# Relaxation of the kinematic dynamo equations

Lauri Hitruhin  Department of Mathematics and Statistics
University of Helsinki, P.O. Box 68, 00014 Helsingin yliopisto, Finland
<EMAIL_ADDRESS>and  Sauli Lindberg  Department of Mathematics and
Statistics
University of Helsinki, P.O. Box 68, 00014 Helsingin yliopisto, Finland
<EMAIL_ADDRESS>
###### Abstract.

We compute the exact relaxation and $\Lambda$-convex hull of the kinematic
dynamo equations and show that they coincide. We also find the relaxation in
the stationary case.

L.H. was supported by the Academy of Finland projects #332671 and SA-1346562.
S.L. was supported by the ERC Advanced Grant 834728.

##  1\. Introduction

_Dynamo theory_ studies the mechanisms by which electrically conducting fluids
generate and sustain the magnetic fields of celestial bodies [2, 8]. In
typical astrophysical applications, the magnetic Reynolds number is very
large, and so one often studies the idealised limit of a perfect conductor [8,
p. 9]. We also assume perfect conductivity in this paper. Dynamo theory is
divided into two parts.

In the simpler _kinematic dynamo model_ , the interaction of the magnetic
field and the fluid is described via the induction equation and Gauss’s law
for magnetism,

(1)
$\partial_{t}\mathbf{B}+\nabla\times(\mathbf{B}\times\mathbf{u})=0,\qquad\nabla\cdot\mathbf{B}=0.$

In (1), $\mathbf{B}$ is the magnetic field and $\mathbf{u}$ is the fluid
velocity. Sometimes, the incompressibility condition $\nabla\cdot\mathbf{u}=0$
is considered as a part of the kinematic model. In the _nonlinear dynamo
model_ , (1) is extended to the full ideal incompressible MHD
(magnetohydrodynamic) equations by adding the Cauchy Momentum equation with
Lorentz force and incompressibility,

(2) $\partial_{t}\mathbf{u}+(\mathbf{u}\cdot\nabla)\mathbf{u}=-\nabla
p+(\nabla\times\mathbf{B})\times\mathbf{B},\qquad\nabla\cdot\mathbf{u}=0.$

The kinematic dynamo equations (1) also allow one to study very general
$\mathbf{u}$ regardless of constitutive laws, compressibility, viscosity,
external forcing and so on [14]. Besides ideal or viscous MHD, one can e.g.
set $\mathbf{u}$ to be the electron fluid velocity in Hall MHD [13] or the
velocity field in Moffatt’s magnetic relaxation equations [4, 25] or magneto-
friction [29].

From a mathematical viewpoint, kinematic dynamo equations are also a useful
toy model for MHD. While the highly difficult analysis of the back-propagation
of the Lorentz force on the fluid is avoided, some of the salient features of
the MHD model are still retained. In particular, one of the classical
conserved quantities of MHD, magnetic helicity
$\mathcal{H}_{M}(t)\mathrel{:\mkern-0.25mu=}\int_{\Omega}\mathbf{A}(x,t)\cdot\mathbf{B}(x,t)\,\textup{d}x$,
is conserved by weak solutions $\mathbf{B},\mathbf{u}\in L^{3}_{t,x}$ of (1)
e.g. under periodic boundary conditions on $\Omega=\mathbb{T}^{3}$ (see [16,
17] for more information). Magnetic helicity constrains the dynamics, in
particular via _Arnold’s inequality_
$\int_{\Omega}\left\lvert\mathbf{B}(x,t)\right\rvert^{2}\,\textup{d}x\geq
c\int_{\Omega}\mathbf{A}(x,t)\cdot\mathbf{B}(x,t)\,\textup{d}x$ which keeps
magnetic energy bounded away from zero if the initial field $\mathbf{B}_{0}$
has non-zero magnetic helicity. By [15, Theorem 2.2], conservation even
extends to a macroscopically averaged version, the so-called _relaxation_ of
(1). In the relaxation, solutions of (1) are replaced by weak limits of
solutions, as we next make precise.

In the Tartar framework [28], the kinematic dynamo equations are decoupled
into the conservation laws

(3)
$\nabla\cdot\mathbf{B}=0,\qquad\partial_{t}\mathbf{B}+\nabla\times\mathbf{E}=0$

(Gauss’s law for magnetism and Faraday’s law of induction) and the
constitutive law $\mathbf{E}=\mathbf{B}\times\mathbf{u}$ (ideal Ohm’s law)
which is codified into the condition that $(\mathbf{B},\mathbf{u},\mathbf{E})$
takes values in the _constraint set_ $K=\\{(B,u,E):E=B\times u\\}$. One can
also encode more information into the constraint set; one can e.g. prescribe
kinetic and magnetic energy densities by letting $r,s>0$ and considering the
normalised set

$K_{r,s}=\\{(B,u,E)\colon E=B\times u,\,\left\lvert
u\right\rvert=r,\,\left\lvert B\right\rvert=s\\}.$

The relaxation of $K_{r,s}$ can be defined as the smallest set
$\widetilde{K_{r,s}}$ such that whenever solutions of (3) take values in
$K_{r,s}$, their weak limits take values in $\widetilde{K_{r,s}}$. Another,
potentially larger (but in our case equal) variant is the smallest superset of
$K_{r,s}$ that is closed under weak convergence for solutions of (3),
essentially following Tartar [28]. The relaxation describes macroscopic
averages of solutions of (1) [28].

We discuss the motivation for studying the relaxation of (1). The bare fields
$\mathbf{B}$ and $\mathbf{u}$ can be neither observed [1] nor predicted [9]
effectively, and so most turbulence theories concentrate on various averaged
quantities which are much better reproducible [9]. Lax [22] has suggested weak
limits as a possible deterministic substitute for ensemble averaging, and the
idea was developed further in [3]. For a recent systematic review on the topic
see [12]. In the relaxation, the averaging is performed at all length scales
simultaneously, whereas e.g. coarse-graining of MHD via spatial filtering [1]
leads to subscale stresses which depend on the filtering resolution and need
to be modelled (e.g. by eddy viscosity models [9, p. 401]).

The relaxation gives geometric insights which remain hidden in conventional
averaging methods. As an example, in the kinematic dynamo equations, the
orthogonality $\mathbf{B}\cdot\mathbf{E}=0$ extends to the relaxation even
though the constitutive law $\mathbf{E}=\mathbf{B}\times\mathbf{u}$ breaks
down; this plays a key role in showing that $L^{3}_{t,x}$ is the optimal
integrability under which magnetic helicity is conserved [16]. It is, at any
rate, a challenging problem to find suitable physically motivated selection
criteria which pick a unique subsolution (i.e. a triple
$(\mathbf{B},\mathbf{u},\mathbf{E})$ which solves (1) and take values in
$\widetilde{K_{r,s}}$) under given initial/boundary (or other) conditions. For
such a criterion in the case of the Muskat problem with flat interface
(”maximal mixing”) see [23, 26, 27].

The relaxation is also used in convex integration, a mathematical technique
which produces a weak solution from a subsolution via an $h$-principle [11].
The use of convex integration in fluid mechanics was pioneered by De Lellis
and Székelyhidi in [10]. By a refined quantitative version of the
$h$-principle one can produce solutions whose averages are dictated by the
subsolution (a rigorous statement was proved by Castro, Faraco and Mengual in
[6]). Furthermore, instabilities in fluid dynamics, which result in a
turbulent evolution intractable with the classical theory, have been
successfully modelled by these techniques; see for example [5, 7] for
Rayleigh-Taylor under IPM, [19, 20] for Rayleigh-Taylor under inhomogeneous
Euler, [18, 24] for Kelvin-Helmoltz and the references contained therein.

Precise information on the relaxation is imporant in identifying the
boundary/initial conditions under which convex integration can be run in the
Tartar framework [27]. It is a natural meta-conjecture in fluid dynamics that
the relaxation coincides with the $\Lambda$-convex hull $K_{r,s}^{\Lambda}$,
i.e., the set of points which cannot be separated from $K_{r,s}$ by a
$\Lambda$-convex function; see §2 for the relevant definitions. We prove the
conjecture in the case of the kinematic dynamo equations. Theorem 1 is also a
step towards computing the exact relaxation of the full MHD system; a
qualitative characterisation was given in [15, Theorem 6.7].

######  Theorem 1.

The relaxation, first laminate and $\Lambda$-convex hull of the kinematic
dynamo equations coincide and can be written as

$\displaystyle K_{r,s}^{\Lambda}=$ $\displaystyle\\{(B,u,E)\colon\left\lvert
B\right\rvert\leq r,\;\left\lvert u\right\rvert\leq s,$ (4) $\displaystyle
B\cdot E=0,\;\left\lvert E-B\times u\right\rvert\leq\sqrt{(r^{2}-\left\lvert
B\right\rvert^{2})(s^{2}-\left\lvert u\right\rvert^{2})}\\}.$

To close the introduction, we mention that (1) also gives the relaxation of
kinematic dynamo equations in the stationary and incompressible cases (as the
wave cone remains unchanged). However, in the incompressible, stationary case,
the wave cone (and, _a fortiori_ , the relaxation) are strictly smaller. We
compute the corresponding relaxation in §4.

##  2\. Preliminaries

We recall some definitions from the theory of differential inclusions and
refer to [21] for more information. The wave cone $\Lambda$ consists of the
directions $(\bar{B},\bar{u},\bar{E})\in\mathbb{R}^{9}$ such that plane waves
$(\mathbf{B},\mathbf{u},\mathbf{E})(\mathbf{x},t)\mathrel{:\mkern-0.25mu=}h((\mathbf{x},t)\cdot\xi)(\bar{B},\bar{u},\bar{E})$,
$h\in C^{\infty}(\mathbb{R})$, solve (3) for some
$\xi\in(\mathbb{R}^{3}\setminus\\{0\\})\times\mathbb{R}$. The wave cone of the
(non-stationary as well as stationary) kinematic dynamo equations has been
computed in [15, Lemma 5.2]:

######  Proposition 2.

$\Lambda=\\{(B,u,E)\colon B\cdot E=0\\}$.

Given any compact set $C\subset\mathbb{R}^{3}$, the _laminates_
$C^{k,\Lambda}$, $k\in\mathbb{N}_{0}$, of $C$ are defined as follows:

$\displaystyle C^{0,\Lambda}\mathrel{:\mkern-0.25mu=}C,$ $\displaystyle
C^{k+1,\Lambda}\mathrel{:\mkern-0.25mu=}\\{(\lambda
z_{1}+(1-\lambda)z_{2}\colon z_{1},z_{2}\in
C^{k,\Lambda},\,z_{1}-z_{2}\in\Lambda,\,\lambda\in[0,1]\\}.$

The _lamination convex hull_ of $C$ is defined as

$C^{lc,\Lambda}\mathrel{:\mkern-0.25mu=}\cup_{k=0}^{\infty}C^{k,\Lambda}.$

Recall also that a function $G\colon\mathbb{R}^{3}\to\mathbb{R}$ is said to be
_$\Lambda$ -convex_ if $t\mapsto G(z_{0}+tz)\colon\mathbb{R}\to\mathbb{R}$ is
convex for every $z_{0}\in\mathbb{R}\times\mathbb{R}^{3}\times\mathbb{R}^{3}$
and $z\in\Lambda$. The _$\Lambda$ -convex hull_ $C^{\Lambda}$ consists of
points $z\in\mathbb{R}^{n}$ that cannot be separated from $C$ by a
$\Lambda$-convex function. More precisely, $z\notin C^{\Lambda}$ if and only
if there exists a $\Lambda$-convex function $G$ such that $G|_{C}\leq 0$ but
$G(z)>0$. We have $C^{\Lambda}\supset C^{lc,\Lambda}$.

##  3\. The proof of Theorem 1

We prove Theorem 1 through a series of lemmas, starting by the inclusion
”$\subset$” in (1). For this we need the following $\Lambda$-affine and
convex, and hence also $\Lambda$-convex, functions.

######  Lemma 3.

The function

$G_{1}(B,u,E)\mathrel{:\mkern-0.25mu=}B\cdot E$

is $\Lambda$-affine and vanishes in $K_{r,s}^{\Lambda}$.

###### Proof.

The $\Lambda$-affinity follows directly from Proposition 2. Since $G_{1}$
vanishes in $K_{r,s}$, it also vanishes in $K_{r,s}^{\Lambda}$. ∎

In Theorem 1, the $\Lambda$-affine function $G_{1}$ gives the correct bounds
for the directions of the vector $E-B\times u$, and to bound the length we
need the following convex function.

######  Lemma 4.

The function

$G_{2}(B,u,E)\mathrel{:\mkern-0.25mu=}\max_{0\leq\alpha\leq
1}[\alpha(\left\lvert B\right\rvert^{2}-r^{2})+(1-\alpha)(\left\lvert
u\right\rvert^{2}-s^{2})+2\sqrt{\alpha(1-\alpha)}\left\lvert B\times
u-E\right\rvert]$

is convex and vanishes in $K_{r,s}$. (Thus, $G_{2}\leq 0$ in
$K_{r,s}^{\Lambda}$.)

###### Proof.

Clearly $G_{2}|_{K_{r,s}}=0$. Denoting $G_{2}=\max_{0\leq\alpha\leq
1}H_{\alpha}$, it then suffices to fix $\alpha\in[0,1]$ and show that
$H_{\alpha}$ is convex. Fix $z_{0}=(B_{0},u_{0},E_{0})$ and $z=(B,u,E)$. Let
$t\in\mathbb{R}$. Then

$\displaystyle H_{\alpha}(z_{0}+tz)-H_{\alpha}(z_{0})$ $\displaystyle\geq
ct+(\alpha\left\lvert B\right\rvert^{2}+(1-\alpha)\left\lvert
u\right\rvert^{2}-2\sqrt{\alpha(1-\alpha)}\left\lvert B\times
u\right\rvert)t^{2}\geq ct$

for some constant $c\in\mathbb{R}$ which is independent of $t$, and so
$H_{\alpha}$ is convex. ∎

The functions from Lemmas 3 and 4 yield the inclusion ”$\subset$” in (1):

######  Lemma 5.

Every triple $(B,u,E)\in K_{r,s}^{\Lambda}$ satisfies

(5) $\left\lvert B\right\rvert\leq r,\quad\left\lvert u\right\rvert\leq
s,\quad\left\lvert B\times u-E\right\rvert\leq\sqrt{(r^{2}-\left\lvert
B\right\rvert^{2})(s^{2}-\left\lvert u\right\rvert^{2})},\quad B\cdot E=0.$

###### Proof.

Suppose $(B,u,E)\in K_{r,s}^{\Lambda}$. Now $\left\lvert
B\right\rvert^{2}-r^{2}\leq G_{2}(B,u,E)\leq 0$ and $\left\lvert
u\right\rvert^{2}-s^{2}\leq G_{2}(B,u,E)\leq 0$. For the third inequality note
that if $\left\lvert B\right\rvert=r$, then Proposition 4 gives, for every
$\alpha\in(0,1)$, $\left\lvert B\times
u-E\right\rvert\leq[\alpha/(1-\alpha)]^{1/2}(r^{2}-\left\lvert
b\right\rvert^{2})$, so that $\left\lvert B\times u-E\right\rvert=0$. The case
$\left\lvert u\right\rvert=s$ is similar, and so we assume that $\left\lvert
B\right\rvert<r$ and $\left\lvert u\right\rvert<s$. Choose
$\alpha=(s^{2}-\left\lvert u\right\rvert^{2})/(s^{2}+r^{2}-\left\lvert
B\right\rvert^{2}-\left\lvert u\right\rvert^{2})$ in Proposition 4. Now
$G_{\alpha}(B,u,E)\leq 0$, that is, $\left\lvert B\times
u-E\right\rvert\leq[(r^{2}-\left\lvert B\right\rvert^{2})(s^{2}-\left\lvert
u\right\rvert^{2})]^{1/2}$. The equality $B\cdot E=0$ follows immediately from
Proposition 3. ∎

Note that by the convexity of $G_{2}$ and Jensen’s inequality, the three
inequalities in (5) also extend to various other averaging processes (whereas
the imporant equality $B\cdot E=0$, which is at the core of magnetic helicity
conservation [17], breaks down).

Our next task is to prove the inclusion ”$\supset$” in (1), i.e., the
sharpness of the upper bound (5). We start with the case $E=B\times u$.

######  Lemma 6.

Suppose $\left\lvert B\right\rvert\leq r$ and $\left\lvert u\right\rvert\leq
s$. Then $(B,u,B\times u)\in K_{r,s}^{1,\Lambda}$.

###### Proof.

Choose $\bar{B},\bar{u}\in\textup{span}\\{B,u\\}^{\perp}$ with
$\bar{B}\times\bar{u}=0$,
$\left\lvert\bar{B}\right\rvert^{2}=r^{2}-\left\lvert B\right\rvert^{2}$ and
$\left\lvert\bar{u}\right\rvert^{2}=s^{2}-\left\lvert u\right\rvert^{2}$. Then

$\displaystyle(B,u,B\times u)$
$\displaystyle=\frac{1}{2}(B+\bar{B},u+\bar{u},(B+\bar{B})\times(u+\bar{u}))$
$\displaystyle+\frac{1}{2}(B-\bar{B},u-\bar{u},(B-\bar{B})\times(u-\bar{u}))\mathrel{=\mkern-0.25mu:}\frac{z_{1}+z_{2}}{2}$

and $z_{1}-z_{2}=(\bar{B},\bar{u},B\times\bar{u}+\bar{B}\times u)\in\Lambda$.
∎

The case $\left\lvert B\right\rvert<r$, $\left\lvert u\right\rvert<s$,
$E-B\times u\neq 0$ requires more work. In Lemma 7 we give a somewhat explicit
characterisation of the first laminate $K^{1,\Lambda}_{r,s}$; the proof of
formula (1) is then completed by showing that whenever $E\perp B$ is as
indicated in Theorem 1, we can write $E=B\times
u+\bar{B}\times\bar{u}/[\left\lvert\bar{B}\right\rvert\left\lvert\bar{u}\right\rvert]$
for some $\bar{B},\bar{u}$ satisfying (7)–(9).

######  Lemma 7.

Suppose $\left\lvert B\right\rvert<r$ and $\left\lvert u\right\rvert<s$. Then
$(B,u,E)\in K^{1,\Lambda}_{r,s}$ if and only if there exist
$\bar{B},\bar{u}\neq 0$ such that

(6) $\displaystyle E=B\times u+\sqrt{(r^{2}-\left\lvert
B\right\rvert^{2})(s^{2}-\left\lvert
u\right\rvert^{2})}\frac{\bar{B}\times\bar{u}}{\left\lvert\bar{B}\right\rvert\left\lvert\bar{u}\right\rvert},$
(7) $\displaystyle\left\lvert\bar{B}\right\rvert^{2}=\frac{r^{2}-\left\lvert
B\right\rvert^{2}}{s^{2}-\left\lvert
u\right\rvert^{2}}\left\lvert\bar{u}\right\rvert^{2}=4(r^{2}-\left\lvert
B\right\rvert^{2}\sin^{2}\alpha_{B,\bar{B}}),$ (8) $\displaystyle\left\lvert
B\right\rvert\cos\alpha_{B,\bar{B}}=\sqrt{\frac{r^{2}-\left\lvert
B\right\rvert^{2}}{s^{2}-\left\lvert u\right\rvert^{2}}}\left\lvert
u\right\rvert\cos\alpha_{u,\bar{u}},$ (9) $\displaystyle
B\cdot\bar{B}\times\bar{u}=0,$

where we set $\left\lvert B\right\rvert\sin\alpha_{B,\bar{B}}=\left\lvert
B\right\rvert\cos\alpha_{B,\bar{B}}=0$ if $B=0$ and similarly for $u$.

###### Proof.

A general $\Lambda$-convex combination of two elements of $K_{r,s}$, with
$\left\lvert B\right\rvert<r$ and $\left\lvert u\right\rvert<s$, is of the
form

(10) $\displaystyle\lambda V_{1}+\mu V_{2}$
$\displaystyle=\lambda(B+\mu\bar{B},u+\mu\bar{u},(B+\mu\bar{B})\times(u+\mu\bar{u}))$
$\displaystyle+\mu(B-\lambda\bar{B},u-\lambda\bar{u},(B-\lambda\bar{B})\times(u-\lambda\bar{u}))$
$\displaystyle=(B,u,B\times u+\lambda\mu\bar{B}\times\bar{u}),$

where $0\leq\lambda\leq 1$, $\lambda+\mu=1$,

(11) $\left\lvert B+\mu\bar{B}\right\rvert=r,\quad\left\lvert
B-\lambda\bar{B}\right\rvert=r,\quad\left\lvert
u+\mu\bar{u}\right\rvert=s,\quad\left\lvert
u-\lambda\bar{u}\right\rvert=s,\quad B\cdot\bar{B}\times\bar{u}=0.$

(In particular, $\bar{B},\bar{u}\neq 0$.)

We intend to show that (11) is equivalent to (7)–(9). We first assume that
(11) holds and aim to prove (7). Squaring and subtracting on both sides of the
first two equations in (11) and moving terms (and recalling that
$\lambda^{2}-\mu^{2}=\lambda-\mu$),

(12)
$2B\cdot\bar{B}=(\lambda-\mu)\left\lvert\bar{B}\right\rvert^{2}=(2\lambda-1)\left\lvert\bar{B}\right\rvert^{2}.\qquad\Longrightarrow\qquad\lambda=\frac{1}{2}+\frac{B\cdot\bar{B}}{\left\lvert\bar{B}\right\rvert^{2}}.$

We next rewrite

$\displaystyle r^{2}$ $\displaystyle=\left\lvert
B+\mu\bar{B}\right\rvert^{2}=\left\lvert
B\right\rvert^{2}-(2B\cdot\bar{B}-\left\lvert\bar{B}\right\rvert^{2})\frac{B\cdot\bar{B}}{\left\lvert\bar{B}\right\rvert^{2}}+\frac{(2B\cdot\bar{B}-\left\lvert\bar{B}\right\rvert^{2})^{2}}{4\left\lvert\bar{B}\right\rvert^{2}}$
(13) $\displaystyle=\left\lvert
B\right\rvert^{2}+\frac{\left\lvert\bar{B}\right\rvert^{4}-4(B\cdot\bar{B})^{2}}{4\left\lvert\bar{B}\right\rvert^{2}}$

and similarly $\lambda=1/2+u\cdot\bar{u}/\left\lvert\bar{u}\right\rvert^{2}$
and $s^{2}=\left\lvert
u\right\rvert^{2}+[\left\lvert\bar{u}\right\rvert^{4}-4(u\cdot\bar{u})^{2}]/[4\left\lvert\bar{u}\right\rvert^{2}]$.
We conclude that

(14) $\frac{r^{2}-\left\lvert
B\right\rvert^{2}}{\left\lvert\bar{B}\right\rvert^{2}}=\frac{\left\lvert\bar{B}\right\rvert^{4}-4(B\cdot\bar{B})^{2}}{4\left\lvert\bar{B}\right\rvert^{4}}=\lambda\mu=\frac{\left\lvert\bar{u}\right\rvert^{4}-4(u\cdot\bar{u})^{2}}{4\left\lvert\bar{u}\right\rvert^{4}}=\frac{s^{2}-\left\lvert
u\right\rvert^{2}}{\left\lvert\bar{u}\right\rvert^{2}}.$

Now (14) immediately implies the first equality in (7), and the second one
follows from (3):

$\left\lvert\bar{B}\right\rvert^{2}=4\left(r^{2}-\left\lvert
B\right\rvert^{2}+\frac{(B\cdot\bar{B})^{2}}{\left\lvert\bar{B}\right\rvert^{2}}\right)=4(r^{2}-\left\lvert
B\right\rvert^{2}\sin^{2}\alpha_{B,\bar{B}}).$

Finally, (8) follows from (12), the equality
$\lambda=1/2+u\cdot\bar{u}/\left\lvert\bar{u}\right\rvert^{2}$ and (14):

$\left\lvert
B\right\rvert\cos\alpha_{B,\bar{B}}=\frac{B\cdot\bar{B}}{\left\lvert\bar{B}\right\rvert}=\frac{\left\lvert\bar{B}\right\rvert}{\left\lvert\bar{u}\right\rvert}\frac{u\cdot\bar{u}}{\left\lvert\bar{u}\right\rvert}=\sqrt{\frac{r^{2}-\left\lvert
B\right\rvert^{2}}{s^{2}-\left\lvert u\right\rvert^{2}}}\left\lvert
u\right\rvert\cos\alpha_{u,\bar{u}}.$

Conversely, if (7)–(9) hold, we intend to show that (11) holds with the choice
$\lambda=1/2+B\cdot\bar{B}/\left\lvert\bar{B}\right\rvert^{2}$. First note
that $\lambda\in[0,1]$ since (7) yields

$\frac{\left\lvert
B\cdot\bar{B}\right\rvert^{2}}{\left\lvert\bar{B}\right\rvert^{2}}=\frac{\left\lvert
B\right\rvert^{2}-\left\lvert
B\right\rvert^{2}\sin^{2}\alpha_{B,\bar{B}}}{\left\lvert\bar{B}\right\rvert^{2}}\leq\frac{r^{2}-\left\lvert
B\right\rvert^{2}\sin^{2}\alpha_{B,\bar{B}}}{\left\lvert\bar{B}\right\rvert^{2}}=\frac{1}{4}.$

Then (3) holds, and similarly $\left\lvert B-\lambda\bar{B}\right\rvert=r$ and
$\left\lvert u+\mu\bar{u}\right\rvert=\left\lvert
u-\lambda\bar{u}\right\rvert=s$. The identity (6) is obtained by noting that
(14) yields

$\lambda\mu=\frac{\sqrt{(r^{2}-\left\lvert
B\right\rvert^{2})(s^{2}-\left\lvert
u\right\rvert^{2})}}{\left\lvert\bar{B}\right\rvert\left\lvert\bar{u}\right\rvert}.$

∎

Recall that our aim is to prove (1). When $\left\lvert B\right\rvert\leq r$,
$\left\lvert u\right\rvert\leq s$ and $\bar{E}\in\bar{B}(0,1)\setminus\\{0\\}$
with $B\cdot\bar{E}=0$, our aim is, therefore, to find $\bar{B}$ and $\bar{u}$
satisfying
$\bar{B}\times\bar{u}/(\left\lvert\bar{B}\right\rvert\left\lvert\bar{u}\right\rvert)=\bar{E}$
along with (7)–(9).

###### Proof of (1).

Suppose first $B\neq 0$. Now $\bar{B},\bar{u}\neq 0$ satisfy
$\bar{B}\times\bar{u}/[|\bar{B}||\bar{u}|]=\bar{E}$ if and only if

(15)
$\bar{B},\bar{u}\in\textup{span}\\{B,B\times\bar{E}\\},\qquad\sin\alpha_{\bar{B},\bar{u}}=|\bar{E}|.$

We therefore consider pairs $(\bar{B},\bar{u})$ which satisfy (15). Given a
direction $\bar{B}/\left\lvert\bar{B}\right\rvert$, note that
$\bar{u}/\left\lvert\bar{u}\right\rvert$ is uniquely fixed by (15) once we
choose $\alpha_{\bar{B},\bar{u}}\in[0,2\pi)$ to be minimal. Also note that
$\\{B,\bar{E},B\times\bar{E}\\}$ is an orthogonal basis of $\mathbb{R}^{3}$.

Our task is to show that (8) can be satisfied simultaneously with
$\sin\alpha_{\bar{B},\bar{u}}=\left\lvert\bar{E}\right\rvert$; then (7) is
obtained simply by scaling $\bar{B}$ and $\bar{u}$ and (9) holds automatically
since $\bar{B},\bar{u}\in\textup{span}\\{B,B\times\bar{E}\\}$.

Aiming to solve (8), we define $G\colon[0,2\pi)\to\mathbb{R}$ by

$G(\alpha_{B,\bar{B}})\mathrel{:\mkern-0.25mu=}\left\lvert
B\right\rvert\cos\alpha_{B,\bar{B}}-\sqrt{\frac{r^{2}-\left\lvert
B\right\rvert^{2}}{s^{2}-\left\lvert u\right\rvert^{2}}}\left\lvert
u\right\rvert\cos\alpha_{u,\bar{u}}.$

Denoting $\alpha_{B,\bar{B}_{1}}=\pi/2$ and $\alpha_{B,\bar{B}_{2}}=3\pi/2$,
we have $\cos\alpha_{B,\bar{B}_{1}}=\cos\alpha_{B,\bar{B}_{2}}=0$ and
$\bar{u}_{2}/|\bar{u}_{2}|=-\bar{u}_{1}/|\bar{u}_{1}|$. Therefore,
$G(\pi/2)=-G(3\pi/2)$. By Bolzano’s theorem, $G(\alpha_{B,\bar{B}})=0$ for
some $\alpha_{B,\bar{B}}\in[\pi/2,3\pi/2]$, that is, (8) holds.

We finish the proof by covering the case $B=0$. We choose
$\bar{u}\in\\{u,\bar{E}\\}^{\perp}\neq\\{0\\}$ and then select $\bar{B}\neq 0$
such that
$\bar{B}\times\bar{u}/(|\bar{B}|\left\lvert\bar{u}\right\rvert)=\bar{E}$.
Again, (8) is satisfied, (7) follows by scaling and (9) is immediate. ∎

###### Proof of Theorem 1.

We have proved (1) above, and the proof shows that (1) also holds when
$K_{r,s}^{\Lambda}$ is replaced by $K_{r,s}^{1,\Lambda}$. The proof of Theorem
1 is completed once we show that the set described in (1) coincides with the
relaxation $\widetilde{K_{r,s}}$.

For the proof of the fact that $\widetilde{K_{r,s}}\supset
K_{r,s}^{1,\Lambda}$ see [28, pp. 162–163]. On the other hand, the inclusion
$\widetilde{K_{r,s}}\subset K_{r,s}^{\Lambda}$ holds since
$K_{r,s}^{\Lambda}=G_{1}^{-1}\\{0\\}\cap G_{2}^{-1}(-\infty,0]$ and the
quadratic $\Lambda$-affine function $G_{1}$ and the convex function $G_{2}$
are lower semicontinuous on sequences of solutions of (3) (see [28, Corollary
13]). ∎

##  4\. The relaxation of the stationary model under incompressibility

The $\Lambda$-convex hull $K_{r,s}^{\Lambda}$ remains unchanged if we
incorporate the incompressibility condition $\nabla\cdot\mathbf{u}=0$ or
consider stationary solutions; in each case, the wave cone is given by
Proposition 2. For completeness, we also compute the $\Lambda$-convex hull for
the stationary kinematic dynamo equations under incompressibility. In this
case, the wave cone, which we denote by $\Lambda_{s}$, is smaller.

######  Proposition 8.

$\Lambda_{s}=\\{(B,u,E)\colon B\cdot E=u\cdot E=0\\}$.

###### Proof.

First suppose $B\cdot E=u\cdot E=0$. Our aim is to find
$\xi\in\mathbb{R}^{3}\setminus\\{0\\}$ such that

(16) $B\cdot\xi=u\cdot\xi=0,\qquad E\times\xi=0.$

First, if $B\times u\neq 0$, then $\\{B,u,B\times u\\}$ is a basis of
$\mathbb{R}^{3}$. Thus $B\cdot E=u\cdot E=0$ implies $E=k\,B\times u$ for some
$k\in\mathbb{R}$, so that we can choose $\xi=B\times u$. If $B\times u=0$ and
$E\neq 0$, we set $\xi=E$. If $B\times u=E=0$, we choose any
$\xi\in\\{B,u\\}^{\perp}\setminus\\{0\\}$.

Suppose then $(B,u,E)\in\Lambda$ so that (16) holds for some $\xi\neq 0$. Now
$E\times\xi=0$ yields $E=k\xi$ for some $k\in\mathbb{R}$. Thus $B\cdot
E=u\cdot E=0$ by (16). ∎

The corresponding relaxation is characterised as follows:

######  Theorem 9.

The relaxation, first laminate and $\Lambda_{s}$-convex hull of the stationary
kinematic dynamo equations under incompressibility coincide and can be written
as

$\displaystyle K_{r,s}^{\Lambda_{s}}=$
$\displaystyle\\{(B,u,E)\colon\left\lvert B\right\rvert\leq r,\;\left\lvert
u\right\rvert\leq s,$ (17) $\displaystyle B\cdot E=0,\;u\cdot
E=0,\;\left\lvert E-B\times u\right\rvert\leq\sqrt{(r^{2}-\left\lvert
B\right\rvert^{2})(s^{2}-\left\lvert u\right\rvert^{2})}\\}.$

The proof of Theorem 9 is almost identical to that of Theorem 1. Below we
briefly indicate the necessary changes. First, the extra condition $u\cdot
E=0$ of the wave cone leads to a new $\Lambda$-affine function:

######  Proposition 10.

The function $G_{3}(B,u,E)\mathrel{:\mkern-0.25mu=}u\cdot E$ is
$\Lambda_{s}$-affine and vanishes in $K_{r,s}^{\Lambda_{s}}$.

Lemmas 3–4 clearly hold when $\Lambda$ is replaced by $\Lambda_{s}$. The
inclusion ”$\subset$” in (9) is then proved just like Lemma 5. It remains to
cover the inclusion ”$\supset$” in (9) when $\Lambda$ is replaced by
$\Lambda_{s}$.

First, Lemma 6 and its proof extend to $\Lambda_{s}$ verbatim. In Lemma 7 and
its proof, the only necessary change is adding the condition
$u\cdot\bar{B}\times\bar{u}=0$ to formulas (9) and (11). The proof of (9)
requires a bit more modification, as we indicate below.

###### Proof of (9).

When $\left\lvert B\right\rvert\leq r$, $\left\lvert u\right\rvert\leq s$ and
$\bar{E}\in\bar{B}(0,1)\setminus\\{0\\}$ with $B\cdot\bar{E}=u\cdot\bar{E}=0$,
our aim is to find $\bar{B},\bar{u}\neq 0$ such that
$\bar{B}\times\bar{u}/[|\bar{B}||\bar{u}|]=\bar{E}$, (7)–(9) hold and
$u\cdot\bar{B}\times\bar{u}=0$.

If $B\times u=0$, we can follow the proof of (1) verbatim. Suppose, therefore,
that $B\times u\neq 0$. Now $\bar{B},\bar{u}$ satisfy
$\bar{B}\times\bar{u}/[|\bar{B}||\bar{u}|]=\bar{E}$ if and only if

(18)
$\bar{B},\bar{u}\in\textup{span}\\{B,u\\},\qquad\sin\alpha_{\bar{B},\bar{u}}=|\bar{E}|.$

Again, we consider pairs $(\bar{B},\bar{u})$ which satisfy (18) and need to
show that (8) can be satisfied simultaneously with
$\sin\alpha_{\bar{B},\bar{u}}=|\bar{E}|$. Just like in the proof of (1), the
function $G\colon[0,2\pi)\to\mathbb{R}$,
$G(\alpha_{B,\bar{B}})\mathrel{:\mkern-0.25mu=}\left\lvert
B\right\rvert\cos\alpha_{B,\bar{B}}-\sqrt{(r^{2}-\left\lvert
B\right\rvert^{2})/(s^{2}-\left\lvert u\right\rvert^{2})}\left\lvert
u\right\rvert\cos\alpha_{u,\bar{u}}$ has a zero, which proves the claim. ∎

Acknowledgments. We thank Daniel Faraco for useful discussions.

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# EZInterviewer: To Improve Job Interview Performance with Mock Interview
Generator

Mingzhe Li∗† Peking University<EMAIL_ADDRESS>,  Xiuying Chen  CBRC,
KAUST
CEMSE, KAUST<EMAIL_ADDRESS>,  Weiheng Liao  Made by DATA
<EMAIL_ADDRESS>,  Yang Song  BOSS Zhipin NLP Center
<EMAIL_ADDRESS>,  Tao Zhang  BOSS Zhipin<EMAIL_ADDRESS>,
Dongyan Zhao  Peking University<EMAIL_ADDRESS>and  Rui Yan‡ Gaoling School
of AI
Renmin University of China<EMAIL_ADDRESS>
(2023)

###### Abstract.

Interview has been regarded as one of the most crucial step for recruitment.
To fully prepare for the interview with the recruiters, job seekers usually
practice with mock interviews between each other. However, such a mock
interview with peers is generally far away from the real interview experience:
the mock interviewers are not guaranteed to be professional and are not likely
to behave like a real interviewer. Due to the rapid growth of online
recruitment in recent years, recruiters tend to have online interviews, which
makes it possible to collect real interview data from real interviewers. In
this paper, we propose a novel application named EZInterviewer, which aims to
learn from the online interview data and provides mock interview services to
the job seekers. The task is challenging in two ways: (1) the interview data
are now available but still of low-resource; (2) to generate meaningful and
relevant interview dialogs requires thorough understanding of both resumes and
job descriptions. To address the low-resource challenge, EZInterviewer is
trained on a very small set of interview dialogs. The key idea is to reduce
the number of parameters that rely on interview dialogs by disentangling the
knowledge selector and dialog generator so that most parameters can be trained
with ungrounded dialogs as well as the resume data that are not low-resource.
Specifically, to keep the dialog on track for professional interviews, we pre-
train a knowledge selector module to extract information from resume in the
job-resume matching. A dialog generator is also pre-trained with ungrounded
dialogs, learning to generate fluent responses. Then, a decoding manager is
finetuned to combine information from the two pre-trained modules to generate
the interview question. Evaluation results on a real-world job interview
dialog dataset indicate that we achieve promising results to generate mock
interviews. With the help of EZInterviewer, we hope to make mock interview
practice become easier for job seekers.

EZInterviewer, mock interview generation, knowledge-grounded dialogs, online
recruitment, low-resource deep learning

* Both authors contributed equally to this research.
$\dagger$ Work done during an internship at BOSS Zhipin.
$\ddagger$ Corresponding author: Rui Yan (ruiyan@ruc.edu.cn).

††journalyear: 2023††copyright: acmlicensed††conference: Proceedings of the
Sixteenth ACM International Conference on Web Search and Data Mining; February
27-March 3, 2023; Singapore, Singapore††booktitle: Proceedings of the
Sixteenth ACM International Conference on Web Search and Data Mining (WSDM
’23), February 27-March 3, 2023, Singapore, Singapore††price: 15.00††doi:
10.1145/3539597.3570476††isbn: 978-1-4503-9407-9/23/02††ccs: Computing
methodologies Natural language generation

##  1\. Introduction

To make better preparations, job seekers practice mock interviews, which aims
to anticipate interview questions and prepare them for what they might get
asked in their real turn. However, the outcome of such an approach is
unsatisfactory, since those “mock interviewers” do not have interview
experience themselves, and do not know what the real recruiters would be
interested in. Mock Interview Generation (MIG) represents a plausible solution
to this problem. Not only makes interviews more cost-effective, but mock
interview generators also appear to be feasible, since much can be learned
about the job seekers from their resumes, as can the job itself from the job
description (JD). An illustration of MIG task is shown in Figure 1.

Figure 1.  An example of the Mock Interview Generation task. Based on the
candidate’s work experience and the current dialog on the experience of web
page development, the system generates an interview question “If a product
needs a three-level classification selection, which component would you use
and how to achieve it?”.

There are two main challenges in this task. One is that the knowledge-grounded
interviews are extremely time-consuming and costly to collect. Without a
sufficient amount of training data, the performance of such dialog generation
models drops dramatically (Zhao et al., 2020). The second challenge is to make
the knowledge-grounded dialog relevant to the candidate resume, job
description, and previous dialog utterances. This makes MIG a complex task
involving text understanding, knowledge selection, and dialog generation.

In this paper, we propose EZInterviewer, a novel mock interview generator,
with the aim of making interviews easier to prepare. The key idea is to train
EZInterviewer in a low-resource setting: the model is first pre-trained on
large-scale ungrounded dialogs and resume data, and then fine-tuned on a very
small set of resume-grounded interview dialogs. Specifically, the knowledge
selector consists of a resume encoder to encode the resume, and a key-value
memory network with mask self-attention mechanism, responsible for selecting
relevant information in the resume to focus on to help generate the next
interview utterance. The dialog generator also has two components, a context
encoder which encodes the current dialog context, and a response decoder,
responsible for generating the next dialog utterance without knowledge from
the resumes. This knowledge-insensitive dialog generator is coordinated with
the knowledge selector by a decoding manager that dynamically determines which
component is activated for utterance generation.

It is noted that the number of parameters in the decoding manager can be
small, therefore it only requires a small number of resume-grounded interview
dialogs. Extensive experiments on real-world interview dataset demonstrate the
effectiveness of our model.

To summarize, our contributions are three-fold:

$\bullet$ We introduce a novel Mock Interview Generation task, which is a
pilot study of intelligent online recruitment with potential commercial
values.

$\bullet$ To address the low-resource challenge, we propose to reduce the
number of parameters that rely on interview dialogs by disentangling knowledge
selector and dialog generator so that the majority of parameters can be
trained with large-scale ungrounded dialog and resume data.

$\bullet$ We propose a novel model to jointly process dialog contexts,
candidate resumes, and job descriptions and generate highly relevant,
knowledge-aware interview dialogs.

##  2\. Related Work

Multi-turn response generation aims to generate a response that is natural and
relevant to the entire context, based on utterances in its previous turns.
(Zhang et al., 2019) concatenated multiple utterances into one sentence and
utilized RNN encoder or Transformer to encode the long sequence, simplifying
multi-turn dialog into a single-turn dialog. To better model the relationship
between multi-turn utterances, (Gao et al., 2020; Chen et al., 2020)
introduced interaction between utterances after encoding each utterance.

As human conversations are almost always grounded with external knowledge, the
absence of knowledge grounding has become one of the major gaps between
current open-domain dialog systems and real human conversations (Fu et al.,
2019; Zhang et al., 2021; Niu et al., 2021). A series of work (Li et al.,
2019; Tian et al., 2020) focused on generating a response based on the
interaction between context and unstructured document knowledge, while a few
others (Xu et al., 2020; Liu et al., 2020a) introduced knowledge graphs into
conversations. These models, however, usually under-perform in a low-resource
setting.

To address the low resource problem, (Li et al., 2020b) proposed to enhance
the context-dependent cross-lingual mapping upon the pre-trained monolingual
BERT representations. (Tae et al., 2020) extended the meta-learning algorithm,
which utilized knowledge learned from high-resource domains to boost the
performance of low-resource unsupervised neural machine translation. Different
from the above methods, (Zhao et al., 2020) proposed a disentangled response
decoder in order to isolate parameters that depend on knowledge-grounded
dialogs from the entire generation model. Our model takes a step further,
taking into account the changes in attention on knowledge in multi-turn dialog
scenarios.

##  3\. Model

###  3.1. Problem Formulation

For an input multi-turn dialog context $U=\\{u_{1},u_{2},\dots,u_{m}\\}$
between a job candidate and an interviewer, where $u_{i}$ represents the
$i$-th utterance, we assume there is a ground truth textual interview question
$Y=\\{y_{1},y_{2},\dots,y_{n}\\}$. $m$ is the utterance number in the dialog
context and $n$ is the total number of words in question $Y$. In the $i$-th
utterance, $u_{i}=\\{x^{i}_{1},x^{i}_{2},\dots,x^{i}_{T_{u}^{i}}\\}$.
Meanwhile, there is a candidate resume
$R=\\{(k_{1},v_{1}),(k_{2},v_{2}),\dots,(k_{T_{r}},v_{T_{r}})\\}$
corresponding to the candidate in the interview, which has $T_{r}$ key-value
pairs, and each of which represents an attribute in the resume. For the job-
resume matching pre-training task, there is an external job description
$J=\\{j_{1},j_{2},\dots,j_{T_{j}}\\}$, which has $T_{j}$ words. The goal is to
generate an interview question $Y^{{}^{\prime}}$ that is not only coherent
with the dialog context $U$ but also pertinent to the job candidate’s resume
$R$.

Figure 2.  Overview of EZInterviewer, which consists of three parts: (1)
Knowledge Selector selects salient knowledge information from the candidate
resume; (2) Dialog Generator predicts the next word without knowledge of
resumes; (3) Decoding Manager coordinates the output from knowledge selector
and dialog generator to produce the interview question.

###  3.2. System Overview

In this section, we propose our Low-resource Mock Interview Generator
(EZInterviewer) model, which is divided into three parts as shown in Figure 2:

$\bullet$ Dialog Generator predicts the next word of a response based on the
prior sub-sequence. In our model, we pre-train it by large-scale ungrounded
dialogs.

$\bullet$ Knowledge Selector selects salient knowledge information from the
candidate resume for interview question generation. In our model, we augment
the ability of the knowledge selector by employing it to perform job-resume
matching.

$\bullet$ Decoding Manager coordinates the output from knowledge selector and
dialog generator to predict the interview question.

It is important to note that to train an EZInterviewer model, two pre-train
techniques are employed. Firstly, we pre-train the knowledge selector in a
job-matching task. This is because while it is hard to attend to appropriate
content in a resume just on its own, the salient information in a resume can
be identified in a job-resume matching task (Yan et al., 2019; Le et al.,
2019). Secondly, the context encoder and response decoder of the dialog
generator are pre-trained with a large scale of ungrounded dialogs, so as to
predict the next word of response based on the prior sub-sequence. Finally,
the decoding manager, which relies on a few parameters, coordinates the two
components to generate knowledge grounded interview utterance.

###  3.3. Dialog Generator

Context Encoder. Instead of processing the dialog context as a flat sequence,
we employ a hierarchical encoder (Chen et al., 2022) to capture intra- and
inter-utterance relations, which is composed of a local sentence encoder and a
global context encoder. For the sentence encoder, to model the semantic
meaning of the dialog context, we learn the representation of each utterance
$u_{i}$ by a self-attention mechanism (SAM) initialized by BERT (Devlin et
al., 2019):

(1) $\displaystyle h^{i}_{j}=\text{SAM}_{\text{u}}(e(x^{i}_{j}),h^{i}_{*}).$

We extract the state at “[cls]” position to denote the utterance state,
abbreviated as $h^{i}$. Apart from the local information exchange in each
utterance, we let information flow across multi-turn context:

(2) $\displaystyle h^{c}_{t}=\text{SAM}_{\text{c}}(h^{t},h^{c}_{*}),$

where $h^{c}_{t}$ denotes the hidden state of the $t$-th utterance in
$\text{SAM}_{\text{c}}$.

Response Decoder. Response decoder is responsible for understanding the
previous dialog context and generates the response without the knowledge of
resume information (Li et al., 2022). Our decoder also follows the style of
Transformer.

Concretely, we first apply the self-attention on the masked decoder input,
obtaining $d_{t}$. Based on $d_{t}$ we compute the cross-attention scores over
previous utterances:

(3)
$\displaystyle\alpha^{c}_{t}=\text{ReLU}([d_{t}W_{d}(h^{c}_{i}W_{h})^{T}]).$

The attention weights $\alpha^{c}_{t}$ is then used to obtain the context
vectors as $c_{t}=\sum_{i=1}^{m}\alpha^{c}_{t}h^{c}_{i}$. The context vectors
$c_{t}$, treated as salient contents of various sources, are concatenated with
the decoder hidden state $d_{t}$ to produce the distribution over the target
vocabulary:

(4) $\displaystyle
P_{v}^{w}=\text{Softmax}\left(W_{o}\left[d_{t};c_{t}\right]\right).$

Pre-training process. While interview dialogs are hard to come by, online
conversation is abundant on the internet, and can be easily collected. Hence,
we pre-train the dialog generator on ungrounded conversations. Concretely,
during pre-training process, we employ the context encoder to first encode the
multi-turn previous dialog context. Then, at the $t$-th decoding step, we use
the response decoder to predict the $t$-th word in the response. We set the
loss as the negative log likelihood of the target word $y_{t}$:

(5) $Loss_{g}=-\frac{1}{n}\textstyle\sum^{n}_{t=1}\log P^{w}_{v}(y_{t}).$

###  3.4. Knowledge Selector

Resume Encoder. As shown in Figure 2, a resume contains several key-value
pairs $(k_{i},v_{i})$. Most of key and value fields include a single word or a
phrase such as “skills” or “gender”, and we can obtain the feature
representation through an embedding matrix. Concretely, for each key or value
field with a single word or a phrase, we establish a corresponding resume
embedding matrix $e_{r}^{i}$ that is different from the previous one. Then we
use the resume embedding matrix to map each field word $k_{i}$ or $v_{i}$ into
to a high-dimensional vector space, denoted as $e_{r}^{i}(k_{i})$ or
$e_{r}^{i}(v_{i})$. For fields with more than one word such as “work
experience” or “I used to…”, we denote them as
$v_{i}=(v_{i}^{1},...v_{i}^{l_{i}})$, where $l_{i}$ denotes the word number of
the current field. We first process them through the previous word embedding
matrix $e$, then there is an $\rm{\text{SAM}_{R}}$, similar with
$\rm{\text{SAM}_{u}}$ in Section 3.3, to model the temporal interactions
between words:

(6) $\displaystyle
h^{r_{i}}_{t}=\text{SAM}_{\text{R}}(e(v_{i}^{j}),h^{r_{i}}_{t-1}).$

We use the last hidden state of the $\text{SAM}_{\text{R}}$, _i.e.,_
$h^{r_{i}}_{l_{i}}$ to denote the overall representation for field $v_{i}$.

For brevity, in the following sections, we use $h^{k}_{i}$ and $h^{v}_{i}$ to
denote the encoded key-value pair $(k_{i},v_{i})$ in the resume.

Masked Self-attention. Traditional self-attention can be used to update
representation of each resume item due to its flexibility in relating two
elements in a distance-agnostic manner (Li et al., 2020a). However, as shown
in (Liu et al., 2020b), too much knowledge incorporation may divert the
representation from its correct meaning, which is called knowledge noise (KN)
issue. In our scenario, the information in the resume is divided into several
parts, _i.e.,_ basic personal information, work experiences and extended work,
each of which contains variable number of items. The items within each part
are closely connected, while different parts can be considered as different
domains, and the interaction may introduce a certain amount of noise. To
overcome this problem, we introduce a visible matrix, in which items belonging
to the same part are visible to each other, while the visibility degree
between items is determined by the cosine similarity of semantic
representations, _i.e.,_ $C_{i,j}=\text{cos\\_sim}(h^{v}_{i},h^{v}_{j})$.
Then, the scaled dot-product masked self-attention is defined as:

(7) $\displaystyle\alpha_{i,j}$
$\displaystyle=\frac{\exp\left((h^{k}_{i}W_{q})C_{i,j}(h^{k}_{j}W_{k})^{T}\right)}{\sum\nolimits_{n=1}^{T_{r}}\exp\left((h^{k}_{i}W_{q})C_{i,n}(h^{k}_{j}W_{k})^{T}\right)},$
(8) $\displaystyle\hat{h}^{v}_{i}$
$\displaystyle=\sum\nolimits_{j=1}^{T_{r}}\frac{\alpha_{i,j}h^{v}_{j}}{\sqrt{d}},$

where $d$ stands for hidden dimension and $C$ is the visible matrix.
$\hat{h}^{v}_{i}$ is then utilized as the updated resume value representation.

Key-Value Memory Network. The goal of key matching is to calculate the
relevance between each attribute of the resume and the previous dialog
context. Given dialog context $h^{i}$, for the $j$-th attribute pair
$(k_{j},v_{j})$, we calculate the probability of $h^{i}$ over $k_{j}$, _i.e.,_
$P(k_{j}|h^{i})$, as the matching score $\beta_{i,j}$. To this end, we exploit
the context representation $h^{i}$ to calculate the matching score:

(9) $\displaystyle\beta_{i,j}$
$\displaystyle=\frac{\exp\left(h^{i}W_{a}h^{k}_{j}\right)}{\sum\nolimits_{n=1}^{T_{r}}\exp\left(h^{i}W_{a}h^{k}_{n}\right)}.$

Since context representation $h^{i}$ and resume key representation $h^{k}_{j}$
are not in the same semantic space, we use a trainable key matching parameter
$W_{a}$ to transform these representations into a same space.

As the relevance between context $h^{i}$ and each pair in the resume table
$(k_{j},v_{j})$, the matching score $\beta_{i,j}$ can help to capture the most
relevant pair for generating a correct question. Therefore, as shown in
Equation 10, the knowledge selector reads the information $M_{i}$ from KVMN
via summing over the stored values, and guides the follow-up response
generation, so we have:

(10) $\displaystyle M_{i}$
$\displaystyle=\sum\nolimits_{j=1}^{T_{r}}\beta_{i,j}\hat{h}^{v}_{j},$

where $\hat{h}^{v}_{j}$ is the representation of value $v_{j}$, and
$\beta_{i,j}$ is the matching score between dialog context $h^{i}$ and key
$k_{j}$.

Pre-training Process. In practice, the resume knowledge contains a variety of
professional and advanced scientific concepts such as “Web front-end”, “HTML”,
and “CSS”. These technical terms are difficult to understand for people not
familiar with the specific domain, not to mention for the model that is not
able to access a large-scale resume-grounded dialog dataset. Hence, it would
be difficult for the knowledge selector to understand the resume content and
previous context about the resume, so as to select the next resume pair to
focus on.

On the other hand, we notice that in job-resume matching task, it is crucial
to capture the decisive information in the resume to perform a good matching.
For example, recruiters may tend to hire the candidate with particular
experiences among several candidates with similar backgrounds (Yan et al.,
2019). Intuitively, the key-value pair that is important for job-resume
matching is also the key factor to consider in a job interview. Hence, if we
can let the model learn the salient information in the resume by performing
the job-resume matching task on large-scale job-resume data, then it would
also bring benefits for selecting salient information in interview question
generation.

Concretely, we use the job description to attend to the resume to perform a
job-resume matching task, as a pre-training process for knowledge selector
module. As shown in Figure 2, the Job Encoder encodes the job description by a
$\text{SAM}_{\text{jd}}$:

(11) $\displaystyle h^{jd}_{i}=\text{SAM}_{\text{jd}}(e(j_{i}),h^{jd}_{i-1}),$

where $j_{i}$ denotes the $i$-th word in the job description, and $e(j_{i})$
is mapped by the previous embedding matrix $e$. We use the final hidden state
of the $\text{SAM}_{\text{jd}}$, _i.e.,_ $h^{jd}_{T_{j}}$ as the overall
representation for the description, abbreviated as $h^{jd}$. $h^{jd}$ plays a
similar part as the context representation $h^{i}$, which first attends to the
keys in the resume, and then is used to “weightedly” read the values in the
resume. We use $m_{jd}$ to denote the weighted read result.

In the training process, we first pre-train the knowledge selector by job-
resume matching task, which can be formulated as a classification problem (Qin
et al., 2018). The objective is to maximize the scores of positive samples
while minimizing that of the negative samples. Specifically, we concatenate
$h^{jd}$ and $m_{jd}$ since vector concatenation for matching is known to be
effective (Severyn and Moschitti, 2015). Then the concatenated vector is fed
to a multi-layer, fully-connected, feed-forward neural network, and the job-
resume matching score $s_{jr}$ is obtained as:

(12) $\displaystyle s_{jr}=\sigma\left(F_{s}([h^{jd};m_{jd}]])\right),$

where $[;]$ denotes concatenation operation, and the outputs are the
probabilities of successfully matching. We use the job-resume pairs in
interviews as positive samples, and then use the job-resume pairs without
interviews as negative instances.

After pre-training, the job description is replaced by the context
representations, while the key matching and value combination processes remain
the same. We use a knowledge memory $M$ to store the selection result, where
each slot stores the value combination result $M_{i}$ in Equation 10.

###  3.5. Decoding Manager

The decoding manager is supposed to generate the proper word based on the
knowledge memory and the response decoder. Our idea is inspired by an
observation on the nature of interview dialogs: despite the fact that a dialog
is based on the resume, words and utterances in the dialog are not always
related to resume. Therefore, we postulate that formation of a response can be
decomposed into two uncorrelated actions: (1) selecting a word according to
the context to make the dialog coherent (corresponding to the dialog
generator); (2) selecting a word according to the extra knowledge memory to
ground the dialog (corresponding to the knowledge selector). The two actions
can be independently performed, which becomes the key reason why the large
resume-job matching and ungrounded dialog datasets, although seemingly
unrelated to interview dialogs, can be very useful in an MIG task.

Note that in Section §3.4, we store the selected knowledge $M_{i}$ in a
knowledge memory $M$. To select a word based on it, similar to the response
decoder, we use $d_{t}$ to attend to each slot of knowledge memory, and we can
obtain the knowledge context vector $g^{k}_{t}$ and the output decoder state
$d^{ko}_{t}$.

The response decoder and knowledge selector are controlled by the decoding
manager with a “fusion gate” to decide how much information from each side
should be focused on at each step of interview question prediction.

(13) $\displaystyle\gamma_{f}$
$\displaystyle=\sigma\left(F_{m}(d_{t})\right),$

where $d_{t}$ is the $t$-th decoder hidden state. Then, the probability to
predict word $y_{t}$ can be formulated as:

(14) $\displaystyle d^{o}_{t}$
$\displaystyle=\gamma_{f}d^{wo}_{t}+(1-\gamma_{f})d^{ko}_{t},$ (15)
$\displaystyle P_{v}$
$\displaystyle=\text{softmax}\left(W_{v}d^{o}_{t}+b_{v}\right).$

As for the optimization goal, generation models that use one-hot distribution
optimization target always suffer from the over-confidence issue, which leads
to poor generation diversity (Wang et al., 2021). Hence, aside from the ground
truth one-hot label $P$, we also propose a soft target label $P^{w}_{v}$ (see
Equation 4), which is borrowed from the pre-trained Dialog Generator in
Section 3.3. Forcing the decoding manager to simulate the pre-trained decoder
can help it learn the context of the interview dialog. We combine the one-hot
label with the soft label by an editing gate $\lambda$, as shown in Figure 2.
Concretely, a smooth target distribution $P^{\prime}$ is proposed to replace
the hard target distribution $P$ as:

(16) $\displaystyle P^{\prime}=\lambda P+(1-\lambda)P^{w}_{v}.$

where $\lambda\in[0,1]$ is an adaption factor, $P^{w}_{v}$ is obtained from
Equation 4, and $P$ is the hard target as one-hot distribution which assigns a
probability of 1 for the target word $y_{t}$ and 0 otherwise.

##  4\. Experimental Setup

###  4.1. Dataset

In this paper, we conduct experiments on a real-world dataset provided by
“Boss Zhipin” 111https://www.zhipin.com, the largest online recruiting
platform in China. To protect the privacy of candidates, user records are
anonymized with all personal identity information removed. The dataset
includes 12,666 resumes, 8,032 job descriptions, and 49,214 interview dialog
utterances. The statistics of the dataset is summarized in Table 1. We then
tokenize each sentence into words with the benchmark Chinese tokenizer toolkit
“JieBa” 222https://github.com/fxsjy/jieba.

To pre-train the knowledge selector module, we use a job-resume matching
dataset (Yan et al., 2019), again from “Boss Zhipin”. The training set and the
validation set include 355,000 and 1,006 job-resume pairs, respectively. To
pre-train dialog generator, we choose Weibo dataset (Chan et al., 2019), which
includes a massive number of multi-turn conversations collected from
“Weibo”333https://www.weibo.com. The data includes 2,990,000 context-response
pairs for training and 5,000 pairs for validation. The details are also
summarized in Table 1.

Table 1. Statistics of the datasets used in the experiments. Statistics | Values
---|---
Interview Dialog Dataset
Total number of resumes | 12,666
Total number of dialog utterances | 49,214
Avg turns # per dialog context | 4.47
Avg words # per utterance | 13.18
Job-Resume Dataset
Key-value pairs # per resume | 22
Avg words # per work experience in resume | 72.80
Avg words # per self description in resume | 51.13
Avg words # per job description | 74.26
Ungrounded Dialog Dataset
Total number of context-response pairs | 2,995,000
Avg turns # per dialog context | 4
Avg words # per utterance | 15.15
Table 2. Comparing model performance on full dataset: automatic evaluation metrics. | BLEU-1 | BLEU-2 | BLEU-3 | BLEU-4 | Extrema | Average | Greedy | Dist-1 | Dist-2 | Entity F1 | Cor
---|---|---|---|---|---|---|---|---|---|---|---
_Knowledge-insensitive dialog generation_ |  |  |  |  |  |  |  |  |
Transformer (Vaswani et al., 2017) | 0.5339 | 0.3811 | 0.2836 | 0.2530 | 0.4859 | 0.7673 | 0.6803 | 0.0928 | 0.3157 | 0.3606 | 0.2711
BERT (Devlin et al., 2019) | 0.5671 | 0.3864 | 0.2735 | 0.2583 | 0.4861 | 0.7669 | 0.6792 | 0.0947 | 0.3558 | 0.3711 | 0.2894
DialoGPT (Zhang et al., 2019) | 0.5722 | 0.4015 | 0.3004 | 0.2697 | 0.4858 | 0.7670 | 0.6814 | 0.1001 | 0.3620 | 0.3843 | 0.3002
T5-CLAPS (Lee et al., 2021) | 0.5846 | 0.4126 | 0.3020 | 0.2783 | 0.4837 | 0.7851 | 0.6674 | 0.0970 | 0.3702 | 0.3549 | 0.2870
_Knowledge-aware dialog generation_ |  |  |  |  |  |  |  |  |
TMN (Dinan et al., 2019) | 0.5437 | 0.3891 | 0.2963 | 0.2630 | 0.4841 | 0.7655 | 0.6811 | 0.0996 | 0.3299 | 0.3830 | 0.2652
ITDD (Li et al., 2019) | 0.5484 | 0.4009 | 0.2929 | 0.2656 | 0.4833 | 0.7650 | 0.6859 | 0.1055 | 0.3703 | 0.3661 | 0.2715
DiffKS (Zheng et al., 2020) | 0.5617 | 0.3898 | 0.2776 | 0.2441 | 0.4826 | 0.7830 | 0.6752 | 0.0937 | 0.3612 | 0.3672 | 0.2750
DRD (Zhao et al., 2020) | 0.5711 | 0.4001 | 0.2914 | 0.2548 | 0.4824 | 0.7813 | 0.6783 | 0.0867 | 0.3661 | 0.3825 | 0.2883
DDMN (Wang et al., 2020) | 0.5693 | 0.4065 | 0.2968 | 0.2694 | 0.4831 | 0.7655 | 0.6811 | 0.0944 | 0.3640 | 0.3754 | 0.2869
Persona (Fu et al., 2022) | 0.5532 | 0.3829 | 0.2715 | 0.2377 | 0.4823 | 0.7822 | 0.6783 | 0.0911 | 0.3598 | 0.3833 | 0.2928
EZInterviewer | 0.6106 | 0.4320 | 0.3284 | 0.2917 | 0.4893 | 0.7884 | 0.6886 | 0.1071 | 0.3747 | 0.3927 | 0.3145
No Pre-train | 0.5738 | 0.4029 | 0.2929 | 0.2599 | 0.4846 | 0.7833 | 0.6831 | 0.0981 | 0.3673 | 0.3819 | 0.3007
w/o KM | 0.5795 | 0.4127 | 0.3069 | 0.2754 | 0.4847 | 0.7841 | 0.6762 | 0.0979 | 0.3685 | 0.3803 | 0.3010
w/o KS | 0.5775 | 0.4122 | 0.3067 | 0.2746 | 0.4781 | 0.7668 | 0.6787 | 0.1003 | 0.3691 | 0.3848 | 0.2994
w/o LS | 0.6007 | 0.4232 | 0.3176 | 0.2821 | 0.4869 | 0.7863 | 0.6832 | 0.0969 | 0.3664 | 0.3902 | 0.3127

###  4.2. Comparisons

We compare our proposed model against traditional knowledge-insensitive dialog
generation baselines, and knowledge-aware dialog generation baselines.

$\bullet$ Knowledge-insensitive dialog generation baselines:

Transformer (Vaswani et al., 2017): is based solely on attention mechanisms.
BERT (Devlin et al., 2019): initializes Transformer with BERT as the encoder.
DialoGPT (Zhang et al., 2019): proposes a large, tunable neural conversational
response generation model trained on more conversation-like exchanges.
T5-CLAPS (Lee et al., 2021): generates samples for contrastive learning by
adding small and large perturbations, respectively.

$\bullet$ Knowledge-aware dialog generation baselines:

TMN (Dinan et al., 2019): is built upon a transformer architecture with an
external memory hosting the knowledge. ITDD (Li et al., 2019): incrementally
encodes multi-turn dialogs and knowledge and decodes responses with a
deliberation technique. DiffKS (Zheng et al., 2020): utilizes the differential
information between selected knowledge in multi-turn conversation for
knowledge selection. DRD (Zhao et al., 2020): tackles the low-resource
challenge with pre-training techniques using ungrounded dialogs and documents.
DDMN (Wang et al., 2020): dynamically keeps track of dialog context for multi-
turn interactions and incorporates KB knowledge into generation. Persona (Fu
et al., 2022): introduces personal memory into knowledge selection to address
the personalization issue.

###  4.3. Implementation Details

We implement our experiments in TensorFlow (Abadi et al., 2016) on an NVIDIA
GTX 1080 Ti GPU. For our model and all baselines, we follow the same setting
as described below. We truncate input dialog to 100 words with 20 words in
each utterance, as we did not find significant improvement when increasing
input length from 100 to 200 tokens. The minimum decoding step is 10, and the
maximum step is 20. The word embedding dimension is set to 128 and the number
of hidden units is 256. Experiments are performed with a batch size of 256,
and the vocabulary is comprised of the most frequent 50k words. We use Adam
optimizer (Kingma and Ba, 2015) as our optimizing algorithm. We selected the 5
best checkpoints based on performance on the validation set and report
averaged results on the test set. Note that for better performance, our model
is built based on BERT, and the decoding process is the same as Transformer
(Vaswani et al., 2017). Finally, due to the limitation of time and memory,
small settings are used in the pre-trained baselines.

###  4.4. Evaluation Metrics

To evaluate the performance of EZInterviewer against baselines, we adopt the
following metrics widely used in existing studies.

Overlap-based Metric. Following (Li et al., 2021), we utilize BLEU score
(Papineni et al., 2002) to measure n-grams overlaps between ground-truth and
generated response. In addition, we apply Correlation (Cor) to calculate the
words overlap between generated question and job description, which measures
how well the generated questions line up with the recruitment intention.

Embedding Metrics. We compute the similarity between the bag-of-words (BOW)
embeddings of generated results and reference to capture their semantic
matching degrees (Gu et al., 2019). In particular we adopt three metrics: 1)
Greedy, _i.e.,_ greedily matching words in two utterances based on cosine
similarities; 2) Average, cosine similarity between the averaged word
embeddings in two utterances (Mitchell and Lapata, 2008); 3) Extrema, cosine
similarity between the largest extreme values among the word embeddings in the
two utterances (Forgues et al., 2014).

Table 3. Human evaluation results on: Readability (Read), Informativeness (Info), Meaningfulness (Mean), Usefulness (Use), Relevance (Rel), and Coherence (Coh). Model | Dialog-level | Interview-level
---|---|---
Read | Info | Mean | Use | Rel | Coh
DiffKS | 1.79 | 2.01 | 1.87 | 2.03 | 1.99 | 2.10
DDMN | 1.97 | 1.83 | 1.63 | 2.12 | 2.14 | 1.91
DRD | 2.05 | 2.11 | 2.09 | 2.08 | 2.17 | 2.02
EZInterviewer | 2.42▲ | 2.51▲ | 2.39▲ | 2.46▲ | 2.57▲ | 2.38▲

Distinctness. The distinctness score (Li et al., 2016) measures word-level
diversity by calculating the ratio of distinct uni-gram and bi-grams in
generated responses.

Entity F1. Entity F1 is computed by micro-averaging precision and recall over
knowledge-based entities in the entire set of system responses, and evaluates
the ability of a model to generate relevant entities to achieve specific tasks
from the provided knowledge base (Wang et al., 2020). The entities we use are
extracted from an entity vocabulary provided by “Boss Zhipin”.

Human Evaluation Metrics. We further employ human evaluations aside from
automatic evaluations. Three well-educated annotators from different majors
are hired to evaluate the quality of generated responses, where the evaluation
is conducted in a double-blind fashion. In total 100 randomly sampled
responses generated by each model are rated by each annotator on both dialog
level and interview level. We adopt the Readability (is the response
grammatically correct?) and Informativeness (does the response include
informative words?) to judge the quality of the generated responses on the
dialog level. On the interview level, we adopt Meaningfulness (is the
generated question meaningful?), Usefulness (is the question worth the job
candidate preparing in advance?), Relevance (is the question relevant to the
resume?) and Coherence (is the generated text coherent with the context?) to
assess the overall performance of a model and the quality of user experience.
Each metric is given a score between 1 and 3 (1 = bad, 2 = average, 3 = good).

##  5\. Experimental Result

###  5.1. Overall Performance

Automatic evaluation. The comparison between EZInterviewer and state-of-the-
art generative baselines is listed in Table 2.

We take note that the knowledge-aware dialog generation models outperform
traditional dialog models, suggesting that utilizing external knowledge
introduces advantages in generating relevant response. We also notice the pre-
train based model DRD outperforms other baselines, showing that initializing
parameters by pre-training on large-scale data can lead to a substantial
improvement in performance. It is worth noting some models achieve better
Entity F1 but a lower BLEU score; this suggests that those models tend to copy
necessary entity words from the knowledge but are not able to use them
properly.

EZInterviewer outperforms baselines on all automatic metrics. Firstly, our
model improves BLEU-1 by 6.92% over DRD. On the Distinctness metric Dist-1,
our model outperforms DialoGPT by 6.99%, suggesting that the generated
interview questions are diversified and personalized with different
candidates’ resumes. Moreover our model attains a good score of 0.3927 on
entity F1, which evaluates the degree to which the generated question is
grounded on the knowledge base. Finally, Cor score of 0.3145 suggests the
questions generated by EZInterviewer is in line with the job description,
hence reflect the intention of the recruiters. Overall the metrics demonstrate
that our model successfully learns an interviewer’s points of interest in a
resume, and incorporates this knowledge into interview questions properly.

Human evaluation. The results of human evaluations on all models are listed in
Table 3. EZInterviewer is the top performer on all the metrics. Specifically,
our model outperforms DiffKS by 35.20% on Readability, suggesting that
EZInterviewer manages to reduce the grammatical errors and improve the
readability of the generated response. As for the Informativeness metric, our
model scores 0.68 higher than DDMN. This indicates that EZInterviewer captures
salient information in the resume. On the interview level, EZInterviewer’s
Usefulness score is 18.27% better than DRD, demonstrating its capabilities to
help job seekers to pick the right questions to prepare. On Relevance metric,
our model outperforms all baselines by a considerable margin, suggesting that
the generated questions are closely related to the interview process. Our
model also performs better than other baselines in Meaningfulness and
Coherence metrics, suggesting the overall higher quality of our model.

The above results demonstrate the competence of EZInterviewer in producing
meaningful and useful interview questions whilst keeping the interview dialog
flowing smoothly, just like a human recruiter. Note that the average kappa
statistics of human evaluation are 0.51 and 0.48 on dialog level and interview
level, respectively, which indicates moderate agreement between annotators. To
prove the significance of these results, we also conduct the two-tailed paired
student t-test between our model and DRD (row with shaded background). The
statistical significance of observed differences is denoted using ▲(or ▼) for
strong (or weak) significance for $\alpha=0.01$. Moreover, we obtain an
average p-value of $5\times 10^{-6}$ and $3\times 10^{-4}$ for both levels,
respectively.

Figure 3.  Visualization of key matching between dialog context and selected
resume keys, _i.e.,_ work experiment (Exp), self description (Desc), skills
(Ski), work years (Year), expected position (Pos), school (Sch), and major
(Maj). $U_{i}$ denotes the $i$-th utterance.

###  5.2. Ablation Study

We conduct an ablation study to assess the contribution of individual
components in the model. The results are shown in Table 2.

To verify the effectiveness of knowledge memory, we omit the knowledge
selection of dialog context history and directly use the last utterance
representation to select knowledge. The results (see row w/o KM) confirm that
employing each turn of historical dialog to select knowledge and saving it in
memory contribute to generating better responses. To confirm whether selecting
knowledge helps with the response generation process, we remove it from the
model, then simply add the representation of each utterance with all resume
values, and store it into the memory. This results in a drop of 5.42% in
BLEU-1 (see row w/o KS), suggesting that selecting resume knowledge is
beneficial in response generation.

Table 4. Translated interview questions generated by baselines and
EZInterviewer: an example.  denotes information extracted and words generated
by knowledge selector, whereas  denotes words generated from dialog generator.

Resume | Interview
---|---
Gender | Male |  Job Description: The main content of this work includes design and development based on the React front-end framework. It requires the ability to efficiently complete front-end development work and serve customers well.
Age | 28
Education | Undergraduate
Major | Computer Science
Work Years | 10 |  Context: U1: Have you been engaged in front-end development work before? U2: Yes, I am good at Vue, Node.js and some other skills. U3: Okay, so do you have any React related experience? U4: Yes, I have more than 10 years of work experience.
Expected Position | Front-end Engineer
Low Salary | 5
High Salary | 6
Skills | Vue, Node.js, Java
Experience |  I was engaged in front-end design and was responsible for the project development based on the React front-end framework and participated in the system architecture process. |  Ground Truth: So can you introduce a React related project you have done? DDMN: What other front-end frameworks would you use? DRD: Hello, can you tell us about your previous work? EZInterviewer: Well, can you introduce the experience based on React framework?

Figure 4.  Automatic evaluation metrics of DDMN, DRD and EZInterviewer on
training data of different scales.

###  5.3. Analysis of Knowledge Selector

In Section § 3.4, we introduce the selecting mechanism of knowledge selector,
where the final attention (matching) score is obtained in Equation 9. To study
what specific information is attended by the knowledge selector, and whether
the selected information is suitable for the next interview question, we
conduct a case study to visualize the matching score produced by the knowledge
selector, as shown in Table 5.2 and Figure 3. The first utterance in the
history is “Have you been engaged in front-end development work before?”, and
the knowledge selector learns that this utterance focuses on the work
experience in the resume. Accordingly, the fourth utterance “I have more than
10 years of work experience.” pays more attention to work years and work
experience than other items in the resume. This demonstrates that the
knowledge selector learns which item in the resume to focus on when generating
each utterance. Hence, when we want to ask the candidate to “introduce a React
related project”, the knowledge selector focuses on the work experience in the
resume and generates the mock interview question.

###  5.4. Impact of Training Data Scales

To understand how our model and baseline models perform in a low-resource
scenario, we first evaluate them on the full training dataset, then on smaller
portions of the training dataset. Figure 4 presents the performance of the
models, DDMN, DRD, and EZInterviewer, on the full, 1/2, 1/4, 1/8 and 1/10 of
the training dataset (data scale), respectively. It is observed that as the
size of training dataset reduces, DDMN suffers a massive drop across all
metrics, whereas the scores of pre-training based models, _i.e.,_ DRD and
EZInterviewer, stay relatively stable. This demonstrates pre-training as an
effective strategy to tackle the low-resource challenge. Moreover, our model
outperforms DRD on all data scales, demonstrating the superiority of our
model. Figure 4 shows EZInterviewer eventually achieves the best performance
on all metrics and outperforms (albeit slightly), with only 1/10 training data
against all state-of-the-art baselines trained with the full training dataset.

###  5.5. Case Study

Table 5.2 presents a translated example of EZInterviewer and baseline models.
We observe that the question from EZInterviewer not only catches the context,
but also expands the conversation with proper knowledge. This is highlighted
in color codes: pink-colored words, _i.e.,_ “experience” and “React
framework”, are what knowledge selector extracts from resume knowledge,
whereas blue-colored words, _i.e.,_ “Well, can you introduce the…” and “based
on”, which closely connect to the context, are generated by dialog generator.
In contrast, the questions from the baselines respond to the dialog but fail
to make connection with the resume knowledge.

##  6\. Conclusion

In this paper, we conduct a pilot study for the novel application of
intelligent online recruitment, namely EZInterviewer, which aims to serve as
mock interviewers for job-seekers. The mock interview is generated with
thorough understanding of the candidate’s resume, the job requirements, the
previous utterances in the context, as well as the selected knowledge for
grounded interviews. To address the low-resource challenge, EZInterviewer is
trained on a very small set of interview dialogs. The key idea is to reduce
the number of parameters that rely on interview dialogs by disentangling the
knowledge selector and dialog generator so that most parameters can be trained
with ungrounded dialogs as well as the resume data that are not low-resource.
We conduct extensive experiments to demonstrate the effectiveness of the
proposed solution EZInterviewer. Our model achieves the best results using
full training data as well as small subsets of the training data in terms of
various metrics such as BLEU, embedding based similarity and diversity, as
well as human judgments. In particular, the human evaluation indicates that
our solution EZInterviewer can provide satisfactory mock interviews to help
the job-seekers prepare the real interview, making the interview preparation
process easier.

## Acknowledgments

We would like to thank the anonymous reviewers for their constructive
comments. This work was supported by National Natural Science Foundation of
China (NSFC Grant No. 62122089). Rui Yan is supported by Beijing Academy of
Artificial Intelligence (BAAI).

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