# Quark stars in the pure pseudo-Wigner phase

Li-Qun Su<EMAIL_ADDRESS>Department of physics, Nanjing
University, Nanjing 210093, China  Chao Shi<EMAIL_ADDRESS>Department of
Nuclear Science and Technology, Nanjing University of Aeronautics and
Astronautics, Nanjing 210016, China  Yong-Feng Huang<EMAIL_ADDRESS>School of
Astronomy and space science, Nanjing University, Nanjing 210023, China  Yan
Yan<EMAIL_ADDRESS>School of Mathematics and physics, Changzhou
University,Changzhou 213164, China  Cheng-Ming Li<EMAIL_ADDRESS>School
of physics and Microelectronics, Zhengzhou University, Zhengzhou 450001, China
Hongshi Zong<EMAIL_ADDRESS>Department of physics, Nanjing University,
Nanjing 210093, China  Department of physics, Anhui Normal University, Wuhu
241000, China  Nanjing Proton Source Research and Design Center, Nanjing
210093, China  Joint Center for Particle, Nuclear Physics and Cosmology,
Nanjing 210093, China

###### Abstract

In this paper, the equation of state (EOS) of deconfined quark stars is
studied in the framework of the two-flavor NJL model, and the self-consistent
mean field approximation is employed by introducing a parameter $\alpha$
combining the original Lagrangian and the Fierz-transformed Lagrangian,
$\mathcal{L}_{R}=(1-\alpha)\mathcal{L}+\alpha\mathcal{L}_{F}$, to measure the
weights of different interaction channels. It is believed that the
deconfinement of phase transition happens along with the chiral phase
transition. Thus, due to the lack of description of confinement in the NJL
model, the vacuum pressure is set to confine quarks at low densities, which is
the pressure corresponding to the critical point of chiral phase transition.
We find that deconfined quark stars can reach over two-solar-mass, and the bag
constant therefore shifts from $(130~{}\mathrm{MeV})^{4}$ to
$(150~{}\mathrm{MeV})^{4}$ as $\alpha$ grows. In addition, the tidal
deformability $\Lambda$ is yielded ranging from 253 to 482 along with the
decrease of $~{}\alpha$, which satisfies the astronomical constraint of
$\Lambda<800$ for 1.4-solar-mass neutron stars.

##  I Introduction

Neutron stars are some of the densest manifestations of massive objects in the
universe. They are natural and ideal astrophysical laboratories for testing
theories of strongly interacting matter. With the help of pulsar observations,
the maximum mass of observed neutron stars is inceasing, i.e., PSR J0348 +
0432 Antoniadis  _et al._ (2013) with 2.01$\pm$0.04 $M_{\odot}$, and therefore
the equation of state (EOS) is constrained since soft EOS can not support
massive stars. Besides, the measurement of radius of neutron stars is obtained
with x-ray observations Bogdanov  _et al._ (2019); Riley  _et al._ (2019);
Capano  _et al._ (2020), such as $11.0^{+0.9}_{-0.6}~{}$km for 1.4-solar-mass
compact stars in Ref. Capano  _et al._ (2020). After the detection of the
Gravitational Wave, i.e., the GW170817 of Binary Neutron Star (BNS) merging
event Abbott and Abbott (2017), the tidal deformability provides an extra
constraint on the EOS, $\Lambda<800$ for the 1.4-solar-mass compact stars,
which rules out many stiff EOS with large tidal deformabilities. Considering
these strict constraints, it is commonly believed that neutron stars encompass
“normal” stars Wu  _et al._ (2013); Yan  _et al._ (2012); Li  _et al._ (2017,
2018a, 2018b) and “strange quark matter” stars Witten (1984). The concept of
strange quark matter stars originated from Witten’s strange quark matter
hypothesis. However, so far, Witten’s strange quark matter hypothesis has
neither been proven nor falsified.

Recently, in Ref. Holdom  _et al._ (2018), after taking into account of the
bulk effect, baryonic matter with only u and d quarks can also be the stable
ground state for the baryon number $A>300$. Thus, the non-strange quark matter
can be the ground state of baryonic matter rather than the strange quark
matter Zhao  _et al._ (2019a); Wang  _et al._ (2019a); Terazawa (1989a, b);
Bodmer (1971); Zhang (2020). Therefore, it is extremely interesting to study
whether there are neutron stars composed of deconfined non-strange quark
matter in nature. The main purpose of this paper is to discuss the possibility
of such neutron stars.

In this paper, the two-flavor Nambu–Jona-Lasinio(NJL) model Chodos  _et al._
(1974); Alford  _et al._ (2005); Roberts and Williams (1994); Roberts and
Schmidt (2000); Maris and Roberts (2003); Cloet and Roberts (2014); Zhao  _et
al._ (2014); Wang  _et al._ (2015); Xu  _et al._ (2015); Asakawa and Yazaki
(1989); Klevansky (1992) will be used to describe the quark matter in neutron
stars, and it is established to manifest the spontaneous breaking of chiral
symmetry, where the effective quark mass is acquired at low densities. As the
chemical potential increases, the chiral phase transition occurs and the
chiral symmetry begins to restore, which is associated with the fact that the
hadronic matter transits to the quark matter, corresponding to the Nambu phase
and the pseudo-Wigner phase respectively Xu  _et al._ (2018); Li  _et al._
(2019). The pressure of nuclear matter then reads Zong and Sun (2008a, b),

$\displaystyle P(\mu)=P(\mu=0)+\int_{0}^{\mu}\rho(\mu^{\prime})d\mu^{\prime},$

where $\rho(\mu^{\prime})$ represents the particle number density at chemical
potential $\mu^{\prime}$. The vacuum pressure $P(\mu=0)$ is set as $-B$ (bag
constant) to confine quark matter at zero chemical potential. The EOS of
strong interaction matter can then be obtained. In our previous works of Refs.
Li  _et al._ (2020); Zhao  _et al._ (2019a); Wang  _et al._ (2019a), an
improved NJL model is engaged to derive the EOS of the neutron star, which
ranges from Nambu phase to pseudo-Wigner phase in our calculation. The
calculated neutron stars have nucleonic matter on its surface, which
transforms into deconfined quark matter in their cores. So they are indeed
hybrid stars rather than pure quark stars. Following the hadron-quark duality,
people can in principle use equivalent quark degree of freedom to describe the
properties of strongly interacting matter in low-density regions. This means
that in principle we can uniformly describe the strong interaction matter EOS
from low density to high density with equivalent quark degree of freedom. In
this work, we discuss the possibility of neutron stars being pure quark stars,
i.e., with quark matter of pseudo-Wigner phase on its surface, hence the
hadronic matter is totally absent.

Additionally, in lattice simulations, a single transition is observed at
temperature. It indicates that the deconfinement transition and the chiral
transition may happen at the same time Ejiri and Yamada (2013). Hence, the
confinement and chiral phase transition are normally considered along with
each other at finite densities as well, even though there is no strong
evidence for such relations. For these reasons, quark stars should consist of
quark matter with only the pseudo-Wigner phase, and the EOS is contributed
only with the chemical potential over the critical point $\mu_{c}$ of the
chiral phase transition. We should note that the pure deconfined quark matter
has tremendous pressure which is required to suppress for stability.
Fortunately, the non-trivial QCD vacuum is able to support the huge pressure,
just like quarks confined by the vacuum as hadrons, and, in conclusion, the
rest parts of the pressure which are contributed from the chemical potential
below $\mu_{c}$ is redefined as the vacuum pressure. In this case, quark
matter becomes a self-constrained system, and the astronomical observations do
not eliminate the possibilities of deconfined quark stars.

In this paper, a self-consistent NJL model is employed, which is recently
proposed in Refs. Yu  _et al._ (2020); Yang  _et al._ (2019); Zhao  _et al._
(2019b); Arbañil and Malheiro (2015); Wang  _et al._ (2019a, b, a, a). In this
model, the parameter $\alpha$ is introduced to evaluate the competition
between the vector interaction channel and the scalar interaction channel
Altland and Simons (2010). With the increase of $\alpha$, corresponding to the
stronger repulsive interactions between quarks, quark matter can support large
pressures, and therefore a massive quark star over two-solar mass becomes
possible. In addition, the proper-time regularization is adopted for the
reason that the three-momentum cutoff has disadvantages in dealing with the
quark matter at large densities, since the central chemical potential for
massive quark stars is usually beyond the cutoff.

This paper is organized as follows: In Section II, the self-consistent NJL
model is adopted and the gap equation with the parameter $\alpha$ is given. In
Section III, the effective quark mass and particle number density are
presented with the proper-time regularization. In Section IV, the EOS in the
framework of the NJL model is calculated under the condition of electric
charge neutrality. In Section V, the TOV equation is solved and the mass-
radius relations of quark stars with different parameter $\alpha$ are shown.
In Section VI, the tidal deformability of quark stars is calculated which is
found to satisfy the constraints from astronomical observations. In Section
VII, our conclusions are presented.

##  II THE SELF-CONSISTENT NJL MODEL

The standard Lagrangian of two-flavor NJL model with chemical potential is,

$\displaystyle\mathcal{L}=\bar{\psi}(i\gamma^{\mu}\partial_{\mu}-m_{0}+\mu\gamma^{0})\psi+G[(\bar{\psi}\psi)^{2}+(\bar{\psi}i\gamma_{5}\bm{\tau}\psi)^{2}],$
(1)

where $G$ is the four-fermion coupling constant, $m_{0}$ is the current quark
mass matrix and$~{}\bm{\tau}~{}$is the Pauli matrix. The mean field
approximation is employed Buballa (2005),

$\displaystyle\mathcal{L}_{\mathrm{eff}}=\bar{\psi}(i\gamma^{\mu}\partial_{\mu}-m_{0}+2G\sigma+\mu\gamma^{0})\psi-G\sigma^{2},$
(2)

here $~{}\sigma=\langle\bar{\psi}\psi\rangle~{}$ is the quark condensate. The
effective quark mass is obtained as

$\displaystyle M=m_{0}-2G\sigma.$ (3)

The quark condensate is derived as

$\displaystyle\sigma=\langle\bar{\psi}\psi\rangle=\int\dfrac{d^{4}p}{(2\pi)^{4}}\mathrm{Tr}[S(p)],$
(4)

here $S(p)$ represents the dressed quark propagator, and the trace operates on
the Dirac, flavor and color space. Under the circumstances of zero temperature
and finite chemical potential, the quark condensate reduces to:

$\displaystyle\sigma=-2N_{c}N_{f}\int\dfrac{d^{3}p}{(2\pi)^{3}}\dfrac{M}{E_{p}}\big{[}1-\theta(\mu-
E_{p})\big{]}.$ (5)

where $E_{p}=\sqrt{p^{2}+M^{2}}$.

For four-fermion intraction channels, the direct and exchange interactions are
associated with the Fierz-transformation, and the Fierz-transformation in the
framework of the NJL model is

$\displaystyle\mathscr{F}=$
$\displaystyle\dfrac{1}{8\mathrm{N_{c}}}[2(\bar{\psi}\psi)^{2}+2(\bar{\psi}i\gamma_{5}\bm{\tau}\psi)^{2}-2(\bar{\psi}\bm{\tau}\psi)^{2}-2(\bar{\psi}i\gamma_{5}\psi)^{2}$
$\displaystyle-4(\bar{\psi}\gamma^{\mu}\psi)^{2}-4(\bar{\psi}i\gamma^{\mu}\gamma_{5}\psi)^{2}+(\bar{\psi}\sigma^{\mu\nu}\psi)^{2}-(\bar{\psi}\sigma^{\mu\nu}\bm{\tau}\psi)^{2}],$
(6)

where color terms are neglected. We only keep the scalar and the vector
interaction channels, which is most crucial for the EOS of neutron stars.
Because the original Lagrangian and Fierz-transformed Lagrangian are
identical, they can be combined at any proportions. The weighting factor
$\alpha$ reflects the competition between the scalar interaction channels and
the vector interaction channels,

$\displaystyle\mathcal{L}_{R}=(1-\alpha)\mathcal{L}+\alpha\mathcal{L}_{F},$
(7)

here $\mathcal{L}$ is the original Lagrangian and $\mathcal{L}_{F}$ is the
Fierz-transformed Lagrangian. However, this situation is different when we
employ the mean field approximation, because the Fierz-transformation and
mean-field approximation are not commutative:

$\displaystyle M=m_{0}-2G(1-\alpha+\dfrac{\alpha}{4N_{c}})\sigma,$ (8)
$\displaystyle\mu^{\prime}=\mu-\dfrac{G^{\prime}\alpha}{N_{c}\pi^{2}}\langle\psi^{\dagger}\psi\rangle,$
(9)

where
$~{}G^{\prime}=\dfrac{2G(1+\dfrac{1}{4N_{c}})}{(1-\alpha+\dfrac{\alpha}{4N_{c}})}~{}$.

It is clear that the gap equations are affected by $\alpha$ Wang  _et al._
(2019c), and the vector interaction between quarks becomes dominating with the
increase of $\alpha$. The “correct” choice of ratio can be motivated only by
physical reasoning, not by plain mathematics Altland and Simons (2010). We
will show that the parameter $\alpha$ is crucial for the EOS of quark stars,
and can be constrained by astronomical observations.

##  III PROPER-TIME REGULARIZATION

As indicated by the divergence of quark condensates, the three-momentum cutoff
is usually adopted. If the quark matter can support at least two-solar mass,
the densities inside the quark star are extremely large and therefore the
central chemical potential of quark stars may reach over 700 MeV. In this
region, the $\theta$ function of the equation is malfunctioning, because the
chemical potential is beyond the cutoff of the momentum. We have the
mathematical identity of

$\displaystyle\dfrac{1}{A^{n}}=\dfrac{1}{\Gamma(n)}\int_{0}^{\infty}d\tau\tau^{n-1}e^{-\tau
A}\rightarrow\dfrac{1}{\Gamma(n)}\int_{\tau_{UV}}^{\infty}d\tau\tau^{n-1}e^{-\tau
A},$ (10)

where $\tau_{UV}$ is set to regularize the divergence. Thus the proper-time
regularization is employed and the quark condensate now can be altered as

$\displaystyle\sigma$
$\displaystyle=-2N_{c}N_{f}\Big{[}\int\dfrac{d^{3}p}{(2\pi)^{3}}M\int_{\tau_{UV}}^{\infty}d\tau\dfrac{1}{\sqrt{\pi\tau}}e^{-\tau
E_{p}^{2}}$
$\displaystyle~{}~{}~{}~{}~{}-\int\dfrac{d^{3}p}{(2\pi)^{3}}\dfrac{M}{E_{p}}\theta(\mu-
E_{p})\Big{]}$
$\displaystyle=-2N_{c}N_{f}\Big{[}\int_{\tau_{UV}}^{\infty}d\tau\dfrac{M}{\tau^{2}}e^{-\tau
E_{p}^{2}}-\int\dfrac{d^{3}p}{(2\pi)^{3}}\dfrac{M}{E_{p}}\theta(\mu-
E_{p})\Big{]}.$ (11)

In this study, the parameters will be fixed as
$\mathrm{m_{0}}=5.0~{}\mathrm{MeV},~{}\mathrm{\tau_{UV}}=1092^{-2}~{}\mathrm{MeV}$
and $\mathrm{G^{\prime}}=3.086\times 10^{-6}~{}\mathrm{MeV}^{-2}$, which fit
the pion decay constant and the pion mass at zero temperature and chemical
potential. The solutions of the gap equations are presented in Fig. 1. The
effective quark mass is 199.73 MeV at zero chemical potential, which is
resulted from the dynamical breaking of chiral symmetry and it describes the
hadronic matter in quark degrees of freedom, corresponding to the Nambu phase.
As a result, the quark condensate associated with the breaking of chiral
symmetry reflects strong interactions between quarks in the non-trivial QCD
vacuum.

Figure 1: The effective quark mass as a function of the chemical potential.

With the increase of chemical potentials, the effective quark mass declines
and the chiral symmetry starts to restore. When the chemical potential of
quark matter is large enough, the effective quark mass shifts to the current
quark mass due to asymptotic freedom, corresponding to the pseudo-Wigner
phase.

The chiral susceptibility is calculated by Halasz  _et al._ (1998)

$\displaystyle\chi=\dfrac{\partial\sigma}{\partial m_{0}}.$ (12)

As is shown in Fig. 2, the finite and smooth peak of the chiral susceptibility
manifests that the phase transition is the crossover. Besides, the critical
value of the chemical potential is
$\mu_{c}=313,~{}316,~{}319,~{}325,~{}337,~{}375~{}\mathrm{MeV}$ for
$\alpha=0.5,~{}0.6,~{}0.7,~{}0.8,~{}0.9,~{}1.0$, respectively.

Figure 2: The chiral susceptibility as a function of the chemical potential
for different $\alpha$.
Figure 3: The baryon number density as a function of the chemical potential
for different $\alpha$.

Furthermore, the quark number density at zero temperature is derived as

$\displaystyle\langle\psi^{\dagger}\psi\rangle=2N_{c}N_{f}\int\dfrac{d^{3}p}{(2\pi)^{3}}\theta(\mu-
E_{p}).$ (13)

The baryon number densities for different $\alpha$ are shown in Fig. 3. It
entails that the quark matter appears at $~{}\mu_{0}=200~{}\mathrm{MeV}$. At a
fixed chemical potential, we can see that the baryon number density decreases
as $\alpha$ increases. When the chemical potential reaches 800 MeV, the baryon
number density is
$1.69,~{}1,53,~{}1.35,~{}1.15,~{}0.89,~{}0.54~{}\mathrm{fm}^{-3}$ for
$\alpha=0.5,~{}0.6,~{}0.7,~{}0.8,~{}0.9,~{}1.0$, respectively.

##  IV THE EQUATION OF STATE

The self-consistent NJL model with the proper-time regularization has been
presented. However, a real quark star has to be in $\beta$ equilibrium, and
the electric charge neutrality should be satisfied. Thus, the boundary
conditions are included as

$\displaystyle\mu_{d}$ $\displaystyle=\mu_{u}+\mu_{e},$ (14)
$\displaystyle\dfrac{2}{3}\rho_{u}$
$\displaystyle=\dfrac{1}{3}\rho_{d}+\rho_{e},$ (15)

where $\rho_{u},~{}\rho_{d},~{}\rho_{e}$ are the quark number densities of
u-quarks, d-quarks and electrons. Under these conditions, the chiral
susceptibility is re-calculated and the results is shown in Fig. 4. Comparing
with Fig. 2, the chiral transitions are still crossover, and the peaks are at
$\mu_{c}=288,~{}291,~{}295,~{}302,~{}316,~{}359~{}\mathrm{MeV}~{}$ for
$\alpha=0.5,~{}0.6,~{}0.7,~{}0.8,~{}0.9,~{}1.0$, respectively. We see that the
critical points of chiral phase transition become smaller when the charge
neutrality is taken into account.

Figure 4: The chiral susceptibility as a function of the chemical potential
for different $\alpha$, when electric and chemical equilibrium conditions are
satisfied.

For the reason of the asymmetry between u-quark and d-quark in the beta
equilibrium system, the pressure of quark matter is given as Zong and Sun
(2008a, b):

$\displaystyle P(\mu_{u},\mu_{d})=$ $\displaystyle
P(\mu_{u}=0,\mu_{d}=0)+\int_{0}^{\mu_{u}}\rho_{u}(\tilde{\mu}_{u},\mu_{d}=0)d\tilde{\mu}_{u}$
$\displaystyle+\int_{0}^{\mu_{d}}\rho_{d}(\mu_{u},\tilde{\mu}_{d})d\tilde{\mu}_{d}.$
(16)

The first term on the right-hand side is the contribution of the quark matter
at zero temperature and zero chemical potential, corresponding to the vacuum
pressure. In the case of the MIT bag model, the pressure of the quark matter
is defined as:

$\displaystyle P(\mu)=P_{\mathrm{free}}(\mu)-B,$ (17)

where $\mathrm{P_{free}}(\mu)$ refers to the pressure contributed from free
quark matter. In this case, the bag constant $B$ is set to keep free quarks
constrained inside the bag, which is due to the fact that free quarks are
never observed. Thus the quark matter should have negative pressure at zero
chemical potential.

When the pressure of quark matter in the framework of the MIT bag model is
beyond zero along with the increase of chemical potentials, free quarks are no
longer trapped inside the bag. Thus, the bag constant is yielded as the
pressure corresponding to the critical point of deconfinement,

$\displaystyle
B=P_{\mathrm{free}}(\mu_{de})=\int_{0}^{\mu_{de}}\rho_{\mathrm{free}}(\tilde{\mu})d\tilde{\mu},$
(18)

where $\mu_{de}$ represents the chemical potential under which free quark
matter appears. In the case of the NJL model, the bag constant should have the
same physical meaning. It is assumed that the chiral phase transition and
deconfinement transition happen roughly at the same time Ejiri and Yamada
(2013). As a result, we redefine the bag constant as the pressure when chiral
phase transition begins so as to make sure that quark matter on the surface
starts to appear with zero pressure. Then we have

$\displaystyle P(\mu_{c})=P(\mu=0)+\int_{0}^{\mu_{c}}\rho d\tilde{\mu}=0,$
$\displaystyle P(\mu=0)=-B=-\int_{0}^{\mu_{c}}\rho d\tilde{\mu}.$ (19)

In conclusion, the pressure of the quark matter in the framework of the two-
flavor NJL model is derived as:

$\displaystyle P(\mu_{u},\mu_{d})=$
$\displaystyle-B+\int_{0}^{\mu_{u}}\rho_{u}(\tilde{\mu}_{u},\mu_{d}=0)d\tilde{\mu}_{u}$
$\displaystyle+\int_{0}^{\mu_{d}}\rho_{d}(\mu_{u},\tilde{\mu}_{d})d\tilde{\mu}_{d},$
(20)

where$~{}B=\sum_{f}\int_{0}^{\tilde{\mu}_{c}}\rho_{f}d\tilde{\mu}_{f}$.

In Table 1, the bag constant is listed for different $\alpha$. We see that it
only weakly depends on $\alpha$. It means that the bag constant is relatively
stable for varying $\alpha$, which is consistent with the ranges in Ref. Song
_et al._ (1992); Lu  _et al._ (1998).

Table 1: The bag constant $\alpha$ | $0.5$ | $0.6$ | $0.7$ | $0.8$ | $0.9$ | $1.0$
---|---|---|---|---|---|---
$B^{1/4}$ $(\mathrm{MeV})$ | 132.918 | 133.37 | 134.672 | 136.94 | 141.09 | 151.91

By considering the thermodynamic relation, we have Benvenuto and Lugones
(1995); Yan  _et al._ (2012):

$\displaystyle\varepsilon=-P+\rho_{u}\mu_{u}+\rho_{d}\mu_{d}.$ (21)

The total pressure and energy density includes the contributions from
electrons:

$\displaystyle P_{tot}=P_{quark}+\dfrac{\mu_{e}^{4}}{12\pi^{2}},$
$\displaystyle\varepsilon_{tot}=\varepsilon_{quark}+\dfrac{\mu_{e}^{4}}{4\pi^{2}}.$
(22)

Figure 5: The equation of state.

The relation between pressure and energy density (EOS) is illustrated in Fig.
5. It shows that the parameter $\alpha$ associated with the vector
interactions makes the quark matter able to support larger pressures. Besides,
the stiffness of the quark star is indicated by the slope of the EOS curve. In
Fig. 6, we plot the square of the sound velocity, i.e.

$\displaystyle C^{2}=\dfrac{dP}{d\varepsilon}.$ (23)

As the parameter $\alpha$ increases, the square of the sound velocity rises as
well. In other words, the vector interaction channel in the NJL model makes
the EOS stiffer.

Figure 6: The square of the sound velocity as a function of the chemical
potential.

##  V THE TOV EQUATION

Now that the EOS has been achieved, the Tolman-Oppenheimer-Volkoff equations
(TOV) can be adopted to calculate the structure of the compact star (as G = c
= 1):

$\displaystyle\dfrac{dP}{dr}=-\dfrac{(\varepsilon+P)(M+4\pi
r^{3}P)}{r(r-2M)},$ (24) $\displaystyle\dfrac{dM}{dr}=4\pi r^{2}\varepsilon.$
(25)

The mass-radius relation is obtained by solving these differential equations.
The results are illustrated in Fig. 7. We see that the parameter $\alpha$
advances the maximum mass of the quark star but decreases the radius. When
$\alpha$ equals to one, the maximum mass reaches 2.06-solar mass with a radius
of 9.9 km (Table 2). It agrees with the observational constraints on the
radius of less than 13.6 and 13.76 km Fattoyev  _et al._ (2018); Annala  _et
al._ (2018).

Figure 7: The mass-radius relation of quark stars. Table 2: The mass-radius relation for the maximum mass star $~{}\alpha~{}$ |  $~{}M_{max}/M_{\odot}~{}$ |  $~{}R(km)~{}$
---|---|---
0.5 |  1.91 |  10.7
0.6 |  1.92 |  10.5
0.7 |  1.92 |  10.3
0.8 |  1.94 |  10.2
0.9 |  1.98 |  10.1
1.0 |  2.06 |  9.9

##  VI THE TIDAL deformability

The tidal deformability is derived by the tidal Love number $k_{2}$, and the
Love number measures the distortion of the shape of the surface of a star by
an external gravity. In the unit $G=c=1$, the relation between $k_{2}$ and the
tidal deformability is:

$\displaystyle k_{2}=\dfrac{3}{2}\Lambda(\dfrac{M}{R})^{5}.$ (26)

The $l=2$ tidal Love number $k_{2}$ is calculated by Hinderer  _et al._
(2010):

$\displaystyle k_{2}=$ $\displaystyle\dfrac{8}{5}C^{\prime
5}(1-2C^{\prime})^{2}\big{[}2+2C^{\prime}(y-1)-y\big{]}$
$\displaystyle\times\big{\\{}2C^{\prime}[6-3y+3C^{\prime}(5y-8)]$
$\displaystyle+4C^{\prime 3}[13-11y+C^{\prime}(3y-2)+2C^{\prime 2}(1+y)]$
$\displaystyle+3(1-2C^{\prime})^{2}[2-y+2C^{\prime}(y-1)]ln(1-2C^{\prime})\big{\\}}^{-1},$
(27)

where $C^{\prime}=M/R$ is the compactness of the quark star and

$\displaystyle y=\dfrac{R\beta(R)}{H(R)}-\dfrac{4\pi
R^{3}\varepsilon_{0}}{M},$ (28)

where $\varepsilon_{0}$ represents the surface energy density of the quark
star. The dimentionless value $y$ is obtained by two differential equations

$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\dfrac{dH}{dr}=\beta,$
(29) $\displaystyle\dfrac{d\beta}{dr}=$
$\displaystyle\dfrac{2H}{1-2M/r}\big{\\{}-2\pi\big{[}5\varepsilon+9P+d\varepsilon/dP(\varepsilon+P)\big{]}$
$\displaystyle+\dfrac{3}{r^{2}}+\dfrac{2}{1-2M/r}(\dfrac{M}{r^{2}}+4\pi
rP)^{2}\big{\\}}$ $\displaystyle+\dfrac{2\beta}{r-2M}[-1+\dfrac{M}{r}+2\pi
r^{2}(\varepsilon-P)].$ (30)

As $r$ $\rightarrow 0$, $H(r)=a_{0}r^{2}$ and $\beta(r)=2a_{0}r$. $a_{0}$ can
be any number, because we only concern about the ratio between H and $\beta$.
In Table 3, the compactness, $k_{2}$ and the tidal deformability $\Lambda$ for
the 1.4-solar-mass quark star are listed for different $\alpha$.

Table 3: Tidal deformability for 1.4-solar-mass quark star $~{}\alpha~{}$ |  $~{}M/R~{}$ |  $~{}k_{2}~{}$ |  $~{}\Lambda~{}$
---|---|---|---
0.5 |  0.1875 |  0.1675 |  481.812
0.6 |  0.1903 |  0.1667 |  445.75
0.7 |  0.1936 |  0.1603 |  392.556
0.8 |  0.1974 |  0.1561 |  347.466
0.9 |  0.2010 |  0.1542 |  313.1
1.0 |  0.2083 |  0.1487 |  252.886

The binary neutron star merger event GW170817 Abbott and Abbott (2017) gives
us constraints on $\Lambda$. For a 1.4 $M_{\odot}$ star, the tidal
deformability should be $\Lambda<800$ Bauswein  _et al._ (2019). Table 3 shows
that this requirement is satisfied by our quark star model.

##  VII SUMMARY AND CONCLUSIONS

In this paper, the EOS of deconfined quark stars is studied in the framework
of the NJL model. The mass-radius relation and the tidal deformability are
obtained. The self-consistent NJL model is employed by introducing the
weighting factor $\alpha$ of the Fierz-transformed Lagrangian. To obtain a
stiffer quark star, a larger $\alpha$ will be needed. Additionally, the
proper-time regularization is used due to the large density in the core of
quark stars, where the three-momentum cutoff assumption fails. Because the
past definition of the bag constant $B$ is not always suitable for the EOS of
quark stars, for an analogy to the MIT bag model, we set the bag constant as
the pressure of the quark matter that is undergoing the chiral phase
transition along with the phase transition of the deconfinement.

The bag constant $B$ is found to vary from $(132.918~{}\mathrm{MeV})^{4}$ to
$(151.91~{}\mathrm{MeV})^{4}$ when $\alpha$ shifts from 0.5 to 1.0 under
electric charge neutrality and $\beta$ equilibrium. By solving the TOV
equations, the mass-radius relations for different $\alpha$ are yielded. It is
shown that as $\alpha$ increases, the maximum mass of quark stars grows but
the corresponding radius decreases. In particular, the maximum mass for
$\alpha=1.0$ reaches 2.06-solar mass, which meets the observational
requirement of PSR J1614-2230 ($M=1.98\pm 0.017M_{\odot}$) and PSR J0348 +
0432 ($M=2.01\pm 0.04M_{\odot}$). For a 1.4-solar-mass quark star, the derived
radius is around 10 km, which is smaller than 13.6 km and 13.67 km as required
by observations Fattoyev  _et al._ (2018); Annala  _et al._ (2018). Besides,
the tidal deformability of quark stars in our framework ranges from 252.886 to
481.812, which also matches the observational constraints of $\Lambda<800$ for
a 1.4 $M_{\odot}$ compact star Bauswein  _et al._ (2019). Our study indicates
that deconfined quark stars may exist in the universe.

## Acknowledgements

We thank the anonymous referee for helpful comments and suggestions. This work
is supported in part by the National Natural Science Foundation of China
(under Grants No. 12075117, No. 11535005, No. 11905104, No. 12005192, No.
11873030, No. 12041306, No. U1938201, and No. 11690030), by National SKA
Program of China No. 2020SKA0120300, by China Postdoctoral Science Foundation
(Grant No. 2020M672255 and No. 2020TQ0287), by the Strategic Priority Research
Program of the Chinese Academy of Sciences (“multi-waveband Gravitational-Wave
Universe”, Grant No. XDB23040000), and by Nation Major State Basic Research
and Development of China (2016YFE0129300). In the end, we would like to thank
Chao Shi for the constructive comments and reading through the manuscript.

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# Image Reconstruction Without Explicit Priors

###### Abstract

We consider solving ill-posed imaging inverse problems without access to an
explicit image prior or ground-truth examples. An overarching challenge in
inverse problems is that there are many undesired images that fit to the
observed measurements, thus requiring image priors to constrain the space of
possible solutions to more plausible reconstructions. However, in many
applications it is difficult or potentially impossible to obtain ground-truth
images to learn an image prior. Thus, inaccurate priors are often used, which
inevitably result in biased solutions. Rather than solving an inverse problem
using priors that encode the explicit structure of any one image, we propose
to solve a set of inverse problems jointly by incorporating prior constraints
on the collective structure of the underlying images.The key assumption of our
work is that the ground-truth images we aim to reconstruct share common, low-
dimensional structure. We show that such a set of inverse problems can be
solved simultaneously by learning a shared image generator with a low-
dimensional latent space. The parameters of the generator and latent embedding
are learned by maximizing a proxy for the Evidence Lower Bound (ELBO). Once
learned, the generator and latent embeddings can be combined to provide
reconstructions for each inverse problem. The framework we propose can handle
general forward model corruptions, and we show that measurements derived from
only a few ground-truth images ($O(10)$) are sufficient for image
reconstruction without explicit priors.

Index Terms—  Inverse problems, computational imaging, prior models,
generative networks, Bayesian inference

##  1 Introduction

In imaging inverse problems, the goal is to recover the ground-truth image
from corrupted measurements, where the measurements and image are related via
a forward model: $y=f(x)+\eta$. Here, $y$ are our measurements, $x$ is the
ground-truth image, $f$ is a known forward model, and $\eta$ is noise. Such
problems are ubiquitous and include denoising [1], super-resolution [2],
compressed sensing [3, 4], and phase retrieval [5]. These problems are often
ill-posed: there are many images that are consistent with the observed
measurements. Thus, traditionally structural assumptions on the image are
required to reduce the space of possible solutions. We encode these structural
assumptions in an image generation model (IGM), which could take the form of
either an implicit or explicit image prior.

In order to define an IGM it is necessary to have knowledge of the structure
of the underlying image distribution. If ground-truth images are available,
then an IGM can be learned directly [6, 7]. However, this approach requires
access to an abundance of clean data and there are many scientific
applications (e.g., geophysical imaging and astronomical imaging) where we do
not have access to ground-truth images. Collecting ground-truth images in
these domains can be extremely invasive, time-consuming, expensive, or even
impossible. For instance, how should we define an IGM for black hole imaging
without having ever seen a direct image of a black hole or knowing what one
should look like? Moreover, classical approaches that utilize hand-crafted
IGMs [8] are prone to human bias [9].

In this work, we show how one can solve a set of ill-posed image
reconstruction tasks without access to an explicit image prior. The key
insight of our work is that knowledge of common structure across multiple
problems is sufficient regularization alone. In particular, we suppose we have
access to a collection of measurements $\\{y^{(i)}\\}_{i=1}^{N}$ which are
observed through a forward model $y^{(i)}:=f(x^{(i)})+\eta^{(i)}$. An
important assumption we make is that the ground-truth images
$\\{x^{(i)}\\}_{i=1}^{N}$ are drawn from the same distribution (unknown a
priori) and share common, low-dimensional structure. This assumption is
satisfied in a number of applications where ground-truth images are not
available. For instance, although we might not know what a black hole looks
like, we might expect it to be similar in appearance over time. Our main
result is that one can exploit this common structure by jointly inferring 1) a
single generator $G_{\theta}$ and 2) $N$ low-dimensional latent distributions
$q_{\phi^{(i)}}$, such that the distribution induced by the push-forward of
$q_{\phi^{(i)}}$ through $G_{\theta}$ approximates the posterior
$p(x|y^{(i)})$ for each example $i\in[N]$.

##  2 Approach

Fig. 1:  We use a set of $N$ measurements $\\{y^{(i)}\\}_{i=1}^{N}$ from $N$
different ground-truth images to infer $\\{\phi^{(i)}\\}_{i=1}^{N}$, the
parameters of the latent distributions, and $\theta$, the parameters of the
generative network $G$. All inferred parameters are colored as blue and the
loss is given by Equation (4). Here, $G\sharp P$ denotes the push-forward of
distribution $P$ induced by $G$.

In this work, we propose to solve a set of inverse problems without access to
an IGM by assuming that the set of ground-truth images have common, low-
dimensional structure. Other works have considered solving linear inverse
problems without an IGM [10, 11], but assume that one has access to multiple
independent measurements of a single, static ground-truth image, limiting
their applicability to many real-world problems. In contrast, in our work we
do not require multiple measurements of the same ground-truth image.

###  2.1 Motivation for ELBO as a model selection criterion

In order to accurately learn an IGM, we motivate the use of the Evidence Lower
Bound (ELBO) as a loss by showing that it provides a good criterion for
selecting a prior model. Suppose we are given noisy measurements from a single
image: $y=f(x)+\eta$. In order to reconstruct the image $x$, we traditionally
first require an IGM $G$ that captures the distribution $x$ was sampled from.
A natural approach would be to find or select the model $G$ that maximizes the
model posterior distribution $p(G|y)\propto p(y|G)p(G).$ That is, conditioned
on the noisy measurements, find the model of highest likelihood. Unfortunately
computing $p(y|G)$ is intractable, but we show that it can be well
approximated using the ELBO.

To motivate our discussion, consider estimating the conditional posterior
$p(x|y,G)$ by learning the parameters $\phi$ of a variational distribution
$q_{\phi}(x)$. Observe that the definition of the KL-divergence followed by
using Bayes’ theorem gives

$\displaystyle D_{\operatorname*{KL}}(q_{\phi}(x)\;\|\;p(x|y,G))=\log p(y|G)$
$\displaystyle-\mathbb{E}_{x\sim q_{\phi}(x)}[\log p(y|x,G)+\log p(x|G)-\log
q_{\phi}(x)].$

The ELBO of an IGM $G$ given measurements $y$ is defined by

$\displaystyle\operatorname*{ELBO}(G,q_{\phi};y)$
$\displaystyle:=\mathbb{E}_{x\sim q_{\phi}(x)}[\log p(y|x,G)+\log p(x|G)$
$\displaystyle-\log q_{\phi}(x)].$ (1)

Rearranging the above equation and using the non-negativity of the KL-
divergence, we see that we can lower bound the model posterior as

$\displaystyle\log p(G|y)\geqslant\operatorname*{ELBO}(G,q_{\phi};y)+\log
p(G)-\log p(y).$ (2)

Note that $-\log p(y)$ is independent of the parameters of interest, $\phi$.
If the variational distribution $q_{\phi}(x)$ is a good approximation to the
posterior $p(x|y,G)$, $D_{\mathrm{KL}}\approx 0$ so maximizing $\log p(G|y)$
with respect to $G$ is approximately equivalent to maximizing
$\operatorname*{ELBO}(G,q_{\phi};y)+\log p(G)$.

Each term in the ELBO objective encourages certain properties of the IGM $G$.
In particular, the first term, $\mathbb{E}_{x\sim q_{\phi}(x)}[\log
p(y|x,G)]$, requires that $G$ should lead to an estimate that is consistent
with our measurements $y$. The second term, $\mathbb{E}_{x\sim
q_{\phi}(x)}[\log p(x|G)]$, encourages images sampled from $q_{\phi}(x)$ to
have high likelihood under our model $G$. The final term is the entropy term,
$\mathbb{E}_{x\sim q_{\phi}(x)}[-\log q_{\phi}(x)]$, which encourages a $G$
that leads to “fatter” minima that are less sensitive to small changes in
likely images $x$ under $G$.

###  2.2 ELBOProxy

Some IGM are explicit (e.g., Gaussian image prior), which allows for direct
computation of $\log p(x|G)$. In this case, we can optimize the ELBO defined
in Equation (1) directly and then perform model selection. However, an
important class of IGMs that we are interested in are those given by deep
generative networks. Such IGMs are not probabilistic in the usual Bayesian
interpretation of a prior, but instead implicitly enforce structure in the
data. Moreover, terms such as $\log p(x|G)$ can only be computed directly if
we have an injective map [12]. This architectural requirement limits the
expressivity of the network. Hence, we instead consider a proxy of the ELBO
that is especially helpful for deep generative networks. Suppose our image
generation model is of the form $x=G(z)$ where $G$ is a generative network and
$z$ is a latent vector. Introducing a variational family for our latent
representations $z\sim q_{\phi}(z)$ and using $\log p(z|G)$ in place of $\log
p(x|G)$, we arrive at the following proxy of the ELBO:

$\displaystyle\operatorname*{ELBOProxy}(G,q_{\phi};y)$
$\displaystyle:=\mathbb{E}_{z\sim q_{\phi}(z)}[\log p(y|G(z))$
$\displaystyle+\log p(z|G)-\log q_{\phi}(z)].$ (3)

When $G$ is injective and $q_{\phi}(x)$ is the push-forward of $G$ through
$q_{\phi}(z)$, then this proxy is exactly the ELBO in Eq. (1). While $G$ may
not necessarily be injective, we show empirically that the ELBOProxy is a
useful criterion for selecting such models.

Toy example: To illustrate the use of the $\operatorname*{ELBOProxy}$ as a
model selection criterion, we conduct the following experiment that asks
whether the $\operatorname*{ELBOProxy}$ can identify the best model from a
given set of plausible IGMs. For this experiment, we use the MNIST dataset
[13] and consider two inverse problems: denoising and phase retrieval. We
train a generative model $G_{c}$ on each class $c\in\\{0,1,2,\dots,9\\}$.
Hence, $G_{c}$ is learned to generate images from class $c$ via $G_{c}(z)$
where $z\sim\mathcal{N}(0,I)$. Then, given noisy measurements $y_{c}$ of a
single image from class $c$, we ask whether the generative model $G_{c}$ from
the appropriate class would achieve the best $\operatorname*{ELBOProxy}$. For
denoising, our measurements are $y_{c}=x_{c}+\eta_{c}$ where
$\eta_{c}\sim\mathcal{N}(0,\sigma^{2}I)$ and $\sigma=\sqrt{0.5}$. For phase
retrieval, $y_{c}=|\mathcal{F}(x_{c})|+\eta_{c}$ where $\mathcal{F}$ is the
Fourier transform and $\eta_{c}\sim\mathcal{N}(0,\sigma^{2}I)$ with
$\sigma=\sqrt{0.05}$.

We construct $5\times 10$ arrays for each problem, where in the $i$-th row and
$j$-th column, we compute the $-\operatorname*{ELBOProxy}$ obtained by using
model $G_{{j-1}}$ to reconstruct images from class $i-1$. We calculate
$\operatorname*{ELBOProxy}(G_{c},q_{\phi_{c}};y_{c})$ by parameterizing
$q_{\phi_{c}}$ with a Normalizing Flow and optimizing network weights
$\phi_{c}$ to maximize (3). Results are shown in Fig. 2. We note that all of
the correct models are chosen in both denoising and phase retrieval. We also
note some interesting cases where the $\operatorname*{ELBOProxy}$ values are
similar for certain cases, such as when recovering the $3$ or $4$ image when
denoising. For example, when denoising the $4$ image, both $G_{4}$ and $G_{9}$
achieve comparable $\operatorname*{ELBOProxy}$ values. By carefully inspecting
the noisy image of the $4$, one can see that both models are reasonable given
the structure of the noise.

Fig. 2: We consider two inverse problems: denoising and phase retrieval. Left:
the two leftmost columns correspond to the ground truth image $x_{c}$ and the
noisy measurements $y_{c}$. Center: in each row, we show the means of the
distribution induced by the push-forward of $G_{j}$ and each latent
distribution $z\sim q_{\phi_{j}}$ for $j\in\\{0,\dots,9\\}$. Right: each row
of the array corresponds to the $-\operatorname*{ELBOProxy}$ achieved by each
model in reconstructing the images. Here, lower is better. Boxes highlighted
in green correspond to the best $-\operatorname*{ELBOProxy}$ values in each
row. In all cases, the correct model was chosen.

###  2.3 Learning the IGM to solve inverse problems

As the previous section illustrates, the $\operatorname*{ELBOProxy}$ provides
a good criterion for choosing an appropriate IGM from noisy measurements.
Here, we consider the task of learning the IGM $G$ directly from corrupted
data. We consider the setting where we have access to a collection of $N$
measurements $y^{(i)}=f(x^{(i)})+\eta^{(i)}$ for $i\in[N]$. The key assumption
we make is that common, low-dimensional structure is shared across the
underlying images $\\{x^{(i)}\\}_{i=1}^{N}$.

We propose to find a shared generator $G_{\theta}$ with weights $\theta$ along
with latent distributions $q_{\phi^{(i)}}$ that can be used to reconstruct the
full posterior of each image $x^{(i)}$ from its corresponding noisy
measurements $y^{(i)}$. This approach is illustrated in Fig. 1. Having the
generator be shared across all images helps capture their common collective
structure. Each corruption, however, could induce its own complicated image
posteriors. Hence, we assign each measurement $y^{(i)}$ its own latent
distribution to capture the differences in their posteriors. While the learned
distribution may not necessarily be the true image posterior (as we are
optimizing a proxy of the ELBO), it still captures a distribution of images
that fit to the observed measurements.

More explicitly, given a set of measurements $\\{y^{(i)}\\}_{i=1}^{N}$, we
optimize the $\operatorname*{ELBOProxy}$ from Equation (3) to jointly infer a
generator $G_{\theta}$ and variational distributions
$\\{q_{\phi^{(i)}}\\}_{i=1}^{N}$:

$\displaystyle\max_{\theta,\phi^{(i)}}$
$\displaystyle\frac{1}{N}\sum_{i=1}^{N}\operatorname*{ELBOProxy}(G_{\theta},q_{\phi^{(i)}};y^{(i)})+\log
p(G_{\theta}).$ (4)

The expectation in this objective is approximated via Monte Carlo sampling. In
terms of choices for $\log p(G_{\theta})$, we can add additional
regularization to promote particular structures, such as smoothness. Here, we
consider having sparse neural network weights as a form of implicit
regularization and use dropout during training to represent $\log
p(G_{\theta})$ [14].

Once a generator $G_{\theta}$ and variational parameters
$\\{\phi^{(i)}\\}_{i=1}^{N}$ have been learned, we solve the $i$-th inverse
problem by simply sampling $\hat{x}^{(i)}=G_{\theta}(\hat{z}^{(i)})$ where
$\hat{z}^{(i)}\sim q_{\phi^{(i)}}$. Producing multiple samples for each
inverse problem can help visualize the range of uncertainty under the learned
IGM $G_{\theta}$, while taking the average of these samples empirically
provides clearer estimates with better metrics in terms of PSNR or MSE. We
report PSNR outputs in our subsequent experiments.

##  3 Experimental Results

We now consider solving a set of inverse problems via the framework described
in 2.3. For each of these experiments, we use a multivariate Gaussian
distribution to parameterize each posterior distribution $q_{\phi^{(i)}}$, and
we use a Deep Decoder [15] with $6$ layers of size $150$, a latent size of
$40$, and a dropout of $10^{-4}$ as the IGM. Each multivariate Gaussian
distribution is parameterized by a mean and covariance matrix
$\\{\mu^{(i)},L^{(i)}L^{{(i)}^{T}}+\varepsilon I\\}_{i=1}^{N}$, with
$\varepsilon=10^{-3}$ is added to the covariance matrix to help with stability
of the optimization.

Denoising: We show results on denoising noisy images of a single face from the
PubFig [16] dataset in Fig. 3. The measurements are defined by $y=x+\eta$
where $\eta\sim\mathcal{N}(0,\sigma^{2}I)$ with an SNR of $\sim$15 dB. Our
method is able to remove much of the noise and recovers small scale features,
even with only 95 examples. Our reconstructions also substantially outperform
the baseline methods AmbientGAN [17], Deep Image Prior (DIP) [18], and
regularized maximum likelihood using total variation (TV-RML), as shown in
Fig. 3. Unlike DIP, our method does not seem to overfit and does not require
early stopping. Our method also does not exhibit noisy artifacts like those
seen in AmbientGAN results.

Fig. 3: Denoising baseline comparisons. We compare to AmbientGAN, DIP, and TV-
RML with weight $\lambda$ and report the average PSNR across all 95
reconstructions. We show both early stopping and full training results using
DIP. Our results exhibit higher PSNR than all other baselines while
maintaining distinct features of the ground-truth images. Fig. 4:
Visualization of the intrinsic resolution of the EHT compressed sensing
measurements. The EHT measures sparse spatial frequencies of the image (i.e.,
components of the image’s Fourier transform). In order to generate the ground
truth image (c), all frequencies in the entire domain of (a) must be used.
Restricting spatial frequencies to the ones in (a) and (b)’s green circle
generates the target (d). The EHT samples a subset of this region, indicated
by the sparse black samples in (b). Naively recovering an image using only
these frequencies results in the dirty image (e), which is computed by
$A^{H}y$. The 2D spatial Fourier frequency coverage represented with $(u,v)$
positions is referred to as the UV coverage.

Compressed sensing: We consider a compressed sensing problem inspired by
astronomical imaging of black holes with the Event Horizon Telescope (EHT):
suppose we are given access to measurements of the form $y=Ax+\eta,\
\eta\sim\mathcal{N}(0,\sigma^{2}I)$ where $A\in\mathbb{C}^{p\times n}$ is a
low-rank compressed sensing matrix arising from interferometric telescope
measurements. This problem is ill-posed and requires the use of priors or
regularizers to recover a reasonable image [9]. Moreover, it is impossible to
acquire ground-truth images of black holes, so any explicit prior defined a
priori will exhibit human bias. However, we can assume that although the black
hole evolves, it will not change drastically from day-to-day.

We show results on 60 frames from a video of a simulated evolving black hole
target [19, 20] with an SNR of $\sim$32 dB in Fig. 5. The reference “target
image” is the ground-truth filtered with a low-pass filter that represents the
maximum resolution intrinsic to the EHT array (visualized and explained in
Fig. 4). Our method is not only able to reconstruct the large scale features
of the ground-truth images without any aliasing artifacts, but also achieves a
level of super-resolution.

Fig. 5: Compressed sensing with a video of a black hole. We demonstrate our
method described in Section 2.3 using 60 images from an evolving black hole
target. Here we show the ground-truth, dirty ($A^{H}y$, see Fig. 4), and mean
reconstruction images, respectively. We also show the unwrapped space × time
image, which is taken along the overlaid white ring illustrated in the T=1
ground-truth image. The bright-spot’s temporal trajectory of our
reconstruction matches that of the ground-truth.

Non-Convex Phase Retrieval: Here we demonstrate our approach on solving non-
convex phase retrieval. Our measurements are described by
$y=|\mathcal{F}(x)|+\eta$ where $\mathcal{F}(\cdot)$ is a complex-valued
linear operator (either Fourier or undersampled Gaussian measurements) and
$\eta\sim\mathcal{N}(0,\sigma^{2}I)$.

Fig. 6 shows results from a set of $N=150$ measurement examples of the MNIST
8’s class with an SNR of $\sim$52 dB. The true posterior is multi-modal due to
the phase ambiguity and non-convexity of the problem. Our reconstructions have
features similar to the digit $8$, but contain artifacts in the Fourier case.
These artifacts are only present in the Fourier case due to additional
ambiguities (i.e., flipping and spatial-shifts), which lead to a more complex
posterior [21].

Fig. 6: Phase retrieval with MNIST 8’s. We shown an example of our method
applied to the challenging non-convex problems of Fourier and complex Gaussian
phase retrieval.

##  4 Conclusion

In this work, we showcased how one can solve a set of inverse problems without
an IGM by leveraging common structure present across the underlying ground-
truth images. We demonstrated that even with a small set of corrupted
measurements, one can jointly solve these inverse problems by directly
learning an IGM that maximizes a proxy of the ELBO. Overall, our work
showcases the possibilities of solving inverse problems in a “prior-free”
fashion, free from human bias typical of ill-posed image reconstruction. We
believe our approach can aid automatic discovery of novel structure from
scientific measurements without access to clean data, leading to potentially
new avenues for scientific discovery.

Acknowledgements This work was sponsored by NSF Award 2048237 and 1935980, an
Amazon AI4Science Partnership Discovery Grant, and the Caltech/JPL President’s
and Director’s Research and Development Fund (PDRDF). This research was
carried out at the Jet Propulsion Laboratory and Caltech under a contract with
the National Aeronautics and Space Administration and funded through the
PDRDF.

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