# Impact of chiral asymmetry and magnetic field on passage of an energetic
test parton in a QCD medium

Ritesh Ghosh<EMAIL_ADDRESS>School of Physical Sciences, National
Institute of Science Education and Research, An OCC of Homi Bhabha National
Institute, Jatni, Khurda 752050, India  Mohammad Yousuf Jamal
<EMAIL_ADDRESS>School of Physical Sciences, Indian Institute of
Technology Goa, Ponda 403401, Goa, India  Manu Kurian<EMAIL_ADDRESS>RIKEN
BNL Research Center, Brookhaven National Laboratory, Upton, New York 11973,
USA

###### Abstract

We study the dependence of collisional energy loss of a test parton moving
with a high velocity on the chiral imbalance and magnetic field in the QCD
medium. A semi-classical approach is adopted to estimate the parton energy
loss that takes into account the back-reaction on the parton due to the
polarization effects of the QCD medium while traversing through the medium. We
find that the motion of the parton is sensitive to the chiral asymmetry in the
medium. Further, we investigate the effect of magnetic field-induced
anisotropy on the energy transfer between the moving parton and the medium.
Our results show that the energy loss of the parton is strongly influenced by
the strength of the magnetic field as well as the relative orientation of the
motion of the parton and the direction of the magnetic field in the medium.

##  I Introduction

The heavy-ion collision (HIC) experiments at Relativistic Heavy Ion Collider
(RHIC) and Large Hadron Collider (LHC) provide a unique opportunity to produce
and investigate the properties of an extremely hot and dense state of matter:
the Quark-Gluon Plasma (QGP), which is believed to resemble the state of the
universe shortly after the Big Bang Back  _et al._ (2005); Arsene  _et al._
(2005); Aamodt  _et al._ (2010); Fukushima  _et al._ (2021). The success of
relativistic viscous hydrodynamics in describing the evolution of the QGP
medium is considered one of the theoretical breakthroughs in this area of
research Gale  _et al._ (2013); Heinz and Snellings (2013). The dissipative
processes and the associated transport parameters rely on the understanding of
the non-equilibrium physics of the QGP. Much research has been devoted to the
extraction of the transport coefficients by employing the hydrodynamics
approach and data fitting methods to the observables associated with the final
state particles Romatschke and Romatschke (2007); Ryu  _et al._ (2015). The
data-driven methods are currently getting wider attention in this field as the
physics of the QGP is entering a high-precision science era Bernhard  _et al._
(2019); Everett  _et al._ (2021); Heffernan  _et al._ (2023).

The class of hard probes such as jets, heavy quarks, etc offers another
possible way to characterize the properties of the QGP. These energetic
patrons penetrate and travel through the QGP and lose their energy due to
several interactions in the medium. Therefore, proper knowledge of the energy
loss of a fast-moving parton is essential for the quantitative description of
the jet quenching phenomenon. The theoretical modeling of the energy loss
mechanism of the energetic partons and its dependence on various observables
of collision experiments play a significant role in elucidating the underlying
physics. Several efforts have been made to understand the transport properties
of the QGP through the investigation of jets while considering both the
collisional and radiative processes of the energetic parton Mustafa (2005);
Sarkar  _et al._ (2018); Liu and Rapp (2020); Mazumder  _et al._ (2014);
Prakash  _et al._ (2021); Shaikh  _et al._ (2021); Dai  _et al._ (2022). The
heavy quarks are another promising candidates that can probe the evolution
history of the QGP as they are created in the very initial stages of the HIC
van Hees  _et al._ (2006); Das  _et al._ (2009, 2014); Cao  _et al._ (2019);
Song  _et al._ (2020); Kurian  _et al._ (2020); Sebastian  _et al._ (2023).
The motion of the energetic partons and their energy loss depend on the
properties of the medium. There have been some attempts to explore the parton
interaction with the non-equilibrium plasma or unstable medium Dumitru  _et
al._ (2008); Carrington  _et al._ (2015); Jamal  _et al._ (2021); Hauksson
_et al._ (2022); Carrington  _et al._ (2022); Ruggieri  _et al._ (2022);
Hauksson and Iancu (2023).

The chiral anomaly and parity-violating effects have recently attracted
substantial theoretical and experimental interest in the study of the QGP and
strong interaction. The interaction of the chiral fermions/quarks with the
nontrivial gluonic field produces asymmetry between left- and right-handed
chiral fermions that yield chiral imbalance Kharzeev  _et al._ (2016a).
Furthermore, the recent observations Acharya  _et al._ (2020); Adam  _et al._
(2019) and the associated studies Das  _et al._ (2017); Jiang  _et al._ (2022)
of the enhanced directed flow of heavy flavor mesons revealed the existence of
a strong magnetic field (for reviews on the strong magnetic field in collision
experiments, see Refs. Miransky and Shovkovy (2015); Huang (2016); Hattori and
Huang (2017); Kharzeev  _et al._ (2016b)) in the initial stages of the HICs.
This generated magnetic field breaks the rotational symmetry of the medium,
and induces anisotropy in the produced medium that, in turn, affects the
thermodynamic and transport properties of the QCD medium as the charged
particle motion is influenced by the field Karmakar  _et al._ (2019a); Hattori
_et al._ (2022); Das  _et al._ (2022); Dash  _et al._ (2020); Wang and
Shovkovy (2022); Kurian  _et al._ (2019); Ghosh  _et al._ (2019); Gowthama
_et al._ (2021); Rath and Patra (2020); Wang  _et al._ (2020); Das  _et al._
(2020); Ghosh and Kurian (2023). Refs. Fukushima  _et al._ (2016); Singh  _et
al._ (2020); Mazumder  _et al._ (2022) showed that the charm quark momentum
diffusion critically depends on the preferred direction and strength of the
magnetic field. The magnetic field, together with the chiral imbalance, leads
to novel phenomena such as Chiral Magnetic Effect (CME) and is one of the
active areas of ongoing studies in contemporary physics Fukushima  _et al._
(2008); Kharzeev and Son (2011); Sadofyev and Isachenkov (2011); Kharzeev  _et
al._ (2016b); An  _et al._ (2022). Both the magnetic field and chiral
imbalance are expected to influence the passage of partons through the medium,
as the momentum broadening and energy loss of energetic partons depend on the
properties of the medium. It is an exciting task to study the impact of chiral
asymmetry and magnetic field on the energy loss pattern of the moving parton
in the medium, and this sets the motivation for the present study.

A semi-classical approach is utilized to set up the formalism for energy loss
experienced by an energetic parton that incorporates its interactions with the
chromodynamic fields in the QCD medium. The motion of the test parton in
plasma is described with Wong’s equations by treating it as a classical
particle with ${SU(N_{c})}$ color charge. The change in the color field
configuration due to the passage of the parton is embedded through the
linearized Yang-Mills equations. We consider three different choices of
plasma, namely (i) isotropic, (ii) chiral asymmetric, and (iii) magnetized QCD
medium. While traversing the medium, the back-reaction exerted on the parton
medium is taken into account by analyzing the polarization effects of the
medium. We observe that the chiral asymmetry affects the parton energy loss
mechanism. Further, we show that the presence of the magnetic field suppresses
the energy loss. Another crucial finding is the magnetic field-induced
anisotropy in the parton energy loss, which indicates that the relative
orientation of the parton’s motion and the magnetic field’s direction strongly
influences the energy loss.

We organize the manuscript as follows. In section (II), we present the
formalism for the energy loss of an energetic parton moving in three different
choices of QCD medium. Section (III) is devoted to the results and
discussions. Finally, we summarize the analysis with an outlook in section
(IV).

Notations and conventions: In the present analysis, we consider
$c=k_{B}=\hbar=1$, $g_{\mu\nu}={\text{diag}}\,(1,-1,-1,-1)$, $N_{c}=3$, and
$N_{f}=2$. The quantity $q_{f}$ denotes the particle’s electric charge with
flavor $f$. A four-vector is defined as $X^{\mu}=(x^{0},|\bf{x}|)$ with the
component of the three-vector is described with the Latin indices $x^{i}$
where $i=(1,2,3)$.

##  II Formalism

A highly energetic test parton, while passing through the QGP medium, loses
energy due to its interactions with the color fields. The energy loss
experienced by the parton can be measured through the work of the retarding
forces acting on the parton in the medium from the induced chromo-electric
field that generated due to its motion. The dynamics of the test parton in
chromodynamic fields can be described with Wong’s equations Wong (1970) that
in the Lorentz covariant form given as,

$\displaystyle\frac{dX^{\mu}(\tau)}{d\tau}$ $\displaystyle=V^{\mu}(\tau),$ (1)
$\displaystyle\frac{dP^{\mu}(\tau)}{d\tau}$
$\displaystyle=g_{s}q^{a}(\tau)F^{\mu\nu}_{a}(X(\tau)){V}_{\nu}(\tau),$ (2)
$\displaystyle\frac{dq^{a}(\tau)}{d\tau}$
$\displaystyle=-g_{s}f^{abc}V_{\mu}(\tau)A^{\mu}_{b}(X(\tau))q_{c}(\tau),$ (3)

with $\tau$, $X^{\mu}(\tau)$, $V^{\mu}(\tau)$ and $P^{\mu}(\tau)$ as the
proper time, position, velocity, and momentum of the test parton. Here,
$g_{s}$ denotes the strong running coupling constant; $q_{a}$ is the parton
color charge; $F^{\mu\nu}$ represents chromodynamic field tensor; $A^{\mu}$ is
the gauge potential; $f^{abc}$ describes the structure constant of
${SU}(N_{c})$ group; and $a$ defines the color index with
$a=1,2,\dots,N_{c}^{2}-1$. We follow two assumptions to solve Wong’s
equations; first, we choose the gauge condition $V_{\mu}A^{\mu}_{a}(X)=0$,
which indicates the gauge potential vanishes on the particle’s trajectory
Jiang  _et al._ (2016); Carrington  _et al._ (2015). Secondly, we consider
that the energy of the moving parton is comparatively much larger than its
energy loss in the medium Jiang  _et al._ (2016); Chakraborty  _et al._
(2007). Next, considering $\mu=0$ component in Eq. (2) and replacing proper
time with time $t=\gamma\tau$, we get,

$\displaystyle\dfrac{d{\it E}}{dt}$ $\displaystyle=$ $\displaystyle
g_{s}q^{a}{{\bf v}}\cdot{{\textbf{E}}_{a}}(t,{\bf x}={\bf v}t),$ (4)

where ${\textbf{E}}_{a}(t,{\bf x}={\bf v}t)$ is the chromo-electric field
induced due to the motion of the parton with the energy, $E$ and velocity,
${\bf v}$. The energy loss can also be described in terms of the color current
${\bf j}_{a}$ generated by the energetic moving parton as,

$\displaystyle\frac{d{\it E}}{dt}$ $\displaystyle=$ $\displaystyle\int
d^{3}{\bf x}\,{{\textbf{E}}_{a}}(t,{\bf x})\cdot{\bf j}_{a}(t,{\bf x}),$ (5)

with ${\bf j}_{a}(t,{\bf x})=g_{s}q^{a}{{\bf v}}\delta^{(3)}({\bf x}-{\bf
v}t)$. The form of ${\textbf{E}}_{a}$ generated due to the test parton motion
in the medium can be obtained by solving the linearized Yang-Mills equation.
Employing the conventional way, i.e., Fourier transforming the linearized
differential equation to algebraic forms, the induced chromo-electric can be
obtained as Thoma and Gyulassy (1991); Chakraborty  _et al._ (2006),

$\displaystyle{\text{E}}^{i}_{a}(K)=i\omega\Delta^{ij}(K)j^{j}_{a}(K),$ (6)

where $K^{\mu}=({\omega,{\bf k}})$. The gluon propagator $\Delta^{ij}(K)$ and
the external current $j^{j}_{a}(K)$ take the following forms Yousuf Jamal and
Chandra (2019),

$\displaystyle\Delta^{ij}=[(|{\bf
k}|^{2}-\omega^{2})\delta^{ij}-k^{i}k^{j}+\Pi^{ij}(K)]^{-1},$ (7)
$\displaystyle j^{j}_{a}(K)=\frac{ig_{s}q^{a}v^{j}}{\omega-{\bf k}\cdot{\bf
v}+i0^{+}}.$ (8)

Here, $\Pi^{ij}(K)$ is the gluon self-energy tensor that captures the medium
effects. Using Eq. (7) and Eq. (8) in Eq. (6), one can obtain the induced
field, and that, in turn, gives the change of parton energy in the Fourier
space. Next, to get back to the coordinate space, we perform the inverse
Fourier transformation, which after averaging over color indices and
completing $\omega-$integration using the Residue theorem (as the integrand
has a pole at $\omega={\bf k}\cdot{\bf v}$) can be written as,

$\displaystyle\bigg{\langle}{\frac{d{E}}{dx}}\bigg{\rangle}=i\frac{1}{|{\bf
v}|}g_{s}^{2}C_{F}v^{i}v^{j}\int{\frac{d^{3}{\bf
k}}{(2\pi)^{3}}\omega\Delta^{ij}},$ (9)

where $C_{F}$ is the Casimir invariant of $SU(N_{c})$ and ${\omega={\bf
k}\cdot{\bf v}}$. Notably, the change of energy of the parton depends on its
velocity and the gluon propagator. The latter has a strong dependence on the
choice of the medium. Depending on the medium properties and initial
conditions, the energy change can be negative or positive. If parton loses
energy while interacting with the medium,
$-\big{\langle}{\frac{d{E}}{dx}}\big{\rangle}$ should be positive; otherwise,
if in some special cases, the test parton gains energy,
$-\big{\langle}{\frac{d{E}}{dx}}\big{\rangle}$ should be negative Chakraborty
_et al._ (2007); Jamal  _et al._ (2021). Next, we shall discuss the gluon
propagator (that depends of the gluon self-energy or the dielectric
permittivity of the medium) and the energy loss mechanism for three different
systems.

###  II.0.1 Moving parton in isotropic medium

For an isotropic medium, the gluon self-energy tensor can be decomposed in
terms of longitudinal projection operator
$B^{\text{ij}}=\frac{k^{i}k^{j}}{|{\bf k}|^{2}}$ and transverse projection
operator $A^{\text{ij}}=\delta^{\text{ij}}-\frac{k^{i}k^{j}}{|{\bf k}|^{2}}$
as,

$\displaystyle\Pi^{ij}(K)=A^{ij}~{}\Pi_{T}(K)+B^{ij}~{}\Pi_{L}(K),$ (10)

where $\Pi_{L}$ and $\Pi_{T}$ are the longitudinal and transverse form
factors, respectively. The $\Pi^{ij}(K)$ can be related to the induced current
in the medium via linear response theory. The current induced in the medium
can be quantified in terms of the deviation of the medium particle
distribution functions that can be obtained by solving the Boltzmann-Vlasov
transport equation. Hence, the form factors can be obtained by solving the
transport equation and employing linear response theory, and can be expressed
as Bellac (2011); Mustafa (2022); Carrington  _et al._ (2021),

$\displaystyle\Pi_{T}(K)={m_{D}^{2}}\frac{\omega^{2}}{2|{\bf
k}|^{2}}\bigg{[}1-\frac{K^{2}}{2\omega|{\bf k}|}\ln\left(\frac{\omega+|{\bf
k}|}{\omega-|{\bf k}|}\right)\bigg{]},$ (11)
$\displaystyle\Pi_{L}(K)=-m_{D}^{2}\frac{\omega^{2}}{k^{2}}\left[1-\frac{\omega}{2k}\ln\left(\frac{\omega+k}{\omega-k}\right)\right],$
(12)

where $K^{2}=\omega^{2}-|{\bf k}|^{2}$ and
$m_{D}^{2}=(N_{f}+2N_{c})\frac{g_{s}^{2}T^{2}}{6}$ is the Debye screening
mass. Employing Eq. (10) in Eq. (7), the gluon propagator for an isotropic
medium can be written as,

$\displaystyle\Delta^{ij}=\frac{1}{C_{T}}~{}A^{ij}+\frac{1}{C_{L}}~{}B^{ij},$
(13)

with $C_{T}=-\omega^{2}+|{\bf k}|^{2}+\Pi_{T}$ and
$C_{L}=-\omega^{2}+\Pi_{L}$. By substituting Eq. (13) in Eq. (9), the loss of
the moving parton in an isotropic medium can be obtained as,

$\displaystyle-\bigg{\langle}{\frac{d{E}}{dx}}$
$\displaystyle\bigg{\rangle}=\frac{\alpha_{s}C_{F}}{2\pi^{2}|{\bf v}|}\int
d^{3}{\bf k}\frac{\omega}{|{\bf
k}|^{2}}\bigg{\\{}\omega^{2}\text{Im}\Big{(}-\omega^{2}+\Pi_{L}\Big{)}^{-1}$
$\displaystyle+\left(|{\bf k}|^{2}|{\bf
v}|^{2}-\omega^{2}\right)\text{Im}\Big{(}{-\omega^{2}+k^{2}+\Pi_{T}}\Big{)}^{-1}\bigg{\\}}_{\omega={\bf
k}\cdot{\bf v}}.$ (14)

The medium effects are incorporated through the form factors of the gluon
self-energy and coupling constant. We can rewrite Eq.(14) (as estimated in
other parallel studies Thoma and Gyulassy (1991); Jamal and Mohanty (2021);
Prakash and Jamal (2023)) in terms of the longitudinal $\epsilon_{L}(K)$ and
transverse $\epsilon_{T}(K)$ components of the dielectric permittivity
$\epsilon^{ij}(K)$ of the medium as,

$\displaystyle-\bigg{\langle}{\frac{d{E}}{dx}}\bigg{\rangle}$
$\displaystyle=\frac{\alpha_{s}C_{F}}{2\pi^{2}|{\bf v}|}\int d^{3}{\bf
k}\frac{\omega}{|{\bf
k}|^{2}}\bigg{\\{}\text{Im}\Big{(}{\epsilon_{L}(K)}\Big{)}^{-1}$
$\displaystyle+\left(|{\bf k}|^{2}{|{\bf
v}|}^{2}-\omega^{2}\right)\text{Im}\Big{(}{\omega^{2}\epsilon_{T}(K)-|{\bf
k}|^{2}}\Big{)}^{-1}\bigg{\\}}_{\omega={\bf k}\cdot{\bf v}},$

where the relation between gluon self-energy and dielectric permittivity is
taken as,

$\displaystyle\epsilon^{ij}(K)=\delta^{ij}-\frac{1}{\omega^{2}}\Pi^{\text{ij}}(K).$
(16)

It is important to emphasize that the Eq. (14) and Eq. (LABEL:eq:de1) describe
the energy loss due to the polarization effects of the medium, $i.e$, the
change in parton energy due to its interaction with collective excitations of
the medium. For the numerical calculation, we took the upper limit of the
integration $k_{\text{max}}\sim E$, $i.e.$, the initial energy of the parton.

###  II.0.2 Moving parton in chiral imbalance medium

The asymmetry between right-handed and left-handed fermions can be quantified
in terms of the chiral chemical potential, $\mu_{5}\equiv\mu_{R}-\mu_{L}$. The
chiral plasma with a finite $\mu_{5}$ can be described within the kinetic
theory with Berry curvature terms Stephanov and Yin (2012); Son and Yamamoto
(2012); Chen  _et al._ (2016); Akamatsu and Yamamoto (2013). In the small
gauge field $A^{\mu}$, using the linear response theory, we can express
$\Pi^{\mu\nu}$ in terms of induced current density in the chiral medium, which
can be obtained from the linearized transport equation with Berry curvature
correction. The parity-violating terms are entering through the Berry
curvature terms, and $\Pi^{ij}$ with finite $\mu_{5}$ can be described as
Akamatsu and Yamamoto (2013),

$\displaystyle\Pi^{ij}(K)=\Pi^{ij}_{+}(K)+\Pi^{ij}_{-}(K),$ (17)

where $\Pi^{ij}_{+}(K)$ and $\Pi^{ij}_{-}(K)$ are parity even and parity odd
parts of self-energy respectively, and can be defined as,

$\displaystyle\Pi^{ij}_{+}(K)=\frac{m_{D}^{2}\omega}{4\pi}\int d{\bf
u}\frac{u^{i}u^{j}}{U\cdot K+i\epsilon},$ (18)
$\displaystyle\Pi^{ij}_{-}(K)=\frac{\mu_{5}g_{s}^{2}}{4\pi^{2}}i\epsilon^{ijk}k^{k}\bigg{(}1-\frac{\omega^{2}}{|{\bf
k}|^{2}}\bigg{)}\bigg{[}1-\frac{\omega}{2|{\bf k}|}\ln\left(\frac{\omega+|{\bf
k}|}{\omega-|{\bf k}|}\right)\bigg{]}.$ (19)

Here, $U^{\mu}=(1,\bf{u})$ with $\bf{u}=\frac{\bf{k}}{|{\bf k}|}$, the indices
$i,j,k$ represent the spatial components, and the Debye mass in the chiral
medium takes the form as,

$\displaystyle
m_{D}^{2}=(N_{f}+2N_{c})\frac{g_{s}^{2}T^{2}}{6}+N_{f}\frac{g_{s}^{2}}{2\pi^{2}}(\mu_{R}^{2}+\mu_{L}^{2}).$
(20)

Due to the parity-violating term, an additional projector operator is required
for the tensorial decomposition of $\Pi^{ij}$ in the chiral medium as compared
to the isotropic case. At finite $\mu_{5}$, gluon propagator can be written as
Akamatsu and Yamamoto (2013),

$\displaystyle\Delta^{ij}=\frac{C_{T}}{C_{T}^{2}-C_{A}^{2}}~{}A^{ij}+\frac{1}{C_{L}}~{}B^{ij}-\frac{C_{A}}{C_{T}^{2}-C_{A}^{2}}~{}C^{ij},$
(21)

where $C^{ij}=i\epsilon^{ijk}\frac{k^{k}}{|{\bf k}|}$ is the anti-symmetric
tensor and $C_{A}$ that denotes the partity violating component of the self-
energy takes the form as,

$\displaystyle C_{A}\equiv\Pi_{A}=\mu_{5}\frac{g_{s}^{2}|{\bf
k}|}{4\pi^{2}}\bigg{(}1-\frac{\omega^{2}}{|{\bf
k}|^{2}}\bigg{)}\bigg{[}1-\frac{\omega}{2|{\bf k}|}\ln\left(\frac{\omega+|{\bf
k}|}{\omega-|{\bf k}|}\right)\bigg{]}.$ (22)

Notably, $C_{A}$ is proportional to chiral chemical potential and in the limit
$\mu_{5}=0$, Eq. (21) reduce back to the result of isotropic case as described
in Eq. (13). Energy loss of the parton can be influenced by the chiral
asymmetry in the medium and can be quantified in terms $\Delta^{ij}$.
Employing Eq. (21) in Eq. (9), the energy loss of the parton in a chiral
imbalance plasma can be written as,

$\displaystyle\bigg{\langle}{\frac{d{E}}{dx}}\bigg{\rangle}=$
$\displaystyle\frac{ig^{2}_{s}C_{F}}{{|{\bf v}|}}\int\frac{d^{3}{\bf
k}}{(2\pi)^{3}}\frac{\omega}{|{\bf
k}|^{2}}\bigg{\\{}\omega^{2}\frac{1}{C_{L}}+\Big{(}|{\bf k}|^{2}{|{\bf
v}|}^{2}-\omega^{2}\Big{)}$
$\displaystyle\times\frac{C_{T}}{C_{T}^{2}-C_{A}^{2}}-i|{\bf k}|~{}{\bf
v}\cdot({\bf v}\times{\bf
k})\frac{C_{A}}{C_{T}^{2}-C_{A}^{2}}\bigg{\\}}_{\omega={\bf k}\cdot{\bf v}}.$
(23)

By substituting the forms of $C_{L}$, $C_{T}$, and $C_{A}$ in Eq. (23) we
obtain,

$\displaystyle-\bigg{\langle}{\frac{d{E}}{dx}}\bigg{\rangle}=$
$\displaystyle\frac{\alpha_{s}C_{F}}{2\pi^{2}{|{\bf v}|}}\int d^{3}{\bf
k}\frac{\omega}{|{\bf
k}|^{2}}\bigg{\\{}\omega^{2}\text{Im}\Big{(}-\omega^{2}+\Pi_{L}\Big{)}^{-1}$
$\displaystyle+\left(|{\bf k}|^{2}{|{\bf
v}|}^{2}-\omega^{2}\right)\text{Im}\Big{(}\frac{-\omega^{2}+|{\bf
k}|^{2}+\Pi_{T}}{(-\omega^{2}+|{\bf
k}|^{2}+\Pi_{T})^{2}-\Pi_{A}^{2}}\Big{)}\bigg{\\}}_{\omega={\bf k}\cdot{\bf
v}}$ (24)

The energy loss term that originates from contraction of the anti-symmetric
projection operator $C^{ij}$ given in Eq. (21) vanishes, however, the chiral
effects still enter through the Debye mass and partity violating scalar
component of the self-energy $\Pi_{A}$. In the case of vanishing chiral
chemical potential, Eq. (24) reduces back to the form of test parton energy
loss expression for an isotropic case as defined in Eq. (14).

###  II.0.3 Moving partons in magnetized QCD medium

Figure 1:  One loop gluon self-energy. The double line denotes the modified
quark propagator in the presence of a strong magnetic field

The magnetic field is a source of anisotropy in the system, which will be
reflected in the medium’s QCD thermodynamics and transport processes. The
motion of the test parton will be affected by the strength ${\bf B}$ and
direction of the magnetic field ${\bf n}$ in the medium. In the magnetized
medium, the tensor basis for a symmetric second-rank tensor will be different
from that of an isotropic medium due to the inclusion of the additional
magnetic field vector. Using vectors $k^{i}$ and $n^{i}$ with $n^{2}=1$, we
can construct the spacial part of the gluon self-energy. In the analysis, we
consider various cases with different directions of the test parton velocity
${\bf v}$ to analyze the impact of the relative orientation of ${\bf v}$ and
${\bf n}$ on the parton energy loss. In the presence of a magnetic field,
$\Pi^{ij}$ can be decomposed in terms of basis tensors which depend on $k^{i}$
and $n^{i}$ as,

$\displaystyle\Pi^{ij}=a\,N^{ij}+b\,B^{ij}+c\,R^{ij}+d\,Q^{ij},$ (25)

where the projection operators can be defined as follows,

$\displaystyle
N^{ij}=-\frac{\hat{k}^{i}\tilde{n}^{j}+\hat{k}^{j}\tilde{n}^{i}}{\sqrt{\tilde{n}^{2}}},$
(26) $\displaystyle B^{ij}=\frac{k^{i}k^{j}}{|{\bf k}|^{2}},$ (27)
$\displaystyle R^{ij}=-\delta^{ij}+\frac{k^{i}k^{j}}{|{\bf
k}|^{2}}-\frac{\tilde{n}^{i}\tilde{n}^{j}}{\tilde{n}^{2}},$ (28)
$\displaystyle Q^{ij}=\frac{\tilde{n}^{i}\tilde{n}^{j}}{\tilde{n}^{2}}.$ (29)

Figure 2:  Temperature dependence of the energy loss of a fast-moving parton
in a chiral imbalance medium (left panel). The ratio of energy loss in a
chiral medium to that of an isotropic medium is plotted as a function of
temperature (right panel).

In the presence of a magnetic field, defining $\tilde{n}^{i}=A^{ij}n^{j}$
vector to describe the tensor basis is convenient. Here, $a$, $b$, $c$, and
$d$ are Lorentz-invariant form factors and can be obtained from the following
relations,

$\displaystyle a=\frac{1}{2}N^{ij}\Pi^{ij},$ (30) $\displaystyle
b=B^{ij}\Pi^{ij},$ (31) $\displaystyle c=R^{ij}\Pi^{ij},$ (32) $\displaystyle
d=Q^{ij}\Pi^{ij}.$ (33)

The form factors can be calculated from the one-loop gluon self-energy. The
fermion loop is shown in Fig. 1, which is affected by the magnetic field. We
are considering the strong magnetic field limit with the energy hierarchy
$T<\sqrt{|q_{f}B|}$. The form factors containing the gluon loop, ghost loop,
and magnetic field modified quark loop contributions are given as Karmakar
_et al._ (2019b),

$\displaystyle a=$
$\displaystyle\sum_{f}\frac{g_{s}^{2}q_{f}B}{4\pi^{2}}e^{{-k_{\perp}^{2}}/{2q_{f}B}}~{}\frac{\omega\,k_{z}}{\omega^{2}-k_{z}^{2}}\sqrt{\frac{k_{\perp}^{2}K^{4}}{\omega^{2}\,|{\bf
k}|^{4}}},$ (34) $\displaystyle b=$
$\displaystyle\frac{N_{c}\,g_{s}^{2}T^{2}}{3}\bigg{(}\frac{K^{4}}{\omega^{2}\,|{\bf
k}|^{2}}\bigg{)}\left[1-\mathcal{T}_{K}(\omega,|{\bf k}|)\right]$
$\displaystyle-\sum_{f}\frac{g_{s}^{2}q_{f}B}{4\pi^{2}}\bigg{(}\frac{K^{4}}{\omega^{2}\,|{\bf
k}|^{2}}\bigg{)}e^{{-k_{\perp}^{2}}/{2q_{f}B}}~{}\frac{k_{z}^{2}}{\omega^{2}-k_{z}^{2}},$
(35) $\displaystyle c=$
$\displaystyle\frac{N_{c}\,g_{s}^{2}T^{2}}{3}\frac{1}{2}\left[\frac{\omega^{2}}{|{\bf
k}|^{2}}-\frac{K^{2}}{|{\bf k}|^{2}}\mathcal{T}_{K}(\omega,|{\bf k}|)\right],$
(36) $\displaystyle d=$
$\displaystyle\frac{N_{c}\,g_{s}^{2}T^{2}}{3}\frac{1}{2}\left[\frac{\omega^{2}}{|{\bf
k}|^{2}}-\frac{K^{2}}{|{\bf k}|^{2}}\mathcal{T}_{K}(\omega,|{\bf k}|)\right]$
$\displaystyle+\sum_{f}\frac{g_{s}^{2}q_{f}B}{4\pi^{2}}\bigg{(}\frac{K^{4}}{\omega^{2}\,|{\bf
k}|^{2}}\bigg{)}e^{{-k_{\perp}^{2}}/{2q_{f}B}}~{}\frac{k_{z}^{2}}{\omega^{2}-k_{z}^{2}},$
(37)

where the angular integral reads as $\mathcal{T}_{K}(\omega,|{\bf
k}|)=\int\frac{d\Omega}{4\pi}\frac{\omega}{\omega-\mathbb{k}\cdot\hat{p}}$
with $\hat{p}$ is unit vector along $\bf{p}$, $i.e.$, $\hat{p}=\bf{p}/|\bf p|$
and $k_{\perp}=\sqrt{k_{x}^{2}+k_{y}^{2}}$. The general structure of the gluon
effective propagator in the magnetized medium can be written as,

$\displaystyle\Delta^{ij}=C_{1}\,B^{ij}+C_{2}\,R^{ij}+C_{3}\,Q^{ij}+C_{4}\,N^{ij},$
(38)

where $C_{1,..4}$ are related to the form factors as,

$\displaystyle C_{1}=\frac{K^{2}-d}{(K^{2}-b)(K^{2}-d)-a^{2}},$ $\displaystyle
C_{2}=\frac{1}{K^{2}-c},$ $\displaystyle
C_{3}=\frac{K^{2}-b}{(K^{2}-b)(K^{2}-b)-a^{2}},$ $\displaystyle
C_{4}=\frac{a}{(K^{2}-b)(K^{2}-d)-a^{2}}.$ (39)

Employing Eq. (38) in Eq. (9), we can estimate the parton energy change in the
magnetized medium as,

$\displaystyle\bigg{\langle}{\frac{d{E}}{dx}}\bigg{\rangle}=$
$\displaystyle\frac{ig_{s}C_{F}}{{|{\bf v}|}}\int\frac{d^{3}{\bf
k}}{(2\pi)^{3}}{\omega}\Bigg{\\{}C_{1}\frac{\omega^{2}}{|{\bf
k}|^{2}}-C_{2}\bigg{(}{|{\bf v}|}^{2}-\frac{\omega^{2}}{|{\bf k}|^{2}}$
$\displaystyle+\frac{1}{\tilde{n}^{2}}({\bf v}\cdot{\bf
n})^{2}+2\frac{k_{z}\omega}{\tilde{n}^{2}\,|{\bf k}|^{2}}({\bf v}\cdot{\bf
n})+\frac{k_{z}^{2}\,\omega^{2}}{|{\bf k}|^{4}\tilde{n}^{2}}\bigg{)}$
$\displaystyle+\frac{C_{3}}{\tilde{n}^{2}}\bigg{(}({\bf v}\cdot{\bf
n})^{2}+2\frac{k_{z}\omega}{|{\bf k}|^{2}}({\bf v}\cdot{\bf
n})+\frac{k_{z}^{2}\,\omega^{2}}{|{\bf k}|^{4}}\bigg{)}$
$\displaystyle-\frac{2C_{4}}{\sqrt{\tilde{n}^{2}}}\frac{\omega}{|{\bf
k}|}\bigg{(}({\bf v}\cdot{\bf n})+\frac{k_{z}\,\omega}{|{\bf
k}|^{2}}\bigg{)}\Bigg{\\}}.$ (40)

Here, we define $({\bf k}\cdot{\bf n})=k_{z}$ as the magnetic field is fixed
along the $\hat{z}-$direction in the present analysis. The term $({\bf
v}\cdot{\bf n})$ describes the relative orientation of the parton with respect
to the direction of the magnetic field and has a direct influence on the
energy loss. This can give rise to anisotropy in the energy loss of the parton
in the magnetized medium. In addition to that, the magnetic field effects are
entering through the form factors as described in Eqs. (34)-(37). For the
perpendicular case, where the parton is moving transverse to the direction of
the magnetic field, $i.e.$, $({\bf v}\cdot{\bf n})=0$, Eq. (40) can be further
simplified as,

$\displaystyle\bigg{\langle}{\frac{d{E}}{dx}}\bigg{\rangle}=$
$\displaystyle\frac{ig_{s}C_{F}}{{|{\bf v}|}}\int\frac{d^{3}{\bf
k}}{(2\pi)^{3}}{\omega}\Bigg{\\{}C_{1}\frac{\omega^{2}}{|{\bf
k}|^{2}}-C_{2}\Big{(}{|{\bf v}|}^{2}-\frac{\omega^{2}}{|{\bf k}|^{2}}\Big{)}$
$\displaystyle+(C_{3}-C_{2})\frac{k_{z}^{2}\,\omega^{2}}{|{\bf
k}|^{4}\tilde{n}^{2}}-\frac{2C_{4}}{\sqrt{\tilde{n}^{2}}}\frac{k_{z}\,\omega^{2}}{\,|{\bf
k}|^{3}}\Bigg{\\}}.$ (41)

Next, we shall discuss the results obtained from various sections.

##  III Results

Our primary findings are the energy loss of a test parton in the presence of
chiral imbalance and a strong magnetic field in the QGP medium. The impact of
chiral asymmetry and medium temperature over the energy loss of the fast-
moving parton within the massless limit is depicted in Fig. 2 (left panel). In
this figure, the imbalance between right-handed and left-handed fermions is
measured using a chiral chemical potential $\mu_{5}=\mu_{R}-\mu_{L}$. For the
quantitative estimation, we consider
$\mu_{L}^{2}+\mu^{2}_{R}=\frac{1}{2}(\mu_{5}^{2}+\mu_{V}^{2})$ with
$\mu_{V}=\mu_{R}+\mu_{L}$. The induced current due to the passage of the
parton depends on the chirality of the medium and is reflected in the energy
loss mechanism as described in Eq. (24). The passage of the energetic parton
is influenced by the chiral effects of the medium that enter through the
medium’s screening mass and parity-violating component of the self-energy
$\Pi_{A}$ as defined in Eq. (20) and Eq. (22), respectively. Similar to the
case of an isotropic medium, the parton energy loss in the chiral imbalance
medium increases with an increase in the temperature of the medium. We observe
an increment in the energy loss in the presence of the chiral asymmetry
compared to the isotropic QGP medium with vanishing chemical potential,, which
has a strong dependence on the values of $\mu_{R}$ and $\mu_{L}$. The ratio of
energy loss of a massless parton in the chiral imbalance medium to the
isotropic medium is shown in Fig. 2 (right panel). It is observed that the
chiral effects of the medium on the parton energy loss goes down with the
increase in temperature, whereas the impact is more visible in the low-
temperature regime.

The energy loss of a parton in a left- and right-hand dominated medium at a
fixed temperature is plotted in Fig. 3. The region above and below the
diagonal line $\mu_{R}=\mu_{L}$ represent the left- and right-hand fermions
dominated regions, respectively. Notably, the energy loss is symmetric at a
fixed temperature in both regions. This is because the parton energy loss in
chiral medium remains as a parity conserving term as
$-\big{\langle}{\frac{d{E}}{dx}}\big{\rangle}$ depends on $\mu_{5}^{2}$ as
described in Eq. (24). It is seen that the parton energy loss increases with
the chemical potential. The curved lines that represent constant energy loss
contours indicate that $-\big{\langle}{\frac{d{E}}{dx}}\big{\rangle}$ in a
chiral medium is not only depending on the value of $\mu_{R}$ and $\mu_{L}$
but also the difference between them.

Figure 3:  Energy loss of a massless energetic parton in the chiral medium at
$T=200$ MeV. Curved lines denote constant energy loss contours and
$\big{\langle}{\frac{d{E}}{dx}}\big{\rangle}$ is described with the unit
[GeV/fm]. Figure 4: Momentum dependence of the parton energy loss in a
magnetized QCD medium for various choices of angle between parton velocity and
magnetic field in the medium ($\theta_{k}$). The mass of fast moving parton is
chosen as $m=1.25$ GeV.

In Fig. 4, we illustrated the momentum behavior of anisotropy generated due to
the strong magnetic field to the energy loss mechanism. It is more relevant to
the case of heavy quarks than the massless test parton, as they expect to
witness a strong magnetic field in the initial stages of the collision. Here,
we target the charm quark energy loss case with $m=1.25$ GeV, $T=250$ MeV, and
$eB=0.3$ GeV2 to ensure the strong field approximation. It is crucial to
underline that the heavy quark does not undergo Landau-level dynamics owing to
its large mass and initial momentum in the medium. However, the QCD medium
properties and hence, the polarization effects will change with the inclusion
of the magnetic field. They can indirectly affect charm quark energy loss in
the magnetized medium. We observe that the energy loss of the charm quark
traveling through a magnetized medium is sensitive to the magnetic field in
the plasma. Notably, the energy loss strongly depends on the relative
direction of the charm quark’s velocity and magnetic field in the medium. The
impact is more pronounced at the high momentum regime. It is seen that the
energy loss is more when the motion is transverse to the direction of the
magnetic field.

The dependence of the strength of the magnetic field and temperature on the
energy loss of a massless parton is shown in Fig. 5. We fix the relative
orientation of parton motion and magnetic field by choosing
$\theta_{k}=\pi/2$. For an energetic parton, the magnetic field suppresses the
energy loss compared to that in the thermal medium due to the dimensionally
reduced Landau level motion of the charged fermions in the plasma. The impact
of the strength of the magnetic field is more pronounced in the low-
temperature regimes. As the temperature rises, the suppression of parton
energy loss that arises from the Landau kinematics is not sensitive to the
strength of the magnetic field. The observation holds true for other choices
of $\theta_{k}$ as well.

Figure 5: Energy loss (in [GeV/fm]) of an energetic massless parton as a
function of temperature and magnetic field.

##  IV Summary and Outlook

In conclusion, we have studied the energy loss of a fast-moving parton for
three scenarios within the QGP medium, viz., the isotropic, chiral asymmetric,
and strongly magnetized QGP medium. To do so, we employed Wong’s equations
combined with linearized Yang-Mills equations that describe the back-reaction
exerted on the energetic parton by the medium while traversing through it. In
the isotropic case, it is found that the energy loss increases with the
temperature of the medium. It further increases with the momentum of the test
parton. Similar trends are found for the chiral asymmetric medium as well.
Moreover, the energy loss turned out to be sensitive to the chiral chemical
potential, especially at the low-temperature regimes. We further investigated
the heavy parton energy loss in a magnetized QCD medium. It is noticed that
the magnetic field induces anisotropy in the medium that, in turn, suppresses
the energy loss. Furthermore, the energy loss is found to have a strong
dependence on the relative orientation of the parton’s velocity and the
magnetic field. It is seen parton loses more energy when it travels
transversely to the direction of the magnetic field compared to other
directions.

The current focus is on the parton energy loss due to the polarization effects
of the medium. To estimate the complete energy loss of the test parton in the
medium, this should be combined with the contribution which originates from
the hard elastic scattering of the parton with the constituents of the medium
that may have a strong dependence on the magnetic field in the medium. Another
interesting direction is to set up the formalism by considering the
fluctuations in a magnetized QCD medium and exploring the possibility of
energy gain of the parton. These are interesting aspects to explore shortly.

## Acknowledgments

We thank Mayank Singh for helpful discussions and feedback. R.G. acknowledges
financial support from the Department of Atomic Energy, India. M.Y.J would
like to acknowledge the SERB-NPDF (National postdoctoral fellow) fellowship
with File No. PDF/2022/001551. M.K acknowledges the support from Special
Postdoctoral Researchers Program of RIKEN.

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# Partially Shared Semi-supervised Deep Matrix Factorization with Multi-view
Data

Haonan Huang2, Naiyao Liang23, Wei Yan2, Zuyuan Yang25, Weijun Sun24
{mrhaonan<EMAIL_ADDRESS><EMAIL_ADDRESS>{yangzuyuan,
<EMAIL_ADDRESS>2Guangdong Key Laboratory of IoT Information Technology,
Guangdong University of Technology, Guangzhou, China 3Key Laboratory of
iDetection and Manufacturing-IoT, Ministry of Education, Guangzhou, China
4Guangdong-Hong Kong-Macao Joint Laboratory for Smart Discrete Manufacturing,
Guangzhou, China 5Author to whom any correspondence should be addressed.

###### Abstract

Since many real-world data can be described from multiple views, multi-view
learning has attracted considerable attention. Various methods have been
proposed and successfully applied to multi-view learning, typically based on
matrix factorization models. Recently, it is extended to the deep structure to
exploit the hierarchical information of multi-view data, but the view-specific
features and the label information are seldom considered. To address these
concerns, we present a partially shared semi-supervised deep matrix
factorization model (PSDMF). By integrating the partially shared deep
decomposition structure, graph regularization and the semi-supervised
regression model, PSDMF can learn a compact and discriminative representation
through eliminating the effects of uncorrelated information. In addition, we
develop an efficient iterative updating algorithm for PSDMF. Extensive
experiments on five benchmark datasets demonstrate that PSDMF can achieve
better performance than the state-of-the-art multi-view learning approaches.
The MATLAB source code is available at
https://github.com/libertyhhn/PartiallySharedDMF.

###### Index Terms:

Multi-view learning; Deep matrix factorization; Semi-supervised learning;
Partially shared structure.

##  I Introduction

In practical applications, real-world data can be described from different
views, that is the so called multi-view data. For instance, an image can be
described by several characteristics, e.g., shape, color, texture and so on.
Because multi-view representation learning can exploit the implicit-dependent
structure of multiple views and improve the performance of the learning tasks.
In recent years, multi-view representation learning has attracted increasing
research attention in machine learning [1, 2, 3].

Over the past decade, lots of researches on multi-view learning have emerged.
In particular, Non-negative Matrix Factorization (NMF) [4, 5] as one of the
most popular high-dimensional data processing algorithms, have been widely
used for clustering multi-view data [7, 6]. NMF-based multi-view clustering
methods have shown to generate superior clustering results which are easy to
interpret [8, 9]. Considering the relations between view-specific
(uncorrelated) and common information (correlated), partially shared NMF-based
multi-view learning approaches are proposed [10, 11].

Figure 1: The general framework of the Deep Multi-View Clustering. Data of the
same class (_shape_) will become more compact and generate more discriminative
representation as the number of decomposition layers increases.

Nevertheless, most existing methods are single layer structures, which are
hard to extract hierarchical structural information [12] of multi-view data.
With the development of deep learning, Wang et al. proposed DCCA to extract
the hidden hierarchical information in 2-view data [13]. Zhao et al. [14]
extended single-view deep matrix factorization [15] into multi-view and
proposed graph regularized deep multi-view clustering, which can eliminate the
interference in multi-view data. Huang et al. [16] designed a novel robust
deep multi-view clustering model to learn the hierarchical semantics without
hyperparameters. For clarity, the general deep matrix factorization framework
for multi-view data as illustrated in Fig. 1. However, most existing deep
matrix factorization multi-view methods are merely considering the common
information and ignoring the effect of view-specific information in each
individual view. Besides, they are formulated as an unsupervised learning
problem and inapplicable when partially labeled data is available. Actually,
researchers have found that integrating such label information can produce a
considerable improvement in learning performance [19, 18, 17].

To this end, we propose a novel deep multi-view clustering methods called
Partially Shared Semi-supervised Deep Matrix Factorization (PSDMF). In our
method, correlated and uncorrelated features of multi-view data are both
considered by partially shared approaches, in which the latent representation
of each view is divided into the common part and the view-specific part. A
robust sparse regression term with $L_{2,1}$ norm is adopted to integrate the
partial label information of the labeled data. Besides, to respect the
intrinsic geometric relationship and avoid the parameters problem in different
views, we apply graph regularization and auto-weighted strategy to PSDMF. An
efficient iterative updating algorithm with pre-training scheme is designed
for PSDMF. We summarize our major contributions as follows:

  * •

This paper proposes an semi-supervised deep multi-view clustering model to
improve the performance of traditional unsupervised deep matrix factorization
methods by using label regression learning approach.

  * •

To respect the common information and view-specific features of multi-view
data, we propose a partially shared deep matrix factorization method to
jointly exploit two kinds of information and learn a comprehensive final layer
representation of multi-view data.

  * •

Local invariant graph regularization and the auto-weighted strategy are
introduced to preserve the intrinsic geometric structure in each view and
further boost the quality of output representation.

Figure 2: Illustration of the work flow of the proposed Partially Shared Semi-
supervised Deep Matrix Factorization.

The rest of this paper is organized as follows. In Section II, we give a brief
review of related works and describe the proposed PSDMF. Section III present
an efficient algorithm to solve the optimization problem. In Section IV, we
report the experimental results on five real-world datasets. Finally, Section
V concludes this paper. Table. I summarizes the general notations in this
article for reader’s convenience.

TABLE I: Notations. Notation | Description
---|---
$P$ | The number of views
$m$ | The number of layers
$k$ | The layer sizes
$N$ | The number of samples
$N_{l}$ | The number of labeled samples
$N_{u}$ | The number of unlabeled samples
$K_{s}$ | The dimension of view-specific encoding matrix
$K_{c}$ | The dimension of shared encoding matrix
$\mathbf{X}^{p}$ | The data matrix of the $p$-th view
$\mathbf{U}_{i}^{p}$ | The hidden matrix in the $i$-th layer of $p$-th view
$\mathbf{V}$ | The partially shared factor latent representation matrix
$\mathbf{V}_{m}^{p}$ | The $m$-th layer partially shared factor of the $p$-th view
$\mathbf{W}$ | The regression weight matrix

##  II Methodology

###  II-A Overview of Deep Matrix Factorization

Semi-NMF is not only a useful data dimensionality reduction technique but also
beneficial to data clustering [20]. Motivated by deep neural network
structures, Trigeorgis et al. [15] proposed a multi-layer structure Semi-NMF
called Deep Semi-NMF to exploit data complex hierarchical information with
implicit lower-level hidden attributes. The Deep Semi-NMF decomposes dataset
$\mathbf{X}$ hierarchically, and the process can be formulated as:

$\displaystyle\mathbf{X}$
$\displaystyle\approx\mathbf{U}_{1}\mathbf{V}_{1}^{+}$ (1)
$\displaystyle\mathbf{X}$
$\displaystyle\approx\mathbf{U}_{1}\mathbf{U}_{2}\mathbf{V}_{2}^{+}$
$\displaystyle\vdots$ $\displaystyle\mathbf{X}$
$\displaystyle\approx\mathbf{U}_{1}\ldots\mathbf{U}_{m}\mathbf{V}_{m}^{+}$

where $m$ is the numbers of layers, $\mathbf{U}_{i}$ is the $i$-th layer
hidden matrix, $\mathbf{V}_{i}^{+}$ denotes the $i$-th layer low-dimensional
representation matrix and we use the notation $+$ to state that a matrix
contains only non-negative elements. The loss function of Deep Semi-NMF is:

$\min_{\mathbf{U}_{i},\mathbf{V}_{m}}\left\|\mathbf{X}-\mathbf{U}_{1}\cdot\cdot\cdot\mathbf{U}_{m}\mathbf{V}_{m}\right\|_{F}^{2}\text{
s.t. }\mathbf{V}_{m}\geq 0.$ (2)

The Semi-NMF and Deep Semi-NMF mentioned above can be regarded as single view
algorithms. For multi-view data, Zhao et al. [14] presented a new Deep Semi-
NMF framework with graph regularization (DMVC) which can eliminate the
negative interference in the multi-source data and obtain an effective
consensus representation in the final layer. Given data $\mathbf{X}$
consisting of $P$ views, denoted by
$\mathbf{X}=\\{\mathbf{X}^{1},\mathbf{X}^{2},...,\mathbf{X}^{P}\\}$, the loss
function of DMVC is:

$\displaystyle\min_{\mathbf{Z}_{i}^{p},\mathbf{H}_{i}^{p},\mathbf{H}_{m},\alpha^{p}}$
$\displaystyle\sum_{p=1}^{P}\left(\alpha^{p}\right)^{r}\left(\left\|\mathbf{X}^{p}-\mathbf{U}_{1}^{p}\mathbf{U}_{2}^{p}\ldots\mathbf{U}_{m}^{p}\mathbf{V}_{m}\right\|_{\mathrm{F}}^{2}\right.$
(3)
$\displaystyle\left.\quad+\beta\operatorname{tr}\left(\mathbf{V}_{m}\mathbf{L}^{p}\mathbf{V}_{m}^{T}\right)\right)$
s.t. $\displaystyle\mathbf{V}_{i}^{p}\geq 0,\mathbf{V}_{m}\geq
0,\sum_{p=1}^{P}\alpha^{p}=1,\alpha^{p}\geq 0$

where $\alpha^{p}$ denotes the weighting coefficient of the $p$-th view and
$r$ is a important hyperparameter to control the weights distribution.
$\mathbf{L}^{p}$ is the graph Laplacian of the graph for view $p$ and $\beta$
is used to adjust the contribution of the graph constrains. The details of how
to construct a graph matrix will be discussed in the next subsection. However,
as an unsupervised method, DMVC cannot make use of partial prior knowledge of
data (e.g. labels). Besides, DMVC only concerns common representation of
multi-view data and ignores the view-specific features. In this paper, we
propose a novel semi-supervised deep matrix factorization model to address
such challenging problems.

###  II-B Partially Shared Semi-supervised Deep Matrix Factorization

To make full use of priori knowledge and learn low-dimensional factors with
powerful discrimination, motivated by recently proposed label regression
learning technique [10, 17], we combine the following $L_{2,1}$-norm
regularized regression in our model:

$\min\left\|\mathbf{W}^{T}\mathbf{V}_{l}-\mathbf{Y}\right\|_{F}^{2}+\gamma\|\mathbf{W}\|_{2,1}.$
(4)

where $\mathbf{V}_{l}\in\mathbb{R}^{K\times N_{l}}$ denotes the labeled part
of representation matrix $\mathbf{V}$ and
$\mathbf{V}_{u}\in\mathbb{R}^{K\times N_{u}}$ is the unlabeled part as shown
in Fig. 2 (i.e., $\mathbf{V}=[\mathbf{V}_{l},\mathbf{V}_{u}]$).
$\mathbf{W}\in\mathbb{R}^{K\times C}$ is the regression coefficient matrix and
the $L_{2,1}$-norm is used to enforce it sparse in rows. Consider we have a
priori knowledge that those samples share the same label, a binary weight
matrix $\mathbf{Y}\in\mathbb{R}^{C\times N}$ is constructed by following rule:

$\mathbf{Y}_{cn}=\left\\{\begin{array}[]{ll}{1}&{\text{ if }n\text{-th data
point belongs to the }c\text{-th class}}\\\ {0}&{\text{ otherwise
}}\end{array}\right.$ (5)

The effectiveness of graph regularization technique has been shown in recent
research work [28, 14] and it is able to keep the geometric structure of data
when the dimension is changed. Similar to the DMVC, we construct a local graph
Laplacian matrix to preserve the local geometrical structure of the each view
$\mathbf{X}^{p}$. As introduced in [14], the graph binary weight matrix
$\mathbf{S}^{p}$ is constructed in _k_ -nearest neighbor (_k_ -NN) fashion.
Formally, the regularization term $\mathcal{R}^{p}$ is calculated:

$\displaystyle\mathcal{R}^{p}$
$\displaystyle=\frac{1}{2}\sum_{j,q=1}^{n}\left\|\mathbf{v}_{j}^{p}-\mathbf{v}_{q}^{p}\right\|^{2}\mathbf{S}_{jq}^{p}$
(6)
$\displaystyle=\sum_{j=1}^{n}\mathbf{v}_{j}^{p}\mathbf{v}_{j}^{p}\mathbf{D}_{jj}^{p}-\sum_{j,q=1}^{n}\mathbf{v}_{j}^{p}\mathbf{v}_{l}^{p}\mathbf{S}_{jq}^{p}$
$\displaystyle=\operatorname{Tr}\left(\mathbf{V}^{p}\mathbf{D}^{p}(\mathbf{V}^{p})^{T}\right)-\operatorname{Tr}\left(\mathbf{V}^{p}\mathbf{S}^{p}(\mathbf{V}^{p})^{T}\right)$
$\displaystyle=\operatorname{Tr}\left(\mathbf{V}^{p}\mathbf{L}^{p}(\mathbf{V}^{p})^{T}\right)$

where $\mathbf{D}_{jj}^{p}=\sum_{q}\mathbf{S}_{jq}^{p}$.
$\mathbf{L}^{p}=\mathbf{D}^{p}-\mathbf{S}^{p}$ denotes the graph Laplacian
matrix for each view data.

Different from the existing deep matrix factorization multi-view methods, we
use partially shared strategy to jointly exploit view-specific and common
features. The partially shared latent representation matrix
$\mathbf{V}\in\mathbb{R}^{K\times N}$ and $K=K_{s}\times P+K_{c}$. The common
factor ratio $\lambda=K_{c}/(K_{s}+K_{c})$. The final $m$-th layer partially
shared factor $\mathbf{V}_{m}^{p}\in\mathbb{R}^{(K_{s}+K_{c})\times N}$ is
divided into four parts: the labeled and unlabeled view-specific encoding
matrix (i.e. $\mathbf{V}_{sl}^{p}\in\mathbb{R}^{K_{s}\times N_{l}}$ and
$\mathbf{V}_{su}^{p}\in\mathbb{R}^{K_{s}\times N_{u}}$), the labeled and
unlabeled shared encoding matrix (i.e.
$\mathbf{V}_{cl}\in\mathbb{R}^{K_{c}\times N_{l}}$ and
$\mathbf{V}_{cu}\in\mathbb{R}^{K_{c}\times N_{u}}$). We propose a general
partially shared deep matrix factorization framework as follows:

$\displaystyle\min\sum_{p=1}^{P}\left\|\mathbf{X}^{p}-\left[\mathbf{U}_{1s}^{p},\mathbf{U}_{1c}^{p}\right]\cdot\cdot\cdot\left[\mathbf{U}_{ms}^{p},\mathbf{U}_{mc}^{p}\right]\left[\begin{array}[]{c}\mathbf{V}_{sl}^{p},\mathbf{V}_{su}^{p}\\\
\mathbf{V}_{cl},\mathbf{V}_{cu}\end{array}\right]\right\|_{F}^{2}$ (7)
$\displaystyle\text{ s.t.
}\mathbf{V}_{sl}^{p},\mathbf{V}_{su}^{p},\mathbf{V}_{cl},\mathbf{V}_{cu}\geq
0$

where $\mathbf{U}_{is}^{p}$ and $\mathbf{U}_{ic}^{p}$ denote the hidden view-
specific matrix and common matrix, respectively. To simplify the problem, we
use
$\mathbf{V}_{m}^{p}=[\mathbf{V}_{s}^{p};\mathbf{V}_{c}]=\left[[\mathbf{V}_{sl}^{p},\mathbf{V}_{su}^{p}];[\mathbf{V}_{cl},\mathbf{V}_{cu}]\right]$,
$\mathbf{U}_{i}^{p}=[\mathbf{U}_{is}^{p};\mathbf{U}_{ic}^{p}]$, by combining
Eq.(4), Eq.(6) and Eq.(7), we get the cost function of PSDMF as:

$\displaystyle\min_{\mathbf{U}_{i}^{p},\mathbf{V}_{m}^{p},\mathbf{W}}O=$
$\displaystyle\sum_{p=1}^{P}\left(\alpha^{p}\left\|\mathbf{X}^{p}-\mathbf{U}_{1}^{p}\cdot\cdot\cdot\mathbf{U}_{m}^{p}\mathbf{V}_{m}^{p}\right\|_{F}^{2}\right.$
(8)
$\displaystyle\left.+\mu\operatorname{tr}\left(\mathbf{V}_{m}^{p}\mathbf{L}^{p}\left(\mathbf{V}_{m}^{p}\right)^{T}\right)\right)$
$\displaystyle+\beta\left(\left\|\mathbf{W}^{T}\mathbf{V}_{l}-\mathbf{Y}\right\|_{F}^{2}+\gamma\|\mathbf{W}\|_{2,1}\right)$
$\displaystyle\text{ s.t. }\mathbf{V}_{m}^{p}\geq 0,\forall p$

where $\alpha^{p}$ is the weighting coefficient for the $p$-th view and it has
a great influence on the effect of the model. In DMVC [14], the weights
distribution smooth is determined by parameter $r$, but $r$ needs to be
searched in a large range manually, which makes it to adjust. To avoid this
problem, inspired by [25, 26], we use the auto-weighted strategy that obtains
the value of $\alpha^{p}$ based on the distance between the data and the
decomposition matrices, as follows:

$\alpha^{p}=\frac{1}{2\left\|\mathbf{X}^{p}-\mathbf{U}_{1}^{p}\cdot\cdot\cdot\mathbf{U}_{m}^{p}\mathbf{V}_{m}^{p}\right\|_{F}}$
(9)

##  III Optimization

To expedite the approximation of the matrices in our proposed model, we
conduct the pre-training [24] by decomposing each view
$\mathbf{X}^{p}\approx\mathbf{U}_{1}^{p}\mathbf{V}_{1}^{p}$ by minimizing the
Semi-NMF
$\left\|\mathbf{X}^{p}-\mathbf{U}_{1}^{p}\mathbf{V}_{1}^{p}\right\|_{F}^{2}$,
where $\mathbf{U}_{1}^{p}\in\mathbb{R}^{M^{p}\times k_{1}}$ and
$\mathbf{V}_{1}^{p}\in\mathbb{R}^{k_{1}\times N}$. Then the factor matrix
$\mathbf{V}_{1}^{p}$ is further decomposed as
$\mathbf{V}_{1}^{p}\approx\mathbf{U}_{2}^{p}\mathbf{V}_{2}^{p}$ by minimizing
the
$\left\|\mathbf{V}_{2}^{p}-\mathbf{U}_{2}^{p}\mathbf{V}_{2}^{p}\right\|_{F}^{2}$,
where $\mathbf{U}_{2}^{p}\in\mathbb{R}^{k_{1}\times k_{2}}$,
$\mathbf{V}_{1}^{p}\in\mathbb{R}^{k_{2}\times N}$ and $k_{1},k_{2}$ denote the
dimensions for layer 1 and layer 2. The process will be repeated till all of
the layers are pre-trained. The optimization of Semi-NMF can be derived
following a similar process as described in [20]. To save the space, we omit
the updating rules here.

Update rule for hidden matrix $\mathbf{U}_{i}^{p}$: By fixing
$\mathbf{V}_{m}^{p}$ and $\mathbf{W}$, we minimize the objective function (8)
with respect to $\mathbf{U}_{i}^{p}$. Letting the partial derivative $\partial
O/\mathbf{U}_{i}^{p}=0$, we can obtain

$\mathbf{U}_{i}^{p}=\left(\Phi_{i-1}^{T}\Phi_{i-1}\right)^{-1}\left(\Phi_{i-1}^{T}\mathbf{X}^{p}\mathbf{U}_{im}^{T}\right)\\\
\left(\mathbf{U}_{im}\mathbf{U}_{im}^{T}\right)^{-1}$ (10)

where
$\Phi_{i-1}=\mathbf{U}_{1}^{p}\mathbf{U}_{2}^{p}\ldots\mathbf{U}_{i-1}^{p}$
and
$\mathbf{U}_{im}=\mathbf{U}_{i+1}^{p}\ldots\mathbf{U}_{m}^{p}\mathbf{V}_{m}^{p}$.

Update rule for regression weight matrix $\mathbf{W}$: Following [27], the
derivative of objective function (8) with respect to $\mathbf{W}$ is as
follows:

$\frac{\partial
O}{\mathbf{W}}=2\beta\left(\mathbf{V}_{l}(\mathbf{V}_{l}^{T}\mathbf{W}-\mathbf{Y}^{T})+\gamma\mathbf{E}\mathbf{W}\right)$
(11)

where $\mathbf{E}$ is a diagonal matrix with
$e_{ii}=1/2\|\mathbf{w}_{i}\|_{2}$. According to the optimization theory, set
$\partial O/\mathbf{W}=0$, then we can obtain the update rule for $\mathbf{W}$
is

$\mathbf{W}=\left(\mathbf{V}_{l}\mathbf{V}_{l}^{T}+\gamma\mathbf{E}\right)^{-1}\mathbf{V}_{l}\mathbf{Y}^{T}.$
(12)

Update rule for partially shared factor matrix $\mathbf{V}_{m}^{p}$: As
illustrated in Fig. 2, $\mathbf{V}_{m}^{p}$ can be divided into four parts:
$\mathbf{V}_{sl}^{p},\mathbf{V}_{su}^{p},\mathbf{V}_{cl}$ and
$\mathbf{V}_{cu}$. Similarly, each data matrix $\mathbf{X}^{p}$ is divided
into two parts $\mathbf{X}_{l}^{p}$ and $\mathbf{X}_{u}^{p}$; each Laplacian
Graph matrix $\mathbf{L}^{p}$ is divided into two parts $\mathbf{L}_{l}^{p}$
and $\mathbf{L}_{u}^{p}$.

For the constraint $\mathbf{V}_{m}^{p}\geq 0$, we introduce the Lagrangian
multiplier $\eta$ as follows:

$\displaystyle\mathcal{L}\left(\mathbf{V}_{m}^{p}\right)$
$\displaystyle=\sum_{p=1}^{P}\left(\alpha^{p}\left\|\mathbf{X}^{p}-\Phi_{m}\mathbf{V}_{m}^{p}\right\|_{\mathrm{F}}^{2}+\mu\operatorname{tr}\left(\mathbf{V}_{m}^{p}\mathbf{L}^{p}(\mathbf{V}_{m}^{p})^{T}\right)\right.$
(13)
$\displaystyle\left.+\beta\left\|\mathbf{W}^{T}\mathbf{V}_{l}-\mathbf{Y}\right\|-tr(\eta^{T}\mathbf{V}_{m}^{p})\right)$

where $\Phi_{m}=\mathbf{U}_{1}^{p}\mathbf{U}_{2}^{p}\ldots\mathbf{U}_{m}^{p}$.
Accordingly, $\Phi_{m}$ is divided into two parts $\Phi_{ms}$ and $\Phi_{mc}$.
For the convenience of writing, we denote that
$\mathbf{A}^{p}=[\Phi_{ms},\Phi_{mc}][\mathbf{V}_{sl}^{p};\mathbf{V}_{cl}]$,
$\mathbf{B}^{p}=[\Phi_{ms},\Phi_{mc}][\mathbf{V}_{su}^{p};\mathbf{V}_{cu}]$.

The zero gradient condition of $\mathcal{L}\left(\mathbf{V}_{m}^{p}\right)$
with respect to $\mathbf{V}_{sl}^{p},\mathbf{V}_{su}^{p},\mathbf{V}_{cl}$ and
$\mathbf{V}_{cu}$, respectively, we have

$\displaystyle\frac{\partial\mathcal{L}}{\partial\mathbf{V}_{sl}^{p}}=$
$\displaystyle\alpha^{p}\Phi_{ms}^{T}\left(\mathbf{A}^{p}-\mathbf{X}_{l}^{p}\right)+\mu\mathbf{V}_{s}^{p}\mathbf{L}_{l}^{p}+\beta\mathbf{F}_{s}^{p}-\eta_{sl}=0$
(14) $\displaystyle\frac{\partial\mathcal{L}}{\partial\mathbf{V}_{su}^{p}}=$
$\displaystyle\alpha^{p}\Phi_{ms}^{T}\left(\mathbf{B}^{p}-\mathbf{X}_{u}^{p}\right)+\mu\mathbf{V}_{s}^{p}\mathbf{L}_{u}^{p}-\eta_{su}=0$
$\displaystyle\frac{\partial\mathcal{L}}{\partial\mathbf{V}_{c1}}=$
$\displaystyle\sum_{p=1}^{P}\alpha^{p}\Phi_{mc}^{T}\left(\mathbf{A}^{p}-\mathbf{X}_{l}^{p}\right)+\mu\mathbf{V}_{c}^{p}\mathbf{L}_{l}^{p}+\beta\mathbf{F}_{c}-\eta_{cl}=0$
$\displaystyle\frac{\partial\mathcal{L}}{\partial\mathbf{V}_{cu}}=$
$\displaystyle\sum_{p=1}^{P}\alpha^{p}\Phi_{mc}^{T}\left(\mathbf{B}^{p}-\mathbf{X}_{u}^{p}\right)+\mu\mathbf{V}_{c}^{p}\mathbf{L}_{u}^{p}-\eta_{cu}=0$

where
$\mathbf{F}=\mathbf{W}(\mathbf{W}^{T}\mathbf{V}_{l}-\mathbf{Y})=[\mathbf{F}_{s}^{1};...;\mathbf{F}_{s}^{p};\mathbf{F}_{c}]$.
Following a similar proof to [14], using the Karush-Kuhn-Tucker condition, we
can formulate the updating rules for $\mathbf{V}_{m}^{p}$:

$\displaystyle\mathbf{V}_{sl}^{p}=\mathbf{V}_{sl}^{p}\odot$ (15)
$\displaystyle\sqrt{\frac{\alpha^{p}[\Phi_{ms}^{T}\mathbf{A}^{p}]^{-}+\alpha^{p}[\Phi_{ms}^{T}\mathbf{X}_{l}^{p}]^{+}+\mu[\mathbf{V}_{s}^{p}\mathbf{L}_{l}^{p}]^{-}+\beta[\mathbf{F}_{s}^{p}]^{-}}{\alpha^{p}[\Phi_{ms}^{T}\mathbf{A}^{p}]^{+}+\alpha^{p}[\Phi_{ms}^{T}\mathbf{X}_{l}^{p}]^{-}+\mu[\mathbf{V}_{s}^{p}\mathbf{L}_{l}^{p}]^{+}+\beta[\mathbf{F}_{s}^{p}]^{+}}}$
$\displaystyle\mathbf{V}_{su}^{p}=\mathbf{V}_{su}^{p}\odot$
$\displaystyle\sqrt{\frac{\alpha^{p}[\Phi_{ms}^{T}\mathbf{B}^{p}]^{-}+\alpha^{p}[\Phi_{ms}^{T}\mathbf{X}_{u}^{p}]^{+}+\mu[\mathbf{V}_{s}^{p}\mathbf{L}_{u}^{p}]^{-}}{\alpha^{p}[\Phi_{ms}^{T}\mathbf{B}^{p}]^{+}+\alpha^{p}[\Phi_{ms}^{T}\mathbf{X}_{u}^{p}]^{-}+\mu[\mathbf{V}_{s}^{p}\mathbf{L}_{u}^{p}]^{+}}}$
$\displaystyle\mathbf{V}_{cl}=\mathbf{V}_{cl}\odot$
$\displaystyle\sqrt{\frac{\sum\limits_{p=1}^{P}\alpha^{p}\left([\Phi_{mc}^{T}\mathbf{A}^{p}]^{-}+[\Phi_{ms}^{T}\mathbf{X}_{l}^{p}]^{+}\right)+\mu[\mathbf{V}_{c}^{p}\mathbf{L}_{l}^{p}]^{-}+\beta[\mathbf{F}_{c}]^{-}}{\sum\limits_{p=1}^{P}\alpha^{p}\left([\Phi_{ms}^{T}\mathbf{A}^{p}]^{+}+[\Phi_{ms}^{T}\mathbf{X}_{l}^{p}]^{-}\right)+\mu[\mathbf{V}_{c}^{p}\mathbf{L}_{l}^{p}]^{+}+\beta[\mathbf{F}_{c}]^{+}}}$
$\displaystyle\mathbf{V}_{cu}=\mathbf{V}_{cu}\odot$
$\displaystyle\sqrt{\frac{\sum\limits_{p=1}^{P}\alpha^{p}[\Phi_{mc}^{T}\mathbf{B}^{p}]^{-}+\sum\limits_{p=1}^{P}\alpha^{p}[\Phi_{ms}^{T}\mathbf{X}_{u}^{p}]^{+}+\mu[\mathbf{V}_{c}^{p}\mathbf{L}_{u}^{p}]^{-}}{\sum\limits_{p=1}^{P}\alpha^{p}[\Phi_{ms}^{T}\mathbf{B}^{p}]^{+}+\sum\limits_{p=1}^{P}\alpha^{p}[\Phi_{ms}^{T}\mathbf{X}_{u}^{p}]^{-}+\mu[\mathbf{V}_{c}^{p}\mathbf{L}_{u}^{p}]^{+}}},$

where $[\mathbf{H}]^{+}$ and $[\mathbf{H}]^{-}$ denote a matrix that all the
negative elements are replaced by $0$ and all the positive elements are
replaced by $0$, respectively. That is,

$\forall
k,j[\mathbf{H}]_{kj}^{+}=\frac{|\mathbf{H}_{kj}|+\mathbf{H}_{kj}}{2},[\mathbf{H}]_{kj}^{-}=\frac{|\mathbf{H}_{kj}|-\mathbf{H}_{kj}}{2}.$
(16)

Algorithm 1 Optimization algorithm of PSDMF

0: Multi-view Data $\\{\mathbf{X}^{p}\\}_{p=1}^{P}$, parameters
$\mu,\beta,\gamma,\lambda,K$, layer sizes $k$

0: Partially shared latent representation $\mathbf{V}$, the regression weight
matrix $\mathbf{W}$

1: Initialize

2: Construct partially label matrix $\mathbf{Y}$ via Eq. (5)

3: for all layers in each view do

4: $(\mathbf{U}_{i}^{p},\mathbf{V}_{i}^{p})$$\leftarrow$ Semi-
NMF$(\mathbf{V}_{i-1}^{p},\boldsymbol{k}_{i})$

5: end for

6: while not converged do

7: Update $\mathbf{W}$ via Eq. (12)

8: for $p=1$ to $P$ do

9: Update $\alpha^{p}$ via Eq. (9)

10: for $i=1$ to $m$ do

11: Update $\mathbf{U}_{i}^{p}$ via Eq. (10)

12: end for

13: Update $\mathbf{V}_{m}^{p}$ via Eq. (15)

14: end for

15: end while

TABLE II: Results on five datasets( mean $\pm$ standard deviation). Higher value indicates better performance and the highest values are in boldface. Datasets |  | DMVC | GMC | lLSMC | LMVSC | DICS | MvSL | PSLF | GPSNMF |  Ours
---|---|---|---|---|---|---|---|---|---|---
Extended |  ACC | 50.34$\pm$0.07 | 43.38$\pm$0.00 | 53.41$\pm$1.29 | 36.62$\pm$0.00 | 47.54$\pm$3.58 | 19.64$\pm$1.81 | 43.79$\pm$4.90 | 69.56$\pm$8.04 |  87.38$\pm$3.45
Yale B |  NMI | 49.97$\pm$0.14 | 44.90$\pm$0.00 | 53.40$\pm$0.64 | 28.09$\pm$0.00 | 50.05$\pm$5.11 | 11.82$\pm$2.88 | 32.14$\pm$4.85 | 60.44$\pm$9.12 |  83.53$\pm$2.44
|  Purity | 50.49$\pm$0.07 | 43.69$\pm$0.00 | 53.45$\pm$1.26 | 42.77$\pm$0.00 | 51.39$\pm$3.82 | 20.52$\pm$1.66 | 44.10$\pm$4.36 | 69.56$\pm$8.04 |  87.38$\pm$3.45
Prokaryotic |  ACC | 53.50$\pm$2.23 | 49.55$\pm$0.00 | 52.31$\pm$6.01 | 57.53$\pm$0.00 | 56.07$\pm$11.76 | 35.68$\pm$2.51 | 46.49$\pm$6.63 | 63.45$\pm$2.59 |  66.83$\pm$2.94
|  NMI | 3.28$\pm$0.66 | 19.34$\pm$0.00 | 16.49$\pm$8.59 | 13.37$\pm$0.00 | 12.63$\pm$9.88 | 1.05$\pm$0.41 | 1.69$\pm$1.02 | 18.46$\pm$5.48 |  21.08$\pm$4.21
|  Purity | 57.31$\pm$0.65 | 58.44$\pm$0.00 | 63.08$\pm$4.15 | 62.94$\pm$0.00 | 60.14$\pm$5.16 | 57.03$\pm$0.07 | 57.46$\pm$0.73 | 63.45$\pm$2.59 |  66.53$\pm$2.94
Caltech101 |  ACC | 54.60$\pm$0.43 | 69.20$\pm$0.00 | 51.98$\pm$2.61 | 69.47$\pm$0.00 | 50.95$\pm$10.61 | 46.42$\pm$4.14 | 76.88$\pm$3.09 | 89.77$\pm$2.54 |  90.24$\pm$3.08
-7 |  NMI | 37.60$\pm$5.79 | 65.95$\pm$0.00 | 53.22$\pm$2.22 | 46.68$\pm$0.00 | 53.08$\pm$7.71 | 51.59$\pm$2.81 | 40.94$\pm$7.75 | 74.53$\pm$4.10 |  75.52$\pm$5.36
|  Purity | 79.55$\pm$4.64 | 88.47$\pm$0.00 | 85.61$\pm$1.64 | 77.07$\pm$0.00 | 85.28$\pm$3.70 | 83.73$\pm$1.49 | 77.56$\pm$4.12 | 90.38$\pm$2.33 |  91.12$\pm$2.03
Caltech101 |  ACC | 51.72$\pm$1.72 | 45.64$\pm$0.00 | 45.03$\pm$4.16 | 42.08$\pm$0.00 | 35.75$\pm$3.83 | 42.01$\pm$2.86 | 55.75$\pm$10.35 | 77.33$\pm$3.34 |  79.74$\pm$3.73
-20 |  NMI | 52.18$\pm$1.49 | 48.09$\pm$0.00 | 61.21$\pm$1.14 | 49.64$\pm$0.00 | 55.99$\pm$2.58 | 60.48$\pm$1.45 | 44.85$\pm$5.77 | 67.48$\pm$3.52 |  71.24$\pm$3.06
|  Purity | 70.23$\pm$0.92 | 55.49$\pm$0.00 | 76.95$\pm$0.79 | 50.84$\pm$0.00 | 69.32$\pm$2.47 | 76.16$\pm$1.17 | 62.55$\pm$4.94 | 77.81$\pm$2.60 |  79.74$\pm$2.84
MSRCV1 |  ACC | 40.76$\pm$2.37 | 74.76$\pm$0.00 | 69.91$\pm$1.79 | 64.76$\pm$0.00 | 51.91$\pm$6.61 | 78.07$\pm$5.83 | 82.96$\pm$4.51 | 72.17$\pm$4.03 |  86.24$\pm$5.69
|  NMI | 27.60$\pm$3.17 | 70.09$\pm$0.00 | 62.37$\pm$2.71 | 58.81$\pm$0.00 | 50.48$\pm$11.29 | 68.32$\pm$4.69 | 75.04$\pm$3.49 | 67.41$\pm$4.34 |  76.14$\pm$3.42
|  Purity | 45.14$\pm$2.41 | 79.05$\pm$0.00 | 72.86$\pm$1.38 | 69.05$\pm$0.00 | 58.09$\pm$8.68 | 78.52$\pm$5.73 | 82.96$\pm$4.51 | 74.92$\pm$4.44 |  86.24$\pm$4.45

Until now, we have all the update rules done. Updates iterations repeatedly
until convergence. The overall optimization process of PSDMF is outlined in
Algorithm 1, where the ”Semi-NMF” procedure performs the pre-train as
described earlier. Once the optimal partially shared latent representation
matrix $\mathbf{V}$ and regression weight matrix $\mathbf{W}$ are obtained,
cluster label $y=\arg\max_{c}y_{c,i}$ where
$\boldsymbol{y}_{i}=\mathbf{W}^{T}\boldsymbol{v}_{i}$.

##  IV Experiments

To comparatively study the performance of PSDMF, we consider several state-of-
the-art baselines as stated bellow:
Multi-view clustering via deep matrix factorization (DMVC) [14], graph-based
multi-view clustering (GMC) [21], latent multi-view subspace clustering
(lLSMC) [23], large-scale multi-view subspace clustering (LMVSC) [22], multi-
view discriminative learning via joint non-negative matrix factorization
(DICS) [18], graph-regularized multi-view semantic subspace learning (MvSL)
[19], partially shared latent factor learning (PSLF) [10] and semi-supervised
multi-view clustering with graph-regularized partially shared non-negative
matrix factorization (GPSNMF) [17]. Among them, four methods are unsupervised
(i.e., DMVC, GMC, lLSMC, LMVSC) and the others are semi-supervised.

###  IV-A Datasets and Evaluation Metric

We perform experiments on five benchmark datasets: Extended Yale B,
Prokaryotic, MSRCV1, Caltech101-7 and 20. More detailed information about
these data is shown in Table III.

TABLE III: Statistics of datasets used in experiments. Dataset | Dimension of Views |  Sizes |  Classes
---|---|---|---
Extended Yale B | 2500/3304/6750 |  650 |  10
Prokaryotic | 393/3/438 |  551 |  4
Caltech101-7/20 | 48/40/254/1984/512/928 |  1474/2386 |  7/20
MSRCV1 | 24/576/512/256/254 |  210 |  7

For evaluation metrics, we use Accuracy (ACC), Normalized Mutual Information
(NMI) and Purity to comprehensively measure the clustering performance in our
experiments. The formal definitions of these metrics, omitted here to save
space, can be found in [9].

###  IV-B Experiment Setup

Following [17], for PSLF, GPSNMF and PSDMF, the dimension of partially shared
latent representation $K$ and the common factor ratio $\lambda$ are set to 100
and 0.5, respectively. Thus, $K_{c}+K_{s}\times P=100$ and
$K_{c}/(K_{s}+K_{c})=0.5$. In addition, they all predicate the labels with
regression matrix $\mathbf{W}$. Other than these three algorithms, all the
above methods are evaluated by the classic clustering algorithm, K-means. In
terms of semi-supervised methods (i.e. DICS, MvSL, PSLF, GPSNMF and PSDMF), we
denote $10\%$ as the proportions of the labeled data. Following [14], the
layer sizes of deep structure models (i.e. DMVC, PSDMF) are set as $[100,50]$.

For all compared methods, 10 repeated times are run to reduce the randomness
caused by initialization, and report their average scores and standard
deviations. As to the compared methods, source codes are obtained from the
authors’ websites, and we set their parameters to the optimal value if they
have.

###  IV-C Performance Evaluation

The performances of multi-view clustering is shown in Table II. From the
experiment results, we have drawn some conclusions as following: In general,
semi-supervised algorithms are superior to unsupervised algorithms. For
example, our PSDMF significantly improves the performance compared with DMVC
all over the five datasets, even when only $10\%$ label information is used.
For the dataset Extended Yale B, we raise the performance bar by around $18\%$
in ACC, $13\%$ in NMI and $18\%$ in Purity. On average, we improve the state-
of-the-art GPSNMF by more than $16\%$, which means multi-layer structure
methods have clustering advantages over single-layer methods. Through the
partially shared deep representation, PSDMF eliminates the influence of
undesirable factors by considering both view-specific and common features, and
keeps the important information in the final representation layer. In
different datasets experiments, our model achieves better performances than
the state-of-the-art methods, which demonstrate the robustness of ours.

Figure 3: The performance of PSDMF in terms of ACC with different value of
parameters $\mu$ and $\beta$. (a) Extended Yale B; (b) Prokaryotic; (c)
Caltech101-7; (d) Caltech101-20; (e) MSRCV1.

###  IV-D Parameters Sensitivity

There are three nonnegative essential parameters (i.e. $\mu,\beta,\gamma$) in
our proposed model. $\mu$ controls the smoothness of the partially shared
factor. $\gamma$ and $\beta$ controls the sparsity of the weight matrix
$\mathbf{W}$ and balances the relation between the regression term and data
reconstruction, respectively. Following [10, 17], $\gamma=10$ usually obtains
better performance. As is shown in Fig. 3, we test the sensitivity of the
parameters $\mu$ and $\beta$ in terms of clustering results, which are tuned
in the range of $\\{0.001,0.005,0.01,0.05,0.1,0.5\\}$ and
$\\{0.01,0.1,1,10\\}$, respectively. We can observe that the parameters affect
the experimental results in different datasets, which indicates that $\mu$ and
$\beta$ play an important role in PSDMF. In practice, we fix $\mu=0.1$ and
$\beta=10$ as default in our experiments.

##  V conclusion

In this paper, we introduced a novel semi-supervised deep matrix factorization
model for multi-view learning, called PSDMF, which is able to learn a
comprehensive partially shared latent final-layer representation for multi-
view data. Through the partially shared multi-layer matrix factorization
structure, our method is capable of exploiting view-specific and common
features among different views, simultaneously. Benefitting from the label
regression term, it can incorporate information from the labeled data.
Furthermore, we utilize graph regularization and auto-weighted strategy to
preserve the intrinsic geometric structure of the data. An iterative
optimization algorithm is developed to deal with the PSDMF. Experimental
results show that the proposed model achieves superior performances.

## Acknowledgment

This work was supported by the National Natural Science Foundation of China
under Grant Nos. 61722304, 61801133, and 61803096, in part by the Guangdong
Science and Technology Foundation under Grant Nos. 2019B010118001,
2019B010121001, and 2019B010154002, National Key Research and Development
Project, China under Grant No. 2018YFB1802400.

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