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|  Nikos G. Evgenidis received the Diploma (5 years) in Electrical and
Computer Engineering from the Aristotle University of Thessaloniki, Greece, in
2022, where he is currently pursuing his PhD. He is also a member of the
Wireless and Communications and Information Processing (WCIP) group. His major
research interests include semantic communications, over-the-air computing,
non-orthogonal multiple access, machine learning and optimization theory.
---|---
|  Nikos A. Mitsiou (Graduate Student Member, IEEE) received the Diploma
Degree (5 years) in Electrical and Computer Engineering from the Aristotle
University of Thessaloniki (AUTH), Greece, in 2021, where he is currently
pursuing his PhD with the Department of Electrical and Computer Engineering.
He is a member of the Wireless and Communications & Information Processing
(WCIP) Group. He was an Exemplary Reviewer of the IEEE Wireless Communications
Letters in 2022 (top 3% of reviewers). His research interests include
optimization theory, and machine learning with application to wireless
networks.
---|---
|  Vasiliki I. Koutsioumpa received the Diploma (5 years) in Electrical and
Computer Engineering from the Aristotle University of Thessaloniki, Greece, in
2023, where she is currently pursuing her PhD. She is also a member of the
Wireless Communications and Information Processing (WCIP) Group. Her major
research interests include next-generation multiple access schemes, satellite
networks and optimization theory.
---|---
|  Sotiris A. Tegos  (Senior Member, IEEE) received the Diploma (5 years) and
Ph.D. degrees from the Department of Electrical and Computer Engineering,
Aristotle University of Thessaloniki, Thessaloniki, Greece, in 2017 and 2022,
respectively. Since 2022, he is a Postdoctoral Fellow at the Wireless
Communications and Information Processing (WCIP) Group, Aristotle University
of Thessaloniki, Thessaloniki, Greece, and at the Department of Applied
Informatics, University of Macedonia, Thessaloniki, Greece. Since 2023, he is
also a Postdoctoral Fellow at the Department of Electrical and Computer
Engineering, University of Western Macedonia, Kozani, Greece. In 2018, he was
a visiting researcher at the Department of Electrical and Computer
Engineering, Khalifa University, Abu Dhabi, UAE. His current research
interests include multiple access in wireless communications, optical wireless
communications, and reconfigurable intelligent surfaces. He is a Working Group
Member of the Newfocus COST Action “European Network on Future Generation
Optical Wireless Communication Technologies”. He serves as an Editor for IEEE
Communications Letters. He received the Best Paper Award in 2023 Photonics
Global Conference (PGC). He was an exemplary reviewer in IEEE Wireless
Communications Letters in 2019 and 2022 (top 3% of reviewers).
---|---
|  Panagiotis D. Diamantoulakis  (Senior Member, IEEE) received the Diploma (5
years) and the Ph.D. degree from the Department of Electrical and Computer
Engineering, Aristotle University of Thessaloniki, Thessaloniki, Greece, in
2012 and 2017, respectively. Since 2022 he is a Postdoctoral Fellow with the
Department of Applied Informatics, University of Macedonia, Thessaloniki,
Greece. Since 2017, he has been a Postdoctoral Fellow with Wireless
Communications and Information Processing (WCIP) Group, AUTH and since 2021,
he has been a Visiting Assistant Professor with the Key Lab of Information
Coding and Transmission, Southwest Jiaotong University, Chengdu, China. His
research interests include optimization theory and applications in wireless
networks, optical wireless communications, and goal-oriented communications.
He is a Working Group Member of the Newfocus COST Action “European Network on
Future Generation Optical Wireless Communication Technologies.” He serves as
an Editor of IEEE Open Journal of the Communications Society, Physical
Communications (Elsevier), and Frontiers in Communications and Networks, while
during 2018-2023 he has been an Editor of IEEE Wireless Communications
Letters. He was also an Exemplary Editor of the IEEE Wireless Communications
Letters in 2020, and an Exemplary Reviewer of the IEEE Communications Letters
in 2014 and the IEEE Transactions on Communications in 2017 and 2019 (top 3%
of reviewers).
---|---
|  George K. Karagiannidis  (Fellow, IEEE) is currently Professor in the
Electrical & Computer Engineering Dept. of Aristotle University of
Thessaloniki, Greece and Head of Wireless Communications & Information
Processing (WCIP) Group. He is also Faculty Fellow in the Cyber Security
Systems and Applied AI Research Center, Lebanese American University. His
research interests are in the areas of Wireless Communications Systems and
Networks, Signal processing, Optical Wireless Communications, Wireless Power
Transfer and Applications and Communications & Signal Processing for
Biomedical Engineering. Dr. Karagiannidis was in the past Editor in several
IEEE journals and from 2012 to 2015 he was the Editor-in Chief of IEEE
Communications Letters. From September 2018 to June 2022 he served as
Associate Editor-in Chief of IEEE Open Journal of Communications Society.
Currently, he is the Editor-in-Chief of IEEE Transactions on Communications.
Recently, he received three prestigious awards: The 2021 IEEE ComSoc RCC
Technical Recognition Award, the 2018 IEEE ComSoc SPCE Technical Recognition
Award and the 2022 Humboldt Research Award from Alexander von Humboldt
Foundation. Dr. Karagiannidis is one of the highly-cited authors across all
areas of Electrical Engineering, recognized from Clarivate Analytics as
Highly-Cited Researcher in the nine consecutive years 2015-2023.
---|---


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##  Appendix A Worst-Case Lower Bounds

In this section we describe a general reduction from the Tseitin formula to
the Perfect Matching formula as it appeared in [15] for Polynomial Calculus.
We then observe that this reduction also works for the SoS and bounded depth
Frege proof systems.

Starting from a graph $G$ such that the Tseitin formula $\tau(G)$ is hard for
a proof system $P$, we want to craft a graph $H$ so that $\mathsf{PM}(H)$ is
hard for $P$. To simplify the presentation, let us assume that $G$ is
$d$-regular. As we are interested in unsatisfiable instances, i.e., when $G$
has an odd number of vertices, we may assume that $d$ is even.

The graph $H$ is a “blow-up” (or “lift”) of $G$: each vertex in $V(G)$ is
lifted to a clique of $d+1$ vertices and each lifted edge connects a single
pair of vertices from the corresponding cliques. If we denote the lifted
vertices of $v\in V(G)$ by
$\mathsf{lift}(v)=\\{(v,\star),(v,1),\ldots,(v,d)\\}$, we add for each edge
$\\{u,v\\}\in E(G)$, where $v$ is the $i$th neighbor of $u$ and $u$ is the
$j$th neighbor of $v$, an edge $\\{(u,i),(v,j)\\}$. An illustration of the
construction of $H$ can be found in Figure 4.

(a) The graph $G$. (b) The lift of a single vertex.

(c) The lifted graph $H$.

Figure 4: An illustration of the blow-up construction, starting from a
4-regular graph.

For intuition, let us describe how we would obtain a satisfying assignment to
the Perfect Matching Formula from a hypothetical satisfying assignment to the
Tseitin Formula. Set the lifted edges to the same value as they are set to in
the Tseitin Formula. Now observe that each charge is odd and hence there is an
even number of vertices left that are not matched yet in each
$\mathsf{lift}(v)$, for $v\in V(G)$. As the vertices in $\mathsf{lift}(v)$
form a clique, we can select a perfect matching on these unmatched vertices to
obtain a satisfying assignment to the Perfect Matching Formula.

Buss et al. [15] showed that Polynomial Calculus can simulate this reduction.

######  Theorem A.1 ([15]).

There are graphs $G$ on an odd number of vertices $n$ and maximum degree
$\Delta(G)=5$ such that Polynomial Calculus over any field of characteristic
different from $2$ requires degree $\Theta(n)$ to refute $\mathsf{PM}(G)$.

What remains is to check that this reduction also gives Perfect Matching
worst-case lower bounds for Sum-of-Squares and bounded depth Frege. This
straightforward verification is carried out in the following two sections.

###  A.1 Sum-of-Squares

A nice property of the Sum-of-Squares system is that if the variables for a
formula $\mathcal{Q}$ can be expressed as well-behaved low-degree polynomials
in the variables of another formula $\mathcal{P}$ for which a pseudo-
expectation exists, then a pseudo-expectation also exists for $\mathcal{Q}$.
This property is well-known but let us state and quickly prove the exact
version we need.

######  Claim 12.

Let $\mathcal{P}\subseteq\mathbb{R}[x_{1},\ldots,x_{n}]$ and
$\mathcal{Q}\subseteq\mathbb{R}[y_{1},\ldots,y_{m}]$ be two systems of
polynomial equations. Let $\tilde{\mathbb{E}}_{\mathcal{Q}}$ be a degree $D$
pseudo-expectation for $\mathcal{Q}$. Suppose there is a function
$f:\\{x_{1},\ldots,x_{m}\\}\to\mathbb{R}[y_{1},\ldots y_{n}]$, mapping the $x$
variables to polynomials in $y$ of degree at most $t$. Extend $f$ to
polynomials by applying the function to each variable individually. If $f$
satisfies that $\tilde{\mathbb{E}}_{\mathcal{Q}}[f(r\cdot p)]=0$, for all
$p\in\mathcal{P}$ and $r\in\mathbb{R}[x_{1},\ldots x_{m}]$ of degree
$\deg(r\cdot p)\leq D/t$, then $\tilde{\mathbb{E}}_{\mathcal{Q}}\circ f$ is a
degree $D/t$ pseudo-expectation for $\mathcal{P}$.

######  Proof A.2.

As $f$ only maps variables we have that
$\tilde{\mathbb{E}}_{\mathcal{Q}}[f(1)]=\tilde{\mathbb{E}}_{\mathcal{Q}}[1]=1$.
Also, we need to check that $\tilde{\mathbb{E}}_{\mathcal{Q}}[f(s^{2})]\geq 0$
for $s\in\mathbb{R}[x_{1},\ldots,x_{m}]$ of degree $\deg(s)\leq D/2t$. As we
apply $f$ individually to each variable, we can write
$\tilde{\mathbb{E}}_{\mathcal{Q}}[f(s^{2})]=\tilde{\mathbb{E}}_{\mathcal{Q}}[\big{(}\sum_{t\in
s}f(t)\big{)}^{2}]\geq 0$, as $\tilde{\mathbb{E}}_{\mathcal{Q}}$ is a degree
$D$ pseudo-expectation.

In order to apply Claim 12 with $\mathcal{Q}=\tau(G)$ and
$\mathcal{P}=\mathsf{PM}(H)$, we need to express each variable from the
Perfect Matching formula as a low degree polynomial in the Tseitin variables.

Let us recall some notation. For a vertex $v\in V(G)$, let $Y_{v}$ be the set
of Tseitin variables corresponding to edges incident to $v$ and denote by
$A_{v}$ all boolean assignments to $Y_{v}$ that satisfy the vertex axiom of
$v$, i.e., assignments that set an odd number of edges to true. For a Tseitin
variable $y_{e}$, where $e\in E(G)$, let $\mathsf{lift}(y_{e})\in E(H)$ denote
the lifted edge variable.

With this notation at hand, let us define the function $f$ to use in Claim 12.
Variables that correspond to lifted edges,
$x_{e}=\mathsf{lift}(y_{e^{\prime}})$ for some $e^{\prime}\in E(G)$, are set
to 1 if and only if $y_{e^{\prime}}$ is set to $1$ and the variables in
$Y_{v}$ are set according to some assignment in $A_{v}$:

$\displaystyle f(x_{e})=\sum_{\begin{subarray}{c}\alpha\in A_{v}\\\
\alpha(y_{e^{\prime}})=1\end{subarray}}\mathbbm{1}_{\\{Y_{v}=\alpha\\}}\enspace.$
(27)

Note that this is a polynomial of degree $\deg(v)=d$ in the $y_{e}$’s. For
each assignment $\alpha\in A_{v}$, set the variables in $\mathsf{lift}(Y_{v})$
according to $\alpha$ and fix a matching $m_{\alpha}$ on the vertices in
$\mathsf{lift}(v)$ not matched by $\alpha$. For any edge
$e\subseteq\mathsf{lift}(v)$, let

$\displaystyle f(x_{e})=\sum_{\begin{subarray}{c}\alpha\in A_{v}\\\ e\in
m_{\alpha}\end{subarray}}\mathbbm{1}_{\\{Y_{v}=\alpha\\}}\enspace.$ (28)

If we apply $f$ individually to each variable, we claim that for
$i\in\\{1,\ldots,d\\}$ and $v\in V(G)$ the polynomial
$f(q^{\mathsf{PM}}_{(v,i)})$ is equal to the Tseitin axiom $q^{\tau}_{v}$:

$\displaystyle f(q^{\mathsf{PM}}_{(v,i)})$
$\displaystyle=\sum_{e\ni(v,i)}f(x_{e})-1$ (29)
$\displaystyle=\sum_{\begin{subarray}{c}\alpha\in A_{v}\\\
\alpha(y_{e^{\prime}})=1\end{subarray}}\mathbbm{1}_{\\{Y_{v}=\alpha\\}}+\sum_{\begin{subarray}{c}\alpha\in
A_{v}\\\
\alpha(y_{e^{\prime}})=0\end{subarray}}\mathbbm{1}_{\\{Y_{v}=\alpha\\}}-1$
(30) $\displaystyle=q^{\tau}_{v}\enspace,$ (31)

using that the $m_{\alpha}$ are matchings. The axioms
$q^{\mathsf{PM}}_{(v,\star)}$ are handled similarly:

$\displaystyle
f(q^{\mathsf{PM}}_{(v,\star)})=\sum_{e\ni(v,\star)}f(x_{e})-1=\sum_{\begin{subarray}{c}\alpha\in
A_{v}\end{subarray}}\mathbbm{1}_{\\{Y_{v}=\alpha\\}}-1=q^{\tau}_{v}\enspace.$
(32)

As $\tilde{\mathbb{E}}_{\tau(G)}$ maps all axioms multiplied by a low degree
polynomial to 0, the same holds for $\tilde{\mathbb{E}}_{\tau(G)}\circ f$ and
we can thus apply Claim 12.

We conclude that if there is a degree $D$ pseudo-expectation
$\tilde{\mathbb{E}}_{\tau(G)}$ for the Tseitin Formula $\tau(G)$, then there
is a degree $D/d$ pseudo-expectation $\tilde{\mathbb{E}}_{\mathsf{PM}(H)}$ for
the Perfect Matching formula over the lifted graph $H$. Using Grigoriev’s
Tseitin lower bound [32] we obtain the following Theorem.

######  Theorem A.3.

There are graphs $G$ on an odd number of vertices $n$ and maximum degree
$\Delta(G)=5$ for which SoS requires degree $\Theta(n)$ to refute
$\mathsf{PM}(G)$.

###  A.2 Bounded Depth Frege

In this section we intend to prove the following theorem.

######  Theorem A.4.

There is a constant $c>0$ such that the following holds. Suppose
$D\leq\frac{c\log n}{\log\log n}$. Then there are graphs $G$ on an odd number
of vertices $n$ and maximum degree $\Delta(G)=5$ such that any depth-$D$ Frege
refutation of $\mathsf{PM}(G)$ requires size $\exp(\Omega(n^{c/D}))$.

As in the previous section we use a function $f$, mapping Perfect Matching
variables to low depth formulas in the Tseitin Variables, to argue that we can
transform a refutation of $\mathsf{PM}(H)$ into a refutation of the Tseitin
formula $\tau(G)$. Assuming that this can be done, we use the following recent
result of Håstad about the Tseitin formula over the grid to obtain Theorem
A.4.

######  Theorem A.5 ([34]).

Suppose that $D\leq\frac{\log n}{59\log\log n}$, then any depth-$D$ Frege
refutation of the Tseitin formula on the $n\times n$ grid requires size
$\exp(\Omega(n^{1/58(D+1)}))$.

In the previous section $f$ mapped to polynomials. As we are now working with
formulas we need to translate the polynomials to formulas. This is
straightforward; reusing notation from the previous section, let

$\displaystyle f(x_{e})=\bigvee_{\begin{subarray}{c}\alpha\in A_{v}\\\
\alpha(y_{e^{\prime}})=1\end{subarray}}\mathbbm{1}_{\\{Y_{v}=\alpha\\}}\enspace,$
(33)

if $x_{e}=\mathsf{lift}(y_{e^{\prime}})$ is a lifted edge. Else let

$\displaystyle f(x_{e})=\bigvee_{\begin{subarray}{c}\alpha\in A_{v}\\\ e\in
m_{\alpha}\end{subarray}}\mathbbm{1}_{\\{Y_{v}=\alpha\\}}\enspace.$ (34)

Suppose there is a depth-$D$ Frege refutation $\pi$ of the Perfect Matching
formula $\mathsf{PM}(H)$. Replace each occurrence of a Perfect Matching
variable $x_{e}$ by $f(x_{e})$ to obtain a depth-$(D+2)$ refutation
$\pi^{\prime}$. We claim that $\pi^{\prime}$ can be massaged into a refutation
of the Tseitin formula $\tau(G)$ of size
$O_{d}(\operatorname{\sf{Size}}(\pi))$.

To this end we need to argue that $f$ maps Perfect Matching axioms to Tseitin
Axioms or tautologies that are derivable in small size and depth. Analoguous
to SoS observe that for all $v\in V(G)$ and $i\in\\{1,\ldots,d\\}$,

$\displaystyle f(\bigvee_{e\ni(v,i)}x_{e})$
$\displaystyle=\bigvee_{e\ni(v,i)}f(x_{e})$ (35)
$\displaystyle=\bigvee_{\begin{subarray}{c}\alpha\in A_{v}\\\
\alpha(y_{e^{\prime}})=1\end{subarray}}\mathbbm{1}_{\\{Y_{v}=\alpha\\}}\vee\bigvee_{\begin{subarray}{c}\alpha\in
A_{v}\\\
\alpha(y_{e^{\prime}})=0\end{subarray}}\mathbbm{1}_{\\{Y_{v}=\alpha\\}}\enspace.$
(36)

Note that that the final formula is equal to the axiom $q^{\tau}_{v}$, up to a
reordering of the terms. As the axioms are over $d$ variables, which is
constant in our case, this formula can be derived from $q^{\tau}_{v}$ in
constant depth and size. The axiom for the vertex $(v,\star)$ is handled in a
similar manner.

Last we need to show that the axioms $\bar{x}_{e}\vee\bar{x}_{e^{\prime}}$,
for edges $e\neq e^{\prime}\in E(H)$ satisfying $e\cap
e^{\prime}\neq\emptyset$, are mapped to a tautology derivable in small size
and depth. If we let $\\{(v,i)\\}=e\cap e^{\prime}$ we can write

$\displaystyle
f(\bar{x}_{e}\vee\bar{x}_{e^{\prime}})=\big{(}\lnot\bigvee_{\beta\in
B}\mathbbm{1}_{\\{Y_{v}=\beta\\}}\big{)}\vee\big{(}\lnot\bigvee_{\gamma\in
C}\mathbbm{1}_{\\{Y_{v}=\gamma\\}}\big{)}\enspace,$ (37)

for disjoint subsets $B,C\subseteq A_{v}$. Observe that this formula is a
tautology and defined on $d$ variables. Thus it is derivable in constant depth
and size dependent on $d$, which is constant in our case.

##  Appendix B Embedding Algorithm

1 $\textnormal{{FixExpansion}}(G,C,A,\beta)$ while  _$G[C]$ is not a
$\beta$-expander_ do

2 $U\leftarrow_{\text{any}}$ subset of $C$ such that $|U|\leq|C|/2$ and
$|N(U,C\setminus U)|<\beta|U|$ ;

3 $C\leftarrow C\setminus U$ ;

4 $A\leftarrow A\cup U$

Algorithm 1 Restores $\beta$-expansion of $G[C]$.

Require : Conditions of Lemma 5.5.

1

2$\textnormal{{EmbedVertex}}(G,r,s,\beta,k)$

3$\gamma\leftarrow\frac{\beta}{3(1+\beta)}$ ;

4 $s\leftarrow\max\\{1/\gamma,s\\}$ ;

5 $r^{\prime}\leftarrow(1+1/\gamma)r$ ;

6 $A,\mathcal{B}\leftarrow\emptyset;C\leftarrow V(G)$ ;

7

8while  _$|\mathcal{B}| <r^{\prime}$_ do

9 $U\leftarrow_{\text{any}}$ subset of $C$ such that $|U|=s$ and $G[U]$ is a
single connected component ;

10 $\mathcal{B}\leftarrow\mathcal{B}\cup\\{U\\}$; $C\leftarrow C\setminus U$ ;

11 $\textnormal{{FixExpansion}}(G,C,A,\gamma)$ ;

12

13 $\mathcal{F}\subseteq\mathcal{B}$ maximal such that
$|\cup_{F\in\mathcal{F}}N(F,C)|<\gamma s|\mathcal{F}|$ ;

14 $\mathcal{B}\leftarrow\mathcal{B}\setminus\mathcal{F}$; $A\leftarrow
A\cup_{F\in\mathcal{F}}F$ ;

15

16 $v\leftarrow_{\text{any}}C$ such that $\deg_{G[C]}(v)\geq r^{\prime}$ ;

17 $F\leftarrow$ a transversal of $\\{\,N(B,C)\mid B\in\mathcal{B}\,\\}$ ;

18 $\\{\,p_{i}\mid i\in[r]\,\\}\leftarrow$ from Lemma 5.3 applied to $G[C]$,
$v$ and $F$ ;

return $\\{v\\}\cup\\{\,V(p_{i})\cup B_{i}\mid i\in[r]\,\\}$ ;

$\triangleright$ Shrink branches appropriately

Algorithm 2 Finds an $(r,s)$-cross in an $\beta$-expander $G$ as in the proof
of Lemma 5.5.

$\textnormal{{UnEmbedVertex}}(A,A^{\prime},B,H,I,x)$
$(v,\mathcal{U})\leftarrow B_{x}$ ;

$\triangleright$ $v$ is the center and $\mathcal{U}$ are the branches of
$B_{x}$

1 $B\leftarrow B\setminus B_{x}$; $I\leftarrow I\setminus x$ ;

2 $W\leftarrow\emptyset$ ;

3

4for all  _$e\in E(H)$ such that $x\in e$ and $e$ is embedded_ do

5 let $U\leftarrow\mathcal{U}$ be the branch adjacent to $B_{e}$ ;

6 $\mathcal{U}\leftarrow\mathcal{U}\setminus U$ ;

7 $B\leftarrow B\setminus B_{e}$ ;

8 $W\leftarrow W\cup U\cup B_{e}$ ;

9

$A\leftarrow A\cup\mathcal{U}$ ;

$\triangleright$ First add to $A$, then to $A^{\prime}$ to maintain the
invariant

10 $A^{\prime}\leftarrow A^{\prime}\cup W\cup\\{v\\}$ ;

Algorithm 3 Remove the embedding of vertex $x$.

1 $\textnormal{{EmbedGraph}}(H,G,\alpha)$ $\beta\leftarrow\alpha/3(1+\alpha)$
;

2 $A,A^{\prime},B\leftarrow\emptyset;C\leftarrow V(G)$ ;

3 $I\leftarrow\emptyset$ ;

4

5while  _$I\neq V(H)$_ do

6 $x\leftarrow_{\text{any}}V(H)\setminus I$ ;

7 $B_{x}\leftarrow\textnormal{{EmbedVertex}}(G[C],\deg_{H}(x),s,\beta,k)$ ;

8 $C\leftarrow C\setminus B_{x}$; $B\leftarrow B\cup B_{x}$; $I\leftarrow
I\cup x$ ;

9 $\textnormal{{FixExpansion}}(G,C,A,\beta)$ ;

10 $\mathcal{U}_{\text{free}}(K)\leftarrow$ branches of the cross $K$ that are
_not_ used to connect to a neighbor ;

11

12 for all  _$\\{x,y\\}\in E(H)$ such that $y\in I$_ do

13 Try;

14 $U_{z}\leftarrow_{\text{any}}\mathcal{U}_{\text{free}}(B_{z})$ such that
$|N(U_{z},C)|\geq\beta|U_{z}|$ for $z\in\\{x,y\\}$ ;

15 Catch no such $U_{z}$ for $z\in\\{x,y\\}$ ;

16 $\textnormal{{UnEmbedVertex}}(A,A^{\prime},B,H,I,z)$ ; continue ;

17

18 $B_{xy}\leftarrow$ odd path from Lemma 5.6 applied to $G[C]$, $N(U_{x},C)$
and $N(U_{y},C)$ ;

19 $C\leftarrow C\setminus B_{xy}$; $B\leftarrow B\cup B_{xy}$ ;

20 $\textnormal{{FixExpansion}}(G,C,A,\beta)$

21

22 return $B$ ;

Algorithm 4 Embeds $H$ in an $\alpha$-expander $G$ as in the proof of Theorem
3.3.


# Hybrid Models for Facial Emotion Recognition in Children

Rafael Zimmer  University of São PauloBrazil<EMAIL_ADDRESS>,  Marcos
Sobral  Federal University of TocantinsBrazil
<EMAIL_ADDRESS>and  Helio Azevedo  Renato Archer
Information Technology CenterBrazil<EMAIL_ADDRESS>
(2023)

###### Abstract.

This paper focuses on the use of emotion recognition techniques to assist
psychologists in performing children´s therapy through remotely robot operated
sessions. In the field of psychology, the use of agent-mediated therapy is
growing increasingly given recent advances in robotics and computer science.
Specifically, the use of Embodied Conversational Agents (ECA) as an
intermediary tool can help professionals connect with children who face social
challenges such as Attention Deficit Hyperactivity Disorder (ADHD), Autism
Spectrum Disorder (ASD) or even who are physically unavailable due to being in
regions of armed conflict, natural disasters, or other circumstances. In this
context, emotion recognition represents an important feedback for the
psychotherapist. In this article, we initially present the result of a
bibliographical research associated with emotion recognition in children. This
research revealed an initial overview on algorithms and datasets widely used
by the community. Then, based on the analysis carried out on the results of
the bibliographical research, we used the technique of dense optical flow
features to improve the ability of identifying emotions in children in
uncontrolled environments. From the output of a hybrid model of Convolutional
Neural Network, two intermediary features are fused before being processed by
a final classifier. The proposed architecture was called HybridCNNFusion.
Finally, we present the initial results achieved in the recognition of
children’s emotions using a dataset of Brazilian children.

Neural Networks, Computer Vision, Emotion Recognition.

††copyright: acmcopyright††journalyear: 2023††submissionid: 27

##  1\. Introduction

Human cognitive development goes through several stages from birth to
maturity. Childhood represents the phase where one acquires the basis of
learning to relate with others and with the world (piaget1952origins, 27).
Unfortunately, the mental development process of a child can be hampered by
mental disorders such as anxiety, stress, obsessive-compulsive behavior or
emotional, sexual or physical abuse (weisz2010evidence, 38). The solution or
reduction of consequences for these afflictions is achieved with therapeutic
processes carried out by professionals in the field of psychology. Due to
limited child maturity, the process involves not only assessment sessions with
the child, but also interviews with parents and educators, observation of the
child in the residential and school environments and data collection through
drawings, compositions, games and other activities (Cuadrado2019, 4),
(WOS:000629356500006, 39).

In this process, leisure resources such as: games, theater activities,
puppets, toys and others gain special prominence and are used as support in
therapy (Moura2000, 7). As a way to contribute to this approach, Embodied
Conversational Agents (ECA) are used as a tool in psycho-therapeutic
applications. Provoost et al. (Provoost2017, 28) performed a scoping review on
the use of ECAs in psychology. After selection, the search revealed 49
references associated with the following mental disorders: autism, depression,
anxiety disorder, post-traumatic stress disorder, psychotic disorder and
substance use. According to the authors, ”ECA applications are very
interesting and show promising results, but their complex nature makes it
difficult to prove that they are effective and safe for use in clinical
practice”. Actually, the strategy suggested by Provoost et al. involves
increasing the evidence base through interventions using low-technology agents
that are rapidly developed, tested, and applied in responsible clinical
practice.

The recognition of emotions during psycho-therapeutic sessions can act as an
aid to the psychology professional involved in the process, with a still big
room for improvement considering the depth of the task at hand
(WOS:000556121100052, 2).

The objective of this work is to discuss and comment on the use of images
generated by cameras in uncontrolled children’s psychotherapy sessions to
classify their emotional state at any given moment in one of the following
basic emotion categories: anger, disgust, fear, happiness, sadness, surprise,
contempt (ekman1984, 9). Given the diversity of Machine Learning algorithms
for emotion recognition tasks overall, correctly addressing our objective is
much more complex than simply choosing the most powerful or recent algorithm
(WOS:000483067100012, 25). For applications in psychology, compared to other
human-centered tasks, the solution has to be almost fail-proof and be able to
function in real uncontrolled scenarios, which is in itself extremely
challenging and therefore raises multiple ethical and morally debatable
questions about the viability of such models (WOS:000697827200029, 16). In
this context, it is important to study and consider the environments for which
a specific algorithm will be used for even before beginning to develop or
train it (WOS:000447336700033, 24).

In Section 2, we briefly discuss the performed bibliographical research on the
state of the art for emotion recognition in children. The training datasets,
as well as the implemented model architecture and produced code are presented
in Sections 3.1 and 3.2 respectively. Results obtained using the suggested
model and conclusions are discussed in Sections 4 and 5.

##  2\.  Bibliographical Research

A bibliographical research was performed to determine the State of The Art
(SOTA) for emotion recognition tasks (FER) in children using computer
algorithms. The search was made using the ”Web of Science” repository
(WEB2023, 37), covering the last 5 years, with the following search key:

(1) $\begin{split}\text{{child* AND emotion AND (recognition OR detection)
AND}}\\\ \text{{(algorithm OR "machine learning" OR "computer
vision")}}\end{split}$

An initial number of 152 references were selected, with a total of 42 accepted
for in-depth reading (39 from the original search, and 3 additional
references). A further reading analysis was done, by tagging each paper
according to a select number of categories, including, but not limited to:
datasets used; age of the patients; psychological procedure adopted; data
format (such as video, photos or scans); algorithm category (deep learning
techniques, pure machine learning, etc). The detailed result of this
categorization can be seen in the spreadsheet available in the Google Drive
(SPREED2023, 40).

###  2.1.  Types of algorithms and datasets

In Fig. 1 we present the main datasets identified during the bibliographical
research. The FER-2013 dataset (Sambare2022, 30) is one of the most used by
researchers with 9 references. We can mention the works by Sreedharan et al.
(Sreedharan2018, 32) which makes use of this dataset for training FER model
using a novel optimisation technique (Grey Wolf optimisation), for instance.

Figure 1. Datasets used for training.

Overall, we found out that Facial Emotion Recognition (FER) algorithms have
had significant improvement in recent years (WOS:000772182600055, 21), driven
by the success of deep learning-based approaches. In Fig. 2 we present the
most frequently used algorithms for emotion recognition. The convolutional
neural network architecture (DL-CNN) was the most used, with 22 references. As
DL-CNN examples, we can cite the works of Haque and Valles (Haque2018, 12) and
Cuadrado et al.(Cuadrado2019, 4). Both of these propose an architecture for a
Deep Convolutional Neural Network for a specific FER task, namely for robot
tutors in primary schools and identifying emotions in children with autism
spectrum disorder.

Figure 2. Algorithms for emotion recognition.

With the demand for high-performance algorithms, numerous novel models, such
as the DeepFace system (Taigman2014, 34) or the Transformer architecture for
sequential features (vaswani2017attention, 35) have also made great steps in
improving the overall accuracy and time efficiency for emotion classification
models.

Among the most popular paradigms currently used for FER, Convolutional Neural
Networks (CNNs) have demonstrated high performance in detecting and
recognizing emotion features from facial expressions in images
(WOS:000697827200029, 16) by applying moving filters over an image, also
called convolution kernels. These models use hierarchical feature extraction
techniques to construct region-based information from facial images, which are
then used for classification. One of the first wide-spread CNN-based models
that have been used for FER is the VGG-16 network, which uses 16 convolution
layers and 3 fully connected layers to classify emotions (vgg, 31). In
addition to CNNs, other models such as Recurrent Neural Networks (RNNs) or a
combination of both have also been proposed for FER.

Overall, FER is an active area of research, and there is ongoing work to
improve the accuracy and robustness of existing solutions.

###  2.2. Classic emotion capture strategies

In Fig. 3 we present the origin of the still emotion pictures present in the
datasets. We can observe that 48.8% of the studies used ”Posed” emotions, such
that the emotions expressed are artificial, and their enactment requested by
an evaluator. As an example of works that use ”Posed” emotions we mention
Sreedharan et al. (Sreedharan2018, 32) which uses the CK+ dataset of posed
emotions, and Kalantarian et. al (Kalantarian2020, 19), in which children with
Autism Spectrum Disorder (ASD) are requested to imitate the emotions shown by
prompts in a mobile game.

The ”Induced” group of emotions contributes to 23.3% of the found papers, for
which we can mention the work of Goulart et al. (Goulart2019, 11), where
children emotions are induced by interaction with a robot tutor and recorded.
Differently from posed emotions, which are obtained by explicitly requesting
participants to imitate the facial expression, induced emotions are implicitly
obtained by showing the participants emotion inducing scenes, such as videos,
photographs and texts.

The ”Spontaneous” group appears in only 16.3% of the studies, possibly due to
the difficulty in capturing emotions in-the-wild (ITW), such as the dataset
discussed in Kahou et. al (Kahou2015, 18), that is, when the individual is not
aware of the purpose or the existence of ongoing video recording or
photography. It is important to note that facial expressions do not completely
correlate to what the individual is feeling, as is the case with posed facial
expressions, but is generally used as an acceptable indicator for emotion,
even when used in combination with other indicators (WOS:000475910800001, 1),
(WOS:000807364500004, 23).

Figure 3. Emotion capture strategies.

###  2.3.  Hybrid Architectures

Models that combine multiple networks into one architecture, called hybrid
models, are becoming increasingly accurate, particularly those that combine
convolutional neural networks (CNNs) and recurrent neural networks (RNNs)
(WOS:000518657800086, 22) for facial emotion recognition (FER) tasks.

In addition, recent research has shown that the integration of recurrent
layers, such as the long-short term memory layer (LSTM layer)
(hochreiter1997long, 14), which processes inputs recursively, making them
particularly useful for capturing the temporal dynamics of facial expressions
and inserting these layers into hybrid models can further improve their
performance (WOS:000756540200001, 15).

Another promising research line for improving FER accuracy is the use of
multiple features such as audio and processed images in addition to facial
color (RGB) images (WOS:000460064700015, 3). These additional features can
provide complementary information that can improve the robustness and accuracy
of the FER system.

However, there are still challenges that need to be addressed, such as how to
effectively fuse multiple features and how to effectively train such time-
consuming models. Anyhow, hybrid models with transformers or LSTMs
classifiers, as well as multiple features are a promising direction for
improving the state of the art in FER (WOS:000756540200001, 15),
(WOS:000783834000109, 29).

##  3\. Methods

###  3.1. Datasets used for Training and Prediction

Considering the need for an architecture that can provide adequate accuracy
and real-time response for predicting emotions in children in uncontrolled
environments, we create the architecture HybridCNNFusion to process the real-
time sequence of frames.

To accomplish the task in hand, we planned on training our model on the two
datasets publicly available with the highest accuracy for FER task in children
(WOS:000556121100052, 2). The datasets used are the FER-2013 (Sambare2022, 30)
and the Karolinska Directed Emotional Faces (kdef, 5). Most datasets for FER
tasks are aimed towards adults and with posed expressions, therefore we
decided to use the ChildEFES (childefes, 26), a private video dataset of
Brazilian children posing emotions for fine-tunning.

###  3.2. HybridCNNFusion Architecture Model

In Fig. 4 we present the elements that make up the HybridCNNFusion
architecture. The first step in building our architecture was to allow the
model to be used in in-the-wild scenarios, by implementing a Haarcascade
(haarcascade, 36) region detection algorithm to center and crop the children
faces.

Figure 4.  HybridCNNFusion architecture. The full implementation is available
here.

These cropped images are then passed to a Convolutional Neural Network (CNN),
specifically the InceptionNet (szegedy2014going, 33), to process the cropped
RGB pixels generated by the Haarcascade algorithm. In parallel, we use the
Gunner Farneback’s algorithm (optical-flow, 10) to retrieve the dense optical
flow values from the current and previous cropped frames. This is made to
allow the network to process the variation in facial muscles and skin movement
over time. The optical flow matrices are then passed to a second CNN,
specifically a variation of the ResNet (he2015deep, 13).

After calculating these two separate features, they are concatenated and used
as input for a final recurrent block, specifically made with layers of LSTM
cells to generate the concatenated intermediary output. This takes advantage
of the sequential nature of the video frames to output a final vector of
predicted probabilities for each emotion. The aforementioned model uses a
technique called Late Fusion (WOS:000756540200001, 15), in which two separate
features are concatenated inside the architecture and used as input for the
final output layers.

The Late fusion technique allows for a better usage of the motion generated by
separate facial Action Units (ekman1978facial, 8) by having to distinctly
trained CNNs, one for raw RGB values (outputed by the InceptionNet) and
another for dense HSV motion matrices (outputed by the ResNet + OpticalFlow
combination). The use of the optical flow features as input for the ResNet
allows for processing sequential information, specifically, that of motion,
through clever manipulation of the raw RGB values.

The step by step used for a single video classification iteration is presented
in the Algorithm 1 section below.

Algorithm 1 HybridCNNFusion pseudo algorithm

Input: $N\times 1920\times 1080$ RGB frames ($\vec{x}_{i}$) and a one-hot-
vector for the emotion label throughout the video ($e$).

Output: $E_{j}$ for each 10 second window of the frames.

Step-by-step:

for $\vec{e_{i}},i=0:M$ do

Cut each vector $\vec{e_{i}}$ using the Haarcascade cropping algorithm to
center the faces,

to $n\times n$ sized images.

Apply the Gunner Farneback’s algorithm to the cropped $\vec{c_{i}}$ frames.

Group the cropped and optical flow features into groups of 30 frames.

Batch input them into two separate CNNs, respectively:

CNNFlow = InceptionNet(3, 8) and CNNRaw = ResNet34(3, 8).

end for

for $group_{j},0:N/30$ do

Concatenate the cropped and optical flow features.

Input the concatenated vectors into a 3-layer LSTM and generate a sequence

of predictions based on the previous emotion probabilities.

Append group emotion label to a sequence of labels for the entire video.

end for

###  3.3. Ethical aspects and considerations of the solution

The task of facial emotion recognition (FER) in children is particularly
challenging due to the ethical issues and the need for a high level of
precision and interpretability.

Most existing FER approaches focus on non-ethically critical situations, such
as customer satisfaction or in controlled lab conditions (WOS:000483067100012,
25). On the other hand, the task of FER in emotionally vulnerable children
requires a much greater level of trustworthiness in accordance to the ethical
constraints of the psychologist-patient relationship (WOS:000647216000001, 6).

This specific research branch of FER tasks demands the ability to accurately
detect and interpret facial expressions in real-time videos of children in in-
the-wild (ITW) situations, all the while ensuring the confidence of the
information being generated (WOS:000523305700001, 20), (WOS:000447336700033,
24).

##  4\. Results

The final model implementation had memory limitations that compromised the
deployment of the HybridCNNFusion architecture. Despite this limitation, the
final model was trained on both the FER2013 and KDEF datasets and fine tuned
on ChildEFES dataset to maximize accuracy. The entire model could not be
entirely fitted through our private dataset, so we measured partial accuracy
for the intermediary models. The InceptionNet had an accuracy of about 70%,
while the ResNet had an accuracy of about 72%. Overall, the model had a speed
averaging 2.5s for a single iteration, for videos averaging 10s of duration.

The input images are cropped to the required size of both networks. The output
consists of a stochastic vector of probabilities predicting one of 7 possible
base emotions, as well as a neutral emotion, totaling 8 possible labels
(ekman1984, 9). Both intermediary CNNs have an output vector of size 32, and
the concatenated feature is a vector with 64 entries. The final output layer
has a size of $(N/30)\times 8$, with $N/30$ equal to the total duration of the
video divided in groups of 30 frames, each group with a separate predicted
emotion label.

##  5\. Conclusion

Considering the technological aspects and the initial results obtained, the
architecture proposed is a continuous push towards identifying children
emotions in in-the-wild conditions, altough not yet fit for real-world usage.

The fusion of dense optical flow features in conjunction with a hybrid CNN and
a recurrent model represents a promising approach in the challenging task of
facial emotion recognition (FER) in children, specifically in uncontrolled
environments. Being a critical need in the field of psychology, this approach
offers a potential solution.

For ethically sensible situations, there are still important metrics that have
to be calculated, such as the Area under the ROC Curve (AOC), which can
indicate whether the model is prone to miss important emotion predictions
within small and specific frames, also called micro-expressions
(WOS:000549823900015, 17).

In fact, there is a large gap on current ethical questions for the task, but
we believe that improving the interpretability of the architecture,
explainability and security of transmission of the processed information
should be the focus of future models and frameworks instead of just the
overall accuracy. This will ensure that the technology can be used safely and
effectively to support the emotional well-being of children.

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