# Coronal Heating Rate in the Slow Solar Wind

Daniele Telloni National Institute for Astrophysics, Astrophysical Observatory
of Torino, Via Osservatorio 20, I-10025 Pino Torinese, Italy  Marco Romoli
University of Florence, Department of Physics and Astronomy, Via Giovanni
Sansone 1, I-50019 Sesto Fiorentino, Italy  Marco Velli Earth, Planetary, and
Space Sciences, University of California, Los Angeles, CA 90095, USA  Gary P.
Zank Center for Space Plasma and Aeronomic Research, University of Alabama in
Huntsville, Huntsville, AL 35805, USA  Department of Space Science, University
of Alabama in Huntsville, Huntsville, AL 35805, USA  Laxman Adhikari Center
for Space Plasma and Aeronomic Research, University of Alabama in Huntsville,
Huntsville, AL 35805, USA  Cooper Downs Predictive Science Inc., San Diego, CA
92121, USA  Aleksandr Burtovoi National Institute for Astrophysics,
Astrophysical Observatory of Arcetri, Largo Enrico Fermi 5, I-50125 Firenze,
Italy  Roberto Susino National Institute for Astrophysics, Astrophysical
Observatory of Torino, Via Osservatorio 20, I-10025 Pino Torinese, Italy
Daniele Spadaro National Institute for Astrophysics, Astrophysical Observatory
of Catania, Via Santa Sofia 78, I-95123 Catania, Italy  Lingling Zhao Center
for Space Plasma and Aeronomic Research, University of Alabama in Huntsville,
Huntsville, AL 35805, USA  Alessandro Liberatore Jet Propulsion Laboratory,
California Institute of Technology, Pasadena, CA 91109, USA  Chen Shi Earth,
Planetary, and Space Sciences, University of California, Los Angeles, CA
90095, USA  Yara De Leo Max Planck Institute for Solar System Research,
Justus-von-Liebig-Weg 3, D-37077 Göttingen, Germany  University of Catania,
Department of Physics and Astronomy, Via Santa Sofia 64, I-95123 Catania,
Italy  Lucia Abbo National Institute for Astrophysics, Astrophysical
Observatory of Torino, Via Osservatorio 20, I-10025 Pino Torinese, Italy
Federica Frassati National Institute for Astrophysics, Astrophysical
Observatory of Torino, Via Osservatorio 20, I-10025 Pino Torinese, Italy
Giovanna Jerse National Institute for Astrophysics, Astronomical Observatory
of Trieste, Località Basovizza 302, I-34149 Trieste, Italy  Federico Landini
National Institute for Astrophysics, Astrophysical Observatory of Torino, Via
Osservatorio 20, I-10025 Pino Torinese, Italy  Gianalfredo Nicolini National
Institute for Astrophysics, Astrophysical Observatory of Torino, Via
Osservatorio 20, I-10025 Pino Torinese, Italy  Maurizio Pancrazzi National
Institute for Astrophysics, Astrophysical Observatory of Torino, Via
Osservatorio 20, I-10025 Pino Torinese, Italy  Giuliana Russano National
Institute for Astrophysics, Astronomical Observatory of Capodimonte, Salita
Moiariello 16, I-80131 Napoli, Italy  Clementina Sasso National Institute for
Astrophysics, Astronomical Observatory of Capodimonte, Salita Moiariello 16,
I-80131 Napoli, Italy  Vincenzo Andretta National Institute for Astrophysics,
Astronomical Observatory of Capodimonte, Salita Moiariello 16, I-80131 Napoli,
Italy  Vania Da Deppo National Research Council, Institute for Photonics and
Nanotechnologies, Via Trasea 7, I-35131 Padova, Italy  Silvano Fineschi
National Institute for Astrophysics, Astrophysical Observatory of Torino, Via
Osservatorio 20, I-10025 Pino Torinese, Italy  Catia Grimani University of
Urbino Carlo Bo, Department of Pure and Applied Sciences, Via Santa Chiara 27,
I-61029 Urbino, Italy  National Institute for Nuclear Physics, Section in
Florence, Via Bruno Rossi 1, I-50019 Sesto Fiorentino, Italy  Petr Heinzel
Czech Academy of Sciences, Astronomical Institute, Fričova 298, CZ-25165
Ondřejov, Czechia  University of Wrocław, Centre of Scientific Excellence –
Solar and Stellar Activity, ul. Kopernika 11, PL-51-622 Wrocław, Poland  John
D. Moses National Aeronautics and Space Administration, Headquarters,
Washington, DC 20546, USA  Giampiero Naletto University of Padua, Department
of Physics and Astronomy, Via Francesco Marzolo 8, I-35131 Padova, Italy
Marco Stangalini Italian Space Agency, Via del Politecnico snc, I-00133 Roma,
Italy  Luca Teriaca Max Planck Institute for Solar System Research, Justus-
von-Liebig-Weg 3, D-37077 Göttingen, Germany  Michela Uslenghi National
Institute for Astrophysics, Institute of Space Astrophysics and Cosmic Physics
of Milan, Via Alfonso Corti 12, I-20133 Milano, Italy  Arkadiusz Berlicki
University of Wrocław, Astronomical Institute, Kopernika 11, PL-51622 Wrocław,
Poland  Roberto Bruno National Institute for Astrophysics, Institute for Space
Astrophysics and Planetology, Via del Fosso del Cavaliere 100, I-00133 Roma,
Italy  Gerardo Capobianco National Institute for Astrophysics, Astrophysical
Observatory of Torino, Via Osservatorio 20, I-10025 Pino Torinese, Italy
Giuseppe E. Capuano National Institute for Astrophysics, Astrophysical
Observatory of Catania, Via Santa Sofia 78, I-95123 Catania, Italy  University
of Catania, Department of Physics and Astronomy, Via Santa Sofia 64, I-95123
Catania, Italy  Chiara Casini National Research Council, Institute for
Photonics and Nanotechnologies, Via Trasea 7, I-35131 Padova, Italy  Centre of
Studies and Activities for Space “Giuseppe Colombo”, Via Venezia 15, I-35131
Padova, Italy  Marta Casti The Catholic University of America at the National
Aeronautics and Space Administration, Goddard Space Flight Center, Greenbelt,
MD 20771, USA  Paolo Chioetto National Research Council, Institute for
Photonics and Nanotechnologies, Via Trasea 7, I-35131 Padova, Italy  Centre of
Studies and Activities for Space “Giuseppe Colombo”, Via Venezia 15, I-35131
Padova, Italy  Alain J. Corso National Research Council, Institute for
Photonics and Nanotechnologies, Via Trasea 7, I-35131 Padova, Italy  Raffaella
D’Amicis National Institute for Astrophysics, Institute for Space Astrophysics
and Planetology, Via del Fosso del Cavaliere 100, I-00133 Roma, Italy  Michele
Fabi University of Urbino Carlo Bo, Department of Pure and Applied Sciences,
Via Santa Chiara 27, I-61029 Urbino, Italy  National Institute for Nuclear
Physics, Section in Florence, Via Bruno Rossi 1, I-50019 Sesto Fiorentino,
Italy  Fabio Frassetto National Research Council, Institute for Photonics and
Nanotechnologies, Via Trasea 7, I-35131 Padova, Italy  Marina Giarrusso
University of Florence, Department of Physics and Astronomy, Via Giovanni
Sansone 1, I-50019 Sesto Fiorentino, Italy  Silvio Giordano National Institute
for Astrophysics, Astrophysical Observatory of Torino, Via Osservatorio 20,
I-10025 Pino Torinese, Italy  Salvo L. Guglielmino National Institute for
Astrophysics, Astrophysical Observatory of Catania, Via Santa Sofia 78,
I-95123 Catania, Italy  Enrico Magli Politecnico of Turin, Department of
Electronics and Telecommunications, Corso Duca degli Abruzzi 24, I-10129
Torino, Italy  Giuseppe Massone National Institute for Astrophysics,
Astrophysical Observatory of Torino, Via Osservatorio 20, I-10025 Pino
Torinese, Italy  Mauro Messerotti National Institute for Astrophysics,
Astronomical Observatory of Trieste, Località Basovizza 302, I-34149 Trieste,
Italy  University of Trieste, Department of Physics, Via Alfonso Valerio 2,
I-34127 Trieste, Italy  Giuseppe Nisticò University of Calabria, Department of
Physics, Ponte Pietro Bucci Cubo 31C, I-87036 Rende, Italy  Maria G. Pelizzo
University of Padua, Department of Information Engineering, Via Giovanni
Gradenigo 6, I-35131 Padova, Italy  Fabio Reale University of Palermo,
Department of Physics and Chemistry - Emilio Segrè, Piazza del Parlamento 1,
I-90134 Palermo, Italy  National Institute for Astrophysics, Astronomical
Observatory of Palermo, Piazza del Parlamento 1, I-90134 Palermo, Italy  Paolo
Romano National Institute for Astrophysics, Astrophysical Observatory of
Catania, Via Santa Sofia 78, I-95123 Catania, Italy  Udo Schühle Max Planck
Institute for Solar System Research, Justus-von-Liebig-Weg 3, D-37077
Göttingen, Germany  Sami K. Solanki Max Planck Institute for Solar System
Research, Justus-von-Liebig-Weg 3, D-37077 Göttingen, Germany  Thomas Straus
National Institute for Astrophysics, Astronomical Observatory of Capodimonte,
Salita Moiariello 16, I-80131 Napoli, Italy  Rita Ventura National Institute
for Astrophysics, Astrophysical Observatory of Catania, Via Santa Sofia 78,
I-95123 Catania, Italy  Cosimo A. Volpicelli National Institute for
Astrophysics, Astrophysical Observatory of Torino, Via Osservatorio 20,
I-10025 Pino Torinese, Italy  Luca Zangrilli National Institute for
Astrophysics, Astrophysical Observatory of Torino, Via Osservatorio 20,
I-10025 Pino Torinese, Italy  Gaetano Zimbardo University of Calabria,
Department of Physics, Ponte Pietro Bucci Cubo 31C, I-87036 Rende, Italy
Paola Zuppella National Research Council, Institute for Photonics and
Nanotechnologies, Via Trasea 7, I-35131 Padova, Italy  Stuart D. Bale Space
Sciences Laboratory, University of California, Berkeley, CA 94720, USA
Physics Department, University of California, Berkeley, CA 94720, USA  Justin
C. Kasper BWX Technologies, Inc., Washington, DC 20002, USA  Climate and Space
Sciences and Engineering, University of Michigan, Ann Arbor, MI 48109, USA

###### Abstract

This Letter reports the first observational estimate of the heating rate in
the slowly expanding solar corona. The analysis exploits the simultaneous
remote and local observations of the same coronal plasma volume with the Solar
Orbiter/Metis and the Parker Solar Probe instruments, respectively, and relies
on the basic solar wind magnetohydrodynamic equations. As expected, energy
losses are a minor fraction of the solar wind energy flux, since most of the
energy dissipation that feeds the heating and acceleration of the coronal flow
occurs much closer to the Sun than the heights probed in the present study,
which range from $6.3$ to $13.3$ R⊙. The energy deposited to the supersonic
wind is then used to explain the observed slight residual wind acceleration
and to maintain the plasma in a non-adiabatic state. As derived in the
Wentzel-Kramers-Brillouin limit, the present energy transfer rate estimates
provide a lower limit, which can be very useful in refining the turbulence-
based modeling of coronal heating and subsequent solar wind acceleration.

magnetohydrodynamics (MHD) — turbulence — Sun: corona — Sun: evolution — Sun:
fundamental parameters — solar wind

††journal: ApJL

##  1 Introduction

Central to the heating and subsequent acceleration of the coronal plasma is
the identification of the physical mechanisms responsible for transporting the
energy available in the photospheric motions to the corona, where it is
dissipated by raising the temperature to a million degrees, sufficient for the
plasma to overcome the Sun’s gravity and thus expand into the heliosphere,
forming the solar wind (Parker, 1958). It is worth noting that this thermal
driving can only account for the acceleration of the slowest coronal flows,
while for the higher speed streams to be accelerated, an additional source of
energy is required, whose dissipation results in further heating and thus
transition to a faster wind regime. The non-resonant dissipation of low-
frequency MagnetoHydroDynamic (MHD) turbulent fluctuations is to date the most
widely accepted heating mechanism, whether for fast or slow wind.

However, a key element for any heating model, namely the estimate of the
coronal energy transfer rate and its radial dependence, is still missing. This
is basically because coronal measurements of magnetic field and non-thermal
plasma motions (related to turbulence), i.e., Alfvén wave energy, are lacking
(or at best inconclusive). In some fluid models (e.g., Holzer & Axford, 1970;
McKenzie et al., 1995), an ad hoc heating function, which decays exponentially
with height over a prescribed dissipation length scale, is assumed to account
for the energy deposition per unit volume. Other approaches resort to the use
of numerical simulations in order to derive the coronal heating rate profile
(e.g., Dmitruk et al., 2002).

Local single-point measurements of the turbulent energy cascade rate, which is
directly related to the solar wind heating rate, are widely available at
various heliospheric heights. Although a detailed discussion of the turbulent
heating rate estimates in the solar wind is beyond the scope of this article
(the interested reader is referred to the comprehensive review by Marino &
Sorriso-Valvo, 2023), the following works are worth mentioning. Sorriso-Valvo
et al. (2007) first applied the generalized form of the Yaglom law for MHD
turbulence (as derived by Politano & Pouquet, 1998) to Ulysses data (Wenzel et
al., 1992) to derive the energy flux of the turbulent cascade in high-speed
solar streams sampled at $3-4$ au. Zhao et al. (2022) extended this approach
to the first sub-Alfvénic interval observed with Parker Solar Probe (PSP; Fox
et al., 2016) at $0.09$ au, using both incompressible and compressible
formalisms for the equivalent Yaglom law (Andrés et al., 2019), and finding a
higher average energy cascade rate than in the surrounding super-Alfvénic
regions. Adopting the turbulence cascade model developed by Tu (1988) and
exploiting PSP measurements, Wu et al. (2020) derived a formula for relating
the radial scaling of the low-frequency spectral break to the energy supply
rate in the slow solar wind, from $0.17$ to $0.7$ au. These authors found that
at distances greater than $0.25$ au from the Sun, injected and dissipated
energies are of the same order, suggesting that the slow solar wind expands
almost adiabatically. Deriving straightforward expressions for the turbulent
heating rate, Adhikari et al. (2022) reported, from both observational and
theoretical perspectives, on the radial evolution of the energy dissipation
rate from $0.17$ to $0.83$ au, using PSP and Solar Orbiter (SO; Müller et al.,
2020) data. It is true that some authors have exploited remote observations of
the solar corona to probe the turbulent properties of coronal flows and thus
derive constraints or upper limits on the rate of energy deposition in the
corona (Chandran et al., 2009; Cranmer, 2020; Sasikumar Raja et al., 2021).
Yet, all these studies make extensive use of unknown parameter assumptions and
are either highly model-dependent or based on numerical simulations. Although
extremely important, these works do not provide direct, empirical observations
of the coronal heating rate as in the present study.

Quadratures between two spacecraft have already proved useful in linking local
turbulent properties with the fluid parameters of remotely observed coronal
flows (Telloni et al., 2021a, 2022a). However, the special orbital
configuration between PSP and SO spacecraft that occurred on June $01$, $2022$
offers an almost unique opportunity. Indeed, during this quadrature, PSP
entered the portion of the Plane of the Sky (POS) imaged by the Metis
coronagraph (Fineschi et al., 2020; Antonucci et al., 2020) on board SO. As
described in the following, this allows the first ever observational estimate
of the energy deposition rate in the solar corona, from $6.3$ to $13.3$ R⊙,
without having to invoke any model or ad hoc parameter assumptions. Indeed,
the simultaneous remote and in-situ measurements of the same coronal flow,
associated with solving the basic equations for a steady-state, radial plasma
flow, allow an empirical low-speed solar wind model to be derived. In
addition, in the Wentzel-Kramers-Brillouin (WKB) theory approximation, the
energy of the outward Alfvén wave flux and, in turn, the energy transfer rate,
can be extended from the PSP location to the whole range of coronal heights
observed with Metis. This is the aim of the present Letter, whose plan is as
follows: Metis/PSP data and equation description (§ 2), derivation of the
empirical wind and turbulent model along with relevant discussion (§ 3).

##  2 SO/Metis – PSP data and MHD equations

On June $01$, $2022$, at $22$:$40$ UT, when SO was at $0.936$ au, PSP was in
quadrature while orbiting at $13.3$ R⊙. A SO spacecraft roll of $45^{\circ}$
along with an off-pointing of $1$ R⊙ towards the West limb was performed for
PSP to squeeze into the Metis POS, which extended from about $6.3$ to $13.3$
R⊙. Metis observations are supported by a $3$D MHD modeling of the solar
corona developed by Predictive Science Incorporated (PSI; Mikić et al., 2018).
This is based on photospheric magnetic field measurements acquired by the
Helioseismic and Magnetic Imager (HMI; Scherrer et al., 2012) on board the
Solar Dynamics Observatory (SDO; Pesnell et al., 2012) and driven by a heating
mechanism fed by the dissipation of low-frequency Alfvén waves.

Figure 1(a) shows the visible light observations of the coronal polarized
Brightness (pB) obtained with Metis during the quadrature with PSP, marked
with a blue dot, and calibrated according to De Leo et al. (2023). PSP,
$3.6^{\circ}$ below the equatorial plane, was immersed in a diffuse coronal
region devoid of emission-enhanced (i.e., denser) structures. Indeed, Figure
1(b), which displays the Carrington map of the squashing factor $Q$ (Titov et
al., 2011) at $10$ R⊙, clearly discloses that PSP sampled a low-latitude
equatorial coronal hole, which, in this case, is near the outer positive
polarity of a decaying active region from the previous rotation. Due to the
pseudostreamer lobes bracketing the open flux (Figure 1(b)), however, the
solar wind flow impinging on PSP was slow (Wang, 1994), as shown in the top
panel of Figure 1(c), which overall displays PSP plasma and magnetic field
measurements acquired on a $0.5$-hour long interval around the quadrature with
SO/Metis (corresponding to $\pm 0.76^{\circ}$ longitude with respect to the
Metis POS). Specifically, from top to bottom, are the solar wind and Alfvén
speed, magnetic field vector strength and inclination with respect to the
radial direction, and proton density and temperature. This is a case of highly
field-aligned ($\theta_{BR}\sim 11^{\circ}$) and sub-Alfvénic slow ($U\sim
310$ km s${}^{-1}<U_{A}$) solar wind, which is additionally quite homogenous
and stationary.

Finally, Figure 1(d) shows the pB radial profile at the latitude in the PSP
direction (indicated by a white dashed line in Figure 1(a)). Relying on the
approach initially advanced by van de Hulst (1950) and largely used with Metis
observations (e.g., Romoli et al., 2021; Antonucci et al., 2023), the coronal
electron density can be determined by inverting the coefficients of a power-
law fit performed on the observed pB values. The combined effect of the
contamination of the K corona by the F corona for distances larger than $10$
R⊙ (e.g., Howard et al., 2019) and a possible instrumental contribution
appearing as a rise in pB values at the outer edge of the detector Field Of
View (FOV) can in principle be responsible for the flattening of the pB radial
profile at heights above $10$ R⊙ (around the value indicated by the horizontal
line). Accordingly, the power-law fitting has been performed up to heights of
$10$ R⊙ and then extrapolated beyond that (red line in Figure 1(d)).
Anticipating the results presented in the next section, it is already worth
mentioning here that the electron density profile thus derived was checked and
validated with the different technique by Hayes et al. (2001) and was verified
to scale approximately as $r^{-2}$ at large distances from the Sun, at the
outer edge of the Metis FOV (i.e., $r>10$ R⊙).

Figure 1: (a) Metis pB image on June $01$, $2022$ at $22$:$40$ UT with PSP
marked in its POS as a blue dot; the PSP direction is indicated by a white
dashed line. (b) Projection of the PSP position (blue dot) onto the Carrington
map of the squashing factor $Q$ at $10$ R⊙. (c) PSP magnetic field and plasma
time series (from top to bottom: wind bulk and Alfvén speed, magnetic field
intensity and angle with the radial, proton density and temperature) during
the quadrature with SO. (d) Radial profile of the Metis pB observations at the
latitude corresponding to the PSP location (open squares), along with the van
de Hulst (1950) power-law fit (red line); the horizontal line at $1.2\times
10^{-11}$ B$/$B⊙ represents the flattening level probably due to the
contamination by the F corona and instrumental effect.

In order to derive an empirical coronal flow model from the joint SO/Metis –
PSP observations of the solar corona, the basic equations of solar wind theory
need to be solved. A steady, radial flow of an ideal plasma is fully described
by the equations for mass, momentum and energy conservation below:

$\frac{d}{dr}(\rho UA)=0;$ (1) $\rho
U\frac{dU}{dr}=-\frac{dp}{dr}-\rho\frac{GM_{\odot}}{r^{2}};$ (2)
$U\frac{dp}{dr}=-\gamma p\nabla\cdot\mathbf{U}+(\gamma-1)\rho\epsilon,$ (3)

where $\rho$ is the mass density, $\mathbf{U}$ is the solar wind velocity,
$A=f(r)r^{2}$ is the cross-sectional area of the flow tube connecting the
equatorial coronal hole to the heliosphere (with $f(r)$ being the expansion
factor), $p=nk_{B}T$ is the thermal pressure (with $n$, $T$, and $k_{B}$ being
the number density, temperature and Boltzmann constant, respectively), $G$ is
the gravitational constant, $M_{\odot}$ is the solar mass, $\gamma$ is the
adiabatic index, $\epsilon$ is the heating per unit mass, and $r$ is the
radial coordinate. Note that the empirical model assumes only a very hot
thermal plasma at the origin and that no distributed heating occurs due to the
dissipation of turbulence. The heating rate $\epsilon$ is primarily due to
low-frequency nearly-incompressible turbulence and expressed as (Matthaeus et
al., 1999a)

$\epsilon\equiv\frac{1}{2}\alpha\frac{\langle|\mathbf{z}^{+}|^{2}\rangle^{2}\langle|\mathbf{z}^{-}|^{2}\rangle+\langle|\mathbf{z}^{+}|^{2}\rangle\langle|\mathbf{z}^{-}|^{2}\rangle^{2}}{\lambda},$
(4)

where $\alpha$ is the Kármán-Taylor constant (usually around $0.125$, see
Usmanov et al., 2014, and references therein),
$(1/2)\langle|\mathbf{z}^{\pm}|^{2}\rangle$ are the energies associated with
the outward/inward fluctuations given in terms of Elsässer variables,
$\mathbf{z}^{\pm}=\mathbf{u}\pm\frac{\mathbf{b}}{\sqrt{\mu\rho}}$ (with
$\mathbf{u}$ and $\mathbf{b}$ being the velocity and magnetic field
fluctuations around the corresponding mean fields, $\langle\mathbf{U}\rangle$
and $\langle\mathbf{B}\rangle$, respectively, and $\mu$ the magnetic
permeability), $\lambda$ is the turbulence correlation length, and
$\langle...\rangle$ indicates time averaging over the data sample. Assuming
that most of the turbulent energy comes from Alfvénic fluctuations, the system
will be characterized by a balance between kinetic and magnetic energies,
i.e., $\langle|\mathbf{u}|^{2}\rangle\sim\langle|\mathbf{b}|^{2}\rangle$.
Equation (4) can thus be conveniently rewritten as (Adhikari et al., 2017)

$\epsilon=\frac{E_{b}^{3/2}}{\left[C_{k}\log\left(\frac{1}{k_{inj}\lambda_{b}}\right)\right]^{3/2}\lambda_{b}},$
(5)

where $E_{b}=\langle|\mathbf{b}|^{2}\rangle/(\mu\rho)$ and $\lambda_{b}$ are
the energy and correlation length associated with magnetic field fluctuations,
respectively, $C_{k}=1.6$ is the Kolmogorov constant, and $k_{inj}=1.07\times
10^{-9}$ km-1 is the large-scale injection wavenumber corresponding to the
solar rotation frequency (related to each other by the Sun’s rotational
velocity). Equation (5) is derived on the basis of a dimensional analysis of
the power spectral density of the turbulent magnetic energy $E_{b}$, assuming
that it scales with the wavenumber $k$ as $Ak^{-1}$ and
$C_{k}\epsilon^{2/3}k^{-5/3}$ (with $A$ being a constant) in the energy-
containing and inertial ranges, respectively, imposing equality of the
spectrum branches at the frequency break $k_{b}$ and integrating from
$k_{inj}$ to $k_{b}$. The only assumptions underlying the expression in
Equation (5) are thus a Kolmogorov-like (Kolmogorov, 1941) power law at fluid
scales and that the correlation length $\lambda_{b}$ corresponds to the
spectral break $k_{b}$ separating injection and inertial ranges. Although
customarily used (e.g., Telloni et al., 2022b), it is worth noting that it is
a rough estimate of the heating rate that does not take into account, for
example, the anisotropy in the solar wind turbulence fluctuations. In
addition, in the Iroshnikov-Kraichnan picture of turbulence (Iroshnikov, 1963;
Kraichnan, 1965), with a shallower spectrum scaling as $k^{-3/2}$ as usually
observed closer to the Sun (e.g., Chen et al., 2020; Duan et al., 2020;
Telloni et al., 2021b), the value for $\epsilon$ as provided by Equation (5)
would be a factor of $\lambda_{b}^{-1/4}$ larger. Completing Equations (1)–(3)
is the conservation of magnetic flux, $\frac{d}{dr}(BA)=0$.

Integrating the equations for mass and magnetic flux conservation, it follows
that the large-scale trends of the solar wind parameters (i.e., $U$, $\rho$,
and $B$) can be estimated based on the values at some reference point
$r=r_{0}$. Specifically, from (i) the estimates of the coronal mass density
$\rho(r)=0.95m_{p}n_{e}$ (for a fully ionized plasma with $2.5\%$ helium,
Moses et al., 2020, and where $m_{p}$ is the proton mass) as a function of the
heliocentric distance derived from the pB Metis observations, (ii) the PSP
plasma and magnetic field measurements at $r_{0}=13.3$ R⊙, and (iii) the
expansion factor $f(r)$ provided by the PSI $3$D MHD simulations (which
results in only a weak dependence on height, i.e., $f(r)\sim 1$, away from the
strong fields at the base of the corona due to the rooting of the open flux
near a low-lying pseudostreamer-type configuration), it is possible to
empirically evaluate the solar wind speed and magnetic field strength radial
profiles, $U(r)$ and $B(r)$, from $6.3$ to $13.3$ R⊙. The Alfvén speed
$U_{A}(r)$ and the solar wind energy flux $F_{w}(r)$ can be then immediately
quantified in the Metis FOV as $U_{A}=B/\sqrt{\mu\rho}$ and
$F_{w}=\frac{1}{2}\rho U^{3}$. In other words, in this joint SO/Metis – PSP
analysis, Metis provides the radial trends for all the magnetofluid and
turbulence parameters, while PSP, which definitely entered the corona (indeed
the Alfvén mach number $M_{A}=U/U_{A}\sim 0.59$), sets the (absolute) value at
$r_{0}$.

The energy of the magnetic field fluctuations $E_{b}$ and, in turn, the
turbulent cascade (heating) rate $\epsilon$ can be empirically estimated
deeper in the solar corona at the heights observed with Metis, by propagating
the measurements made locally by PSP back to the Metis FOV. For simplicity, it
is assumed the propagation equation for Alfvén waves in an expanding flow
(Velli, 1993), generalized to include a dissipation term
$\mathcal{D}(\langle|\mathbf{z}^{+}|^{2}\rangle^{2}\langle|\mathbf{z}^{-}|^{2}\rangle,\langle|\mathbf{z}^{+}|^{2}\rangle\langle|\mathbf{z}^{-}|^{2}\rangle^{2},\lambda)$
and possible in-situ sources of turbulence $\mathcal{S}$ (Zhou & Matthaeus,
1990; Zank et al., 1996; Wang et al., 2022)

$\frac{\partial\mathbf{z}^{\pm}}{\partial
t}+(\mathbf{U}\pm\mathbf{U_{A}})\cdot\nabla\mathbf{z}^{\pm}+\mathbf{z}^{\mp}\cdot\nabla(\mathbf{U}\mp\mathbf{U_{A}})+\frac{1}{2}(\mathbf{z}^{-}-\mathbf{z}^{+})\nabla\cdot(\mathbf{U_{A}}\mp\frac{1}{2}\mathbf{U})=-\mathcal{D}(-)+\mathcal{S}.$
(6)

If the further simplifying assumption of linearity and a slowly varying
background is made, Equation (6) can be solved in the WKB approximation to
retain an analytically tractable solution. In the presence of a nonuniform but
stationary flow, the wave action flux
$S=\frac{1}{2}\rho|\mathbf{z}^{+}|^{2}/\omega$, namely the wave energy flux
per unit frequency $\omega$, is conserved (see, e.g., Velli, 1993, and
references therein). Assuming an outwardly directed magnetic field, wave
action conservation in the stationary WKB limit translates into

$\nabla\cdot(\mathbf{U}+\mathbf{U_{A}})S=0.$ (7)

Considering that the wave frequency in the absolute (stationary) frame, that
is the wave eigenfrequency $\omega_{0}$, is an invariant, the frequency in the
plasma frame $\omega$ is given by

$\omega=\frac{U_{A}}{U+U_{A}}\omega_{0}.$ (8)

Bearing furthermore in mind the mass flux conservation $\rho Ur^{2}=const$ and
the divergence formula in spherical coordinates for a purely radial vector
$\nabla\cdot\mathbf{F}=\frac{1}{r^{2}}\frac{\partial}{\partial
r}(r^{2}F_{r})$, Equation (7) simplifies to

$\frac{(U+U_{A})^{2}}{UU_{A}}|\mathbf{z}^{+}|^{2}=C,$ (9)

with $C$ a constant. Equation (9) represents, in the WKB limit, a very useful
expression for the radial dependence of the Alfvén wave energy, once estimated
at a reference point $r=r_{0}$ (i.e., at PSP). It predicts that the outwardly
propagating Alfvén mode energy peaks (and, therefore, turbulence is maximum)
where $U=U_{A}$, i.e., at the Alfvén point. In the case of pure Alfvén waves
(i.e., $\sigma_{c}=\pm 1$ and $\sigma_{r}=0$, where $\sigma_{c}$ and
$\sigma_{r}$ are the normalized cross-helicity and residual energy, which
measure the imbalance between outward and inward modes and between kinetic and
magnetic energies, respectively, e.g., Tu & Marsch, 1995), the above equation
can also be used for the energy of the magnetic fluctuations $E_{b}$.
Associating, to a first approximation, the wavelength of turbulence with the
expansion of the flow tube, i.e., $\lambda_{b}\propto\sqrt{A}\sim r$, a
complete set of functional forms for estimating the energy transfer rate at
the coronal heights observed with Metis is provided.

In the PSP $0.5$-hour interval considered in the present analysis (Figure
1(c)), $\sigma_{c}=0.97$ and $\sigma_{r}=-0.10$, suggesting the overwhelming
presence of Alfvén waves in the downwind direction. Furthermore, the power
spectrum of the magnetic field fluctuations may be compatible with the $5/3$
Kolmogorov turbulence (the power-law fit performed at fluid scales to $E_{b}$
in fact returns a spectral index of $1.61\pm 0.04$). It follows that Equation
(5) can be safely back-projected from the PSP location, which is below the
Alfvén point ($U<U_{A}$, see Figure 1(c)), into the Metis FOV and thus the
coronal heating rate directly derived in the solar corona.

##  3 Empirical slow solar wind and WKB-like turbulence model

Figure 2 shows the solar wind (left panels) and turbulence (right panels)
models empirically derived according to the methodological approach described
in the previous section (PSP measurements-based estimates are indicated by a
blue dot). Specifically, Figures 2(a)–(d) display the modeled coronal plasma
flow speed, mass density and energy flux, and coronal magnetic field,
respectively, compared with the corresponding quantities obtained from the PSI
$3$D MHD simulations (red lines). In Figure 2(a), also depicted are the Alfvén
speed (thin lines) and, for comparison, the spherically symmetric expansion of
an isothermal $T=1.5\times 10^{6}$ K corona (Parker, 1958, green line).

Figure 2: Data-based (black) and physics-based (red) modeling of the radial
evolution of solar wind $U$ (thick lines) and Alfvén $U_{A}$ (thin lines)
speed (a), plasma mass density $\rho$ (b), solar wind flux energy $F_{w}$ (c),
magnetic field $B$ (d), turbulent energy $E_{b}$ (e) and correlation length
$\lambda_{b}$ (f), and heating rate $\epsilon$ (g) in the slow coronal flow
jointly observed with SO/Metis and PSP during their quadrature on June $01$,
$2022$. The blue dot in each panel refers to the values obtained with PSP
measurements. The Parker classical model for an expanding isothermal $1.5$ MK
corona is shown as a green line in (a).

Immediately evident is the good agreement, both in radial trends and absolute
values, between the empirical model and the outcome of the MHD simulations.
This allows a first, prompt consideration. In fact, the two models, in
addition to being independently derived, are also the result of somewhat
opposite approaches. The MHD simulation is based on a forward modeling driven
by (remote) photospheric measurements and the dissipation of Alfvénic
turbulence as the mechanism for heating the coronal plasma. The empirical
model obtained in this work, on the other hand, is a backward extrapolation
based on PSP in-situ measurements and the solution of the Euler equations in
the WKB limit. Thus, the good agreement between the two models (for both fluid
and magnetic parameters) indicates, on the one hand, the accuracy of the
present analysis and especially of the Metis observations and, on the other
hand, that the aforementioned assumed physical mechanism underlying the
heating and subsequent acceleration of the coronal wind complies with
observations, again identifying it as the most likely mechanism of coronal
heating (in contrast to the alternative high-frequency resonant dissipative
processes). More quantitatively, the empirically-derived and simulation-based
models differ (on average) by $16\%$ for the flow speed, $12\%$ for the plasma
density and magnetic field, and $37\%$ for the wind energy flux (for which, as
a derived quantity, the discrepancies associated with the fundamental
parameters $U$ and $\rho$ obviously widen). In discussing in greater detail
the expansion rate of the wind, Figure 2(a) clearly shows that, although at
the distances observed with Metis the coronal plasma is already largely
accelerated, some residual acceleration still persists: in fact, the wind
speed increases from $\sim 250$ to $\sim 310$ km s-1 over a range of $7$ R⊙.
Nevertheless, as evidenced by the basically good agreement with Parker’s
model, this acceleration may just be thermally driven, which indicates that
collisionless field-particle interaction mechanisms for heating (and thereby
boosting) the wind are most effective lower than the distances observed with
Metis (as, for instance, shown in Telloni et al., 2023). Rough estimates of
the power-law radial dependence of the physical quantities displayed in
Figures 2(a)–(d), which follow from the continuity equations above and the
experimental finding that $\rho\sim r^{-2.21}$, can be given. It turns out
that $U\sim r^{0.23}$, $F_{w}\sim r^{-1.52}$, and $B\sim r^{-1.98}$ for $6.3$
R⊙ $<r<13.3$ R⊙.

Figures 2(e)–(g) show the empirically modeled turbulence energy, correlation
length and cascade rate, respectively. As a consequence of Equation (9) and
having assumed that $\lambda_{b}\propto\sqrt{A}$, the fluctuating magnetic
energy and the corresponding correlation length decrease from the PSP position
towards the Sun (according to $E_{b}\sim r^{0.45}$ and $\lambda_{b}\sim
r^{0.99}$, respectively). The resulting heating rate in the slow coronal
plasma jointly observed with SO/Metis – PSP keeps a relatively constant value
throughout the considered heliocentric distances ($\epsilon\sim r^{-0.18}$),
in agreement with earlier studies of wind equations and turbulence models
(e.g., Dmitruk et al., 2002), which show that the energy per unit mass does
not fall off rapidly with altitude. In this regard, it is worth noting that
the slight peak in $\epsilon$ observed at about $7-8$ R⊙ is merely due to the
combination of the $E_{b}$ and $\lambda_{b}$ radial trends and, hence, does
not indicate any particular underlying physical process. In fact, the heating
rate is expected to peak much closer in, at the temperature maximum and the
sonic point (around around $2-4$ R⊙), i.e., where plasma heating and
acceleration are most effective. The average (turbulent) heating rate per unit
volume $\overline{\epsilon\rho}=5.5\times 10^{-12}$ J m-3 s-1 is somewhat
lower that the estimates of $10^{-9}-10^{-11}$ J m-3 s-1 recently reported by
Chandran et al. (2009) and Sasikumar Raja et al. (2021) in approximately the
same altitude region ($5-45$ R⊙). This is not surprising, since these authors
provided upper limits for $\epsilon$, whereas those presented here are lower
estimates. In fact, the WKB approximation does not take into account the
undoubted dissipation of $\mathbf{z}^{+}$ modes. As a result, the rate at
which their energy increases with distance has to be lower than the WKB
prediction. Hence, from the PSP position down into the solar corona, this
means that their energy $E_{b}$ would fall off less rapidly, resulting in an
energy transfer rate higher than estimated here in the WKB limit. Moreover,
under the likely scenario that deeper into the corona the spectrum of the
magnetic field fluctuations more closely resembles the turbulence à la
Iroshnikov-Kraichnan, for the above, the energy transfer rate would be larger
by a factor of $\lambda_{b}^{-1/4}$, resulting in an average rate per unit
volume of $5.2\times 10^{-11}$ J m-3 s-1, thus at the lower bound of the range
of values previously reported in the literature.

The coronal energy loss $H$, expressed as a flux, can be estimated by
integrating the heating rate in the Metis FOV, i.e., from $6.3$ to $13.3$ R⊙,

$H=\int\epsilon(r)\rho(r)dr=2.7\times
10^{-2}\,\mathrm{J}\,\mathrm{m}^{-2}\,\mathrm{s}^{-1}.$ (10)

It turns out that $H$ is $33\%$ of the total energy flux of the solar wind
($F_{w}=8.1\times 10^{-2}$ J m-2 s-1). This is expected, since, as mentioned
just above, most of the acceleration has already taken place at the heights
observed with Metis. This energy deposition feeds the observed residual
acceleration of the slow solar wind and heats the non-adiabatic coronal
plasma. Indeed, in a collisionless plasma with no additional heating, based on
the conservation of the Chew-Goldberger-Low (CGL; Chew et al., 1956)
invariants, the total temperature should scale as $r^{-4/3}$ with distance.
This is not observed either in the corona or in interplanetary space (where a
radial scaling greater than that predicted by the adiabatic theory of plasma
thermal dynamics is found, e.g., Zhao et al., 2019), implying that plasma
heating (via turbulence dissipation) occurs, although less significantly even
at distances far beyond where the heating and acceleration processes are most
at work (around the sonic point). It is well known that the deficit in
internal energy implied by an adiabatic expansion persists at least to $1$ au,
and in fact well beyond (Richardson et al., 1995; Matthaeus et al., 1999b).

As a final remark, although the present WKB-like analysis is important as it
provides a model-independent and empirical lower limit of the coronal heating
rate (thanks to the coordinated Metis/SO – PSP observations), it should be
recalled that it is well known that the radial evolution of the turbulent
solar wind can resemble WKB theory even when driven by waves (Roberts, 1989)
or when turbulent dissipation and shear driving is included (Zank et al.,
1996). It is also worth noting that the WKB approximation does not hold in the
expanding solar corona. In fact, the wave-only WKB theory predicts no
dissipation, whereas turbulence has largely been dissipated far below to
accelerate the wind to supersonic speeds. It is therefore evident that a more
realistic description of the turbulence evolution from the subsonic wind to
the PSP location is needed to compare directly with the Metis observations and
thus accurately estimate the (turbulent) heating rate in the solar corona. A
variety of turbulence models (e.g., Verdini et al., 2010; Oughton et al.,
2011) may be useful as extension of the present simplified WKB-like model in
order to explain observations when linked self-consistently with large scale
MHD equations.

Solar Orbiter is a space mission of international collaboration between ESA
and NASA, operated by ESA. D.T. was partially supported by the Italian Space
Agency (ASI) under contract 2018-30-HH.0. G.P.Z., L.A., and L.-L.Z.
acknowledge the partial support of a NASA Parker Solar Probe contract
SV4-84017, an NSF EPSCoR RII-Track-1 Cooperative Agreement OIA-2148653, and a
NASA IMAP grant through SUB000313/80GSFC19C0027. S.B and J.K. acknowledge
support from NASA contract NNN06AA01C. The Metis program is supported by ASI
under contracts to the National Institute for Astrophysics and industrial
partners. Metis was built with hardware contributions from Germany
(Bundesministerium für Wirtschaft und Energie through the Deutsches Zentrum
für Luft- und Raumfahrt e.V.), the Czech Republic (PRODEX) and ESA. The Metis
team thanks the former PI, Ester Antonucci, for leading the development of
Metis until the final delivery to ESA. D.T. also wishes to thank her for the
helpful comments to the paper. The Metis data analyzed in this paper are
available from the PI on request. Parker Solar Probe data was downloaded from
the NASA’s Space Physics Data Facility (https://spdf.gsfc.nasa.gov).

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# Improving Next-Application Prediction with Deep Personalized-Attention
Neural Network

Jun Zhu, Gautier Viaud, Céline Hudelot  Mathématiques et Informatique pour la
Complexité et les Systèmes
CentraleSupélec, Université Paris-Saclay
Gif-sur-Yvette, France
{jun.zhu, gautier.viaud<EMAIL_ADDRESS>
###### Abstract

Recently, due to the ubiquity and supremacy of E-recruitment platforms, job
recommender systems have been largely studied. In this paper, we tackle the
next job application problem, which has many practical applications. In
particular, we propose to leverage next-item recommendation approaches to
consider better the job seeker’s career preference to discover the next
relevant job postings (referred to jobs for short) they might apply for. Our
proposed model, named Personalized-Attention Next-Application Prediction
(PANAP), is composed of three modules. The first module learns job
representations from textual content and metadata attributes in an
unsupervised way. The second module learns job seeker representations. It
includes a personalized-attention mechanism that can adapt the importance of
each job in the learned career preference representation to the specific job
seeker’s profile. The attention mechanism also brings some interpretability to
learned representations. Then, the third module models the Next-Application
Prediction task as a top-$K$ search process based on the similarity of
representations. In addition, the geographic location is an essential factor
that affects the preferences of job seekers in the recruitment domain.
Therefore, we explore the influence of geographic location on the model
performance from the perspective of negative sampling strategies. Experiments
on the public CareerBuilder12 dataset show the interest in our approach.

###### Index Terms:

Next-Application Prediction, Personalized-Attention, Neural Network, Job
Recommendation

##  I Introduction

Recently, the rapid development of E-recruitment has considerably influenced
the human resource management field. In particular, due to the explosion of
recruitment data (i.e., over 3 million jobs are posted on LinkedIn in the U.S.
every month111https://economicgraph.linkedin.com/resources/linkedin-workforce-
report-february-2021), many works have been proposed to build effective job
recommender systems [1]. Like many other domains, Deep Learning (DL) based
recommendation models have been extensively studied in the recruitment domain.
Nevertheless, while mostly studied, there are still important issues with
current approaches, which we summarize below:

  * •

Issue 1: The first issue is related to intrinsically dynamic and constantly
evolving Job-Person interactions, which can be built from job application
records or working histories. Identically, we assume that temporal relations
between application records tend to point towards sequential/session-based
recommendations222In this paper, we describe the recommendation task to
explore the sequential data by a broader term “sequential”. Thus “session” and
“sequential” are interchangeable.. To our knowledge, although various session-
based methods have been proposed in others domains, session-based job
recommendation is less studied. Moreover, another important aspect of the Job-
Person interaction matrix is its sparsity, since most job seekers only apply
for a few job positions or work as a specific occupation.

  * •

Issue 2: The second issue is related to the personalization of
recommendations. Indeed, two job seekers can apply for the same job with
different motivations. As a consequence, the important information in a job is
specific to the job seeker. Thus, it is essential to weigh the jobs
differently in the modeling tasks of job seeker careers. General-purpose
approaches are often unable to capture such specific information. In addition,
the personal context information of the job seeker is of prime importance in
the recruitment domain. For example, the geographic location of a job seeker
often has a substantial impact on the application decision and the
recommendation result. It raises two opposite goals: (i) it is essential to
consider this personal context information when modeling personal preference,
(ii) it also requires the recommender system to distinguish this information
from more core content, i.e., job content, when making recommendations.

  * •

Issue 3: The last issue is due to the evolving nature of the recruitment
domain. In fact, in addition, to constantly updated E-recruitment websites
(i.e., a large number of job postings are added or removed daily, and hundreds
of candidate profiles are created or updated), the domain is still evolving
(i.e., new occupations, formations, and skills). All of this intensifies the
cold-start problem. Therefore, it is impractical to build a static method that
cannot be adjusted quickly as the data constantly changes. We thus need to
avoid retraining a deep model for each change. Moreover, from the perspective
of content analysis methods, due to the sensitivity and privacy of the data in
the recruitment domain, it is not easy to create an unbiased and evolving
labeled dataset [2, 3]. Thus it prevents the use of supervised machine
learning methods.

In this paper, we address the problem of job recommendation under the task of
the next job application prediction. We propose a hybrid Personalized-
Attention Next-Application Prediction model (PANAP) to answer the previous
issues partially. Specifically, for Issue 1, we model the next job application
problem as a sequential recommendation problem, and then compare our proposed
model with different session-based recommendation methods to analyze the
dynamics in the recruitment domain. Our model is composed of three independent
modules. The first module learns job representations from textual job content
and available metadata in an unsupervised way to answer Issue 3. The second
module contains a personalized-attention mechanism to learn the job seeker
representation that can solve the problem of personalized recommendations
mentioned in Issue 2. The third module is used to predict the next-application
that a job seeker will apply for. In order to answer Issue 3, this module is
inspired by the Deep Structured Semantic Model (DSSM) [4] with a training loss
based on representation similarities. Moreover, based on the nature of the
recruitment domain, we propose a specific negative sampling strategy
considering the geographic location factor. To summarize our contributions as
follows:

  * •

A new session-based model, named PANAP, is used for Next-Application
Prediction task. It integrates the personalized-attention mechanism to improve
the recommendation accuracy.

  * •

An extensive experimental study has been conducted in the CareerBuilder12
dataset, allowing us to investigate the effect of leveraging different
metadata types and textual job contents, and the impact of different sampling
strategies on the quality of recommendations.

##  II Related Work

Job recommender systems based on traditional approaches (i.e., collaborative-,
content-based and hybrid methods) have been well-studied [5]. However, these
methods have some important limitations in the recruitment domain. For
example, the sparsity of Job-Person interaction matrices limits the
performance of collaborative-based methods. When the information contained in
job contents is insufficient, or the feature-engineering on contents is
difficult, content-based approaches may become inaccurate. Furthermore, these
methods generally lack the ability to model personal information. Recently, DL
has dramatically revolutionized recommendation systems. As in other domains,
DL-based methods have been applied in the recruitment domain. These works can
be categorized into three classes according to the data they used and the
learning paradigm. (i) Text-based matching models: When labeled Person-Job
matching records are available, the recommendation problem can be seen as a
supervised text matching task to match jobs with candidates automatically.
Many approaches [2, 6, 3] with different architectures have been proposed in
the literature. However, due to the protection of trade secrets and personal
privacy, collecting such labeled records is difficult. (ii) Unsupervised
representation learning models: When matching records are not available, the
recommendation task can be seen as the learning of representations of job and
person in the same embedding space. The recommendation is then modeled as a
top-$K$ search based on these learned embeddings using similarity-based
algorithms. For example, [7] uses graphs to learn job title and skills
representations. [8] proposes a collective multi-view method to learn job
title representations. In these approaches, the different level/view of
information is of prime importance. (iii) Approaches based on career paths:
The methods mentioned above primarily focus on characterizing the person or
the job information. Transitions in career paths are less considered. To
address this problem, some works are proposed, e.g., NEMO [9] explores action
dependencies in career paths through Long Short-Term Memory (LSTM) [10] to
predict the next career move. In [11], they use a hierarchical LSTM to predict
the next potential employer of a person and how long he/she will stay in the
new position. However, predicting the next job application based on
application records received relatively little consideration except [12],
which recommends the next job posting in a $K$-nearest neighbor manner. It is
one of our baselines.

The next-application prediction problem can be considered as a sequential
recommendation problem, which has been extensively studied recently in other
domains, such as news or product recommendation [13, 14, 15, 16]. Recurrent
Neural Networks (RNNs) have been widely studied for the sequential
recommendation since they have demonstrated their effectiveness in processing
sequential data, one representative work is GRU4Rec [17, 18, 19]. The
attention mechanism [20] has shown promising potential in improvements of
accuracy and interpretability, like the vanilla attention-based [14, 21] and
self-attention based one [22, 23].

Different from the previous state-of-the-art in the recruitment domain, in
this paper, we tackle the problem of next-application prediction by leveraging
session-based recommendation models. We further study the effectiveness of
session-based models in the recruitment domain. Moreover, we integrate the
personalized-attention mechanism to model the different informativeness of job
postings for different job seekers, and we take into account the “location”
factor when making recommendations.

##  III Problem Formulation

Let $\mathcal{U}$ be a set of job seekers and $\mathcal{J}$ be a set of job
positions (referred to jobs for short). A job $j\in\mathcal{J}$ can be
represented as
$j=(\text{ID}_{j},\\{w_{1}^{j},\dots,w_{m}^{j}\\},\text{meta}_{j})$, where
$\text{ID}_{j}$ is the unique identifier of job $j$.
$\\{w_{1}^{j},\dots,w_{m}^{j}\\}$ is a sequence of $m$ words, representing the
textual content of job $j$. It can be any text information about the job.
$\text{meta}_{j}$ are associated metadata attributes (i.e., job city).
Similarly, each job seeker $u\in\mathcal{U}$ can be summarized as a tuple
$u=(\text{ID}_{u},\mathcal{H}_{u},\text{meta}_{u})$ with an identifier
$\text{ID}_{u}$, a professional profile $\mathcal{H}_{u}$, and metadata
attributes $\text{meta}_{u}$, like the education background or the
geographical information. Specifically, the professional profile
$\mathcal{H}_{u}$ can be represented either as his/her working experience or
his/her job application record. In this work, we target at the job application
record, and we denote $\mathcal{H}_{u}$ as a job sequence (session) ordered by
time $\mathcal{H}_{u}=\\{j_{1},\dots,j_{n}\\}$, where each $j_{i}$ is a
specific job in $\mathcal{J}$. With the above notations, the Next-Application
Prediction problem can be formally defined as follows:

Given a set of jobs $\mathcal{J}$ and a set of job seeker $\mathcal{U}$, for
each specific job seeker $u\in\mathcal{U}$ who has a job application sequence
$\mathcal{H}_{u}=\\{j_{1},\dots,j_{t}\\}$ at one specific time step $t$.

Predict the next most likely job $j_{t+1}$ that the job seeker $u$ might apply
for, which means maximizing the likelihood
$P(j_{t+1}=j^{+}|\mathcal{H}_{u},\mathcal{J})$ of the actual next-applied job
$j^{+}$ given the application sequence $\mathcal{H}_{u}$.

##  IV The Proposed Model

An overview of our PANAP model is given in Figure 1. We detail our model by
describing these three different modules.

Figure 1: The proposed PANAP framework consists of three parts: (i) Job
Content Representation (blue dashed box) is used for job content
representation learning. (ii) Job Seeker Representation (green dashed box)
uses the personalized-attention mechanism to characterize the career
preference of job seekers based on the Job Application Record. (iii) Next-
Application Predictor (yellow dashed box) utilizes a ranking loss based on the
representation similarity of job and job seeker to train the model.

###  IV-A Job Content Representation

JCR (Job Content Representation), is responsible for learning a representation
of each job $j\in\mathcal{J}$, where
$j=(\text{ID}_{j},\\{w_{1}^{j},\dots,w_{m}^{j}\\},\text{meta}_{j})$.
Specifically, for job $j$, a textual representation
$\bm{h}_{j}\in\mathbb{R}^{d}$ is learned from its textual content
$\\{w_{1}^{j},\dots,w_{m_{j}}^{j}\\}$ with a text encoder:

$\bm{h}_{j}=\text{text\\_encoder}(\\{w_{1}^{j},\dots,w_{m}^{j}\\}),$ (1)

where the text encoder can be some unsupervised textual embedding approaches
(e.g., Word2Vec [24] and Doc2Vec [25]) or pre-trained models (e.g., BERT
[26]). We believe that textual representations generated by different text
encoders will affect the accuracy of the recommendations. It will a research
line in our future work. The metadata attributes $\text{meta}_{j}$ (e.g., city
and state) are respectively embedded into vectors (e.g.,
$\bm{v}_{j}^{\text{city}}$ and $\bm{v}_{j}^{\text{state}}$) through different
trainable embedding matrices (e.g., $\bm{M}^{\text{city}}$ and
$\bm{M}^{\text{state}}$), which will be jointly learned during the training
process. The vector $\bm{v}_{j}^{\text{meta}}$ is then obtained by
concatenation of all vectors,
$\bm{v}_{j}^{\text{meta}}=[\bm{v}_{j}^{\text{city}}\oplus\bm{v}_{j}^{\text{country}}\oplus\dots]$,
where the symbol $\oplus$ represents the concatenation operator. Then the
textual job representation $\bm{h}_{j}$ and job metadata vector
$\bm{v}_{j}^{\text{meta}}$ are combined by using a sequence of Fully Connected
(FC) layers to produce the Job Content Vector
$\bm{v}_{j}\in\mathbb{R}^{d^{\mathcal{J}}}$:

$\bm{v}_{j}=\text{FCs}\big{(}[\bm{h}_{j}\oplus\bm{v}_{j}^{\text{meta}}]\big{)}.$
(2)

It is worth mentioning that the textual job representation $\bm{h}_{j}$ is
trained separately in an unsupervised way or generated by a pre-trained
language model, so that a new job can be added easily, thereby alleviating the
job cold-start problem.

###  IV-B Job Seeker Representation

JSR (Job Seeker Representation), is responsible for learning representations
of job seekers. It requires three inputs: the identifier $\text{ID}_{u}$ of
job seeker $u$, his/her historical applied job sequence
$\mathcal{H}_{u}=\\{j_{1},\dots,j_{n}\\}$333We omit the superscript of $u$
without loss of clarity. and associated metadata attributes $\text{meta}_{u}$.
In addition, $\mathcal{H}_{u}$ can be transformed as a sequence of Job Content
Vectors $\\{\bm{v}_{j_{1}},\dots,\bm{v}_{j_{n}}\\}$, where each
$\bm{v}_{j_{i}}\in\mathbb{R}^{d^{\mathcal{J}}}$ is obtained from JCR module.
Since the same job may have different informativeness for different job
seekers, it contributes differently to characterize career profiles of job
seekers. Then, we learn the job seeker representation with a personalized-
attention mechanism [27]. We first embed the $\text{ID}_{u}$ of job seeker $u$
into a vector $\bm{e}_{u}$ using an identifier embedding matrix
$\mathbf{M}^{\mathcal{U}}\in\mathbb{R}^{|\mathcal{U}|\times d^{s}}$, where
$d^{s}$ denotes the dimension of identifier embedding. Then $\bm{e}_{u}$ is
passed to a dense layer parameterized with
$\bm{W}^{q}\in\mathbb{R}^{d^{\mathcal{U}}\times d^{q}}$ and
$\bm{b}^{q}\in\mathbb{R}^{d^{q}}$ to form the preference query vector
$\bm{q}_{u}=\text{ReLU}(\bm{W}^{q}\times\bm{e}_{u}+\bm{b}^{q})$, where $d^{q}$
is the query dimension. The importance score of $i$-th job for job seeker $u$
is calculated as follows:

$\alpha_{i}=\exp\big{(}\bm{v}_{j_{i}}^{T}\bm{p}_{u}\big{)}/\textstyle\sum_{i^{\prime}=1}^{n}\exp{\big{(}\bm{v}_{j_{i^{\prime}}}^{T}\bm{p}_{u}}\big{)},$
(3)

where
$\bm{p}_{u}=\text{tanh}\big{(}\bm{W}^{a}\times\bm{q}_{u}+\bm{b}^{a}\big{)}$,
with projection parameters $\bm{W}^{a}\in\mathbb{R}^{d^{q}\times
d^{\mathcal{J}}}$ and $\bm{b}^{a}\in\mathbb{R}^{d^{\mathcal{J}}}$. Note that
the attention scores might differ for the same job, as the query vector
depends on the ID of the job seeker. This allows forming a weighted career
preference representation $\bm{h}_{u}$ for job seeker $u$:

$\bm{h}_{u}=\textstyle\sum_{i=1}^{n}\alpha_{i}\bm{v}_{j_{i}}.$ (4)

Similar to JCR module, the metadata vector of job seeker $u$ is represented as
$\bm{v}_{u}^{\text{meta}}=[\bm{v}_{u}^{\text{city}}\oplus\bm{v}_{u}^{\text{education}}\oplus\dots]$
by concatenation. $\bm{v}_{u}^{\text{meta}}$ and $\bm{h}_{u}$ are combined by
using a sequence of FC layers to produce the final job seeker representation:

$\bm{v}_{u}=\text{FCs}\big{(}[\bm{h}_{u}\oplus\bm{v}_{u}^{\text{meta}}]\big{)},$
(5)

where $\bm{v}_{u}$ has a dimension of $d^{\mathcal{U}}$, which is equal to the
dimension of Job Content Vector $\bm{v}_{j}$ (e.g.,
$d^{\mathcal{U}}=d^{\mathcal{J}}$).

###  IV-C Next-Application Predictor

In classical deep recommendation models, the output is usually a probability
vector whose dimension is the number of available items. In our scenario of
job recommendation, due to the evolving nature of the recruitment domain
(Issue 3), such models are not tractable in practice. Thus this dynamic
scenario needs an approach that does not need to retrain the whole network at
each change. As a consequence, inspired by the highly dynamic news
recommendation scenario, we use the ranking loss [13] to train the predictor.
The principle of this loss is to train the predictor to maximize the
similarity between user preferences and positive samples, while minimizing
similarities with negative samples. The idea of the ranking loss comes from
DSSM [4], which is an effective document ranking model and has been leveraged
for the recommendation. In our case, it can immediately recommend a newly
published job as soon as its representation is learned. More precisely, given
the application history $\mathcal{H}_{u}=\\{j_{1},\dots,j_{n}\\}$ of job
seeker $u$, we formulate the above representation generating processes as
$\bm{v}_{u}=JSR(\text{ID}_{u},\mathcal{H}_{u},\text{meta}_{u};\Theta_{1})$,
where each $j_{i}\in\mathcal{H}_{u}$ can be embedded into a Job Content Vector
$\bm{v}_{j_{i}}$ through $JCR(j_{i};\Theta_{2})$, $\Theta_{1}$ and
$\Theta_{2}$ are model parameters. $\bm{v}_{u}$ and $\bm{v}_{j_{i}}$ are
vectors of the same dimension, and we thus can define the relevance score
between job seeker $u$ and job $j$ by the cosine similarity of their
representations:

$sim(u,j_{i})=cos(\bm{v}_{u},\bm{v}_{j_{i}})=\frac{\bm{v}_{u}\cdot\bm{v}_{j_{i}}}{\|\bm{v}_{u}\|\|\bm{v}_{j_{i}}\|}$
(6)

The posterior probability of $j$ being the next-job for $u$ is formulated as
follows:

$P(j|u)=\frac{\exp\big{(}sim(u,j)\big{)}}{\sum_{j^{\prime}\in{j^{+}\cup\mathcal{J}^{-}}}\exp\big{(}sim(u,j^{\prime})\big{)}},$
(7)

where $j^{+}$ is the actual next-application, and $\mathcal{J}^{-}$ represents
the negative sample set of jobs which are not applied by the job seeker during
his active session. The recommender should learn to maximize the similarity
between the content vector $\bm{v}_{j^{+}}$ of $j^{+}$ and the job seeker
preference vector $\bm{v}_{u}$, while minimizing similarities with job vector
$\bm{v}_{j^{-}}$ in the set $\mathcal{J}^{-}$.

$\mathcal{L}_{sim}=-\log\textstyle\prod_{(u,j^{+})}P(j^{+}|\text{ID}_{u},\mathcal{H}_{u},\text{meta}_{u};\Theta_{1},\Theta_{2}).$
(8)

###  IV-D Negative Sampling Strategies

Since our method is trained and evaluated with negative samples as described
in Section IV-C, the negative sample set $\mathcal{J}^{-}$ significantly
influences the model performance. A well-used sampling strategy is the mini-
batch based sampling proposed in [17], which treats the items from the other
training/evaluation sessions in the same mini-batch as negative samples. [19]
extends the mini-batch based strategy by adding additional samples (based on
unity or based on popularity). For some applications, such as news, music, and
video, popularity is also an essential factor that influences user choice
besides personal preference. Therefore, sampling based on popularity is a good
sampling strategy. However, in the recruitment field, unlike these
applications, the choices of job seekers are more influenced by their personal
contexts, such as the geographic location factor. When job seekers make a
choice, they usually first need to consider the job location. They are more
likely to apply for jobs in their cities or other cities not far from their
current locations (i.e., cities in the same state). We will prove this
observation in Section V-A. To solve this problem, our method cooperates with
the “location” metadata attributes (e.g., city, state and country) of job and
job seeker to model their representations, respectively. As a consequence,
these representations contain the “location” information. In addition,
considering the “location” when generating negative samples can enable our
model to learn useful information for more meaningful recommendations. More
experimental details are provided in Section V-B.

##  V Experiments

###  V-A Datasets

We employ CareerBuilder12444https://www.kaggle.com/c/job-recommendation to
evaluate our proposed method. Its statistics are given in Table I. In this
dataset, job seeker has five metadata: City, State, Country, Degree, and
Major. Job metadata are City, State and Country. The textual job content
includes a Job Title, a Job Description and some Job Requirements. From this
initial dataset, we created two datasets: (i) CB12_s like in [12], in which
sessions are created via a time-based split of 30 minutes inactivity
threshold, and we discarded sessions with less than two applications for next-
job prediction purpose. (ii) CB12_l uses all application records during 13
weeks to model the career profile of each job seeker. Thus, CB12_l has longer
sequences that enable us to evaluate the effectiveness of our proposed method.
We further split the last 14 days for testing and the remaining sessions for
training. We filter job applications in the test set that do not belong to the
training set as this enables a better comparison with the approaches, which
can only recommend items that have been used to train the model.

TABLE I: Statistics of datasets, $\\#$S represents the session number, and $\\#$A is the applications number. Avg_SLen is the average application number in sessions. $\\#$Metadata contains the cardinality of each metadata attribute. Dataset | $|\mathcal{U}|$ | $|\mathcal{J}|$ | $\\#$S | $\\#$A | Avg_SLen | $\\#$Metadata
---|---|---|---|---|---|---
City/State/Country/Degree/Major
CB12_s | 111,785 | 207,972 | 165,027 | 638,469 | 3.87 | 8,226/122/33/7/21,224
CB12_l | 137,642 | 239,581 | 137,642 | 772,305 | 5.61 | 8,856/130/37/7/25,201

(a) City

(b) State

Figure 2: The relationship between locations of the job seeker and the applied
job in CareerBuilder12 datasets.

As we described in Section IV-D, the “location” is an essential factor that
needs to be considered in the job recommendation. In order to further prove
this observation, in Figure 2, we illustrate the relationship between
locations of job seekers and applied jobs in CareerBuilder12 datasets. As
shown in Figure 2(b), most job seekers ($92.7\%$ in CB12_s and $93.8\%$ in
CB12_l) have applied for jobs in their states. Among these people, $81.3\%$
and $79.7\%$ of job seekers only consider jobs in their own states. Figure
2(a) shows that only $39.6\%$ and $40.5\%$ of job seekers apply for jobs in
their own cities, because job opportunities in their cities are usually
limited. People are also applying for jobs in other cities in the same state,
as explained above. Therefore, the “location” (the job site or the current
location of the job seeker) is an essential factor to be considered in the job
recommendation.

###  V-B Experimental Settings

In this experiment, job representations $\bm{h}_{j}$ were obtained by Doc2Vec
with dimension $d=300$ via the distributed memory. The dimension $d^{s}$ of
identifier embedding $\bm{e}_{u}$ and the query dimension $d^{q}$ were set to
100. The dimensions $d^{\mathcal{U}}$ and $d^{\mathcal{J}}$ were also set to
300. We applied the dropout technique with a rate of 0.2 to each layer and L2
regularization with rate 1e-4 to parameter weights. The PANAP was trained and
evaluated with a 256 mini-batch size, and we used 15 negative samples for
training and 50 for evaluation. The Adam [28] optimizer with a learning rate
of 5e-4 was used. Metadata attributes with low cardinality ($<=10$) were one-
hot encoded, and high cardinality attributes were represented as trainable
embeddings. We used two FC layers, with Leaky ReLU [29] and tanh activation
functions to combine metadata vectors.

We first explore a sampling strategy inspired by [30], and then propose an
improved sampling strategy for the job recommendation scenario. This improved
strategy considers the current geographic location of the job seeker and the
job site when sampling negative jobs. The two strategies are:

  * •

Strategy 1 (S1)-mini-batch + additional samples: It is proposed in [30], which
adds additional samples with a uniform sampling strategy from a global buffer
of the $N$ most recently applied jobs, when there are not enough negative
samples within the mini-batch. Jobs are uniformly sampled from the “candidate
set” (i.e., jobs within the mini-batch and additional jobs);

  * •

Strategy 2 (S2)-mini-batch + additional samples + location-biased: Different
from Strategy 1, in Strategy 2, we first select the jobs in the same state as
the job seeker from the “candidate set” as negative samples. When there are
not enough negative samples in the current state, we sample jobs in other
states from the “candidate set”.

The baselines used are listed as follows:

  * •

POP: recommends the most applied job;

  * •

Association Rule (AR): is a simplified version of association rule [31] with a
maximum rule size of two;

  * •

Content Similarity (CS): recommends similar jobs based on the cosine
similarity between representations of each applied job and the $N$ most
recently applied jobs in the global buffer;

  * •

Item-kNN (IkNN): recommends jobs that are similar to the last applied job
during the current session as in [17];

  * •

Session-kNN (SkNN) [32]: compares the entire current session with the past
sessions in the training dataset, rather than considering only the last job;

  * •

Vector Multiplication SkNN (V-SkNN) [33]: is a variant of SkNN that emphasizes
jobs more recently interacted within the current session, when computing the
similarities with past sessions (a linear decay function is used);

  * •

VAE_Comb: is the best model proposed in [12];

  * •

GRU4Rec: is the most recent version of GRU4Rec [19];

  * •

Bert4Rec [22]: introduces the bi-directional self-attention model to model
user behavior sequences;

  * •

Two variants of PANAP: (i) LSTM replaces the personalized-attention layer with
a LSTM layer, Figure 3(a), and (ii) PLSTM adds job seeker metadata to each job
representation to generate Personalized job embedding [13] before the LSTM
layer, Figure 3(b).

(a) LSTM

(b) PLSTM

Figure 3: Two variants of PANAP.

The evaluation metrics used in this work are Hit Rate (HR@5) [33], Mean
Reciprocal Rank (MRR@5) [33] and Normalized Discounted Cumulative Gain
(NDCG@5) [34].

##  VI Results

In this section, we present our experimental results (HR@5/MRR@5/NDCG@5). For
all tables shown in this section, the score in bold is the best in each
metric, and the score in the underline is the second best.

###  VI-A Next-Application Prediction

TABLE II: Next-Application Prediction performance. Method | CB12_s | CB12_l
---|---|---
POP | 0.089/0.042/0.054 | 0.093/0.046/0.058
AR | 0.273/0.200/0.218 | 0.202/0.146/0.160
CS | 0.226/0.128/0.152 | 0.214/0.118/0.142
IkNN | 0.274/0.202/0.220 | 0.202/0.147/0.161
SkNN | 0.349/0.247/0.272 | 0.276/0.196/0.216
V-SkNN | 0.349/0.248/0.273 | 0.276/0.196/0.217
VAE_Comb | 0.345/0.239/0.256 | 0.282/0.203/0.224
GRU4Rec | 0.367/0.208/0.230 | 0.403/0.309/0.352
BERT4Rec | 0.373/0.232/0.245 | 0.438/0.311/0.369
PANAP (S2) | 0.691/0.492/0.541 | 0.742/0.543/0.593
LSTM | 0.540/0.339/0.389 | 0.569/0.368/0.418
PLSTM | 0.480/0.292/0.338 | 0.516/0.323/0.371

According to Table II, we have the following observations: (i) Among all
models, POP and CS give the lowerest scores, as they neither model the
personalized preference nor consider the sequential information. Although POP
is often a strong baseline in certain domains, i.e., news and movies, career
preferences are less affected by popularity factors in the recruitment domain.
(ii) Overall, the NN-based methods consistently outperform traditional
methods, demonstrating that NNs are good at modeling sequential information,
and the self-attention mechanism can improve accuracy. (iii) Our proposed
method PANAP (S2) with sampling strategy S2 and its variants LSTM and PLSTM
perform best among all baselines, which indicates the job content and metadata
can effectively improve the recommendation performance, similar observations
can be found in Section VI-C. (iv) PANAP (S2) outperforms LSTM and PLSTM,
which use LSTM to model the sequential information. One possible reason is
that in the recruitment domain, career preferences are less dynamic than other
domains, and application sequences are relative short (3.87 and 5.61 on
average for both sets), thus the advantage of RNN can not be well
demonstrated. This result also demonstrates the advantage of personalized-
attention as different jobs might have different importance for career
preference modeling, and selecting the more critical jobs is useful for
achieving better recommendation performance. Moreover, LSTM is better than
PLSTM, one possible reason is that PLSTM merges the job seeker metadata into
each job in the session, which will weaken the information carried by the job
itself.

###  VI-B Effectiveness of Personalized Attention

In this part, we analyze the effectiveness of the personalized-attention
mechanism using CB12_s dataset. As shown in Table III, the models with
attention mechanism consistently outperform the model without attention, and
our model with the personalized-attention outperforms its variant with vanilla
attention (similar to the global attention used in [11]). Such a result is
probably since the vanilla attention uses a fixed query vector and cannot
adjust to different personal preferences.

TABLE III: Different attention mechanisms. Attention_mechanism | HR@5/MRR@5/NDCG@5
---|---
Personalized-attention | 0.691/0.492/0.541
Vanilla attention | 0.685/0.480/0.531
No attention (avg) | 0.565/0.367/0.416
LSTM | 0.540/0.339/0.389

###  VI-C Effectiveness of Different Features

As shown in Table IV, (i) It is obvious that methods with the additional
information (i.e., content representation or metadata) generally give better
results than what is achievable from the job identifier $\textit{ID}_{j}$ only
(Only_JobID) on CB12_s dataset. (ii) The metadata has a fewer influence on
PANAP (S2) than LSTM (29.1% average reduction between Meta+Content+JobID and
No_Meta on three metrics compared to 41.0%). One possible reason is that PANAP
(S2) utilizes a personalized-attention mechanism to model career preferences,
which already contain personal information. (iii) Since the “location” factor
affects the choice of job seekers, and our sampling strategy is location-
biased. The job metadata, e.g., City, State and Country could give more
relevant information about “location”, so the scores of No_JobMeta are lower
than that of No_SeekerMeta. (iv) PANAP (S2) consistently outperforms LSTM,
even only the job identifier $\textit{ID}_{j}$ is used, which indicates that
PANAP (S2) does capture personal preference. This observation also proves the
advantage of the personalized-attention mechanism in cases where no additional
information is available.

TABLE IV: Different feature combinations. No_Meta means that neither the job seeker metadata nor the job metadata is considered.  Feature | PANAP (S2) | LSTM
---|---|---
Meta+Content+JobID | 0.691/0.492/0.541 | 0.540/0.339/0.389
No_Meta | 0.516/0.334/0.379 | 0.341/0.189/0.226
No_Content | 0.530/0.341/0.386 | 0.449/0.286/0.326
Only_JobID | 0.501/0.312/0.355 | 0.331/0.182/0.218
No_JobMeta | 0.511/0.338/0.381 | 0.377/0.212/0.253
No_SeekerMeta | 0.601/0.399/0.449 | 0.505/0.297/0.349

###  VI-D Negative Sampling Analysis

We examine the performance of two sampling strategies described in Section V-B
on CB12_s dataset. According to Table V, PANAP (S1) outperforms PANAP (S2).

TABLE V: Different negative sampling strategies. Sampling_strategy | HR@5/MRR@5/NDCG@5
---|---
PANAP (S1) | 0.756/0.620/0.680
PANAP (S2) | 0.691/0.492/0.541

To explain these results, we visualize job seeker representations (session
representations) generated from PANAP (S1) and PANAP (S2) in Figure 4. We use
t-SNE [35] to reduce representation dimensions. For illustration purposes, we
categorize each job seeker according to his/her Major. Note that categorizing
job seekers through Major is not the most reasonable way because some job
seekers are currently engaged in occupations that do not match their majors.
Thus, we select three non-similar majors. People with these professional
backgrounds are more likely to engage in related jobs, including Management,
Computer Science and Medical Assistant. We also plot job seeker
representations labeled with State in Figure 4(b) and Figure 4(d) to show the
influences of different sampling strategies. Each color corresponds to one
State. We observe from Figure 4(b) that representations learned by our model
with S1 (i.e., PANAP (S1)) are well-clustered into groups, each corresponds to
a State. This can be used to explain why the accuracy scores of PANAP (S1) are
better than that of PANAP (S2). A reasonable explanation is that S1 does not
consider the “location” factor when generating negative samples. More
specifically, as mentioned in Section IV-D, job seekers are more likely to
apply for jobs located in their cities or other cities in the same state. The
two main reasons why a job seeker does not apply for a job are: the job
content is inappropriate, or the work location is not suitable. If most of the
negative samples are jobs in other states, regardless of the job content,
these negative samples tend to force the model to capture subtle information
between different states rather than different job contents. For instance, a
job seeker in Seattle-Washington has a background of computer science. sales
representative and java developer are two negative samples from Miami-Florida.
Due to the “location” factor, this job seeker may not apply for java
developer, while sales representative is not suitable from both “location” and
job content perspectives. With these negative samples, the model learns to
distinguish between locations (e.g., Seattle-Washington and Miami-Florida)
rather than the contents of jobs (e.g., computer science and sales
representative). Therefore, the learned job seeker representations are not
categorized according to their Major categories, as shown in Figure 4(a).
Instead, they are grouped together by the “location” as in Figure 4(b). To
handle this problem, we propose a location-biased sampling strategy, S2, as
described in Section V-B, which prioritizes jobs in the same state as the job
seeker as negative samples. We visualize the representations learned through
S2 in Figures 4(c) and 4(d) to show the promising results of our proposed
strategy. We observe that the job seekers are grouped by Major in Figure 4(c),
which demonstrates the effectiveness of S2.

(a) Strategy 1: Major

(b) Strategy 1: State

(c) Strategy 2: Major

(d) Strategy 2: State

Figure 4: Visualization of learned representations.

Figure 5: Different numbers of training negative samples $k$.

###  VI-E Number of Negative Samples

This section discusses the influence of the number of negative samples used
for training on prediction accuracy. The experimental results on CB12_s
dataset are shown in Figure 5, where $k$ negative samples are used for
training and 50 samples for evaluation. When $k$ increases (e.g., from 5 to
10), the performance of our model first has a noticeable improvement. This may
be because when $k$ is too small, the information provided by negative samples
is relatively limited, and the model cannot learn useful information. Then,
when $k$ continues to increase (e.g., from 10 to 20), the performance becomes
stable. However, if $k$ is too large, there may not be enough jobs in the same
state. Negative samples in other states become dominant, making it difficult
for the model to identify the valuable information correctly. As explained in
Section VI-D, the model will learn to distinguish the difference of positions,
just like PANAP (S1). As a result, the performance consistently improves.

##  VII Conclusions

In this work, we proposed a personalized-attention model for the Next-
Application Prediction problem, which improves the prediction accuracy. The
experiments confirm that, by incorporating personalized-attention, our method
can better capture the personal career preference than baseline methods. We
investigated the importance of incorporating job content information and
metadata in the recruitment domain. Moreover, considering the “location”
factor when generating negative samples can provide more useful information on
job content.

## Acknowledgment

This work is supported by Randstad corporate research chair in collaboration
with Université Paris-Saclay, CentraleSupélec, MICS. We would like to thank
the Mésocentre computing center of CentraleSupélec and École Normale
Supérieure Paris-Saclay 555http://mesocentre.centralesupelec.fr/ for providing
computing resources.

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