Anna Glazkova, Maksim Glazkov, and Timofey Trifonov 11institutetext:
University of Tyumen, ul. Volodarskogo 6, 625003 Tyumen, Russia 11email:
<EMAIL_ADDRESS>11email<EMAIL_ADDRESS>”Organization of cognitive associative systems” LLC, ul. Gertsena 64, 625000
Tyumen, Russia 22email<EMAIL_ADDRESS>
# g2tmn at Constraint@AAAI2021: Exploiting CT-BERT and Ensembling Learning for
COVID-19 Fake News Detection

Anna Glazkova[0000-0001-8409-6457] 11 Maksim Glazkov[0000-0002-4290-2059] 22
Timofey Trifonov[0000-0001-7996-0044] 1122

###### Abstract

The COVID-19 pandemic has had a huge impact on various areas of human life.
Hence, the coronavirus pandemic and its consequences are being actively
discussed on social media. However, not all social media posts are truthful.
Many of them spread fake news that cause panic among readers, misinform people
and thus exacerbate the effect of the pandemic. In this paper, we present our
results at the Constraint@AAAI2021 Shared Task: COVID-19 Fake News Detection
in English. In particular, we propose our approach using the transformer-based
ensemble of COVID-Twitter-BERT (CT-BERT) models. We describe the models used,
the ways of text preprocessing and adding extra data. As a result, our best
model achieved the weighted F1-score of 98.69 on the test set (the first place
in the leaderboard) of this shared task that attracted 166 submitted teams in
total.

###### keywords:

COVID-Twitter-BERT, social media, fake news, ensembling learning, coronavirus,
infodemic, text classification

##  1 Introduction

Social media is a unique source of information. On the one hand, their low
cost, easy access and distribution speed make it possible to quickly share the
news. On the other hand, the quality and reliability of social media news is
difficult to verify [38]. This is the source of a lot of false information
that has a negative impact on society.

Over the past year, the world has been watching the situation developing
around the novel coronavirus pandemic. The COVID-19 pandemic has become a
significant newsworthy event of 2020. Therefore, news related to COVID-19 are
actively discussed on social media and this topic generates a lot of
misinformation. Fake news related to the pandemic have large-scale negative
social consequences, they provoke huge public rumor spreading and
misunderstanding about the COVID-19 and aggravate effects of the pandemic.
Moreover, recent studies [22] show an increase in symptoms such as anxiety and
depression in connection with the pandemic. This is closely related to the
spread of misinformation, because fake news can be more successful when the
population is experiencing a stressful psychological situation [25]. The
popularity of fake news on social media can rapidly increase, because the
rebuttal is always published too late. In this regard, there is evidence that
the development of tools for automatic COVID-19 fake news detection plays a
crucial role in the regulation of information flows.

In this paper, we present our approach for the Constraint@AAAI2021 Shared
Task: COVID-19 Fake News Detection in English [29] that attracted 433
participants on CodaLab. This approach achieved the weighted F1-score of 98.69
(the first place in the leaderboard) on the test set among 166 submitted teams
in total.

The rest of the paper is organized as follows. A brief review of related work
is given in Section 2. The definition of the task has been summarized in
Section 3, followed by a brief description of the data used in Section 4. The
proposed methods and experimental settings have been elaborated in Section 5.
Section 6 contains the results and error analysis respectively. Section 7 is a
conclusion.

##  2 Related Work

In recent years, the task of detecting fake news and rumors is extremely
relevant. False information spreading involves various research tasks,
including: fact checking [4, 40], topic credibility [15, 41], fake news
spreaders profiling [34], and manipulation techniques detection [8]. Various
technologies and approaches in this field range from traditional machine
learning methods [5, 23, 33], to state-of-the-art transformers [24, 47].

A overview of fake news detection approaches and challenges on social media
has been discussed in [38, 50]. Many scholars have proposed their solutions to
this problem in different subject areas (in particular, [6, 35]). Up to now, a
large number of studies in fake news detection have used supervised methods
including models based on transformers-based architecture [13, 17, 49].

Some recent work have focused on detecting fake news about COVID-19. For
example, the predictors of the sharing of false information about the pandemic
are discussed in [3]. In [44], a novel COVID-19 fact checking algorithm is
proposed that retrieves the most relevant facts concerning user claims about
particular facts. A number of studies have begun to examine COVID-19 fake news
detection methods for non-English languages [10, 14, 48].

In addition, several competitions have been announced over the past year
related to the analysis of posts about COVID-19 on social media [1, 27, 36].

##  3 Task Definition

The task focused on the detection of COVID-19-related fake news in English.
The sources of data was various social-media platforms such as Twitter,
Facebook, Instagram, etc. Formally, the task is described as follows.

  * •

Input. Given a social media post.

  * •

Output. One of two different labels, such as ”fake” or ”real”.

The official competition metric was F1-score averaged across the classes (the
weighted F1-score). The participants were allowed five submissions per team
throughout the test phase.

##  4 Dataset

The dataset [28] provided to the participants of the shared task contains
10,700 manually annotated social media posts divided into training (6420),
validation (2140), and test (2140) sets. The vocabulary size (i.e., unique
words) of the dataset is 37,505 with 5141 common words in both fake and real
news. The dataset contains the post ID, the post, and the corresponding label
which is ”fake” or ”real” (see Table 1).

Table 1: Some examples of fake and real posts Label |  Post
---|---
real |  The CDC currently reports 99031 deaths. In general the discrepancies in death counts between different sources are small and explicable. The death toll stands at roughly 100000 people today
real |  #IndiaFightsCorona: We have 1524 #COVID testing laboratories in India and as on 25th August 2020 36827520 tests have been done : @ProfBhargava DG @ICMRDELHI #StaySafe #IndiaWillWin https://t.co/Yh3ZxknnhZ
fake |  Politically Correct Woman (Almost) Uses Pandemic as Excuse Not to Reuse Plastic Bag https://t.co/thF8GuNFPe #coronavirus #nashville
fake |  Obama Calls Trump’s Coronavirus Response A Chaotic Disaster https://t.co/DeDqZEhAsB

##  5 Our Approach

In this section, we describe the approaches that we evaluated on the
validation data during the validation phase. We used transformer-based models
as they demonstrate state-of-the-art results in most text classification
tasks. We also evaluated the empirical effectiveness of a Linear Support
Vector baseline (Linear SVC) and different text preprocessing techniques and
adding extra data. The results are shown in the next section.

###  5.1 Data Preprocessing

Our approaches to text preprocessing for transformer-based models are various
combinations of the following steps, most of which have been inspired by [18,
42]:

  * •

removing or tokenizing hashtags, URLs, emoji, and mentions using a
preprocessing library for tweet data written in Python [43]. Tokenization
means the replacement of URLs, mentions, and emoji with special tokens, such
as $URL$, $MENTION$, and $HASHTAG$ respectively (for example ”HHS to
distribute $4 billion to #COVID hot spots; $340 million already paid out.
https://t.co/uAj29XA1Y5” (original) $\rightarrow$ ”HHS to distribute $4
billion to $HASHTAG$ hot spots; $340 million already paid out. $URL$”
(tokenizing); ”HHS to distribute $4 billion to hot spots; $340 million already
paid out.” (removing));

  * •

using the Python emoji library to replace the emoji with short textual
description [11], for example :red_heart:, :thumbs_up:, etc.;

  * •

converting hashtags to words (”#COVID” $\rightarrow$ ”COVID”);

  * •

translating in the lowercase.

In the case of the baseline, we translated the text to the lowercase, removed
punctuation and special characters, and then lemmatized the words. Further, we
converted texts into the form of a token counts matrix (a bag of words model).

###  5.2 Models

We experimented with the following transformer-based models:

  * •

BERT [9]. BERT is a language representation model presented by Google, which
stands for Bidirectional Encoder Representations from Transformers. BERT-based
models show great results in many natural language processing tasks. In our
work, we used BERT-base-uncased, which is pretrained on texts from Wikipedia.

  * •

RoBERTa [19]. RoBERTa is a robustly optimized BERT approach introduced at
Facebook. Unlike BERT, RoBERTa removes the Next Sentence Prediction task from
the pretraining process. RoBERTa also uses larger batch sizes and dynamic
masking so that the masked token changes while training instead of the static
masking pattern used in BERT. We experimented with RoBERTa-large.

  * •

COVID-Twitter-BERT [26]. CT-BERT is a transformer-based model, pretrained on a
large corpus of Twitter messages on the topic of COVID-19 collected during the
period from January 12 to April 16, 2020. CT-BERT is optimised to be used on
COVID-19 content, in particular social media posts from Twitter. This model
showed a 10–30% marginal improvement compared to its base model, BERT-large,
on five different specialised datasets. Moreover, it was successfully used for
a variety of natural language tasks, such as identification of informative
COVID-19 tweets [18], sentiment analysis [16], and claims verification [2,
45].

###  5.3 Additional Data

To improve the quality of our approach, we made attempts to add extra data to
the model. For this purpose we used two datasets related to the topic of
COVID-19 fake news:

  * •

CoAID: COVID-19 Healthcare Misinformation Dataset [7]. The dataset includes
4251 health-related fake news posted on websites and social platforms.

  * •

FakeCovid - A Multilingual Cross-domain Fact Check News Dataset for COVID-19
[37]. The dataset contains 5182 fact-checked news articles for COVID-19
collected from January to May 2020.

In our experiments, we added news headlines to the training set.

###  5.4 Experimental Settings

We conducted our experiments on Google Colab Pro (CPU: Intel(R) Xeon(R) CPU @
2.20GHz; RAM: 25.51 GB; GPU: Tesla P100-PCIE-16GB with CUDA 10.1). Each model
was trained on the training set for 3 epochs and evaluated on the validation
set. As our resources are constrained, we used random seeds to fine-tune pre-
trained language models and made attempts to combine them with other
parameters. The models are optimised using AdamW [21] with a learning rate of
2e-5 and epsilon of 1e-8, max sequence length of 128 tokens, and a batch size
of 8. We implemented our models using Pytorch [30] and Huggingface’s
Transformers [46] libraries.

The Linear SVC was implemented with Scikit-learn [31]. For text preprocessing,
we used NLTK [20] and Scikit-learn’s CountVectorizer with a built-in list of
English stop-words and a maximum feature count of 10,000.

##  6 Results and Discussion

###  6.1 Comparison of Models for Fake News Detection

In Table 2, we present the results of our experiments in a step by step
manner. We started with a Linear SVC baseline and then evaluated BERT-based
models using a variety of text preprocessing and adding extra data techniques.
Note that we evaluated our models using F1-score for the fake class while the
official competition metric was the weighted F1-score.

Table 2: Evaluation results Model |  Data preprocessing |  Additional data |  F1-Score (%, for fake class)
---|---|---|---
LinearSVC |  converting into a bag of words |  no |  88.39
BERT |  lowercase |  no |  96.75
RoBERTa |  lowercase |  no |  97.62
RoBERTa |  removing hashtags, URLs, emoji + lowercase |  no |  95.79
RoBERTa |  removing URLs and emoji + converting hashtags to words + lowercase |  no |  95.68
CT-BERT |  lowercase |  no |  98.07
CT-BERT |  tokenizing URLs and mentions + converting emoji to words + lowercase |  no |  97.87
CT-BERT |  converting emoji to words + lowercase |  no |  98.32
CT-BERT |  tokenizing URLs + converting emoji to words + lowercase |  no |  98.42
CT-BERT |  tokenizing URLs + converting emoji to words + lowercase |  FakeCovid |  98.23
CT-BERT |  tokenizing URLs + converting emoji to words + lowercase |  CoAID |  98.37

As can be seen from the table above, CT-BERT models showed absolutely better
results compared to BERT- and RoBERTa-based classifiers. Our work doesn’t
contain a detailed comparative analysis of text preprocessing techniques for
this task. Still we can see that text preprocessing can affect the quality of
fake news detection. For example, there was no evidence that removing emoji
and mentions improve the model results. A clear benefit of converting hashtags
into words could not be identified during this evaluation. Also, as a result
of our experiments, we decided not to tokenize links to other user’s accounts
(mentions). This can be seen in the case of mentions of major news channels
like CNN that tend to indicate that the post is real. The next section of the
model evaluation was concerned with using additional datasets. We noticed that
adding extra data did not show any benefits in our experiments

###  6.2 Final Submissions

As it was mentioned above, the participants of the shared task were allowed
five submissions per team throughout the test phase. Our best model based on
experimental results (subsection 5.1) included the following preprocessing
steps: tokenizing URLs, converting emoji to words, and lowercase. As final
submissions, we used the results of hard voting ensembles of three such models
with random seed values and with different data splitting into training and
validation samples. The final architecture of our solution is shown in Figure
1.

Figure 1: Our approach to COVID-19 fake news detection.

Four of our five submitted models were trained entirely on the dataset
provided by the competition organizers [28]. The last model was trained on the
competition data and on additional datasets [7, 37]. The training details and
the results of our final submissions are summarised in Table 3.

Table 3: Final submissions Place among all submissions |  Submission name |  Weighted F1-score (%) |  Training details
---|---|---|---
1 |  g2tmn_2.csv |  98.69 |  All models were trained both on training and validation sets with no hold out validation. We trained 3 models and then used hard-voting to ensemble their predictions together.
6 |  g2tmn_4.csv |  98.51 |  1000 random posts were used for hold-out validation. Models were trained on all other data. We trained 5 models with random seed values and choose 3 models with the best F1-score performances to ensemble them together.
11 |  g2tmn_1.csv |  98.37 |  Models were trained on the official training set. The validation set was used for hold-out validation. We trained 5 models with random seed values and choose 3 best-performance models to ensemble their predictions.
15 |  g2tmn_3.csv |  98.32 |  This submission was similar to g2tmn_1.csv but it had different seed values.
25 |  g2tmn_5.csv |  98.18 |  1000 random posts were used for hold-out validation. Models were trained on all other official data, CoAID and FakeCovid data. We trained 5 models with random seed values and used 3 best-performance models for ensembling learning.

It can be seen from the data in Table 3 that, with the weighted F1-score, our
model performance is 98.69% (the random seeds are 23, 30, and 42), which was
ranked the first place of the leaderboard of this task.

###  6.3 Error Analysis

Error analysis allows us to further evaluate the quality of the machine
learning model and conduct a quantitative analysis of errors. Figure 2
provides the confusion matrix for our best solution when detecting fake news
about COVID-19 on the test set. As can be seen from the figure, the precision
of our system is slightly higher than its recall. In other words, the false
positive value is greater that false negative. Table 4 shows the examples of
false negative and false positive errors.

We noticed that the type of error is frequently related to the topic of the
post. For example, the model often misclassifies false reports about the
number of people infected. At the same time, true posts related to the
coronavirus vaccine or to political topics can be identified as false.

Figure 2: Confusion matrix of our best-performance ensemble for COVID-19 fake news detection (for the fake class). Table 4: Some examples of misclassified posts True label |  Prediction |  Post
---|---|---
real |  fake |  Scientists ask: Without trial data how can we trust Russia’s #COVID vaccine? https://t.co/gArcUf0Pji https://t.co/0bdcA7lf56
real |  fake |  *DNA Vaccine: injecting genetic material into the host so that host cells create proteins that are similar to those in the virus against which the host then creates antibodies
real |  fake |  Donald Trump has claimed he ”up-played” the seriousness of the coronavirus pandemic - despite admitting earlier this year he had ”wanted to always play it down” https://t.co/wEgnnZzrNW
fake |  real |  Govt has added #Corona disease in all existing mediclaim insurances as a special case #COVID2019India https://t.co/39vpW7tBqq
fake |  real |  As tuberculosis shaped modernism, so COVID-19 and our collective experience of staying inside for months on end will influence architecture’s near future, @chaykak writes. https://t.co/ag34yZckbU
fake |  real |  Northern Ireland was testing for COVID-19 at a rate 10 times that of Scotland reported on 9 May 2020.

##  7 Conclusion

In this work, we propose a simple but effective approach to COVID-19 fake news
detection based on CT-BERT and ensembling learning. Our experiments confirmed
that BERT-based models specialized in the subject area successfully cope with
such tasks and perform high-quality binary classification.

The experimental results showed that our solution achieved 98.69% of the
weighted F1-score on test data and ranked in the first place in the
Constraint@-AAAI2021 shared task. For future work, we can experiment with
different training and data augmentation techniques. We can also apply and
evaluate hybrid models combining BERT-based architectures with other methods
of natural language processing [32, 39].

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# Joint Receive Antenna Selection and Beamforming in RIS-Aided MIMO Systems

Chongjun Ouyang⋆, Ali Bereyhi†, Saba Asaad†, Ralf R. Müller†, and Hongwen
Yang⋆ ⋆School of Information and Communication Engineering, Beijing University
of Posts and Telecommunications
†Institute for Digital Communications (IDC), Friedrich-Alexander-Universität
Erlangen-Nürnberg
Email: ⋆{DragonAim<EMAIL_ADDRESS>†{ali.bereyhi, saba.asaad,
<EMAIL_ADDRESS>
###### Abstract

This work studies a low-complexity design for reconfigurable intelligent
surface (RIS)-aided multiuser multiple-input multiple-output systems. The base
station (BS) applies receive antenna selection to connect a subset of its
antennas to the available radio frequency chains. For this setting, the BS
switching network, uplink precoders, and RIS phase-shifts are jointly
designed, such that the uplink sum-rate is maximized. The principle design
problem reduces to an NP-hard mixed-integer optimization. We hence invoke the
weighted minimum mean squared error technique and the penalty dual
decomposition method to develop a tractable iterative algorithm that
approximates the optimal design effectively. Our numerical investigations
verify the efficiency of the proposed algorithm and its superior performance
as compared with the benchmark.

###### Index Terms:

Multiuser multiple-input multiple-output, receive antenna selection,
reconfigurable intelligent surfaces.

##  I Introduction

Large multiple-input multiple-output (MIMO) technology, often referred to as
massive MIMO, is known to boost the spectral efficiency of wireless channels.
However, its fully-digital implementation with a dedicated radio frequency
(RF) chain at each antenna suffers from expensive hardware costs and excessive
energy consumption. Numerous potential approaches have been introduced to
alleviate this issue over the past years; see [1, 3, 2] and the references
therein. Among them is antenna selection that sets only a small subset of
antennas active in each coherence time. This way it reduces the number of
required RF chains, and hence the overall RF cost without significantly
degrading the performance.

On a parallel track, recent advances in RF micro electromechanical systems
have made programmable meta-surfaces a reality, among which reconfigurable
intelligent surfaces (RISs) have received a growing attention [4]. An RIS is a
planar array comprising low-cost passive reconfigurable reflecting elements
(REs). The main idea behind RIS is to tune the phase responses of REs, such
that the propagation environment is smartly reconfigured. Deploying an RIS
introduces additional degrees of freedom to the system and improves the
wireless link performance, e.g., spectral efficiency [5].

Against this backdrop, it is natural to combine the two technologies of
antenna selection and RIS to improve system performance while reducing the RF
cost; see [6, 8, 7, 9, 10] for some recent studies in this respect. The early
attempt in [6] studied the joint antenna selection and RIS phase-shifts design
of a single-user multiple-input single-output (MISO) case. The result is
extended to the single-user MIMO case in [7]. Extensions to multiuser MISO,
multiuser MIMO (MU-MIMO), and multi-cell settings are further discussed in
[8], [9], and [10], respectively.

###  I-A Contributions

Despite research progresses on this topic, the available literature is still
restricted to some particular scenarios. These restrictions are mainly in two
respects: firstly, these earlier studies only considered single-stream
transmission and ignored the design of precoders at the user terminals (UTs);
Secondly, most existing works, e.g., [7, 9, 10], assume that RE phase-shifts
are continuously tuned. From the implementational viewpoint, this is an
unrealistic assumption, as most proposed realizations of RISs apply quantized
phase-shifts on the receive signal.

In this paper, we consider an RIS-aided MU-MIMO system, where each RE takes
its phase-shift from a discrete set, and each UT sends multiple data streams
to the base station (BS). We maximize the throughput of this system by
proposing a joint design that optimizes the UT precoders, RIS phase-shifts,
and the switching network at the BS. The basic design problem is an NP-hard
problem dealing with integer programming. Starting from this problem, we
propose an iterative algorithm through the following lines of contributions:
1) We propose a penalty dual decomposition (PDD)-based method to tackle the
non-convex joint design problem via capitalizing on the weighted minimum mean
squared error (WMMSE) technique. 2) To further alleviate the complexity, we
next propose an alternative algorithm based on sequential optimization (SO).
This algorithm is shown to reduce the complexity at the expense of performance
losses. This result verifies the efficiency of our PDD-based algorithm and
depicts the complexity-performance trade-off. We verify our low-complexity
design through numerical experiments. Our numerical results demonstrate that
the proposed algorithms significantly outperform the benchmark.

###  I-B Notation

Throughout this paper, scalars, vectors, and matrices are denoted by non-bold,
bold lower-case, and bold upper-case letters, respectively. For the matrix
$\mathbf{A}$, $[\mathbf{A}]_{i,j}$, ${\mathbf{A}}^{\mathsf{T}}$, and
${\mathbf{A}}^{\mathsf{H}}$ denote the $(i,j)$th entry, transpose, and
transpose conjugate of $\mathbf{A}$, respectively. For the square matrix
$\mathbf{B}$, ${\mathbf{B}}^{\frac{1}{2}}$, ${\mathbf{B}}^{-1}$,
${\mathsf{tr}}(\mathbf{B})$. and $\det(\mathbf{B})$ denote the principal
square root, inverse, trace, and determinant of $\mathbf{B}$, respectively.
The notation $[\mathbf{a}]_{i}$ denotes the $i$th entry of vector
$\mathbf{a}$, and $\mathsf{diag}\\{\mathbf{a}\\}$ returns a diagonal matrix
whose diagonal elements are entries of $\mathbf{a}$. The identity matrix, zero
matrix, and all-one vector are represented by $\mathbf{I}$, $\mathbf{0}$, and
$\mathbf{1}$, respectively. The matrix inequalities
${\mathbf{A}}\succeq{\mathbf{0}}$ and ${\mathbf{A}}\succ{\mathbf{0}}$ imply
that $\mathbf{A}$ is positive semi-definite and positive definiteness,
respectively. The operation $\angle s$ extracts the phase of the complex value
$s$. The set $\mathbbmss{C}$ stands for the complex plane and notation
${\mathbbmss{E}}[\cdot]$ represents mathematical expectation. The Hadamard
product is shown by $\odot$, and $[K]$ represents the integer set
$\\{1,\ldots,K\\}$. Finally,
${\mathbf{x}}\sim{\mathcal{CN}}\left({\mathbf{0}},{\mathbf{X}}\right)$ denotes
a circularly symmetric complex Gaussian vector with mean zero and covariance
matrix $\mathbf{X}$.

Figure 1: Illustration of a RIS-aided MU-MIMO system

##  II System Model and Problem Formulation

We consider uplink transmission in a RIS-aided multiuser MIMO setting in which
$K$ multiple-antenna UTs send their encoded messages simultaneously to a BS
with $N$ antennas. A schematic view of the setting is given in Figure 1. The
number of antennas at UT $k\in[K]$ is assumed to be $N_{k}$. The communication
from all the UTs to the BS is aided by an RIS with a set of $M$ REs. It is
assumed that the direct UT-to-BS links are blocked due to unfavorable
propagation conditions.

UT $k\in[K]$ employs the linear precoder
${\mathbf{P}}_{k}\in{\mathbbmss{C}}^{N_{k}\times L_{k}}$ to construct its
transmit signal ${\mathbf{x}}_{k}\in{\mathbbmss{C}}^{N_{k}}$ from its vector
of symbols ${\mathbf{s}}_{k}\in{\mathbbmss{C}}^{L_{k}}$ that is considered to
have zero mean and identity covariance matrix. We consider a standard multiple
access channel (MAC) in which UTs send independent data streams, i.e.,
${\mathbbmss{E}}\left[{\mathbf{s}}_{k}{\mathbf{s}}_{k^{\prime}}^{\mathsf{H}}\right]={\mathbf{0}}$
for $k\neq k^{\prime}$. In this setting, the received signal at the BS is
given by

$\displaystyle{\mathbf{y}}=\mathbf{G}{\bm{\Phi}}\sum\nolimits_{k=1}^{K}\mathbf{H}_{k}{\mathbf{x}}_{k}+\mathbf{n}.$
(1)

The terms appearing in (1) are defined as follows:

  * •

$\mathbf{G}\in{\mathbbmss{C}}^{N\times M}$ is the RIS-to-BS channel matrix.

  * •

$\mathbf{H}_{k}\in{\mathbbmss{C}}^{M\times N_{k}}$ is the matrix of channel
coefficients from UT $k$ to the RIS.

  * •

$\mathbf{n}$ denotes additive white Gaussian noise (AWGN) with mean zero and
variance $\sigma^{2}$, i.e.,
$\mathbf{n}\sim{\mathcal{CN}}\left(\mathbf{0},\sigma^{2}\mathbf{I}\right)$.

  * •

${\bm{\Phi}}\in{\mathbbmss{C}}^{M\times M}$ is the diagonal matrix of RIS
phase-shifts, i.e., ${\bm{\Phi}}={\mathsf{diag}}\left\\{{\bm{\phi}}\right\\}$
for ${\bm{\phi}}=[\phi_{1},\ldots,\phi_{M}]^{\mathsf{T}}$, where
$\phi_{m}={\rm{e}}^{{\rm{j}}\theta_{m}}$ for $m\in[M]$ with $\mathrm{j}$
denoting the imaginary unit and $\theta_{m}$ being the phase-shift applied by
the $m$th RE of the RIS on its received signal.

We assume that REs apply quantized phase-shifts with $Q$ bits of quantization.
This means that $\theta_{m}$ is adjusted to $2^{Q}$ discrete values. The
discrete levels are specified by uniform quantization of the phase interval
$[0,2\pi)$, i.e.,

$\displaystyle\phi_{m}\in{\mathcal{F}}_{Q}=\left\\{{\rm e}^{{\rm
j}2\pi\frac{t-1}{2^{Q}}}:t\in\left[2^{Q}\right]\right\\}.$ (2)

The system operates in the time division duplexing (TDD) mode. The channel
state information (CSI) is hence estimated in the uplink training phase via
pilot sequences. We assume that the pilots are mutually orthogonal and that
the estimation error is negligible. The BS thus learns perfectly the CSI.
Details on channel estimation for RIS-aided MIMO systems are found in [11] and
the references therein.

###  II-A Receive Antenna Selection

The BS has $T<N$ RF chains and hence uses a switching network to select a
subset of receive antennas. This switching network connects the selected
antennas to the available $T$ RF chains at the BS. As a result, the received
signal at the RF front-end of the BS is given by

$\displaystyle{\mathbf{r}}={\mathbf{S}}{\mathbf{y}}={\mathbf{S}}\mathbf{G}{\bm{\Phi}}\sum\nolimits_{k=1}^{K}\mathbf{H}_{k}{\mathbf{x}}_{k}+{\mathbf{S}}{\mathbf{n}},$
(3)

where ${\mathbf{S}}={\\{0,1\\}}^{T\times N}$ is the antenna selection matrix
with

$\displaystyle[{\mathbf{S}}]_{t,n}=\begin{cases}1&{\text{antenna}}~{}n~{}{\text{is
connected to}}~{}{\text{RF chain}}~{}t\\\ 0&{\text{otherwise}}\end{cases}.$

Since $\mathbf{S}$ is a fat permutation matrix, we have
${\mathbf{S}}{\mathbf{S}}^{{\mathsf{H}}}={\mathbf{I}}$.

###  II-B Performance Metric: Sum-Rate

The sum-rate term for this uplink multiuser MIMO setting is given by [12]

$\displaystyle\mathcal{R}=\log_{2}\\!\det\\!\left(\\!{\mathbf{I}}+\sum_{k=1}^{K}\frac{1}{\sigma^{2}}{\mathbf{S}}{\mathbf{G}}{\bm{\Phi}}{\mathbf{H}}_{k}{\mathbf{Q}}_{k}{\mathbf{H}}_{k}^{\mathsf{H}}{\bm{\Phi}}^{\mathsf{H}}{\mathbf{G}}^{\mathsf{H}}{\mathbf{S}}^{\mathsf{H}}\\!\right),$
(4)

where

$\displaystyle{\mathbf{Q}}_{k}={\mathbbmss{E}}\left[{\mathbf{x}}_{k}{\mathbf{x}}_{k}^{\mathsf{H}}\right]={\mathbf{P}}_{k}{\mathbf{P}}_{k}^{\mathsf{H}}\in{\mathbbmss{C}}^{N_{k}\times
N_{k}}$ (5)

is the transmit covariance matrix of UT $k$ subject to the power budget
${\mathsf{tr}}({\mathbf{Q}}_{k})\leq p_{k}$. The sum-rate is achieved via
minimum mean square error (MMSE) estimation along with successive interference
cancellation (SIC), referred to as MMSE-SIC.

The sum-rate expression can be further simplified: by exploiting the structure
of the selection matrix $\mathbf{S}$, we have

$\displaystyle{\mathbf{S}}^{\mathsf{H}}{\mathbf{S}}={\mathsf{diag}}\left\\{{\mathbf{s}}\right\\}\triangleq{\bm{\Delta}}\in{\mathbbmss{C}}^{N\times
N},$ (6)

where $\mathbf{s}=[s_{1},\ldots,s_{N}]^{\mathsf{T}}$ with $s_{n}\in\\{0,1\\}$
representing the activity of antenna $n$, i.e., $s_{n}=1$ if antenna $n$ is
selected, and $s_{n}=0$ otherwise. Noting
${\bm{\Delta}}{\bm{\Delta}}^{\mathsf{H}}={\bm{\Delta}}$ and defining
$\overline{\mathbf{H}}=\left[{\mathbf{G}}{\bm{\Phi}}{\mathbf{H}}_{1}{\mathbf{P}}_{1},\ldots,{\mathbf{G}}{\bm{\Phi}}{\mathbf{H}}_{K}{\mathbf{P}}_{K}\right]$,
we can rewrite (4) as

$\displaystyle\mathcal{R}=\log_{2}\det\left(\mathbf{I}+{\sigma^{-2}}{\bm{\Delta}}\overline{\mathbf{H}}\overline{\mathbf{H}}^{\mathsf{H}}{\bm{\Delta}}^{\mathsf{H}}\right).$
(7)

In the sequel, we consider the sum-rate as the metric which quantifies the
throughput of this system.

###  II-C Problem Formulation

Our ultimate goal is to find the system design that optimizes the throughput.
This means that we strive to jointly design the precoding matrices
${\mathbf{P}}=\\{{\mathbf{P}}_{k}\\}_{k=1}^{K}$111The precoding matrices can
be firstly designed at the BS side and then shared with the UTs via an error-
free feedback link., the phase-shift vector ${\bm{\phi}}$, and the antenna
selection vector $\mathbf{s}$, such that the sum-rate term ${\mathcal{R}}$ is
maximized. Consequently, our design problem is formulated as

$\begin{split}\max_{{\mathbf{s}},{\mathbf{P}},{\bm{\phi}}}~{}{\mathcal{R}}~{}{\rm{s.t.}}~{}&C_{1}:s_{n}\in\left\\{0,1\right\\},{\text{for}}~{}n\in[N],{\mathbf{1}}^{\mathsf{T}}{\mathbf{s}}=T,\\\
&C_{2}:{\mathsf{tr}}\left({\mathbf{P}}_{k}{\mathbf{P}}_{k}^{\mathsf{H}}\right)\leq
p_{k},{\text{for}}~{}k\in[K],\\\
&C_{3}:\phi_{m}\in{\mathcal{F}}_{Q},{\text{for}}~{}m\in[M].\end{split}$
($\mathcal{P}_{1}$)

The design problem in ($\mathcal{P}_{1}$) in its basic form is challenging due
to three main reasons. Firstly, the objective function of ($\mathcal{P}_{1}$)
is of a non-convex form. Secondly, the discrete constraints in $C_{1}$ and
$C_{3}$ reduce ($\mathcal{P}_{1}$) to an integer programming problem (which is
an NP-hard problem). Finally, the presence of the phase-shifts further
complicates the optimization procedure. These challenges make the optimal
design computationally intractable. In the sequel, we develop an efficient
framework to approximate the optimal design via a feasible computational
complexity.

##  III PDD-Based Joint Design

We simplify problem ($\mathcal{P}_{1}$) to a more tractable yet equivalent
form by invoking the WMMSE framework [13]. We then handle the resulting
equivalent problem via the PDD that efficiently addresses nonconvex nonsmooth
problems with coupling equality constraints [14].

###  III-A Reformulation of Problem ($\mathcal{P}_{1}$)

The principle problem ($\mathcal{P}_{1}$) can be converted to a variational
form in which the objective is replaced by a weighted mean squared error
(WMSE) expression. To find this variational form, we regard the objective of
($\mathcal{P}_{1}$) as the data rate (input-output mutual information) of the
hypothetical channel described by
${\mathbf{y}}_{\rm{h}}={\bm{\Delta}}\overline{\mathbf{H}}{\mathbf{x}}_{\rm
h}+{\mathbf{n}}_{\rm h}$. This is an hypothetical Gaussian MIMO channel whose
channel matrix is
${\bm{\Delta}}\overline{\mathbf{H}}\in{\mathbbmss{C}}^{N\times L}$ with
$L=\sum_{k=1}^{K}L_{k}$, and whose AWGN reads ${\mathbf{n}}_{\rm
h}\sim{\mathcal{CN}}\left({\mathbf{0}},\sigma^{2}{\mathbf{I}}\right)$. The
vector ${\mathbf{x}}_{\rm h}\in{\mathbbmss{C}}^{L}$ further denotes an
independent and identically distributed (i.i.d.) Gaussian transmit signal with
mean zero and unit average power, i.e., ${\mathbf{x}}_{\rm
h}\sim{\mathcal{CN}}({\mathbf{0}},{\mathbf{I}})$. Given the output of this
hypothetical Gaussian channel, i.e., ${\mathbf{y}}_{\rm{h}}$, let the matrix
${\mathbf{U}}_{\rm{h}}\in{\mathbbmss{C}}^{N\times L}$ be employed as the
linear receiver to estimate ${\mathbf{x}}_{\rm h}$. In this case, the mean
squared error (MSE) of the linear estimation is given by the trace of MSE
matrix ${\mathbf{E}}_{\rm{h}}$ that is defined as

$\displaystyle{\mathbf{E}}_{\rm{h}}$
$\displaystyle={\mathbbmss{E}}_{{\mathbf{s}}_{\rm h},{\mathbf{n}}_{\rm
h}}\left\\{\left({\mathbf{U}}_{\rm{h}}^{\mathsf{H}}{\mathbf{y}}_{\rm{h}}-{\mathbf{s}}_{\rm
h}\right)\left({\mathbf{U}}_{\rm{h}}^{\mathsf{H}}{\mathbf{y}}_{\rm{h}}-{\mathbf{s}}_{\rm
h}\right)^{\mathsf{H}}\right\\}$ (8a)
$\displaystyle=\left({\mathbf{U}}_{\rm{h}}^{\mathsf{H}}{\bm{\Delta}}\overline{\mathbf{H}}-{\mathbf{I}}\right)\left({\mathbf{U}}_{\rm{h}}^{\mathsf{H}}{\bm{\Delta}}\overline{\mathbf{H}}-{\mathbf{I}}\right)^{\mathsf{H}}+\sigma^{2}{\mathbf{U}}_{\rm{h}}^{\mathsf{H}}{\mathbf{U}}_{\rm{h}}.$
(8b)

Invoking the WMMSE technique [13], we now introduce an auxiliary optimization
variable ${\mathbf{W}}_{\rm{h}}\in{\mathbbmss{C}}^{L\times L}$ to define the
following WMSE minimization problem:

$\begin{split}&\min_{{\mathbf{W}}_{\rm{h}},{\mathbf{U}}_{\rm{h}},{\mathbf{s}},{\bm{\phi}},{\mathbf{P}}}~{}{\mathsf{tr}}\left\\{{\mathbf{W}}_{\rm{h}}{\mathbf{E}}_{\rm{h}}\right\\}-\log\det\left({\mathbf{W}}_{\rm{h}}\right)\\\
&\quad\quad{\rm{s.t.}}~{}C_{1},C_{2},C_{3}.\end{split}$ ($\mathcal{P}_{2}$)

Theorem 1 in [13] indicates that ($\mathcal{P}_{2}$) is a variational form of
the principle problem ($\mathcal{P}_{1}$). This means that the solutions
$\left\\{{\mathbf{s}},{\bm{\phi}},{\mathbf{P}}\right\\}$ for both the problems
are identical. It is worth mentioning that unlike the principle from in
problem ($\mathcal{P}_{1}$), the problem ($\mathcal{P}_{2}$) has an objective
function that is marginally convex over each of the variable
${\mathbf{W}}_{\rm{h}}$, ${\mathbf{U}}_{\rm{h}}$, ${\mathbf{s}}$,
${\bm{\phi}}$, and ${\mathbf{P}}$.

The marginally convex form of the objective in ($\mathcal{P}_{2}$) suggests to
use the alternating optimization (AO) scheme. We however still face the issue
of having discrete constraints $C_{1}$ and $C_{3}$. To tackle this issue, we
define the auxiliary variables
$\overline{\mathbf{s}}=[\overline{s}_{1},\ldots,\overline{s}_{N}]^{\mathsf{T}}$
and ${\mathbf{v}}=[v_{1},\ldots,v_{M}]^{\mathsf{T}}$ which satisfy the
following constraints: ${\overline{s}}_{n}=s_{n}$,
$s_{n}\left(1-{\overline{s}}_{n}\right)=0$, ${\mathbf{v}}={\bm{\phi}}$, and
$v_{m}\in{\mathcal{F}}_{Q}$. We thus can equivalently find the solution of
($\mathcal{P}_{2}$) by solving the following optimization:

$\begin{split}&\min_{{\mathbf{W}}_{\rm{h}},{\mathbf{U}}_{\rm{h}},\overline{\mathbf{s}},{\mathbf{s}},{\bm{\phi}},{\mathbf{v}},{\mathbf{P}}}~{}{\mathsf{tr}}\left\\{{\mathbf{W}}_{\rm{h}}{\mathbf{E}}_{\rm{h}}\right\\}-\log\det\left({\mathbf{W}}_{\rm{h}}\right)\\\
&~{}{\rm{s.t.}}~{}{\overline{s}}_{n}=s_{n},s_{n}\left(1-{\overline{s}}_{n}\right)=0,{\text{for}}~{}n\in[N],{\mathbf{1}}^{\mathsf{T}}{\mathbf{s}}=T,\\\
&\qquad{\bm{\phi}}={\mathbf{v}},{v}_{m}\in{\mathcal{F}}_{Q},{\text{for}}~{}m\in[M],\\\
&\qquad{\mathsf{tr}}\left({\mathbf{P}}_{k}{\mathbf{P}}_{k}^{\mathsf{H}}\right)\leq
p_{k},{\text{for}}~{}k\in[K].\end{split}$ (${\mathcal{P}}_{3}$)

It is worth mentioning that in (${\mathcal{P}}_{3}$), variables $\mathbf{s}$
and $\bm{\phi}$ are only constrained through ${\overline{s}}_{n}=s_{n}$,
$s_{n}\left(1-{\overline{s}}_{n}\right)=0$,
${\mathbf{1}}^{\mathsf{T}}{\mathbf{s}}=T$, and ${\bm{\phi}}={\mathbf{v}}$. To
handle these equality constraints, we resort to the PDD technique.

###  III-B The Proposed PDD-Based Algorithm

The PDD-based algorithm is characterized by an embedded double loop structure
[14]. The inner loop solves the augmented Lagrangian (AL) subproblem while the
outer loop updates the dual variables and the penalty parameters that
correspond to constraint violation. We illustrate these concepts clearly in
the sequel throughout the derivation of the PDD-based algorithm.

PDD converts problem (${\mathcal{P}}_{3}$) into its AL by moving the equality
constraints as a penalty term to the objective function. The AL subproblem
corresponding to (${\mathcal{P}}_{3}$) is given by

$\begin{split}&\min_{{\mathbf{W}}_{\rm{h}},{\mathbf{U}}_{\rm{h}},\overline{\mathbf{s}},{\mathbf{s}},{\bm{\phi}},{\mathbf{v}},{\mathbf{P}}}~{}{\mathsf{tr}}\left\\{{\mathbf{W}}_{\rm{h}}{\mathbf{E}}_{\rm{h}}\right\\}-\log\det\left({\mathbf{W}}_{\rm{h}}\right)+f_{\rho}\\\
&~{}{\rm{s.t.}}~{}{v}_{m}\in{\mathcal{F}}_{Q},{\text{for}}~{}m\in[M],\\\
&~{}\quad~{}~{}{\mathsf{tr}}\left({\mathbf{P}}_{k}{\mathbf{P}}_{k}^{\mathsf{H}}\right)\leq
p_{k},{\text{for}}~{}k\in[K],\end{split}$ (${\mathcal{P}}_{4}$)

where $\rho>0$ is the penalty parameter penalizing the violation of the
equality constraints, and $f_{\rho}$ is given by

$\displaystyle
f_{\rho}=\frac{1}{2\rho}\left[\left({\mathbf{1}}^{\mathsf{T}}{\mathbf{s}}-T+\rho\xi\right)^{2}+\sum_{n=1}^{N}\left[\left(s_{n}-{\overline{s}}_{n}+\rho\mu_{n}\right)^{2}\right.\right.$
$\displaystyle\left.\left.+\left(s_{n}\left(1-{\overline{s}}_{n}\right)+\rho\lambda_{n}\right)^{2}\right]+\sum_{m=1}^{M}|\phi_{m}-v_{m}+\rho\tau_{m}|^{2}\right],$
(9)

with $\xi$, ${\bm{\mu}}=[\mu_{1},\ldots,\mu_{N}]^{\mathsf{T}}$,
${\bm{\lambda}}=[\lambda_{1},\ldots,\lambda_{N}]^{\mathsf{T}}$, and
${\bm{\tau}}=[\tau_{1},\ldots,\tau_{M}]^{\mathsf{T}}$ denting the dual
variables associated with the equality constraints in
${\mathbf{1}}^{\mathsf{T}}{\mathbf{s}}=T$, ${\overline{s}}_{n}=s_{n}$,
$s_{n}\left(1-{\overline{s}}_{n}\right)=0$, and ${\bm{\phi}}={\mathbf{v}}$,
respectively. It is observed that as $\rho\rightarrow 0$, the penalty term is
forced to zero, i.e., equality constraints are enforced. It is shown in [14]
that updating the primal and dual variables, as well as the penalty factor in
an alternating manner, PDD converges to a stationary-point solution.

Starting with the penalized form (${\mathcal{P}}_{4}$), we now discuss the
underlying problems in the inner and outer loops. The solution to the AL
problem (inner loop) is iteratively approximated via the block coordinate
descent (BCD) method. To this end, we minimize the objective function of
(${\mathcal{P}}_{4}$) by sequentially updating ${\mathbf{W}}_{\rm{h}}$,
${\mathbf{U}}_{\rm{h}}$, ${\mathbf{s}}$, $\overline{\mathbf{s}}$,
${\bm{\phi}}$, $\mathbf{v}$, and $\mathbf{P}$ in the inner loop through the
following marginal optimizations.

####  III-B1 Optimizing the Weights

We optimize ${\mathbf{W}}_{\rm{h}}$ while treating the remaining variables as
constants, i.e., we find

$\displaystyle{\mathbf{W}}_{\rm{h}}^{\star}=\operatorname*{argmin}\nolimits_{{\mathbf{W}}_{\rm{h}}}{\mathsf{tr}}\left\\{{\mathbf{W}}_{\rm{h}}{\mathbf{E}}_{\rm{h}}\right\\}-\log\det\left({\mathbf{W}}_{\rm{h}}\right).$
(10)

Using its first-order optimality condition, we get
${\mathbf{W}}_{\rm{h}}^{\star}={\mathbf{E}}_{\rm{h}}^{-1}$.

####  III-B2 Optimizing the Hypothetical Receiver

The marginally optimal ${\mathbf{U}}_{\rm{h}}$ is given by solving a linear
MSE minimization problem whose solution is

$\displaystyle{\mathbf{U}}_{\rm{h}}^{\star}=(\sigma^{2}{\mathbf{I}}+{\bm{\Delta}}\overline{\mathbf{H}}{\overline{\mathbf{H}}}^{\mathsf{H}}{\bm{\Delta}}^{\mathsf{H}})^{-1}{\bm{\Delta}}\overline{\mathbf{H}}.$
(11)

####  III-B3 Optimizing $\overline{\mathbf{s}}$

The marginal optimization for $\\{\overline{s}_{n}\\}_{n=1}^{N}$ decouples
into $N$ scalar optimization problems with solution to the $n$th problem being

$\overline{s}_{n}^{\star}=\operatorname*{argmin}\nolimits_{\overline{s}_{n}\in{\mathbbmss{R}}}\left(\overline{a}_{n}\overline{s}_{n}^{2}-2\overline{b}_{n}\overline{s}_{n}\right).$
(12)

Here, $\overline{a}_{n}=1+s_{n}^{2}$ and
$\overline{b}_{n}=s_{n}+\rho\lambda_{n}+s_{n}^{2}+s_{n}\rho\mu_{n}$. The
optimization in (12) is a quadratic scalar problem whose solution is given by
${\overline{s}}_{n}^{\star}={\overline{b}_{n}}/{\overline{a}_{n}}$.

####  III-B4 Optimizing $\mathbf{s}$

The marginal problem for $\mathbf{s}$ is given by

${\mathbf{s}}^{\star}=\operatorname*{argmin}\nolimits_{{\mathbf{s}}\in{\mathbbmss{R}}^{N}}\left({\mathbf{s}}^{\mathsf{T}}{\mathbf{Q}}{\mathbf{s}}-{\mathbf{s}}^{\mathsf{T}}{\mathbf{g}}\right),$
(13)

where

$\displaystyle{\mathbf{Q}}$
$\displaystyle\triangleq\Re\left\\{\left({\mathbf{U}}_{\rm{h}}{\mathbf{W}}_{\rm{h}}{\mathbf{U}}_{\rm{h}}^{\mathsf{H}}\right)\odot(\overline{\mathbf{H}}\overline{\mathbf{H}}^{\mathsf{H}})\right\\}+{2^{-1}\rho^{-1}}\left({\mathbf{1}}{\mathbf{1}}^{\mathsf{T}}+{\mathbf{I}}\right)$
$\displaystyle\quad+{2^{-1}\rho^{-1}}{{\mathsf{diag}}\left\\{\left(1-{\overline{s}}_{1}\right)^{2},\cdots,\left(1-{\overline{s}}_{N}\right)^{2}\right\\}},$
(14) $\displaystyle{\mathbf{g}}$ $\displaystyle\triangleq
2\Re\\!\left\\{{\mathbf{q}}\right\\}\\!-\\!\frac{1}{\rho}\left(\left(\rho\xi\\!-\\!T\right){\mathbf{1}}\\!+\\!\left(\rho{\bm{\lambda}}\\!-\\!\overline{\mathbf{s}}\right)\\!+\\!\rho\left({\mathbf{1}}\\!-\\!\overline{\mathbf{s}}\right)\\!\odot\\!{\bm{\mu}}\right),$
(15) $\displaystyle{\mathbf{q}}$
$\displaystyle\triangleq\left[[{\mathbf{U}}_{\rm{h}}{\mathbf{W}}_{\rm{h}}\overline{\mathbf{H}}^{\mathsf{H}}]_{1,1},\cdots,[{\mathbf{U}}_{\rm{h}}{\mathbf{W}}_{\rm{h}}\overline{\mathbf{H}}^{\mathsf{H}}]_{N,N}\right]^{\mathsf{T}}.$
(16)

Using the Schur product theorem, we can conclude that
$({\mathbf{U}}_{\rm{h}}{\mathbf{W}}_{\rm{h}}{\mathbf{U}}_{\rm{h}}^{\mathsf{H}})\odot(\overline{\mathbf{H}}\overline{\mathbf{H}}^{\mathsf{H}})\succeq{\mathbf{0}}$
and thus ${\mathbf{Q}}\succ{\mathbf{0}}$, which means that the problem (13) is
a standard convex problem. Using its first-order optimality condition, we
obtain ${{\mathbf{s}}}^{\star}=\frac{1}{2}{\mathbf{Q}}^{-1}{\mathbf{g}}$.

####  III-B5 Optimizing the Phase-Shifts

The marginal problem for ${\bm{\phi}}$ is given by

$\displaystyle{\bm{\phi}}^{\star}=\operatorname*{argmin}\nolimits_{{\bm{\phi}}\in{\mathbbmss{C}}^{M}}~{}({\bm{\phi}}^{\mathsf{H}}\mathbf{B}{\bm{\phi}}-2\Re\\{{\bm{\phi}}^{\mathsf{H}}{\mathbf{b}}\\}).$
(17)

The terms appearing in (17) are defined as follows:

  * •

${\mathbf{b}}=\mathsf{diag}\\{{\mathbf{G}}^{\mathsf{H}}{\bm{\Delta}}^{\mathsf{H}}{\mathbf{U}}_{\rm{h}}{\mathbf{W}}_{\rm{h}}{\mathbf{K}}^{\mathsf{H}}\\}+\frac{1}{2\rho}({\mathbf{v}}-\rho{\bm{\tau}})$.

  * •

${\mathbf{K}}=\left[{\mathbf{H}}_{1}{\mathbf{P}}_{1},\ldots,{\mathbf{H}}_{K}{\mathbf{P}}_{K}\right]$.

  * •

${\mathbf{B}}={\mathbf{B}}_{1}\odot{\mathbf{B}}_{2}^{\mathsf{T}}+\frac{1}{2\rho}{\mathbf{I}}$.

  * •

${\mathbf{B}}_{1}={\mathbf{G}}^{\mathsf{H}}{\bm{\Delta}}^{\mathsf{H}}{\mathbf{U}}_{\rm{h}}{\mathbf{W}}_{\rm{h}}{\mathbf{U}}_{\rm{h}}^{\mathsf{H}}{\bm{\Delta}}{\mathbf{G}}\succeq{\mathbf{0}}$
and
${\mathbf{B}}_{2}={\mathbf{K}}{\mathbf{K}}^{\mathsf{H}}\succeq{\mathbf{0}}$.

Using the Schur product theorem, we can conclude that
${\mathbf{B}}_{1}\odot{\mathbf{B}}_{2}^{\mathsf{T}}\succeq{\mathbf{0}}$, and
thus ${\mathbf{B}}\succ{\mathbf{0}}$. Therefore, problem (17) is an
unconstrained convex quadratic optimization problem. Using its first-order
optimality condition, we find the solution as

$\displaystyle{\bm{\phi}}^{\star}={\mathbf{B}}^{-1}{\mathbf{b}}.$ (18)

####  III-B6 Optimizing $\mathbf{v}$

The marginal optimization for this auxiliary variable $\mathbf{v}$ decouples
into $M$ parallel scalar optimizations with the $m$th subproblem being

$\displaystyle v_{m}^{\star}$
$\displaystyle=\operatorname*{argmin}\nolimits_{v_{m}\in{\mathcal{F}}_{Q}}|\phi_{m}-v_{m}+\rho\tau_{m}|.$
(19)

Using the fact that $|v_{m}|^{2}=1$, we have

$\displaystyle v_{m}^{\star}$
$\displaystyle=\operatorname*{argmax}\nolimits_{v_{m}\in{\mathcal{F}}_{Q}}\Re\\{v_{m}(\phi_{m}+\rho\tau_{m})^{*}\\}={\rm
e}^{{\rm j}{\vartheta}_{m}^{\star}},$ (20)

where ${\vartheta}_{m}^{\star}$ is obtained by mapping
$\angle({\phi_{m}+\rho\tau_{m}})$ to the nearest discrete phase in the set of
discrete phases. With infinite-precision representation of phase-shifts, i.e.,
$Q\rightarrow\infty$, we have $v_{m}^{\star}={\rm{e}}^{{\rm
j}\angle({\phi_{m}+\rho\tau_{m}})}$.

####  III-B7 Optimizing the Precoders

The marginal optimization in terms of ${\mathbf{P}}$ reduces to $K$
subproblems with the $k$th one being

$\displaystyle{\mathbf{P}}_{k}^{\star}\\!=\\!\operatorname*{argmin}\nolimits_{{\mathsf{tr}}({\mathbf{P}}_{k}{\mathbf{P}}_{k}^{\mathsf{H}})\leq
p_{k}}({\mathsf{tr}}({\mathbf{P}}_{k}^{\mathsf{H}}{\mathbf{C}}_{k}{\mathbf{P}}_{k})\\!-\\!2\Re\\{{\mathsf{tr}}({\mathbf{P}}_{k}^{\mathsf{H}}{\mathbf{D}}_{k})\\}),$

where ${\mathbf{C}}_{k}={\mathbf{O}}_{k}{\mathbf{W}}_{\rm
h}{\mathbf{O}}_{k}^{\mathsf{H}}$,
${\mathbf{O}}_{k}={\mathbf{H}}_{k}^{\mathsf{H}}{\bm{\Phi}}^{\mathsf{H}}{\mathbf{G}}^{\mathsf{H}}{\bm{\Delta}}^{\mathsf{H}}{\mathbf{U}}_{\rm
h}$, ${\mathbf{D}}_{k}={\mathbf{O}}_{k}{\mathbf{W}}_{{\rm h},k}^{\mathsf{H}}$,
and ${\mathbf{W}}_{{\rm h},k}$ denotes the submatrix of ${\mathbf{W}}_{{\rm
h}}$ obtained by extracting the elements of the rows with indices from
$(1+\sum_{k^{\prime}=1}^{k-1}L_{k^{\prime}})$ to
$\sum_{k^{\prime}=1}^{k}L_{k^{\prime}}$. This is a standard convex quadratic
optimization subproblem whose solution is given by

$\displaystyle{\mathbf{P}}_{k}^{\star}=({\mathbf{C}}_{k}+\lambda{\mathbf{I}})^{-1}{\mathbf{D}}_{k}.$
(21)

The regularizer $\lambda$ is chosen, such that the complementarity slackness
condition, i.e.,
$\lambda({\mathsf{tr}}({\mathbf{P}}_{k}{\mathbf{P}}_{k}^{\mathsf{H}})-p_{k})=0$,
is satisfied. If
${\mathsf{tr}}({\mathbf{D}}_{k}^{\mathsf{H}}({\mathbf{C}}_{k}+\lambda{\mathbf{I}})^{-2}{\mathbf{D}}_{k})=p_{k}$;
then, $\lambda=0$. Otherwise, we can obtain the solution of $\lambda$ from the
following identity:

$\displaystyle{\mathsf{tr}}({\mathbf{P}}_{k}{\mathbf{P}}_{k}^{\mathsf{H}})={\mathsf{tr}}({\mathbf{D}}_{k}^{\mathsf{H}}({\mathbf{C}}_{k}+\lambda{\mathbf{I}})^{-2}{\mathbf{D}}_{k})=p_{k}.$
(22)

It follows that

$\displaystyle\sum\nolimits_{n_{k}=1}^{N_{k}}\frac{|[{\mathbf{U}}_{{\mathbf{C}}_{k}}{\mathbf{D}}_{k}{\mathbf{D}}_{k}^{\mathsf{H}}{\mathbf{U}}_{{\mathbf{C}}_{k}}^{\mathsf{H}}]_{n_{k},n_{k}}|^{2}}{([{\bm{\Lambda}}_{{\mathbf{C}}_{k}}]_{n_{k},n_{k}}+\lambda)^{2}}=p_{k},$
(23)

where
${\mathbf{C}}_{k}={\mathbf{U}}_{{\mathbf{C}}_{k}}^{\mathsf{H}}{\bm{\Lambda}}_{{\mathbf{C}}_{k}}{\mathbf{U}}_{{\mathbf{C}}_{k}}$
is the eigen-decomposition of ${\mathbf{C}}_{k}$. Since
$[{\bm{\Lambda}}_{{\mathbf{C}}_{k}}]_{n_{k},n_{k}}\geq 0$ for
$n_{k}\in[N_{k}]$, the left-hand side of (23) is a monotonously decreasing
function of $\lambda\geq 0$. Consequently, we can find $\lambda$ by solving
equation (23) using the bisection-based search method.

We next consider the outer loop of the PDD-based algorithm. In this loop, the
dual variables $\\{\xi,{\bm{\mu}},{\bm{\lambda}},{\bm{\tau}}\\}$ and the
penalty factor $\rho$ are updated. These variables are updated by optimizing
the constraint violation quantity that is defined as

$\displaystyle h\triangleq\max_{\forall
n,m}\\{|\phi_{m}\\!-\\!v_{m}|,|{\mathbf{1}}\\!^{\mathsf{T}}{\mathbf{s}}\\!-\\!T|,\left|{\overline{s}}_{n}\\!-\\!s_{n}\right|,\left|s_{n}\\!\left(1\\!-\\!{\overline{s}}_{n}\right)\right|\\}.$
(24)

The optimization of this quantity is discussed in great detail in [14] and is
skipped here, due to lack of space. The outcome of this optimization yield the
update scheme as follows: if $h<\mu$, we update the dual variables via the
recursive equalities

$\displaystyle\xi^{(t+1)}$
$\displaystyle=\xi^{(t)}+{\rho}^{-1}\left({\mathbf{1}}^{\mathsf{T}}{\mathbf{s}}-T\right),$
(25a) $\displaystyle\mu_{n}^{(t+1)}$
$\displaystyle=\mu_{n}^{(t)}+{\rho}^{-1}\left({\overline{s}}_{n}-s_{n}\right),$
(25b) $\displaystyle\lambda_{n}^{(t+1)}$
$\displaystyle=\lambda_{n}^{(t)}+{\rho}^{-1}s_{n}\left(1-{\overline{s}}_{n}\right),$
(25c) $\displaystyle\tau_{m}^{(t+1)}$
$\displaystyle=\tau_{m}^{(t)}+{\rho}^{-1}(v_{m}-\phi_{m}),$ (25d)

where $t$ denotes the outer iteration index. We refer the reader to [14, Table
I, Line 4] for further details. The derived PDD-based algorithm is summarized
in Algorithm 1.

###  III-C Convergence and Complexity Analyses

As shown in [14], Algorithm 1 is guaranteed to converge to a stationary
solution of problem ($\mathcal{P}_{1}$). The computational complexity of the
proposed algorithm can be further characterized in terms of problem
dimensions. To this end, let $I_{\rm{out}}$ and $I_{\rm{in}}$ denote the
numbers of iterations in the outer loop and the inner loop, respectively. The
per-iteration complexity in the outer loop is mainly originated from the BCD
updates in the inner loop. The computational complexity of one inner-loop
iteration is further composed of the complexity of updating variables
$\left\\{{\mathbf{W}}_{\rm{h}},{\mathbf{U}}_{\rm{h}},\bar{\mathbf{s}},{\mathbf{s}},{\bm{\phi}},{\mathbf{v}},{\mathbf{P}}\right\\}$.
It is readily shown that the complexity of marginal optimizations with respect
to ${\mathbf{W}}_{\rm{h}}$, ${\mathbf{U}}_{\rm{h}}$, $\bar{\mathbf{s}}$,
${\mathbf{s}}$, ${\bm{\phi}}$, $\mathbf{v}$, and $\mathbf{P}$ scale with
${\mathcal{O}}\left(L^{3}\right)$, ${\mathcal{O}}\left(N^{3}\right)$,
${\mathcal{O}}\left(N\right)$, ${\mathcal{O}}\left(N^{3}\right)$,
${\mathcal{O}}\left(M^{3}\right)$, ${\mathcal{O}}\left(M\right)$, and
${\mathcal{O}}\left(\sum_{k=1}^{K}N_{k}^{3}\right)$, respectively. Hence, the
overall complexity of Algorithm 1 scales with
${\mathcal{O}}\left(I_{\text{out}}I_{\text{in}}\left(2N^{3}+M^{3}+L^{3}+\sum_{k=1}^{K}N_{k}^{3}\right)\right)$,
which is of a polynomial order.

Algorithm 1 PDD-based algorithm for solving problem ($\mathcal{P}_{1}$)

1: Initialize primary variables
$\left\\{{\mathbf{P}},{\mathbf{v}},{\bm{\phi}},{\mathbf{s}},\overline{\mathbf{s}},{\mathbf{W}}_{\rm{h}},{\mathbf{U}}_{\rm{h}}\right\\}$,
dual variables $\left\\{\xi,{\bm{\lambda}},{\bm{\mu}},{\bm{\tau}}\right\\}$,
iteration index $t=0$, threshold $\mu$, penalty factor $\rho>0$, and scaling
factor $\chi\in(0,1)$

2: repeat

3: repeat

4: Update
$\left\\{{\mathbf{W}}_{\rm{h}},{\mathbf{U}}_{\rm{h}},\overline{\mathbf{s}},{\mathbf{s}},{\bm{\phi}},{\mathbf{v}},{\mathbf{P}}\right\\}$
by the BCD method

5: until convergence

6: Calculate the constraint violation $h$ by (24)

7: if $h<\mu$ then

8: Update the dual variable by (25)

9: else

10: Set $\rho=\chi\rho$

11: end if

12: Set $\mu=\chi h$ and $t=t+1$

13: until convergence

##  IV Low-Complexity Sequential Optimization

The PDD-based algorithm proposed in Section III optimizes the antenna
selection vector $\mathbf{s}$, phase-shift vector $\bm{\phi}$, and transmit
precoders $\\{{\mathbf{P}}_{k}\\}_{k=1}^{K}$ alternately. Although this leads
to a feasible complexity, it can lead to a computational burdensome in many
applications. To this end, we develop an alternative scheme based on
sequential optimization approach in this section. In this respect, we first
design the phase-shifts using an element-wise BCD (EWBCD) method; then, we
select the antennas using a greedy search (GS). We finally obtain the optimal
transmit precoders with the aid of iterative water-filling method. These three
steps are illustrated in the sequel.

####  IV-1 Phase-Shifts Design

At the first step of the SO-based method, we design the phase-shifts by
maximizing the effective channel gain given by

$\displaystyle\mathsf{tr}({\overline{\mathbf{H}}}{\overline{\mathbf{H}}}^{\mathsf{H}})=\sum\nolimits_{k=1}^{K}\mathsf{tr}({\mathbf{G}}{\bm{\Phi}}{\mathbf{H}}_{k}{\mathbf{Q}}_{k}{\mathbf{H}}_{k}^{\mathsf{H}}{\bm{\Phi}}^{\mathsf{H}}{\mathbf{G}}^{\mathsf{H}}).$
(26)

Assuming ${\mathbf{Q}}_{k}={\frac{p_{k}}{N_{k}}}{\mathbf{I}}$ for $k\in[K]$,
the phase-shifts design problem is formulated as follows:

$\displaystyle{\bm{\phi}}^{\star}=\operatorname*{argmax}({\mathsf{tr}}({\bm{\Phi}}^{\mathsf{H}}{\mathbf{G}}^{\mathsf{H}}{\mathbf{G}}{\bm{\Phi}}\widetilde{\mathbf{H}})={\bm{\phi}}^{\mathsf{H}}\hat{\mathbf{H}}{\bm{\phi}}),~{}{\rm{s.t.}}~{}C_{3},$
(27)

where
$\widetilde{\mathbf{H}}=\sum\nolimits_{k=1}^{K}\frac{p_{k}}{N_{k}}{\mathbf{H}}_{k}{\mathbf{H}}_{k}^{\mathsf{H}}$
and
$\hat{\mathbf{H}}=({\mathbf{G}}^{\mathsf{H}}{\mathbf{G}})\odot{\widetilde{\mathbf{H}}}^{\mathsf{T}}$.
To handle the discretely constrained $\bm{\phi}$, we propose to optimize the
elements of $\bm{\phi}$ one by one, resulting in an EWBCD method comprised of
$M$ steps. At the $m$th step, we optimize $\phi_{m}$ while treating
$\phi_{m^{\prime}}$ for $m^{\prime}\neq m$ as constants. This leads to

$\displaystyle{\phi}_{m}^{\star}=\operatorname*{argmax}\nolimits_{\phi_{m}\in{\mathcal{F}}_{Q}}{2\Re\\{\phi_{m}^{*}\varphi_{m}\\}},$
(28)

where $\varphi_{m}=\sum\nolimits_{m^{\prime}\neq
m}[\hat{\mathbf{H}}]_{m^{\prime},m}\phi_{m^{\prime}}$. The solution of this
problem is given by ${\phi}_{m}^{\star}={\rm e}^{{\rm j}\delta_{m}^{\star}}$,
where $\delta_{m}^{\star}$ is obtained by mapping $\angle{\varphi_{m}}$ to the
nearest discrete phase.

####  IV-2 Antenna Selection

In the next step, we optimize the antenna selection vector $\mathbf{s}$ for
the updated phase-shifts. The resulting problem of antenna selection is given
by

$\displaystyle{\mathbf{s}}^{\star}=\operatorname*{argmax}\log_{2}\det(\mathbf{I}+{\sigma^{-2}}{\bm{\Delta}}{\mathbf{X}}{\bm{\Delta}}^{\mathsf{H}}),~{}{\rm{s.t.}}~{}C_{1},$
(29)

where
${\mathbf{X}}={\mathbf{G}}{\bm{\Phi}}\widetilde{\mathbf{H}}{\bm{\Phi}}^{\mathsf{H}}{\mathbf{G}}^{\mathsf{H}}$.
To tackle the discrete constraint, we exploit the GS method. More details
about this method can be found in [2, 3] and are omitted here due to page
limitations.

####  IV-3 Transmit Precoding Design

After obtaining the phase-shifts and antenna selection vector, we find the
optimal transmit precoding matrices by solving

$\displaystyle\max_{\mathbf{P}}~{}\log_{2}\det(\mathbf{I}+\sum\nolimits_{k=1}^{K}{\mathbf{G}}_{{\bm{\phi}},k}{\mathbf{P}}_{k}{\mathbf{P}}_{k}^{\mathsf{H}}{\mathbf{G}}_{{\bm{\phi}},k}^{\mathsf{H}}),~{}{\rm{s.t.}}~{}C_{2},$
(30)

where
${\mathbf{G}}_{{\bm{\phi}},k}=\sigma^{-1}{\bm{\Delta}}{\mathbf{G}}{\bm{\Phi}}{\mathbf{H}}_{k}$.
The optimization in (30) can be optimally solved by the iterative water-
filling method [15].

Based on [2] and [15], we can show that the complexity of sequential
optimizations with respect to ${\bm{\phi}}$, ${\mathbf{s}}$, and $\mathbf{P}$
scale with ${\mathcal{O}}\left(M\right)$, ${\mathcal{O}}\left(NTM\right)$, and
${\mathcal{O}}\left(KT^{3}\right)$, respectively. Hence, the complexity of the
SO-based method scales with ${\mathcal{O}}(NTM+KT^{3})$, which is lower than
the PDD-based method.

(a) Convergence of sum rate.

(b) Constraint violation.

Figure 2: Average convergence performances for $Q=4$ and $\log{p_{k}}=-5$ dBm
($\forall k$).

(a) Sum rate vs. the power budget.

(b) Sum rate vs. the number of quantization bits.

(c) Sum rate vs. the number of RF chains.

Figure 3: Sum-rate performances of the proposed algorithms. The simulation
parameters are given as follows: (a) $N=16$, $T=6$, $M=32$, and $Q=4$; (b)
$N=16$, $T=6$, and $\log{p}=-5$ dBm; (c) $M=32$, $Q=4$, and $\log{p}=-5$ dBm.

##  V Numerical Results

In this section, we verify the effectiveness of the proposed algorithms
through numerical experiments. For simulations, the following parameters are
used unless specified otherwise later: $N_{k}=L_{k}=3$, $p_{k}=p$, for
$k\in[K]$, $\log{\sigma^{2}}=-120$ dBm, $K=4$, $\mu=1$, $\rho=1$, and
$\chi=0.1$. We further generate the channel realizations as follows. Regarding
the large scale fading, we assume that all the composite UT-RIS-BS channels,
i.e., ${\mathbf{G}}{\mathbf{H}}_{k}$ exhibit the same path loss $-120$ dB for
illustration. Meanwhile, for the small scale fading of the UT-to-RIS and RIS-
to-BS channels, we consider the standard Rayleigh fading model. All the
optimization variables are randomly initialized, and all simulation curves are
averaged over $500$ independent channel realizations.

In Figure 2, we first study the convergence behavior of the proposed PDD-based
method. As can be seen from Figure 2(a), the sum-rate rapidly converges to a
stationary value. Figure 2(b) shows the constraint violation in terms of the
number of outer iterations. We observe that the constraint violation reduces
to a threshold $10^{-4}$ in less than $15$ outer iterations, implying that the
solution has essentially met the equality constraints for problem
($\mathcal{P}_{1}$).

We next consider the following benchmark schemes for performance comparison:
1) Random scheme, in which $\mathbf{s}$ and $\bm{\phi}$ are randomly set, and
${\mathbf{Q}}_{k}={\frac{p_{k}}{N_{k}}}{\mathbf{I}}$ for $k\in[K]$. 2) AO-
based scheme, in which $\mathbf{s}$, $\bm{\phi}$, and $\mathbf{P}$ are updated
alternately until convergence. Here, we design $\mathbf{s}$, $\bm{\phi}$, and
$\mathbf{P}$ using the GS-based, EWBCD-based, and iterative water-filling
methods, respectively. This scheme is a generalization of the methods in [6,
8, 10, 7, 9], whose complexity scales with
${\mathcal{O}}(I_{\rm{AO}}(M^{3}+NTM+KT^{3}))$ with $I_{\rm{AO}}$ denotes the
number of AO iterations. It is worth noting that the complexity of the AO-
based method is between the PDD-based and the SO-based methods.

Figure 3 compares the sum-rate performances achieved by the proposed methods
and the benchmark methods. In Figure 3(a), the sum-rate is plotted against the
transmit power budget $p$ for different optimization schemes assuming $M=32$.
From the figure, it is seen that the proposed PDD-based method significantly
outperforms the AO-based one. The SO-based method further achieves almost the
same performance as the AO-based scheme and achieves a significant performance
gain as compared with the random scheme. As mentioned, the SO-based method
involves less complexity than the AO-based scheme. The simulation results
hence imply that for systems with highly restricted computational capacity,
the SO-based method is preferred, whereas in systems with less limitations on
the computational load, the PDD-based method is suggested.

We next set $\log{p}=-5$ dBm and plot the sum-rate against the number of
quantization bits. The results are shown in Figure 3(b) implying that for the
schemes with optimized phase-shifts, the performance degradation caused by
phase-shift quantization is negligible, as the resolution exceeds $4$ bits.
Comparing the proposed algorithms with the random scheme highlights the
significance of the phase-shift design. From Figure 3(b), it is further
observed that the SO-based method outperforms the AO-based scheme for RISs of
smaller size.

Figure 3(c) illustrates the sum-rate versus the number of RF chains for
selected BS antenna numbers. From the figure, one can observe that in all the
presented cases, the increase in the number of RF chains enhances the system
throughput.

##  VI Conclusion

We proposed an iterative algorithm based on the PDD method for joint antenna
selection and beamforming in RIS-aided MU-MIMO systems. An alternative
algorithm with reduced complexity was further developed based on the SO
scheme. To reduce the complexity, we also developed a SO-based algorithm. Our
numerical results imply that the PDD-based method outperforms both the SO-
based and benchmark schemes. The SO-based algorithm moreover tracks closely
the benchmark performance while gaining considerably in terms of complexity.

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