# Gravitational lensing by a Bronnikov-Kim wormhole under a weak-field
approximation and in a strong deflection limit

Naoki Tsukamoto1<EMAIL_ADDRESS>1Department of General Science and
Education, National Institute of Technology, Hachinohe College, Aomori
039-1192, Japan

###### Abstract

We consider gravitational lensing under a weak-field approximation and in a
strong deflection limit by a Bronnikov-Kim wormhole with the same metric as
the one of a wormhole which has been suggested in Einstein-Dirac-Maxwell
theory. The metric approaches into the metric of an extreme charged Reissner-
Nordström black hole in a black hole limit and it becomes the metric of a
spatial Schwarzschild wormhole in an ultrastatic limit. In both of the black
hole limit and the ultrastatic limit, the coefficient of a divergent term and
the constant term of the deflection angle of a light in the strong deflection
limit can be obtained exactly without expanding parameters of the spacetime.
Interestingly, in the both limits to the black hole and the ultrastatic
wormhole, we obtain exactly the same coefficient and constant term in the
strong deflection limit.

##  I Introduction

Gravitational lensing under a weak-field approximation is a useful tool to
find dark and massive objects Schneider_Ehlers_Falco_1992 ;
Schneider_Kochanek_Wambsganss_2006 . Recently, the direct observations of
gravitational waves from compact objects have been reported by LIGO Scientific
and Virgo Collaborations Abbott:2016blz  and the shadow image of a
supermassive black hole candidate at the center of galaxy M87 has been
reported by Event Horizon Telescope Collaboration Akiyama:2019cqa .
Gravitational lensing in a strong gravitational field by compact objects also
can become an important tool to search them in future. Gravitational lensing
in a strong deflection limit Bozza:2002zj  can be used to obtain observables
by a light ray reflected by a photon sphere, which is a sphere formed by
unstable circular light orbits, around a compact object Hagihara_1931 ;
Darwin_1959 ; Atkinson_1965 ; Luminet_1979 ; Ohanian_1987 ; Nemiroff_1993 ;
Frittelli_Kling_Newman_2000 ; Virbhadra_Ellis_2000 ;
Bozza_Capozziello_Iovane_Scarpetta_2001 ; Bozza:2002zj ; Perlick:2003vg ;
Perlick_2004_Living_Rev ; Virbhadra:2008ws ; Bozza_2010 ; Tsukamoto:2016zdu ;
Shaikh:2019jfr ; Shaikh:2019itn ; Tsukamoto:2020uay ; Tsukamoto:2020iez ;
Paul:2020ufc . The strong deflection limit analysis in the Schwarzschild
spacetime was investigated by Bozza et al.
Bozza_Capozziello_Iovane_Scarpetta_2001  and its extensions and alternatives
were investigated by several authors Tsukamoto:2016zdu ; Shaikh:2019jfr ;
Shaikh:2019itn ; Tsukamoto:2020uay ; Tsukamoto:2020iez ; Paul:2020ufc ;
Bozza:2002zj ; Bozza:2002af ; Eiroa:2002mk ; Petters:2002fa ; Eiroa:2003jf ;
Eiroa:2004gh ; Bozza:2004kq ; Bozza:2005tg ; Bozza:2006sn ; Bozza:2006nm ;
Iyer:2006cn ; Bozza:2007gt ; Tsukamoto:2016qro ; Ishihara:2016sfv ;
Tsukamoto:2016oca ; Tsukamoto:2016jzh ; Tsukamoto:2017edq ; Hsieh:2021scb ;
Aldi:2016ntn ; Tsukamoto:2020bjm ; Hsieh:2021scb ; Takizawa:2021gdp ;
Tsukamoto:2021caq ; Aratore:2021usi .

A wormhole is a hypothetical spacetime structure with nontrivial topology in
general relativity Visser_1995  and it can be a black hole mimicker
Damour:2007ap ; Lemos:2008cv ; Tsukamoto:2019ihj . It is known that energy
conditions are violated at the throat of any asymptotically flat, static, and
spherically symmetric wormhole at least if general relativity without a
cosmological constant is assumed Morris:1988cz . Many passable wormholes were
suggested after an Ellis-Bronnikov wormhole Ellis_1973 ; Bronnikov_1973 ;
Martinez:2020hjm  which is known as the earliest passable wormhole. In 2001,
Dadhich et al. considered wormhole metrics with a vanishing Ricci scalar
Dadhich:2001fu  and then Bronnikov and Kim also suggested wormhole metrics
with a vanishing Ricci scalar in a braneworld scenario Bronnikov:2002rn ;
Bronnikov:2003gx . Gravitational lensing by a spatial Schwarzschild wormhole
with the vanishing Ricci scalar Dadhich:2001fu  in the strong deflection limit
Bozza:2002zj  has been investigated in Ref. Tsukamoto:2016zdu .

Recently, a wormhole filled with massless and neutral fermions in Einstein-
Dirac-Maxwell theory was obtained numerically Blazquez-Salcedo:2020czn  and
the existence of a thin shell at the wormhole throat was discussed in Refs.
Bolokhov:2021fil ; Konoplya:2021hsm ; Blazquez-Salcedo:2021udn ;
Danielson:2021aor . Its metric corresponds with the Bronnikov-Kim wormhole as
a simple and analytical metric case Bronnikov:2002rn ; Bolokhov:2021fil ;
Blazquez-Salcedo:2021udn . Its shadow size also was investigated by Bronnikov
et al. Bronnikov:2021liv  and the orbit of a star around the wormhole was
investigated by Jusufi et al. Jusufi:2021lei . Moreover, Churilova et al.
Churilova:2021tgn  discussed the shadow or the apparent size of a photon
sphere of an asymmetric wormhole Konoplya:2021hsm  in a small asymmetry case.

In this paper, we consider gravitational lensing under the weak-field
approximation and in the strong deflection limit by the Bronnikov-Kim wormhole
Bronnikov:2002rn ; Bronnikov:2003gx  with the same metric as the one of the
wormhole which was suggested in Einstein-Dirac-Maxwell theory Blazquez-
Salcedo:2020czn . The metric approaches into the metric of an extreme charged
Reissner-Nordström black hole in a black hole limit and it approaches into the
metric of the spatial Schwarzschild wormhole Dadhich:2001fu  in an ultrastatic
limit.

This paper is organized as follows. The deflection angle of a ray in the
Bronnikov-Kim wormhole spacetime is obtained in Sec. II, a lens equation is
introduced in Sec. III, and observables under a weak-field approximation are
obtained in Sec. IV. We review a Schwarzschild spacetime as a reference in
Sec. V. The deflection angle and observables by the Bronnikov-Kim wormhole in
the strong deflection limit are investigated in Sec. VI and then we discuss
and conclude our results in Sec. VII. We investigate an Arnowitt-Deser-
Misner(ADM) mass in the Appendix. In this paper, we use the units in which the
light speed and Newton’s constant are unity.

##  II Deflection angle in the Bronnikov-Kim wormhole spacetime

The line element of a Bronnikov-Kim wormhole spacetime Bronnikov:2002rn ;
Bronnikov:2003gx  is given by

$ds^{2}=-A(r)dt^{2}+B(r)dr^{2}+r^{2}(d\vartheta^{2}+\sin^{2}\vartheta
d\varphi^{2}),$ (1)

where $A(r)$ and $B(r)$ are defined by

$\displaystyle A(r)\equiv\left(1-\frac{M}{r}\right)^{2}$ (2)

and

$\displaystyle
B(r)\equiv\frac{1}{\left(1-\frac{r_{+}}{r}\right)\left(1-\frac{r_{-}}{r}\right)}=\frac{1}{1-\frac{2Q^{2}}{Mr}+\frac{Q^{2}}{r^{2}}},$
(3)

respectively, where $M$ and $Q$ are positive constants which hold $0<M<Q$, and
where $r_{\pm}$ is defined by

$r_{\pm}\equiv\frac{Q^{2}}{M}\pm\sqrt{\frac{Q^{4}}{M^{2}}-Q^{2}}.$ (4)

There is a wormhole throat at $r=r_{+}$. Note that $M/2<r_{-}<M<r_{+}$ as
shown in Fig. 1. In a black hole limit $Q\rightarrow M$, the metric approaches
into the metric of the extreme charged Reissner-Nordström black hole spacetime
and we obtain $r_{\pm}=M=Q$. There are time translational and axial Killing
vectors $t^{\mu}\partial_{\mu}=\partial_{t}$ and
$\phi^{\mu}\partial_{\mu}=\partial_{\phi}$, since the spacetime has
stationarity and axisymmetry, respectively. If the norm of the time
translational Killing vector is a constant, the spacetime is called
ultrastatic spacetime. Notice that a relation
$M=2Q^{2}r_{+}/(Q^{2}+r_{+}^{2})$. As shown in the Appendix, an ADM mass $m$
is given by

$m=\frac{Q^{2}}{M}.$ (5)

If we consider an ultrastatic limit $M\rightarrow 0$ under a fixed ADM mass
$m$, we obtain the metric of a spatial Schwarzschild wormhole Dadhich:2001fu ;
Tsukamoto:2016zdu  in the following form:

$\displaystyle
ds^{2}=-dt^{2}+\frac{dr^{2}}{1-\frac{2m}{r}}+r^{2}(d\vartheta^{2}+\sin^{2}\vartheta
d\varphi^{2}),$ (6)

which has a vanishing Ricci scalar while it has nonzero components of Ricci
tensors. We can assume $\vartheta=\pi/2$ without loss of generality.

Figure 1: The reduced radial coordinates of a photon sphere
$r_{\mathrm{m}}/M$, throat $r_{+}/M$, and $r_{-}/M$ for given $Q/M$. A (red)
solid line, a (green) dashed curve, a (blue) dotted curve, and a (black) dash-
dotted curve denote $r_{\mathrm{m}}/M$ for $1<Q/M\leq 2/\sqrt{3}$, $r_{+}/M$
for $1<Q/M\leq 2/\sqrt{3}$, $r_{-}/M$ for $1<Q/M$, and
$r_{+}/M=r_{\mathrm{m}}/M$ for $2/\sqrt{3}\leq Q/M$, respectively.

From $k^{\mu}k_{\mu}=0$, where $k^{\mu}\equiv\dot{x}^{\mu}$ is a wave number
vector and the overdot denotes a differentiation with respect to an affine
parameter, the trajectory of a ray is given by

$-A\dot{t}^{2}+B\dot{r}^{2}+r^{2}\dot{\varphi}^{2}=0$ (7)

and it can be expressed by $\dot{r}^{2}+V(r)/E^{2}=0$, where $V(r)$ is an
effective potential defined by

$V(r)\equiv\frac{1}{Br^{2}}\left(b^{2}-\frac{r^{2}}{A}\right),$ (8)

where $b\equiv L/E$ is the impact parameter of the ray and $E\equiv-
g_{\mu\nu}t^{\mu}k^{\nu}=A\dot{t}$, and $L\equiv
g_{\mu\nu}\phi^{\mu}k^{\nu}=r^{2}\dot{\varphi}$ are the conserved energy and
angular momentum of the light ray, respectively. We assume that $L$ and $b$
are positive unless we focus on negative ones. We assume that the closest
distance of the light ray is at $r=r_{0}$. Note that $V_{0}\equiv V(r_{0})=0$
holds. Here and hereinafter, any function with the subscript $0$ denotes the
function at the closest distance $r=r_{0}$. Light rays can exist in a region
for $V(r)\leq 0$. For $1<Q/M<2/\sqrt{3}$, a ray with $b=b_{\mathrm{m}}=4M$ has
$V_{\mathrm{m}}=V_{\mathrm{m}}^{\prime}=0$ and
$V_{\mathrm{m}}^{\prime\prime}<0$ on the photon sphere at
$r=r_{\mathrm{m}}=2M$ and a ray with $b=r_{+}^{2}/(r_{+}-M)$ has
$V(r_{+})=V^{\prime}(r_{+})=0$ and $V^{\prime\prime}(r_{+})>0$ on the throat
acting as an antiphoton sphere, which is a sphere formed by stable circular
light orbits, at $r=r_{+}$.111It is a concern that stable circular light
orbits may cause instability of ultracompact objects because of the slow decay
of linear waves Keir:2014oka ; Cardoso:2014sna ; Cunha:2017qtt . Here and
hereinafter, the prime denotes the differentiation with respect to the radial
coordinate $r$ or the closest distance $r_{0}$ and functions with the
subscript m denote the functions at $r=r_{\mathrm{m}}$ or
$r_{0}=r_{\mathrm{m}}$. The effective potentials are shown in Fig. 2. For
$Q/M>2/\sqrt{3}$, a ray with $b=b_{\mathrm{m}}=r_{+}^{2}/(r_{+}-M)$ has
$V_{\mathrm{m}}=V_{\mathrm{m}}^{\prime}=0$ and
$V_{\mathrm{m}}^{\prime\prime}<0$ on the throat which works as the photon
sphere at $r=r_{+}=r_{\mathrm{m}}$ as shown Fig. 3. For a marginal case
$Q/M=2/\sqrt{3}\sim 1.1547$, a ray with $b=b_{\mathrm{m}}=4M$ has
$V_{\mathrm{m}}=V_{\mathrm{m}}^{\prime}=V_{\mathrm{m}}^{\prime\prime}=0$ and
$V_{\mathrm{m}}^{\prime\prime\prime}=-4/M^{3}<0$ on the throat working as a
marginally unstable photon sphere at $r=r_{\mathrm{m}}=r_{+}=2M$. A light ray
falls into the throat at $r=r_{+}$ for $b<b_{\mathrm{m}}$, it rotates around
the photon sphere at $r=r_{\mathrm{m}}$ infinite times for $b=b_{\mathrm{m}}$,
and it is reflected by the throat for $b>b_{\mathrm{m}}$. In this paper, we
concentrate on light rays in the scattered case with $b>b_{\mathrm{m}}$ and
gravitational lensing in a usual lens configuration.

Figure 2: Effective potentials $V$ with $Q/M=1.05$. Solid (red), dashed
(green), dot-dashed (black) curves denote effective potentials $V$ with
$b/M=1.01b_{\mathrm{m}}/M=4.04$, $b/M=b_{\mathrm{m}}/M=4.00$, and
$b/M=r_{+}^{2}/(r_{+}-M)=4.72$, respectively.  Figure 3: Effective potentials
$V$ with $Q/M=1.2$. Solid (red) and dashed (green) curves denote effective
potentials $V$ with $b/M=1.01b_{\mathrm{m}}/M=4.09$ and
$b/m=b_{\mathrm{m}}/M=4.05$, respectively.

From Eq. (7), we obtain

$A_{0}\dot{t}_{0}^{2}=r_{0}^{2}\dot{\varphi}^{2}_{0}$ (9)

and the impact parameter of the light can be expressed by, as a function of
$r_{0}$,

$b(r_{0})=\frac{L}{E}=\frac{r_{0}^{2}\dot{\varphi}_{0}}{A_{0}\dot{t}_{0}}=\frac{r_{0}}{\sqrt{A_{0}}}=\frac{r_{0}^{2}}{r_{0}-M}.$
(10)

From the equation of the trajectory of the ray (7), the deflection angle of
the ray is given by

$\alpha=I(r_{0})-\pi,$ (11)

where $I(r_{0})$ is defined as

$I(r_{0})\equiv 2\int^{r_{0}}_{\infty}\frac{bdr}{r^{2}\sqrt{-V(r)}}.$ (12)

##  III Lens equation

We assume that an observer O and a source S are on the same side of a throat.
A light ray with an impact parameter $b$ is emitted by S with a source angle
$\phi$, it is reflected with a deflection angle $\alpha$ by a wormhole as a
lens L, and it is observed by O as an image I with an image angle $\theta$. We
assume small angles $\bar{\alpha}\ll 1$, $\theta=b/D_{\mathrm{ol}}\ll 1$, and
$\phi\ll 1$, where $D_{\mathrm{ol}}$ is a distance between O and L and
$\bar{\alpha}$ is an effective deflection angle of the light ray defined by

$\bar{\alpha}\equiv\alpha\quad(\mathrm{mod}\quad 2\pi).$ (13)

We introduce the winding number $N$ of the light, and we can express the
deflection angle as

$\alpha=\bar{\alpha}+2\pi N.$ (14)

Figure 4: Lens configuration. A source S with a source angle $\phi$ emits a
light ray with an impact parameter $b$. It is reflected by a lens L with an
effective deflection angle $\bar{\alpha}$ on a lens plane. An observer O sees
it as an image I with an image angle $\theta$. Note that S and O are the same
side of a wormhole throat. The distances between O and S, between L and S, and
between O and L are denoted by $D_{\mathrm{os}}$, $D_{\mathrm{ls}}$, and
$D_{\mathrm{ol}}$, respectively.

A small-angle lens equation Bozza:2008ev  is obtained from the lens
configuration in Fig. 4 as

$D_{\mathrm{ls}}\bar{\alpha}=D_{\mathrm{os}}(\theta-\phi),$ (15)

where $D_{\mathrm{ls}}$ and $D_{\mathrm{os}}=D_{\mathrm{ol}}+D_{\mathrm{ls}}$
are the distances between L and S and between O and S, respectively.

##  IV Gravitational lensing under a weak-field approximation

Under a weak-field approximation $r_{\mathrm{m}}\ll r_{0}$ or
$b_{\mathrm{m}}\ll b$, since $M\ll r$ and $Q^{2}/M\ll r$ must be satisfied,
the line element (1) can be expanded as

$\displaystyle ds^{2}$ $\displaystyle\sim$
$\displaystyle-\left(1-\frac{2M}{r}\right)dt^{2}+\left(1+\frac{2\gamma
M}{r}\right)dr^{2}$ (16) $\displaystyle+r^{2}(d\vartheta^{2}+\sin^{2}\vartheta
d\varphi^{2})$ $\displaystyle=$
$\displaystyle-\left(1-\frac{2M}{r}\right)dt^{2}+\left(1+\frac{2Q^{2}}{Mr}\right)dr^{2}$
$\displaystyle+r^{2}(d\vartheta^{2}+\sin^{2}\vartheta d\varphi^{2}),$

where $\gamma$ is given by $\gamma\equiv Q^{2}/M^{2}$. From Eq. (8.5.8) in
Ref. Weinberg:1972kfs  and $r_{0}\sim b$, the deflection angle under the weak-
field approximation is given by

$\alpha\sim\frac{4M}{b}\left(\frac{1+\gamma}{2}\right)=\frac{2}{b}\left(M+\frac{Q^{2}}{M}\right)=\frac{2\left(M+m\right)}{b}.$
(17)

Note that the winding number $N$ vanishes under the weak-field approximation.

From Eqs. (14) and $(\ref{eq:weakal})$, $N=0$, and $\theta=b/D_{\mathrm{ol}}$,
the lens equation (15) can be written as

$\theta^{2}-\phi\theta=\theta_{\mathrm{E}0}^{2},$ (18)

where $\theta_{\mathrm{E}0}$ is defined as

$\theta_{\mathrm{E}0}\equiv\sqrt{\frac{2(M+m)D_{\mathrm{ls}}}{D_{\mathrm{os}}D_{\mathrm{ol}}}}.$
(19)

The positive and negative solutions of the lens equation (18) are obtained as
$\theta=\theta_{0\pm}=\theta_{0\pm}(\phi)$, where

$\theta_{0\pm}\equiv\frac{1}{2}\left(\phi\pm\sqrt{\phi^{2}+4\theta_{\mathrm{E}0}^{2}}\right).$
(20)

Here and hereinafter, the upper (lower) sign is chosen for the positive
(negative) image angle with a positive (negative) impact parameter. In a
perfect-aligned case with $\phi=0$, a ring-shaped image, which is called an
Einstein ring, can be observed. The radius of the Einstein ring is given by

$\theta=\theta_{0+}(0)=-\theta_{0-}(0)=\theta_{\mathrm{E}0}.$ (21)

The magnifications for the images are given by

$\displaystyle\mu_{0\pm}$ $\displaystyle\equiv$
$\displaystyle\frac{\theta_{0\pm}}{\phi}\frac{d\theta_{0\pm}}{d\phi}$ (22)
$\displaystyle=$
$\displaystyle\frac{1}{4}\left(2\pm\frac{\phi}{\sqrt{\phi^{2}+4\theta_{\mathrm{E}0}^{2}}}\pm\frac{\sqrt{\phi^{2}+4\theta_{\mathrm{E}0}^{2}}}{\phi}\right)$

and its total magnification becomes

$\displaystyle\mu_{0\mathrm{tot}}$ $\displaystyle\equiv$
$\displaystyle\left|\mu_{0+}\right|+\left|\mu_{0-}\right|$ (23)
$\displaystyle=$
$\displaystyle\frac{1}{2}\left(\frac{\left|\phi\right|}{\sqrt{\phi^{2}+4\theta_{\mathrm{E}0}^{2}}}+\frac{\sqrt{\phi^{2}+4\theta_{\mathrm{E}0}^{2}}}{\left|\phi\right|}\right).$

##  V Gravitational lensing in a strong deflection limit

###  V.1 Deflection angle in a strong deflection limit

We investigate the deflection angle of a ray in a strong deflection limit
$b\rightarrow b_{\mathrm{m}}$ in the following form:

$\displaystyle\alpha$ $\displaystyle=$
$\displaystyle-\bar{a}\log\left(\frac{b}{b_{\mathrm{m}}}-1\right)+\bar{b}$
(24)
$\displaystyle+O\left(\left(\frac{b}{b_{\mathrm{m}}}-1\right)\log\left(\frac{b}{b_{\mathrm{m}}}-1\right)\right),$

where $\bar{a}$ and $\bar{b}$ are obtained from the line element. 222 We
should read the order of a following term $O(b-b_{\mathrm{m}})$ in Ref.
Bozza:2002zj  as
$O\left(\left(b/b_{\mathrm{m}}-1\right)\log\left(b/b_{\mathrm{m}}-1\right)\right)$
as discussed in Refs. Iyer:2006cn ; Tsukamoto:2016qro ; Tsukamoto:2016jzh .

####  V.1.1 For $1<Q/M<2/\sqrt{3}$

In the case of $1<Q/M<2/\sqrt{3}$, the photon sphere of a light ray with
$b=b_{\mathrm{m}}=4M$ is at $r=r_{\mathrm{m}}=2M$. We introduce a variable
Tsukamoto:2016jzh

$z\equiv 1-\frac{r_{0}}{r}$ (25)

and we rewrite $I(r_{0})$ as

$I(r_{0})=\int^{1}_{0}R(z,r_{0})F(z,r_{0})dz,$ (26)

where $R(z,r_{0})$ is given by

$R(z,r_{0})\equiv\frac{2\sqrt{M}r_{0}}{\sqrt{M\left[r_{0}^{2}+Q^{2}(1-z)^{2}\right]+2Q^{2}r_{0}(z-1)}}$
(27)

and $F(z,r_{0})$ is defined by

$F(z,r_{0})\equiv\frac{1}{\sqrt{f(z,r_{0})}},$ (28)

where $f(z,r_{0})$ is given by

$f(z,r_{0})\equiv\frac{r_{0}^{4}}{b^{2}(r_{0})\left[r_{0}+M(z-1)\right]^{2}}-(1-z)^{2}.$
(29)

We expand $f(z,r_{0})$ around $z=0$ and obtain

$f(z,r_{0})=c_{1}(r_{0})z+c_{2}(r_{0})z^{2}+O(z^{3}),$ (30)

where $c_{1}(r_{0})$ and $c_{2}(r_{0})$ are given by

$\displaystyle c_{1}(r_{0})\equiv\frac{2(2M-r_{0})}{M-r_{0}}$ (31)

and

$\displaystyle
c_{2}(r_{0})\equiv\frac{2M^{2}+2Mr_{0}-r_{0}^{2}}{(M-r_{0})^{2}},$ (32)

respectively. Here, we have used Eq. (10). From $c_{1\mathrm{m}}\equiv
c_{1}(r_{\mathrm{m}})=0$ and $c_{2\mathrm{m}}\equiv c_{2}(r_{\mathrm{m}})=2$,
$F(z,r_{0})$ diverges and the leading order of the divergence is $z^{-1}$ in
the strong deflection limit $r_{0}\rightarrow r_{\mathrm{m}}$.

We expand $c_{1}(r_{0})$ and $b(r_{0})$ around $r_{0}=r_{\mathrm{m}}$ as

$\displaystyle
c_{1}(r_{0})=c_{1\mathrm{m}}^{\prime}(r_{0}-r_{\mathrm{m}})+O\left((r_{0}-r_{\mathrm{m}})^{2}\right),$
(33)

where $c_{1\mathrm{m}}^{\prime}=2/M$, and

$b(r_{0})=b_{\mathrm{m}}+\frac{1}{2}b^{\prime\prime}_{\mathrm{m}}(r_{0}-r_{\mathrm{m}})^{2}+O\left((r_{0}-r_{\mathrm{m}})^{3}\right),$
(34)

where $b^{\prime\prime}_{\mathrm{m}}=2/M$ since we use them later.

We separate $I$ into a divergent part $I_{D}$ and a regular part $I_{R}$. The
divergent part is defined by

$I_{D}=\int^{1}_{0}R(0,r_{\mathrm{m}})F_{D}(z,r_{0})dz,$ (35)

where $R(0,r_{\mathrm{m}})$ is given by

$R(0,r_{\mathrm{m}})=\frac{4M}{\sqrt{4M^{2}-3Q^{2}}}$ (36)

and $F_{D}(z,r_{0})$ is defined by

$F_{D}(z,r_{0})\equiv\frac{1}{\sqrt{c_{1}(r_{0})z+c_{2}(r_{0})z^{2}}}.$ (37)

We can integrate $I_{D}$ as Bozza:2002zj ; Tsukamoto:2016jzh

$I_{D}=\frac{2R(0,r_{\mathrm{m}})}{\sqrt{c_{2}(r_{0})}}\log\frac{\sqrt{c_{2}(r_{0})}+\sqrt{c_{1}(r_{0})+c_{2}(r_{0})}}{\sqrt{c_{1}(r_{0})}}.$
(38)

By using Eqs. (33)-(38), $I_{D}$ in a strong deflection limit
$r_{0}\rightarrow r_{\mathrm{m}}$ or $b\rightarrow b_{\mathrm{m}}$ is obtained
as

$I_{D}=-\bar{a}\log\left(\frac{b}{b_{\mathrm{m}}}-1\right)+\bar{a}\log 4,$
(39)

where $\bar{a}$ is given by

$\bar{a}\equiv\frac{\sqrt{2}M}{\sqrt{4M^{2}-3Q^{2}}}.$ (40)

The regular part $I_{R}$ is given by

$I_{R}=\int^{1}_{0}G(z,r_{0})dz,$ (41)

where $G(z,r_{0})$ is defined as

$G(z,r_{0})\equiv R(z,r_{0})F(z,r_{0})-R(0,r_{\mathrm{m}})F_{D}(z,r_{0}).$
(42)

We expand $G(z,r_{0})$ in the power of $r_{0}-r_{\mathrm{m}}$ as

$I_{R}(r_{0})=\sum^{\infty}_{j=0}\frac{1}{j!}(r_{0}-r_{\mathrm{m}})^{j}\int^{1}_{0}\left.\frac{\partial^{j}G}{\partial
r_{0}^{j}}\right|_{r_{0}=r_{\mathrm{m}}}dz$ (43)

and we consider the first term, in which we are interested,

$I_{R}=\int^{1}_{0}G(z,r_{\mathrm{m}})dz.$ (44)

We can obtain $\bar{b}$ as

$\bar{b}=\bar{a}\log 4+I_{R}-\pi.$ (45)

The parameters $\bar{a}$ and $\bar{b}$ are shown in Fig. 5.

Figure 5:  Solid (red) and dashed (green) curves denote the parameters
$\bar{a}$ and $\bar{b}$ in the deflection angle, respectively.

In the extreme charged Reissner-Nordström black hole limit $Q\rightarrow M$,
the parameters are obtained analytically as $\bar{a}=\sqrt{2}\sim 1.41$ and
$\bar{b}=\sqrt{2}\log\left[32(3-2\sqrt{2})\right]-\pi\sim-0.733$ as obtained
in Refs. Tsukamoto:2016jzh ; Tsukamoto:2016oca .

####  V.1.2 For $Q/M>2/\sqrt{3}$

In the case of $Q/M>2/\sqrt{3}$, there is a photon sphere with
$b=b_{\mathrm{m}}=r_{+}^{2}/(r_{+}-M)$ on the throat at
$r=r_{\mathrm{m}}=r_{+}$. Since the factor $R(0,r_{\mathrm{m}})$ diverges at
$z=0$, we rewrite $I$ as

$I(r_{0})=\int^{1}_{0}S(z,r_{0})H(z,r_{0})dz,$ (46)

where $S(z,r_{0})$ and $H(z,r_{0})$ are defined by

$S(z,r_{0})\equiv 2\sqrt{M}r_{0}\left[r_{0}+M(z-1)\right]$ (47)

and

$H(z,r_{0})\equiv\frac{1}{\sqrt{h(z,r_{0})}},$ (48)

respectively, where $h(z,r_{0})$ is given by

$\displaystyle h(z,r_{0})$ $\displaystyle\equiv$
$\displaystyle\left\\{(M-r_{0})^{2}-(z-1)^{2}\left[M(z-1)+r_{0}\right]^{2}\right\\}$
$\displaystyle\times\left\\{M\left[Q^{2}(z-1)^{2}+r_{0}^{2}\right]+2Q^{2}r_{0}(z-1)\right\\}.$

Here, we have used $b(r_{0})=r_{0}^{2}/(r_{0}-M)$. We expand $h(z,r_{0})$
around $z=0$ as

$h(z,r_{0})=\bar{c}_{1}(r_{0})z+\bar{c}_{2}(r_{0})z^{2}+O(z^{3}),$ (50)

where $\bar{c}_{1}(r_{0})$ and $\bar{c}_{2}(r_{0})$ are given by

$\displaystyle\bar{c}_{1}(r_{0})\equiv
2\left(2M^{2}-3Mr_{0}+r_{0}^{2}\right)\left[M\left(Q^{2}+r_{0}^{2}\right)-2Q^{2}r_{0}\right]$

and

$\displaystyle\bar{c}_{2}(r_{0})$ $\displaystyle\equiv$
$\displaystyle-14M^{3}Q^{2}+6r_{0}^{3}\left(M^{2}+Q^{2}\right)-Mr_{0}^{4}$
(52) $\displaystyle-Mr_{0}^{2}\left(6M^{2}+29Q^{2}\right)+38M^{2}Q^{2}r_{0},$

respectively. Since we get
$\bar{c}_{1\mathrm{m}}\equiv\bar{c}_{1}(r_{\mathrm{m}})=0$ and

$\displaystyle\bar{c}_{2\mathrm{m}}\equiv\bar{c}_{2}(r_{\mathrm{m}})=\frac{4Q^{2}\left(Q^{2}-M^{2}\right)}{M^{3}}$
$\displaystyle\times\left[2M^{4}-7M^{2}Q^{2}+(4Q^{2}-5M^{2})Q\sqrt{Q^{2}-M^{2}}+4Q^{4}\right],$

$H(z,r_{0})$ diverges in order of $z^{-1}$ at $z=0$. We expand $\bar{c}_{1}$
and $b$ around $r_{0}=r_{\mathrm{m}}$ as

$\displaystyle\bar{c}_{1}(r_{0})=\bar{c}_{1\mathrm{m}}^{\prime}(r_{0}-r_{\mathrm{m}})+O\left((r_{0}-r_{\mathrm{m}})^{2}\right),$
(54)

where $\bar{c}_{1\mathrm{m}}^{\prime}$ is given by

$\displaystyle\bar{c}_{1\mathrm{m}}^{\prime}$ $\displaystyle=$
$\displaystyle\frac{2(M^{2}-Q^{2})\sqrt{Q^{2}-M^{2}}+3M^{2}Q-2Q^{3}}{M^{2}}$
(55) $\displaystyle\times 4Q(M^{2}-Q^{2}),$

and

$b(r_{0})=b_{\mathrm{m}}+b^{\prime}_{\mathrm{m}}(r_{0}-r_{\mathrm{m}})+O\left((r_{0}-r_{\mathrm{m}})^{2}\right),$
(56)

where $b^{\prime}_{\mathrm{m}}$ is given by

$b^{\prime}_{\mathrm{m}}=\frac{Q^{2}-2Q\sqrt{Q^{2}-M^{2}}}{M^{2}-Q^{2}},$ (57)

respectively, to use them later.

We separate $I$ into a divergent part $I_{d}$ and a regular part $I_{r}$. The
divergent part is defined by

$I_{d}\equiv\int^{1}_{0}S(0,r_{\mathrm{m}})H_{d}(z,r_{0})dz,$ (58)

where $S(0,r_{\mathrm{m}})$ is obtained as

$\displaystyle
S(0,r_{\mathrm{m}})=\frac{2Q\sqrt{Q^{2}-M^{2}}\left(\sqrt{Q^{2}-M^{2}}+Q\right)^{2}}{M^{3/2}}$
(59)

and $H_{d}(z,r_{0})$ is defined by

$H_{d}(z,r_{0})\equiv\frac{1}{\sqrt{\bar{c}_{1}(r_{0})z+\bar{c}_{2}(r_{0})z^{2}}}.$
(60)

The divergent term $I_{d}$ can be integrated as Bozza:2002zj ;
Tsukamoto:2016jzh

$I_{d}=\frac{2S(0,r_{\mathrm{m}})}{\sqrt{\bar{c}_{2}(r_{0})}}\log\frac{\sqrt{\bar{c}_{2}(r_{0})}+\sqrt{\bar{c}_{1}(r_{0})+\bar{c}_{2}(r_{0})}}{\sqrt{\bar{c}_{1}(r_{0})}}.$
(61)

From Eqs. (54)-(61), $I_{d}$ in a strong deflection limit $r_{0}\rightarrow
r_{\mathrm{m}}$ or $b\rightarrow b_{\mathrm{m}}$ is given by

$I_{d}=-\bar{a}\log\left(\frac{b}{b_{\mathrm{m}}}-1\right)+\bar{a}\log\frac{8\sqrt{Q^{2}-M^{2}}-4Q}{\sqrt{Q^{2}-M^{2}}},$
(62)

where $\bar{a}$ is obtained as

$\displaystyle\bar{a}=\frac{(Q+\sqrt{Q^{2}-M^{2}})^{2}}{\sqrt{2M^{4}-7M^{2}Q^{2}+4Q^{4}+(4Q^{2}-5M^{2})Q\sqrt{Q^{2}-M^{2}}}}.$

We define the regular part $I_{r}$ as

$I_{r}\equiv\int^{1}_{0}g(z,r_{0})dz,$ (64)

where $g(z,r_{0})$ is given by

$g(z,r_{0})\equiv S(z,r_{0})H(z,r_{0})-S(0,r_{\mathrm{m}})H_{d}(z,r_{0})$ (65)

and it can be expanded in the power of $r_{0}-r_{\mathrm{m}}$ as

$I_{r}(r_{0})=\sum^{\infty}_{j=0}\frac{1}{j!}(r_{0}-r_{\mathrm{m}})^{j}\int^{1}_{0}\left.\frac{\partial^{j}g}{\partial
r_{0}^{j}}\right|_{r_{0}=r_{\mathrm{m}}}dz.$ (66)

The first term in which we are interested is given by

$I_{r}=\int^{1}_{0}g(z,r_{\mathrm{m}})dz.$ (67)

Thus, we can express $\bar{b}$ as

$\bar{b}=\bar{a}\log\frac{8\sqrt{Q^{2}-M^{2}}-4Q}{\sqrt{Q^{2}-M^{2}}}+I_{r}-\pi.$
(68)

Notice that $\bar{a}$ and $\bar{b}$ are shown in Fig. 5. The ultrastatic limit
$M\rightarrow 0$ under a fixed ADM mass $m$, the deflection angle of the ray
has $\bar{a}=\sqrt{2}\sim 1.41$ and
$\bar{b}=2\sqrt{2}\log\left[4(2-\sqrt{2})\right]-\pi=\sqrt{2}\log\left[32(3-2\sqrt{2})\right]-\pi\sim-0.733$
as obtained by Tsukamoto and Harada Tsukamoto:2016zdu . We notice that the
parameters $\bar{a}$ and $\bar{b}$ are coincident with the ones in the case of
$Q\rightarrow M$.

###  V.2 Observables in the strong deflection limit

We introduce an angle $\theta^{0}_{N}$ defined by

$\alpha(\theta^{0}_{N})=2\pi N$ (69)

and we expand the deflection angle $\alpha(\theta)$ around
$\theta=\theta^{0}_{N}$ to obtain

$\alpha(\theta)=\alpha(\theta^{0}_{N})+\left.\frac{d\alpha}{d\theta}\right|_{\theta=\theta^{0}_{N}}(\theta-\theta^{0}_{N})+O\left((\theta-\theta^{0}_{N})^{2}\right).$
(70)

In the strong deflection limit $b\rightarrow b_{\mathrm{m}}+0$, the deflection
angle is expressed by

$\displaystyle\alpha(\theta)$ $\displaystyle=$
$\displaystyle-\bar{a}\log\left(\frac{\theta}{\theta_{\infty}}-1\right)+\bar{b}$
(71)
$\displaystyle+O\left(\left(\frac{\theta}{\theta_{\infty}}-1\right)\log\left(\frac{\theta}{\theta_{\infty}}-1\right)\right),$

where $\theta_{\infty}\equiv b_{\mathrm{m}}/D_{\mathrm{ol}}$ is the image
angle of the photon sphere, and $\theta^{0}_{N}$ is obtained as, from Eqs.
(69) and (71),

$\theta^{0}_{N}=\left(1+e^{\frac{\bar{b}-2\pi
N}{\bar{a}}}\right)\theta_{\infty}$ (72)

and we obtain, from Eq. (71),

$\left.\frac{d\alpha}{d\theta}\right|_{\theta=\theta^{0}_{N}}=\frac{\bar{a}}{\theta_{\infty}-\theta^{0}_{N}}.$
(73)

The effective deflection angle for the positive solution $\theta=\theta_{N}$
of the lens equation with a positive winding number $N$ is given by, from Eqs.
(14), (69), (70), (72), and (73),

$\bar{\alpha}(\theta_{N})=\frac{\bar{a}}{\theta_{\infty}e^{\frac{\bar{b}-2\pi
N}{\bar{a}}}}(\theta^{0}_{N}-\theta_{N}).$ (74)

By substituting the effective deflection angle (74) into the lens equation
(15), we obtain the positive solution of the lens equation or the image angle
for the positive winding number $N$ as

$\theta_{N}(\phi)\sim\theta^{0}_{N}-\frac{\theta_{\infty}e^{\frac{\bar{b}-2\pi
N}{\bar{a}}}D_{\mathrm{os}}(\theta^{0}_{N}-\phi)}{\bar{a}D_{\mathrm{ls}}},$
(75)

the radius of an Einstein ring with the positive winding number $N$ as

$\theta_{\mathrm{E}N}\equiv\theta_{N}(0)\sim\left(1-\frac{\theta_{\infty}e^{\frac{\bar{b}-2\pi
N}{\bar{a}}}D_{\mathrm{os}}}{\bar{a}D_{\mathrm{ls}}}\right)\theta^{0}_{N},$
(76)

and the difference between the image angle with $N=1$ and the innermost image
angle $\bar{\mathrm{s}}$ as

$\bar{\mathrm{s}}\equiv\theta_{1}-\theta_{\infty}\sim\theta^{0}_{1}-\theta^{0}_{\infty}=\theta_{\infty}e^{\frac{\bar{b}-2\pi}{\bar{a}}}.$
(77)

The magnification of the image with $\theta_{N}(\phi)$ for each positive
winding number $N$ is given by

$\mu_{N}\equiv\frac{\theta_{N}}{\phi}\frac{d\theta_{N}}{d\phi}\sim\frac{\theta^{2}_{\infty}D_{\mathrm{os}}\left(1+e^{\frac{\bar{b}-2\pi
N}{\bar{a}}}\right)e^{\frac{\bar{b}-2\pi
N}{\bar{a}}}}{\phi\bar{a}D_{\mathrm{ls}}},$ (78)

the sum of the magnifications from $N=1$ to $\infty$ is given by

$\displaystyle\sum^{\infty}_{N=1}\mu_{N}\sim\frac{\theta^{2}_{\infty}D_{\mathrm{os}}\left(1+e^{\frac{2\pi}{\bar{a}}}+e^{\frac{\bar{b}}{\bar{a}}}\right)e^{\frac{\bar{b}}{\bar{a}}}}{\phi\bar{a}D_{\mathrm{ls}}\left(e^{\frac{4\pi}{\bar{a}}}-1\right)},$
(79)

and the ratio of the magnification of the image with $N=1$ to the others
$\bar{\mathrm{r}}$ is given by

$\bar{\mathrm{r}}\equiv\frac{\mu_{1}}{\sum^{\infty}_{N=2}\mu_{N}}\sim\frac{\left(e^{\frac{4\pi}{\bar{a}}}-1\right)\left(e^{\frac{2\pi}{\bar{a}}}+e^{\frac{\bar{b}}{\bar{a}}}\right)}{e^{\frac{2\pi}{\bar{a}}}+e^{\frac{4\pi}{\bar{a}}}+e^{\frac{\bar{b}}{\bar{a}}}},$
(80)

where $\sum^{\infty}_{N=2}\mu_{N}$ is

$\displaystyle\sum^{\infty}_{N=2}\mu_{N}\sim\frac{\theta^{2}_{\infty}D_{\mathrm{os}}\left(e^{\frac{2\pi}{\bar{a}}}+e^{\frac{4\pi}{\bar{a}}}+e^{\frac{\bar{b}}{\bar{a}}}\right)e^{\frac{\bar{b}-4\pi}{\bar{a}}}}{\phi\bar{a}D_{\mathrm{ls}}\left(e^{\frac{4\pi}{\bar{a}}}-1\right)}.$
(81)

##  VI Schwarzschild spacetime

We consider a Schwarzschild spacetime to compare observables with ones in the
Bronnikov-Kim spacetime. The line element in the Schwarzschild spacetime is
given by

$ds^{2}=-\left(1-\frac{2m_{\mathrm{S}}}{r}\right)dt^{2}+\frac{dr^{2}}{1-\frac{2m_{\mathrm{S}}}{r}}+r^{2}(d\vartheta^{2}+\sin^{2}\vartheta
d\varphi^{2}),$ (82)

where $m_{\mathrm{S}}$ is its ADM mass.

Under the weak-field approximation, the line element can be expressed by

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-\left(1-\frac{2m_{\mathrm{S}}}{r}\right)dt^{2}+\left(1+\frac{2m_{\mathrm{S}}}{r}\right)dr^{2}$
(83) $\displaystyle+r^{2}(d\vartheta^{2}+\sin^{2}\vartheta d\varphi^{2}).$

If we read the parameters $Q$, $M$, and $m$ in the Bronnikov-Kim spacetime as
$m_{\mathrm{S}}$, we obtain the deflection angle and the observables under the
weak-field approximation in the Schwarzschild spacetime. The deflection angle
of a ray under the weak-field approximation is given by

$\displaystyle\alpha\sim\frac{4m_{\mathrm{S}}}{b}$ (84)

and the radius of the Einstein ring is given by

$\theta_{\mathrm{E}0}=\sqrt{\frac{4m_{\mathrm{S}}D_{\mathrm{ls}}}{D_{\mathrm{os}}D_{\mathrm{ol}}}}.$
(85)

In the strong deflection limit $b\rightarrow
b_{\mathrm{m}}=3\sqrt{3}m_{\mathrm{S}}$, the parameters $\bar{a}$ and
$\bar{b}$ in the deflection angle are obtained as $\bar{a}=1$ and
$\bar{b}=\log[216(7-4\sqrt{3})]-\pi\sim-0.40$, respectively
Bozza_Capozziello_Iovane_Scarpetta_2001 ; Bozza:2002zj .

##  VII Discussion and Conclusion

We have investigated the gravitational lensing by the Bronnikov-Kim wormhole
under the weak-field approximation and in the strong deflection limit. The
Bronnikov-Kim wormhole metric is the same as the one of a wormhole suggested
in Einstein-Dirac-Maxwell theory 333A thin shell at the throat is discussed in
Refs. Bolokhov:2021fil ; Konoplya:2021hsm ; Blazquez-Salcedo:2021udn ;
Danielson:2021aor . in a simple case Blazquez-Salcedo:2020czn ;
Bolokhov:2021fil . The metric becomes the one of an extreme charged Reissner-
Nordström black hole in a limit $Q\rightarrow M$ and the one of a spatial
Schwarzschild wormhole Dadhich:2001fu  in an ultrastatic limit $M\rightarrow
0$ under a fixed ADM mass $m$. The parameter $\bar{b}$ of the Bronnikov-Kim
wormhole has been calculated numerically partly while $\bar{b}$ of the extreme
charged Reissner-Nordström black hole Tsukamoto:2016jzh ; Tsukamoto:2016oca
and the spatial Schwarzschild wormhole Tsukamoto:2016zdu  are obtained
analytically. Interestingly, in both cases of the extreme charged Reissner-
Nordström black hole and the spatial Schwarzschild wormhole, we obtain exactly
the same parameters $\bar{a}=\sqrt{2}$ and
$\bar{b}=\sqrt{2}\log\left[32(3-2\sqrt{2})\right]-\pi$.

Recently, in Ref. Jusufi:2021lei , Jusufi et al. have discussed shadow images
under an assumption that a supermassive compact object at the center of our
Galaxy is the Bronnikov-Kim wormhole. From observational data on the orbit of
S2 star Do:2019txf , they have concluded that the parameters of the metric are
$Q\sim 8\times 10^{6}M_{\odot}$ and $M\sim 4\times 10^{6}M_{\odot}$. Note that
a Schwarzschild black hole with an ADM mass $m_{\mathrm{S}}=4\times
10^{6}M_{\odot}$, which matches the observation of orbit of S2 star, can form
an Einstein ring with a diameter $2\theta_{\mathrm{E}0}\sim 2.86$ arcsecond
and a photon sphere with a diameter $2\theta_{\infty}\sim 51.58$ $\mu$as if we
set distances $D_{\mathrm{os}}=16$ and $D_{\mathrm{ol}}=D_{\mathrm{ls}}=8$
kpc. On the other hand, the Bronnikov-Kim wormhole with the parameters
$Q=8\times 10^{6}M_{\odot}$ and $M=4\times 10^{6}M_{\odot}$ has
$2\theta_{\mathrm{E}0}\sim 4.52$ arcsecond and $2\theta_{\infty}\sim 85.56$
$\mu$as as shown in Table I. Therefore, we would distinguish the Schwarzschild
black hole with ADM mass $m_{\mathrm{S}}=4\times 10^{6}M_{\odot}$ from the
Bronnikov-Kim wormhole with the parameters $Q\sim 8\times 10^{6}M_{\odot}$ and
$M\sim 4\times 10^{6}M_{\odot}$ at the center of our Galaxy if lensed images
under the weak-field approximation and $D_{\mathrm{os}}$ are observed or if
lensed images in the strong deflection limit are observed.

Table 1: Parameters $\bar{a}$ and $\bar{b}$, the diameters of the photon sphere $2\theta_{\infty}$ and the outermost image $2\theta_{\mathrm{E}1}$, the difference of their radii $\bar{\mathrm{s}}=\theta_{\mathrm{E}1}-\theta_{\infty}$, the magnification of the pair of the outermost images $\mu_{1\mathrm{tot}}(\phi)\sim 2\left|\mu_{1}\right|$, and the ratio of the magnification of the outermost image to the others $\bar{\mathrm{r}}=\mu_{1}/\sum^{\infty}_{N=2}\mu_{N}$ in the strong deflection limit for given $Q/M$ are shown. The diameter of an Einstein ring $2\theta_{\mathrm{E}0}$ under the weak-field approximation is also shown. As a reference, the ones of the Schwarzschild spacetime also are considered. We set the parameters $M=m_{\mathrm{S}}=4\times 10^{6}M_{\odot}$, distances $D_{\mathrm{os}}=16$ and $D_{\mathrm{ol}}=D_{\mathrm{ls}}=8$ kpc, and the source angle $\phi=1$ arcsecond.  $Q/M$ |  | $1$ | $1.1$ | $1.2$ | $1.5$ | $1.8$ | $2$ | $2.2$ | $2.5$ | $3$ | Schwarzschild
---|---|---|---|---|---|---|---|---|---|---|---
$\bar{a}$ |  | $1.41$ | $2.33$ | $3.84$ | $1.89$ | $1.66$ | $1.60$ | $1.56$ | $1.52$ | $1.48$ | $1.00$
$\bar{b}$ |  | $-0.733$ | $-3.312$ | $-6.150$ | $-0.820$ | $-0.698$ | $-0.689$ | $-0.690$ | $-0.696$ | $-0.705$ | $-0.400$
$2\theta_{\infty}$ ($\mu$as) |  | $39.71$ | $39.71$ | $40.15$ | $52.30$ | $70.84$ | $85.56$ | $102.0$ | $129.7$ | $184.1$ | $51.58$
$2\theta_{\mathrm{E}1}$ ($\mu$as) |  | $40.00$ | $40.35$ | $41.72$ | $53.51$ | $71.91$ | $86.64$ | $103.1$ | $131.0$ | $185.7$ | $51.65$
$\bar{\mathrm{s}}$ ($\mu$as) |  | $0.14$ | $0.32$ | $0.76$ | $0.60$ | $0.53$ | $0.54$ | $0.57$ | $0.61$ | $0.82$ | $0.03$
$\mu_{1\mathrm{tot}}(\phi)\times 10^{17}$ |  | $3.8$ | $5.4$ | $8.3$ | $17$ | $22$ | $29$ | $37$ | $55$ | $100$ | $1.6$
$\bar{\mathrm{r}}$ |  | $85$ | $14$ | $4.3$ | $28$ | $43$ | $51$ | $56$ | $63$ | $69$ | $535$
$2\theta_{\mathrm{E}0}$ (arcsecond) |  | $2.86$ | $3.01$ | $3.16$ | $3.65$ | $4.17$ | $4.52$ | $4.89$ | $5.45$ | $6.40$ | $2.86$

The qualitative features of the gravitational lensing by the Bronnikov-Kim
wormhole in the strong deflection limit are the same as the ones in the
Schwarzschild spacetime because both cases follow general features of lensing
by a photon sphere. Distinguishing the Bronnikov-Kim wormhole from the
Schwarzschild black hole with only the images with winding numbers $N=1$ and
$N\geq 2$ is a challenging future work since the difference of their
observables in the strong deflection limit is small. The ratio of the
magnifications of the image with $N=1$ to the others $\bar{r}$ and
$\bar{\mathrm{s}}/\theta_{\infty}\sim e^{(\bar{b}-2\pi)/\bar{a}}$ can be used
for careful verifications of the spacetimes in the future since they are not
affected by the details of the lens configuration. Table II shows the
observables of a supermassive object at the center of galaxy M87 with the
parameters $m_{\mathrm{S}}=6.5\times 10^{9}M_{\odot}$ and
$D_{\mathrm{ol}}=16.8$ Mpc estimated in Ref. Akiyama:2019cqa . We set the
parameters $M$ and $Q$ of the Bronnikov-Kim wormhole so that $\theta_{\infty}$
of the wormhole is the same value as $\theta_{\infty}$ of the Schwarzschild
black hole by using $4M=3\sqrt{3}m_{\mathrm{S}}$ for $Q/M<2/\sqrt{3}\sim 1.15$
and $r_{+}^{2}/(r_{+}-M)=3\sqrt{3}m_{\mathrm{S}}$ for $Q/M>2/\sqrt{3}$.

Table 2: In the case of the parameters $m_{\mathrm{S}}=6.5\times 10^{9}M_{\odot}$, distances $D_{\mathrm{os}}=33.6$ and $D_{\mathrm{ol}}=D_{\mathrm{ls}}=16.8$ Mpc, and the source angle $\phi=1$ arcsecond is shown. The parameters $M$ and $Q$ are set so that $\theta_{\infty}$ of the Bronnikov-Kim wormhole is the same value as $\theta_{\infty}$ of the Schwarzschild black hole.  $Q/M$ |  | $1$ | $1.1$ | $1.2$ | $1.5$ | $1.8$ | $2$ | $2.2$ | $2.5$ | $3$ | Schwarzschild
---|---|---|---|---|---|---|---|---|---|---|---
$\bar{a}$ |  | $1.41$ | $2.33$ | $3.84$ | $1.89$ | $1.66$ | $1.60$ | $1.56$ | $1.52$ | $1.48$ | $1.00$
$\bar{b}$ |  | $-0.733$ | $-3.312$ | $-6.150$ | $-0.820$ | $-0.698$ | $-0.689$ | $-0.690$ | $-0.696$ | $-0.705$ | $-0.400$
$2\theta_{\infty}$ ($\mu$as) |  | $39.91$ | $39.91$ | $39.91$ | $39.91$ | $39.91$ | $39.91$ | $39.91$ | $39.91$ | $39.91$ | $39.91$
$2\theta_{\mathrm{E}1}$ ($\mu$as) |  | $40.19$ | $40.56$ | $41.47$ | $40.84$ | $40.51$ | $40.42$ | $40.36$ | $40.31$ | $40.27$ | $39.96$
$\bar{\mathrm{s}}$ ($\mu$as) |  | $0.14$ | $0.32$ | $0.78$ | $0.46$ | $0.30$ | $0.25$ | $0.23$ | $0.20$ | $0.18$ | $0.025$
$\mu_{1\mathrm{tot}}(\phi)\times 10^{17}$ |  | $3.9$ | $5.4$ | $8.2$ | $9.7$ | $7.1$ | $6.2$ | $5.7$ | $5.2$ | $4.7$ | $0.97$
$\bar{\mathrm{r}}$ |  | $85$ | $14$ | $4.3$ | $28$ | $43$ | $51$ | $56$ | $63$ | $69$ | $535$
$2\theta_{\mathrm{E}0}$ (arcsecond) |  | $2.87$ | $3.02$ | $3.15$ | $3.19$ | $3.13$ | $3.09$ | $3.06$ | $3.02$ | $2.98$ | $2.52$

On this paper, we do not treat the gravitational lensing by the marginally
unstable photon sphere on the throat with $Q/M=2/\sqrt{3}$ since the
deflection angle would diverge nonlogarithmically as discussed in Ref.
Tsukamoto:2020uay .

##  Appendix A Arnowitt-Deser-Misner (ADM) mass

We calculate an ADM mass Poisson  of the Bronnikov-Kim wormhole. Under the
weak-field approximation, the line element can be expressed as

$\displaystyle ds^{2}$ $\displaystyle\sim$
$\displaystyle-\left(1-\frac{2M}{r}\right)dt^{2}$ (86)
$\displaystyle+\left(1+\frac{2Q^{2}}{Mr}\right)dr^{2}+r^{2}(d\vartheta^{2}+\sin^{2}\vartheta
d\varphi^{2})$ $\displaystyle=$
$\displaystyle-\left(1-\frac{2M}{r(r_{*})}\right)dt^{2}$
$\displaystyle+\left(1+\frac{2Q^{2}}{Mr_{*}}\right)\left[dr_{*}^{2}+r_{*}^{2}(d\vartheta^{2}+\sin^{2}\vartheta
d\varphi^{2})\right].$

We consider a hypersurface $\Sigma_{t}$ which is the surface of constant $t$.
A unit normal to the hypersurface $\Sigma_{t}$ is given by
$n_{\mu}=-(1-M/r)\partial_{t}$ and an induced metric on the hypersurface is
given by

$h_{ab}dy^{a}dy^{b}=\left(1+\frac{2Q^{2}}{Mr_{*}}\right)\left[dr_{*}^{2}+r_{*}^{2}(d\vartheta^{2}+\sin^{2}\vartheta
d\varphi^{2})\right].$ (87)

Its boundary $S_{t}$ is a two-sphere $r_{*}=R_{*}$ and its unit normal is
given by

$r_{*a}=\left(1+\frac{Q^{2}}{Mr_{*}}\right)\partial_{r_{*}}$ (88)

and an induced metric on the boundary $S_{t}$ is given by

$\sigma_{AB}d\theta^{A}d\theta^{B}=\left(1+\frac{2Q^{2}}{MR_{*}}\right)R_{*}^{2}(d\vartheta^{2}+\sin^{2}\vartheta
d\varphi^{2}).$ (89)

The ADM mass $m$ is obtained as

$\displaystyle
m\equiv-\frac{1}{8\pi}\lim_{S_{\mathrm{t}}\rightarrow\infty}\oint_{S_{\mathrm{t}}}(k-k_{0})\sqrt{\sigma}d^{2}\theta=\frac{Q^{2}}{M},$
(90)

where the extrinsic curvature of $S_{t}$ embedded in $\Sigma_{t}$ is given by

$\displaystyle
k=r^{a}_{*|a}=\frac{2}{R_{*}}\left(1-\frac{2Q^{2}}{MR_{*}}\right),$ (91)

the extrinsic curvature of $S_{t}$ embedded in a flat space is given by

$\displaystyle
k_{0}=\frac{2}{\left(1+\frac{2Q^{2}}{MR_{*}}\right)^{1/2}R_{*}}=\frac{2}{R_{*}}\left(1-\frac{Q^{2}}{MR_{*}}\right),$
(92)

and the leading term of $\sigma$ is given by
$\sigma=R_{*}^{4}\sin^{2}\vartheta$.

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# TabMixer: Excavating Label Distribution Learning with Small-scale Features

###### Abstract

Label distribution learning (LDL) differs from multi-label learning which aims
at representing the polysemy of instances by transforming single-label values
into descriptive degrees. Unfortunately, the feature space of the label
distribution dataset is affected by human factors and the inductive bias of
the feature extractor causing uncertainty in the feature space. Especially,
for datasets with small-scale feature spaces (the feature space dimension
$\approx$ the label space), the existing LDL algorithms do not perform well.
To address this issue, we seek to model the uncertainty augmentation of the
feature space to alleviate the problem in LDL tasks. Specifically, we start
with augmenting each feature value in the feature vector of a sample into a
vector (sampling on a Gaussian distribution function). Which, the variance
parameter of the Gaussian distribution function is learned by using a sub-
network, and the mean parameter is filled by this feature value. Then, each
feature vector is augmented to a matrix which is fed into a mixer with local
attention (TabMixer) to extract the latent feature. Finally, the latent
feature is squeezed to yield an accurate label distribution via a squeezed
network. Extensive experiments verify that our proposed algorithm can be
competitive compared to other LDL algorithms on several benchmarks.

Index Terms—  Label distribution learning, uncertainty augmenting, Gaussian
distribution function, TabMixer.

##  1 Introduction

During the development of machine learning tasks, label distribution learning
(LDL) [1] is an important machine learning paradigm that leverages a function
to map a single instance to a set of labels (labels are represented in the
form of descriptive degrees and the sum of the descriptive degrees is 1).
Unlike the multi-label learning paradigm, the LDL conveys a richer semantic
content in terms of characterizing the instance’s emotions [2, 3] and
estimating the learning task’s uncertainty [4, 5, 6].

Although several classical LDL algorithms [1, 7, 8, 9, 10, 11, 12, 13, 14, 15,
16] are proposed to tackle the task of modeling the feature space into the
label space, these algorithms usually favor an accurate and ample feature
space. Briefly, these algorithms expect to conduct a process of condensing the
representation space rather than augmenting it. Here, we define a lemma that a
feature space of dimension $\approx$ the label space is a small-scale feature
space. A shred of evidence is that almost all the proposed LDL algorithms
report weak performance on benchmark datasets with a large number of labels in
many studies. So far, we draw two questions about this: 1) For the label
space, is it difficult for the comparatively small amount of feature
information to provide the algorithm with effective features to regress an
accurate label distribution? 2) For the feature space, are there artificial
reasons and uncertainty of the feature extractor that cause the low quality of
the feature space? Unfortunately, we cannot parse the existing LDL dataset
because the details of feature processing are blind-boxed. Further, we want to
boost the feature dimension and infer the uncertainty of the feature space by
tapping into expert knowledge is costly. To solve the above two problems, we
propose a feature augmentation technique with uncertainty awareness enforced
on TabMixer (Tabular MLP-Mixer) to learn an LDL dataset with small-scale
features. Note that our network treats tabular data equally and does not
distinguish between logical and continuous values.

Overall, our approach can be grouped into the following learning cohorts.
First, to augment the feature space, an MLP-based sub-network (Learner) is
created to learn the variance of a Gaussian function. This Learner inputs the
raw feature vectors and then assigns a unique variance value to each of the
feature values in the raw feature vectors. Combining the above, we can design
a Gaussian function for each element in the raw feature space by taking the
feature value as the mean value and using Gaussian sampling to obtain a vector
to replace that element (the time seed is fixed in the model training phase).
By now, our input pattern is evolved from 1D to 2D and can be pseudo-
considered as a grayscale map. Subsequently, the augmented feature information
is fed into TabMixer, where each linear layer shortcut in TabMixer is a
convolution operator to capture the local characteristics of the features.
Finally, the output feature map is squeezed by the squeezed network to obtain
an accurate label distribution, where the floodgates of the network use
softmax. The network is trained using the loss function of only L1 and K-L
divergence. We use two standard and a synthetic benchmark to evaluate our
approach and other comparative algorithms, and the experimental results verify
that the proposed algorithm is still robust under fully supervised and noisy
conditions. Furthermore, since there are random sessions in the network
content, we considered a pre-training manner to eliminate this random
consistency. This paper has two key contributions, i) We propose a novel one-
stop feature augmentation-learning solution executed on LDL datasets with
small-scale features. ii) We develop a deep network (TabMixer) that takes into
account both local and global information and a new synthetic dataset.

Fig. 1: Our architecture. Our algorithm aims to regress the label distribution
of a sample using TabMixer, where there are two key approaches, one is to
augment the feature space by modeling uncertainty, another one is to obtain an
accurate labeling distribution by mixed learning, and in addition, randomness
is also considered.

##  2 related work

Label distribution learning. Geng et al. [1] pioneered a new machine learning
paradigm: LDL, which conveys richer semantics by converting labels into
descriptive degrees. Subsequently, numerous studies [1, 7, 8, 9, 10, 11, 12,
13, 14, 15, 16, 4, 5, 6] are opened for LDL tasks, which involve both
applications and pure theory. One of the papers [4] is very interested in
modeling the uncertainty of the label distribution values via deep networks.
Inspired by this, we address the feature space at small scales to model
uncertainty to offer richer materials for downstream models.

Tabular Learning. Recently there is extensive work [17, 18, 19, 20, 21] being
proposed to model on tabular datasets. However, these methods are usually
known for the characteristics of the table’s attributes. Inspired by TransTab
[21], we seek to use MLP to globally model on tabular datasets. Furthermore,
to enhance the modeling capability of the whole model, inductive bias based on
convolutional operators is also fused in the network. The architecture of the
whole model is thanks to MLP-Mixer [22].

##  3 Proposed method

The architecture of our approach is shown in Fig. 1. Our approach can be
described as a two-stage tactic in an end-to-end manner. The first stage is
feature augmentation with uncertainty awareness, which aims at the re-
representation of the input information by embedding prior knowledge. The
purpose of the second stage is to learn the label distribution with the help
of an TabMixer in a new feature space. Furthermore, we introduce the training
strategy (loss functions) of the model and a regularization scheme
(elimination of the random consistency) at the end of this section.

Feature augmentation with uncertainty awareness. Given an input feature space
$\mathcal{X}\in R^{m\times n}$ ($m$ is the number of instances and $n$ is the
dimension of features), we assume the existence of Gaussian noise
$\mathcal{N}$ in this space [23]. In other words, we augment a single feature
value and must consider that the source of this value may be a Gaussian
distribution. The Gaussian function has two key parameters ($\mu$ and
$\sigma$), which can be formalized as:

$\vspace{-1mm}\mathcal{N}=\frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{(x-\mu)^{2}}{2\sigma^{2}}},\quad
x\sim\mathcal{X}.\vspace{-1mm}$

So far, our feature augmentation method with uncertainty is based on this a
priori assumption to provide more material for the downstream network. The
following describes the pipeline for this method.

For a single sample $\mathcal{V}$, we develop a Learner to adapt a variance
$\sigma_{i}$ to each element $\mathcal{V}_{i}$ in this sample. Learner is
consist of three linear layers and three activation layers, each of which
utilizes the ReLU operator except for the last layer. The last layer of the
network layer uses sigmoid and the dimensionality of the output layer is the
same as the input layer. Next, a sampling action is conducted where we need to
construct a Gaussian distribution function $\mathcal{N}_{i}$ for each element
$\mathcal{V}_{i}$. We construct the two parameters of the Gaussian function
$\mathcal{N}_{i}$ using the studied variance $\sigma_{i}$ and the feature
value $\mathcal{V}_{i}$ of this sample, respectively. Then, the execution
adopts operations on these Gaussian functions, the number of sampling points
is consistent with the dimensionality of the samples, and the time seed is
fixed during program implementation. For this newly created sample
$\mathcal{G}_{i}$, it can be represented as follows:

$\mathcal{G}_{i}=\left(\begin{array}[]{cccc}g_{1}^{1}&g_{1}^{2}&\cdots&g_{1}^{t}\\\
g_{2}^{1}&g_{2}^{2}&\cdots&g_{2}^{t}\\\ \vdots&\vdots&\ddots&\vdots\\\
g_{t}^{1}&g_{t}^{2}&\cdots&g_{t}^{t}\end{array}\right),$

where each column represents a vector from the elemental values
$\mathcal{V}_{i}$ of the raw samples $\mathcal{V}$. Furthermore, to model the
long-range dependence between features of the new samples, we introduce an
attention mechanism acting on $\mathcal{G}_{i}$. This attention module
performs an MLP(sigmoid($\mathcal{G}_{i}$)) in the horizontal dimension of
$\mathcal{G}_{i}$, where the output is multiplied by $\mathcal{G}_{i}$.
Finally, the generated feature matrix $\mathcal{G}^{*}$ is used to yield an
accurate label distribution.

Learning label distribution. Up to this point, an issue is raised that since
the dimensionality of the feature signal is transformed from vector to matrix,
a vanilla MLP may be difficult to employ on $\mathcal{G}^{*}$. To address this
problem, we propose a mixer with local attention (TabMixer) to extract the
deep semantics of tabular data. In a nutshell, the network treats each vector
$\mathcal{G}^{*}_{i}$ as a token and relies on a mixed-learning algorithm to
study the correlation between features. To boost the modeling capability of
the whole network, we develop a local attention (LA) block to be co-located
with the MLP as a component (LMResidual) in the network. LA consists of a 2D
convolution and a ReLU, where the convolution kernel is 3, the stride size is
1, and the padding is 1. So, for this module, it can be expressed as follows:

$\text{LMResidual}=\texttt{LA}(x)\times\texttt{MLP}(\texttt{LayerNorm}(x)+x),$

where $x$ is the feature map and the whole operation is a structure of
residuals. Finally, $\mathcal{G}^{*}$ passes through several LMResiduals to
obtain a feature map $x$. The feature map $x$ is fed into a squeezed network
with a squeezing operator to obtain an accurate label distribution. This
squeeze network mainly consists of an Linear and an Softmax, and the squeeze
operator is a mean value function enforced on the vertical axis of the feature
map $x$.

Loss function. We used $L_{1}$ and KL divergence ($L_{KL}$) to optimize the
network. So our loss function can be expressed as follows:

$\text{Loss}=\alpha\times L_{1}+\beta\times L_{KL},$

where $\alpha$ and $\beta$ are parameters. In the experiment, we select
$\alpha$ and $\beta$ to be 1 and 0.5, respectively.

Eliminating random consistency. Foreseeably, there is an uncertainty session
in our model (Gaussian sampling) that may generate pseudo-accuracy in the
training stage to mislead researchers to perform early stops. This is due to
the Gaussian sampling that causes the output distribution of the model
inconsistent with the distribution of the real training set. A typical
solution is to enforce a regularization term on the loss function in the
training phase to remedy the accuracy calculation lemma. In contrast, we seek
to alleviate this problem using a pre-training approach. Specifically, we
remove the feature augmentation algorithm and replace it by copying several
copies of the sample to concatenate it into a matrix $\mathcal{M}$ consistent
with the $\mathcal{G}^{*}$ dimension. This new sample space $\mathcal{M}$ is
fed into TabMixer to pre-train the parameters of the whole network. Pre-
training method is validated in the ablation experiment session.

##  4 EXPERIMENTS

Dataset. To demonstrate the capabilities of the model, we conduct experiments
on three special LDL datasets (Yeast-alpha, Human Gene, and a synthetic
dataset). These datasets are all small-scale feature spaces, as shown in Table
1. The synthetic data is based on the minist dataset and the feature space is
flattened with the help of an Flatten operator, after which the feature space
(28 $\times$ 28) is reduced to 28 using PCA. The label space is unsampled by
building a Gaussian function (mean is the class of handwriting and variance is
0.5). When evaluating the model performance, we used the six metrics proposed
by [1], which are Chebyshev distance (Chebyshev $\downarrow$), Clark distance
(Clark $\downarrow$), Canberra distance (Canberra $\downarrow$), KL divergence
(K-L $\downarrow$), Cosine similarity (Cosine $\uparrow$), and Intersection
similarity (Intersection $\uparrow$), where $\uparrow$ represents the bigger
the better, and $\downarrow$ represents the smaller the better.

Comparative algorithms. We compared the results of our model with five LDL
algorithms: DDH-LDL [14], BFGSLLD [1], LDL-LRR [11], LDLSF [12] and LALOT
[13]. Except for DDH-LDL, the indicators for all datasets of the models are
from [11]. Since DDH-LDL does not release the code under public resources, we
attempt to replicate it using the PyTorch 1.2 platform on a single GPU shader.

Table 1: Statistics of three real-world datasets. Dataset | Examples | Features | Labels
---|---|---|---
Human Gene | 17892 | 36 | 68
Yeast-alpha | 2465 | 24 | 18
Synthetic dataset | 60000 | 28 | 56
Table 2: Parameters on the three datasets. Dataset | Batch size | Learning rate | Epoch
---|---|---|---
Human Gene | 1000 | 0.0002 | 500
Yeast-alpha | 1000 | 0.0002 | 500
Synthetic dataset | 20000 | 0.0003 | 200

Experimental setting. For our algorithm, we put the customized selection of
parameters for each dataset in Table 2. The parameters that may be used
involve batch size, learning rate, and epoch. Our network is stacked with 12
LMResiudal on each dataset, with 512 neurons used on the MLP. Besides, we use
the PyTorch 1.2 framework and AdamW optimizer to train and test a deep network
on the GPU shader. For time seeds, we used Seed(1024) on the PC (Intel(R)
Xeon(R) Gold 6226R CPU @ 2.90GHz) and server (3090RTX) terminals. Datasets are
loaded using number_work = 8 on the PyTorch platform.

Results and discussion. The experimental results for each dataset are
summarized in Table 3, where we take the results of the 10 times 5-fold cross-
validation. The experimental results are reported in the form of
“mean$\pm$std”. Our algorithm performs optimally on six parameters across the
three datasets, particularly on the metrics (Clark, Cosine). Our approach also
achieves competitive results in the remaining metrics. Our method has two key
advantages, firstly it estimates the noise in the feature space and secondly,
the deep network has robust modeling capability. Besides, algorithms with
restricted priors have better performance, such as LDL-LRR. Our method has a
better performance compared to LDL-LRR by taking into account the uncertainty
of the feature space. Although our modeling has an expensive cost, a GPU
acceleration can be enforced in the PyTorch 1.12 to alleviate this problem. We
statistic the time TabMixer executes inference on three datasets, including
CPU and GPU. TabMixer is run on the Yeast-alpha dataset for {0.04s, 0.02s}, on
the Human Gene dataset for {0.06s, 0.02s}, and on the Synthetic dataset for
{0.008s, 0.01s} on the CPU and GPU, respectively.

Table 3: Experimental results on three datasets and the best results are
bolded.

Dataset | Algorithm | Chebyshev $\downarrow$ | Clark $\downarrow$ | Canberra $\downarrow$ | K-L $\downarrow$ | Cosine $\uparrow$ | Intersection $\uparrow$
---|---|---|---|---|---|---|---
Synthetic dataset | Ours | 0.1133$\pm$0.0065 | 0.5012$\pm$0.0014 | 0.9787$\pm$0.0040 |  0.1061$\pm$0.0021 | 0.9596$\pm$0.0072 | 0.8491$\pm$0.0066
DDH-LDL |  0.1541$\pm$0.0621 |  0.6431$\pm$0.2224 |  1.2002$\pm$0.0034 |  0.1451$\pm$0.0122 |  0.9039$\pm$0.0044 |  0.7953$\pm$0.0923
LDL-LRR |  0.1150$\pm$0.0013 |  0.5231$\pm$0.0065 |  0.9996$\pm$0.0093 | 0.0982$\pm$.0031 |  0.9153$\pm$0.0020 |  0.8355$\pm$0.0023
BFGS-LLD |  0.1396$\pm$0.0225 |  0.5856$\pm$0.0494 |  1.1309$\pm$0.1098 |  0.1372$\pm$0.0374 |  0.9108$\pm$.0227 |  0.8153$\pm$0.0263
LALOT |  0.2243$\pm$0.0912 |  1.334$\pm$0.800 |  2.9023$\pm$0.8996 |  3.5516$\pm$0.9542 |  0.7020$\pm$0.1415 |  0.7015$\pm$0.2214
LDLSF |  0.1270$\pm$0.0011 |  0.6227$\pm$0.0112 |  1.0146$\pm$0.0158 |  0.1936$\pm$0.0257 |  0.9220$\pm$0.0028 |  0.8280$\pm$0.0034
Yeast-alpha | Ours | 0.0113$\pm$0.0025 | 0.2044$\pm$0.0071 | 0.6601$\pm$0.1522 |  0.0061$\pm$0.0112 | 0.9948$\pm$0.0087 | 0.9633$\pm$0.0052
DDH-LDL |  0.1627$\pm$0.0529 |  0.5228$\pm$0.1937 |  1.0236$\pm$0.3634 |  0.1177$\pm$0.0638 |  0.8794$\pm$0.0936 |  0.8128$\pm$0.9355
LDL-LRR |  0.0134$\pm$0.0002 |  0.2093$\pm$0.0031 |  0.6791$\pm$0.0100 | 0.0054$\pm$0.0001 |  0.9946$\pm$0.0001 |  0.9625$\pm$0.0005
BFGS-LLD |  0.0135$\pm$0.0001 |  0.2110$\pm$0.0030 |  0.6865$\pm$0.0106 |  0.0055$\pm$0.0001 |  0.9949$\pm$0.0001 |  0.9629$\pm$0.0006
LALOT |  0.0165$\pm$0.0003 |  0.2608$\pm$0.0043 |  0.8544$\pm$0.0150 |  0.0084$\pm$0.0003 |  0.9917$\pm$0.0003 |  0.9526$\pm$0.0008
LDLSF |  0.0139$\pm$.0002 |  0.2164$\pm$0.0021 |  0.6874$\pm$0.0111 |  0.0058$\pm$0.0001 |  0.9943$\pm$0.0001 |  0.9613$\pm$.00004
Human Gene | Ours | 0.0512$\pm$0.0066 | 2.0100$\pm$1.1254 | 13.4451$\pm$0.1152 | 0.2232$\pm$0.0347 | 0.8452$\pm$0.0081 | 0.7949$\pm$0.0088
DDH-LDL |  0.0597$\pm$0.0752 |  2.8830$\pm$0.8514 |  19.9391$\pm$0.9627 |  0.3815$\pm$0.3870 |  0.7313$\pm$0.8355 |  0.6878$\pm$0.8845
LDL-LRR |  0.0532$\pm$0.0011 |  2.1114$\pm$0.0122 |  13.5681$\pm$0.0025 |  0.2365$\pm$0.0049 |  0.8346$\pm$0.0020 |  0.7844$\pm$0.0014
BFGS-LLD |  0.0539$\pm$0.0009 |  2.1270$\pm$0.0141 |  14.5633$\pm$0.1107 |  0.2398$\pm$0.0038 |  0.8328$\pm$0.0018 |  0.7828$\pm$0.0014
LALOT |  0.0573$\pm$0.0012 |  4.0703$\pm$3.6917 |  17.8198$\pm$3.7167 |  0.2956$\pm$0.0059 |  0.8055$\pm$0.0024 |  0.7578$\pm$0.0016
LDLSF |  0.0533$\pm$0.0009 |  2.1295$\pm$0.0209 |  14.5681$\pm$0.0055 |  0.2395$\pm$0.0054 |  0.8332$\pm$0.0028 |  0.7828$\pm$0.0022

Ablation studies. We conduct ablation studies to demonstrate the effectiveness
of feature augmentation and pre-training algorithms on dataset Human Gene. The
results of simple ablation experiments are summarized in Table 4. w/o FA is an
algorithm for removing the feature augmentation (the replacement scheme is a
reference to the augmentation method of the data in the elimination random
consistency section) and w/o PT is defined when the pre-training algorithm is
removed. The demise of the data illustrates the effectiveness of the feature
augmentation and the pre-training algorithms. We conduct 10 times 5-fold
cross-validation on the dataset of the ablation studies. The gray markers
represent optimal performance.

Noise interference. To verify the robustness of our algorithm against the
noise and uncertainty in feature space, we experiment on the dataset Human
Gene with Gaussian noise. We synthesize a dataset on Human Gene by Gaussian
function to enforce different degrees of noise. For feature space on the
dataset, the variance of the Gaussian distribution function is selected from
(0.1, 0.5, and, 1), and the mean value uses 0. The results of simple anti-
noise experiments are summarized in Table 5. Our method still has good
competition up to a noise level of 1, while other algorithms perform noise
disturbances with performance degradation far beyond our method.

Table 4: Results on ablation studies. Measures | Ours | w/o FA | w/o PT
---|---|---|---
Chebyshev $\downarrow$ | 0.0512$\pm$0.0066 |  0.0536$\pm$0.0012 |  0.5249$\pm$0.0130
Clark $\downarrow$ | 2.0100$\pm$1.2554 |  2.1100$\pm$1.1010 |  2.0571$\pm$0.0066
Canberra $\downarrow$ | 13.4451$\pm$0.1152 |  14.1222$\pm$0.1466 |  13.4989$\pm$0.0023
K-L $\downarrow$ | 0.2232$\pm$0.0347 |  0.2281$\pm$0.0051 |  0.2293$\pm$0.0020
Cosine $\uparrow$ | 0.8452$\pm$0.0081 |  0.8440$\pm$0.0070 |  0.8227$\pm$0.0845
Intersection $\uparrow$ | 0.7949$\pm$0.0088 |  0.7799$\pm$0.0088 |  0.7890$\pm$0.0014
Table 5: Results on noise interference. Measures | 0.1 | 0.5 | 1.0
---|---|---|---
Chebyshev $\downarrow$ |  0.0521$\pm$0.0043 |  0.0556$\pm$0.0011 |  0.6149$\pm$0.0100
Clark $\downarrow$ |  2.0400$\pm$1.1133 |  2.1945$\pm$1.1246 |  2.2471$\pm$0.0014
Canberra $\downarrow$ |  13.4661$\pm$0.1217 |  14.3111$\pm$0.1123 |  14.9989$\pm$0.0099
K-L $\downarrow$ |  0.2431$\pm$0.0202 |  0.2599$\pm$0.0053 |  0.2943$\pm$0.0033
Cosine $\uparrow$ |  0.8369$\pm$0.0086 |  0.8179$\pm$0.0070 |  0.7227$\pm$0.0444
Intersection $\uparrow$ |  0.7712$\pm$0.0088 |  0.7411$\pm$0.0071 |  0.6109$\pm$0.0055

##  5 CONCLUSION

How to extract enough information from fewer features is a challenge in LDL.
In this paper, we propose an end-to-end LDL method to address it. Our method
is performed with the help of TabMixter augmentation considering the feature
space uncertainty. The experimental results show that our model achieves good
results on all six metrics. Moreover, our method is further boosted by taking
into account the random consistency factor of the model. Finally, we evaluate
the noise tolerance of the model on the benchmark dataset.

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