 3.2,
while the second training stage of energy-based FL with partial labels is
described in Sec. 3.3. Finally, in Sec. 3.4, we present the prototype-based
nearest-neighbor classifier for few-shot matching.

###  3.1 Preliminaries

FedAvg As a seminal FL model, FedAvg [18] aggregates the model weights
$\\{\theta_{k}\\}_{k=1}^{K}$ as a weighted average. Mathematically, we have

$\theta_{0}=\sum_{k=1}^{K}a_{k}\theta_{k},$ (1)

where $a_{k}=\frac{n_{k}}{n_{tot}}$. The metadata $n_{k}$ is the number of
labeled training examples stored in client $k$ and
$n_{tot}=\sum_{k=1}^{K}n_{k}$ is the total number of training examples.

Energy Function Given a discriminative neural network classifier $f$, the
energy function $E(x;f):\mathbb{R}^{H{\times}W}\rightarrow\mathbb{R}$ maps an
image with shape $H{\times}W$ to a scalar, which is also known as _Helmholtz
free energy_ [15]. The energy is defined as

$E(x;f)=-\tau\log\int_{y}\exp^{\frac{f^{y}(x)}{\tau}},$ (2)

where $f^{y}(x)$ is the logit of the $y^{\mathrm{th}}$ class label and $\tau$
is the temperature parameter.

###  3.2 Federated Self-Supervised Learning

The first training stage consists of FSSL, where a feature extractor
$f_{\theta}$ is pre-trained to learn class-agnostic representations.
Theoretically, multiple existing self-supervised learning frameworks (_e.g.
missing_ [10, 1, 9]) could serve as the local backbone. In this work, however,
due to its lightweight nature, we leverage SimSiam [2]. Let $\theta_{0}^{t}$
denote the aggregated model weights in the PS at the end of the
$t^{\mathrm{th}}$ training round. Thus, at the beginning of the
$t+1^{\mathrm{th}}$ round, the model weights at client $k$ should be
synchronized to $\theta_{0}^{t}$. After the local updates of the
$t+1^{\mathrm{th}}$ round, the local model weights of client $k$ are now
$\theta_{k}^{t+1}$ and, $\theta_{0}^{t+1}$ is computed by applying Eq. 1 on
$\\{\theta_{k}^{t+1}\\}_{k=1}^{K}$.

###  3.3 Energy-Based Federated Learning with Partial Labels

For standard MLC, it is common to use a $C$-dimensional binary vector to
encode the label information for a given input. When all binary entries are
0s, the input does not contain any class of interest. However, with limited
partial labels of UCs, it is difficult to train an MLC model for $\mathcal{C}$
or $\mathcal{C}_{u}$ directly. Instead, we first train an MLC model for CCs
$\mathcal{C}_{c}$. In contrast to previous studies [25, 19], we encode the
label into a $(C_{c}+1)$-dimensional vector, where $C_{c}=|\mathcal{C}_{c}|$.
That is to say, we use an additional dimension (denoted as $0^{\mathrm{th}}$
class444When the additional dimension is 0, the rest of $C_{c}$ dimensions
should have at least one 1; when the additional dimension is 1, $C_{c}$
dimensions should all be 0s.) to specifically determine whether the patient
contains any CCs. Note, this $0^{\mathrm{th}}$ class only reflects the
information on CCs, as we have no label information for the UCs.

Without loss of generality, let us consider a client $k\in\mathcal{K}_{c}$
with only $\mathcal{C}_{c}$ labeled, where $\mathcal{K}_{c}$ denotes the
clients with the CCs. Given an example $x$ in client $k$ with corresponding
partial label $y$, the binary cross-entropy loss is

$\mathcal{L}_{BCE}(x,y)=-\sum_{j\in\\{0\\}\cup\mathcal{C}_{c}}y_{j}\log(f_{\theta}^{j}(x))+(1-y_{j})\log(1-f_{\theta}^{j}(x)),$
(3)

where $f_{\theta}^{j}(x)$ is the probability score for the $j^{\mathrm{th}}$
class. As MLC can be decomposed into multiple binary classification tasks, the
energy of $x$ for class $j$ degenerates to
$E(x;f_{\theta}^{j})=-\tau\log(1+\exp^{\frac{f_{\theta}^{j}(x)}{\tau}})$
(_c.f_. Eq. (2)) and the _joint energy_ [24] of $x$ is then the sum of
energies over all CCs $\mathcal{C}_{c}$:

$E(x,f_{\theta})=\sum_{j\in\\{0\\}\cup\mathcal{C}_{c}}E(x;f_{\theta}^{j})=-\sum_{j\in\\{0\\}\cup\mathcal{C}_{c}}\tau\log(1+\exp^{\frac{f_{\theta}^{j}(x)}{\tau}}).$
(4)

We include a regularization term [17] to penalize the energy of $x$ with a
squared hinge loss:

$\mathcal{L}_{E_{c}}(x)=\lambda\|\max(0,E(x;f_{\theta})-m_{c})\|^{2}_{2},$ (5)

where the margin $m_{c}$ is a hyperparameter chosen empirically to decrease
the energy of $x$ and $\lambda$ is a weight hyperparameter. The final
optimization goal for client $k\in\mathcal{C}_{c}$ is to minimize the sum of
the two losses:

$\mathcal{L}_{c}=\mathcal{L}_{BCE}(x,y)+\mathcal{L}_{E_{c}}(x).$ (6)

For clients with only UC (_i.e. missing_ no CCs), we only minimize a
regularization term:

$\mathcal{L}_{u}=\lambda\|\max(0,m_{u}-E(x;f_{\theta}))\|^{2}_{2},$ (7)

where the margin $m_{u}$ is chosen empirically to increase the energy of $x$.
Note, Eq. (5) and Eq. (7) are both designed to enlarge the _energy gap_
between $\mathcal{C}_{c}$ and $\mathcal{C}_{u}$. We aggregate the models
weights $\\{\theta_{k}\\}_{k=1}^{K}$ via Eq. (1), where
$a_{k}=\frac{n_{k}^{p}}{\sum_{j=1}^{K}n_{j}^{p}}$. The complete pseudo-code is
given in Algorithm 1.

1:Input: $\theta_{0}^{0}$, $\\{\mathcal{P}_{k}\\}_{k=1}^{K}$, $T_{w}$, $T$

2:Output: $\theta_{0}^{T}$

3:for $t=1,2,\cdots,T_{w}$ do $\triangleright$ Warm up

4: for $k\in\mathcal{K}_{c}$ do

5: $\theta_{k}^{t}\leftarrow\theta_{0}^{t-1}$ $\triangleright$ Synchronize
with PS

6: $\theta_{k}^{t}\leftarrow$ local_update$(\theta_{k}^{t})$ $\triangleright$
Eq. (6)

7:
$\theta_{0}^{t}\leftarrow\sum_{k\in\mathcal{K}_{c}}a_{k}^{t}\theta_{k}^{t}$$\triangleright$
Aggregate with Eq. (1)

8:for $t=T_{w}+1,T_{w}+2,\cdots,T$ do

9: for $k=1,2,\cdots,K$ do

10: $\theta_{k}^{t}\leftarrow\theta_{0}^{t-1}$ $\triangleright$ Synchronize
with PS

11: $\theta_{k}^{t}\leftarrow$ local_update$(\theta_{k}^{t})$ $\triangleright$
Eq. (6) or Eq. (7)

12:
$\theta_{0}^{t}\leftarrow\sum_{k=1}^{K}a_{k}^{t}\theta_{k}^{t}$$\triangleright$
Aggregate with Eq. (1)

Algorithm 1 Energy-Based Federated Partially Supervised Training. $T$ is the
total number of rounds. $\theta_{k}^{t}$ denotes the weights stored in client
$k$ at the $t^{\mathrm{th}}$ round.

###  3.4 Prototype-Based Inference

After the federated training in Sec. 3.3, $f_{\theta}$ can be directly used as
an MLC model to predict CCs.555Given the $(C_{c}+1)$-dimensional output
vector, we drop the $0^{\mathrm{th}}$ dimension and only use the
$C_{c}$-dimensional vector as the final prediction. Now, we use the energy
(Eq. (4)) to detect UCs, _i.e. missing_ if the energy of an example is lower
than a threshold666The threshold is chosen to maximize the number of correctly
classified training examples., then the example is deemed to contain no UCs;
if the energy is higher than the threshold, we further match the test example
to the nearest neighbor, given few partially labeled examples. However, due to
the constraint of data regulations, the training data and the test data are
stored in separated clients. Thus, similar to [5], we transfer the metadata of
UCs to the PS. Here, the metadata is the mean of the extracted features. For
class $c\in\mathcal{C}_{u}$, we have
$\bm{\mu}_{c}^{pos}=\frac{\sum_{i=1}^{n_{c}^{pos}}g_{\theta}(x_{i}^{pos})}{n_{c}^{pos}},~{}\bm{\mu}_{c}^{neg}=\frac{\sum_{i=1}^{n_{c}^{neg}}g_{\theta}(x_{i}^{neg})}{n_{c}^{neg}},$
where we use $pos$ and $neg$ to denote the positive and negative examples of
class $c$, respectively, and use $g_{\theta}$ to denote the feature extractor.
Note, $\bm{\mu}_{c}^{pos}$ and $\bm{\mu}_{c}^{neg}$ factually define the
_prototypes_ in the few-shot learning literature [21]. With the _dual_
-prototypes for class $c$, we match the test example to the closer one by
computing the distance between the features of test example and the two
prototypes777Again, this is a simple case. When there are multiple prototypes
collected from different clients for class $c$, majority voting is adopted..

##  4 Experiments

###  4.1 Experimental Setup

To provide empirical insights into the problem and ensure fair comparisons
with the baselines, we share the same experimental setup among all
experiments.

Implementation We explore two network backbones ($f_{\theta}$), ResNet34 (RN)
[11] and DenseNet121 (DN) [12], which are lightweight models commonly used for
FL. We use SimSiam [9], a state-of-the-art SSL framework, to pre-train
$f_{\theta}$ locally, and use a standard Adam [14] optimizer with fixed
learning rate $10^{-3}$ and batch-size 64 for both pre-training and fine-
tuning. We set $\tau=1$ when computing the energy, following [17]. The values
of $m_{c}$ and $m_{u}$ can be chosen around the mean of energy scores from a
model trained without energy-based loss (_e.g. missing_ warm-up stage in this
work) for images from the clients of CCs and the clients of UCs, respectively.
In this work, we simply set $m_{c}=-5$ and $m_{u}=-25$ to intentionally
enlarge the energy gap and we use $\lambda=0.01$. We follow the same data pre-
processing and augmentation procedure as [5]. The synchronization and
aggregation for federated methods are performed every $10$ epochs. For the
second stage, $T_{w}=20$ and $T=100$. All models are implemented in PyTorch
(1.10.1) on an NVIDIA Tesla V100.

Data We use the multi-label dataset ChestX-
ray14888https://nihcc.app.box.com/v/ChestXray-NIHCC [25] and adopt its default
batch splits to ensure reproducibility. Based on the label statistics of the
dataset, we choose _edema_ , _pneumonia_ , and _hernia_ as the three UCs and
use the remaining 11 classes as CCs. Note, most CXR images do not contain any
diseases. We use 6 batches999We use batch 2 to 7. Each batch has $10^{4}$ CXR
images and similar label distributions. to simulate the $K=6$ clients, where
we use the first three batches to simulate the partially labeled datasets for
CCs, where we randomly sample $n_{k}^{p}=5{\times}10^{3}$ images to keep
partial labels for CCs. Each of the remaining three batches contains one of
the three UCs. We sample $10$ positive examples and $90$ negative examples to
simulate the class imbalance for UCs, _i.e. missing_ $n_{k}^{p}=100$. See Fig.
1 for an illustration of the class assignment. From the remaining batches, we
hold out 100 positive and 100 negative examples for each class as the test
set.

###  4.2 Results

Empirical Analysis of FSSL Following previous SSL studies, we examine the FSSL
performance via the _linear classification protocol_ [10, 1, 9]. Similar to
[5], we fix all the weights of $f_{\theta}$ except the last layer and only
fine-tune the last layer on a public pneumonia
dataset101010https://data.mendeley.com/datasets/rscbjbr9sj/2 [13]. In this
dataset, there are three mutually exclusive classes, _normal_ , _bacteria
pneumonia_ , and _virus pneumonia_. We randomly split the images of each class
into two halves as the training and test sets. We use the test accuracy as the
_proxy_ measure to assess the representation learning performance. Firstly, we
provide a counter-intuitive observation that more disease images might not
improve the performance. We create two clients with Chest-Xray14 data, where
one client contains $10^{4}$ images without any diseases and the other client
contains $r{\times}10^{4}$ images with various diseases. The results in Table
2 show that FedAvg does not always benefit from large $r$ and it might be
unnecessary to collect a large number of images with related diseases for pre-
training. Secondly, we examine the impact of the training epochs for federated
SSL in Fig. 2, where we pre-train $f_{\theta}$ on the 6 clients described in
Sec. 4.1. In contrast to the empirical findings collected from general images
[10, 1, 9, 2], more epochs will not lead to diminishing performance gain but
decreasing results. We will use DenseNet121 as the default network and use the
pre-trained weights with 100 epochs in the following experiments.

Epoch | $r=1$ | $r=0.5$ | $r=0.1$
---|---|---|---
RN | DN | RN | DN | RN | DN
100 | 61.69 | 70.47 | 69.34 | 71.94 | 65.92 | 73.81
200 | 62.96 | 71.22 | 68.32 | 71.36 | 65.24 | 74.43
Table 1: Impact of class imbalance on FSSL. We report the mean accuracy over
three random seeds.

Table 2: Impact of the number of training epochs on FSSL.

Evaluation of FedFew Following the experimental setup in Sec. 4.1, we evaluate
FedFew against a few seminal baselines from two aspects, _i.e. missing_ we aim
to achieve high accuracy on UCs while maintaining robust performance on CCs.
The first baseline is a standard MLC model [19], which is trained with FedAvg
on decentralized partially labeled data of all 14 classes and weighted binary
cross-entropy [19]. We use _MLC w/o FSSL_ and _MLC w/ FSSL_ to differentiate
whether $f_{\theta}$ is pre-trained with FSSL. The second baseline is a
hypothetical local nearest-neighbor classifier (_NN_) that does not use
prototypes, where $g_{\theta}$ is either pre-trained with FSSL alone or
further fine-tuned with _MLC w/ FSSL_. Note, _NN_ can not be used in a
federated system due to data regulations. For FedFew, we use _EBM_ to denote
the energy-based loss in the training. The results on UCs are presented in
Table 3 and include the mean accuracy (A), precision (P), recall (R), and F-1
(F) score over three random seeds. Note, standard MLC models fail to detect
any images of the UCs due to extreme class imbalance. With only prototypes (as
only metadata can be transferred to the PS), NNs struggle to improve over
random guessing. FedFew (_w/o EBM_) outperforms the two baselines by a large
margin while _EBM_ further improves the performance of FedFew with higher
precision. Similar to Table 3, we report the mean _area under the receiver
operating characteristic_ (AUROC) for the 11 CCs over three runs in Table 5.
We find that FSSL can boost the performance for standard MLC. While FedFew
(_w/o EBM_) and FedFew (_w/ EBM_) both outperform _MLC w/ FSSL_ , including
_EBM_ slightly improves the performance on CCs. We conjecture that _EBM_ can
play a role as regularization in the federated training. By combining the
results from both Table 3 and Table 5, we conclude that FedFew (_w/ EBM_)
provides a robust solution to the problem of interest.

Analysis on Energy As an ablation study, we visualize the energy density plots
between images with and without UCs in Fig. 5, which demonstrates that
including _EBM_ in the training does increase the energy gap, thus leading to
improved performance.

Distance Metric We consider three distance metrics for the nearest-neighbor
matching is Sec. 3.4, which are cosine distance [23], Euclidean distance [21],
and earth mover’s distance [26] (computed with Sinkhorn-Knopp algorithm [3]).
We choose the cosine distance based on its empirical robustness. It achieves
an average F-1 score over the UCs of 0.67, while the Euclidean distance
obtains 0.6 and the earth mover’s distance obtains 0.43.

Method | Edema | Pneumonia | Hernia
---|---|---|---
A | P | R | F | A | P | R | F | A | P | R | F
MLC w/o FSSL | 0.50 | 0.00 | 0.00 | 0.00 | 0.50 | 0.00 | 0.00 | 0.00 | 0.50 | 0.00 | 0.00 | 0.00
MLC w/ FSSL | 0.50 | 0.00 | 0.00 | 0.00 | 0.50 | 0.00 | 0.00 | 0.00 | 0.50 | 0.00 | 0.00 | 0.00
NN (FSSL) | 0.53 | 0.53 | 0.50 | 0.52 | 0.51 | 0.51 | 0.46 | 0.48 | 0.50 | 0.50 | 0.54 | 0.52
NN (MLC w/ FSSL) | 0.53 | 0.53 | 0.50 | 0.52 | 0.51 | 0.51 | 0.46 | 0.48 | 0.50 | 0.50 | 0.54 | 0.52
FedFew w/o EBM | 0.71 | 0.85 | 0.50 | 0.63 | 0.69 | 0.84 | 0.46 | 0.59 | 0.72 | 0.83 | 0.54 | 0.65
FedFew w/ EBM | 0.75 | 1.00 | 0.50 | 0.67 | 0.73 | 1.00 | 0.46 | 0.63 | 0.77 | 1.00 | 0.54 | 0.70
Table 3: Performance comparison on the UCs. The standard MLC and NN models
fail to predict the UCs. A, P, R, and F denote the mean accuracy, precision,
recall, and F1-score over three random seeds, respectively.

Method | AUROC
---|---
MLC w/o FSSL | 0.6905
MLC w/ FSSL | 0.7227
FedFew w/o EBM | 0.7423
FedFew w/ EBM | 0.7479
Table 4: Performance comparison on the CCs. AUROC denotes the mean AUROC over
three random seeds.

Table 5: Energy gaps between positive and negative cases before (pre) and
after (post) _EBM_ training in Client 4.

##  5 Conclusion

In this work, we raise awareness of an under-explored problem, namely the
learning of underrepresented classes from decentralized partially labeled
medical images. We not only provide a solution to this novel problem but also
provide the first empirical understanding of federated partially supervised
learning with extreme class imbalance, a new research direction on label-
efficient learning.

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