# Joint User Association and Downlink Beamforming for Green Cloud-RANs with
Limited Fronthaul

Zhi Yu, Ke Wang, Hong Ji, Xi Li, Heli Zhang  Key Laboratory of Universal
Wireless Communications, Ministry of Education
Beijing University of Posts and Telecommunications
Beijing, P.R. China
Email: zhiyu, wangke, jihong, lixi<EMAIL_ADDRESS>
###### Abstract

With the explosive growth of smart devices and mobile data traffic, limited
fronthaul capacity has become a notable bottleneck of green communication
access networks, such as cloud radio access networks(C-RANs). In this paper,
we proposed a joint user association and downlink beamforming scheme for green
C-RANs to minimize the network power consumption with the limited fronthaul
links. We first formulate the design problem as a mixed integer nonlinear
programming (MINLP), and then transformed the MINLP problem into a mixed
integer second-order cone program (MI-SOCP) which is a convex program when the
integer variables are fixed. By relaxing the integer variables to continuous
ones, an inflation algorithm, which can be finished within polynomial time,
was proposed to solved the problem. The simulation results are presented to
validate the effectiveness of our proposed algorithm compared with the the
scheme adopted by LTE-A.

###### Index Terms:

Cloud-RAN, user association, downlink beamformer design, green communication,
mixed integer second-order cone programming.

##  I Introduction

Cloud radio access network (C-RAN) has recently been proposed as a promising
network architecture to jointly improve energy efficiency and spectrum
efficiency, and reduce both the capital expenses and operation expenses [1].
In C-RANs, baseband data and channel state information (CSI) are processed in
a central processor called Baseband Unit (BBU) pool and shared among the
densely deployed remote radio heads (RRHs) via fronthaul links, which allows
the RRHs to cooperatively transmit the data to the mobile users (MUs) [2].
Unfortunately, with the dense deployment of RRHs, the overall network power
consumption, including both transmit and circuit power consumption [3], will
increase significantly. Therefore, green C-RAN has quickly attracted wide
attention [4, 5, 6].

The authors in [4] address the problem of downlink beamforming to improve
energy efficiency of C-RANs by using weighted mixed norm minimization. A
three-stage algorithm based on the group-sparsity inducing norm is presented
to minimize the power consumption for multicast C-RANs [5]. Combining
virtualized network resources and virtualized functional entities of baseband
processing, the authors in [6] propose an energy-saving scheme for C-RANs
based on formation of Virtual Base Station. However, these excellent works do
not take into account limitation of fronthaul links, which is becoming the a
bottleneck for realizing the potential performance gain of C-RANs [7].
Moreover, with the huge number of MUs involved in the network, enormous
baseband signals and signaling overheads are required to be transmitted in
fronthaul links, which incurs that the fronthaul limited problem becomes more
severe. Hence, finding a solution of network power minimization problem under
limited fronthaul is critical to achieve green communication and commercial
deployment of C-RANs.

Some other excellent works have been done on energy saving with constrained
fronthaul links in CRANs. The authors in [8] investigate the tradeoff between
total transmit power and sum backhaul capacity over all BSs in a network MIMO
system. To minimize downlink transmit power, a efficient algorithm with
constraints on fronthaul capacity is presented in [9]. For transmit power
minimization, the authors in [10] jointly consider coordinated beamforming and
admission control design in fronthaul constrained C-RANs. Nevertheless, all
these works assume that all the RRHs are involved to cooperatively transmit
data to MUs, i.e., the RRH sleeping mechanism which is closely related to the
user association status is not considered. Therefore, the schemes proposed by
these works can not efficiently reduce the circuit power consumption, which
accounts for a large part of network power consumption [11].

In this paper, we investigate the network power minimization problem in C-RAN
with limited fronthaul. This design problem jointly considers user association
and downlink beamforming, which dominate circuit power consumption and
transmit power consumption, respectively. to be more specific, we first
characterize the problem as a mixed integer nonlinear program (MINLP), which
is hard to be solved because of the features of the user association. Then, we
transform the problem into a mixed integer second-order cone program(MI-SOCP)
by making the objective function plus a sufficient small variable. Finally, we
relax the integer variables to continuous ones and adopt a low complexity
inflation algorithm to obtain the suboptimal solution. Numerical results
verify the effectiveness of the proposed scheme by compared with the scheme
adopted by LTE-A.

The rest of this paper is organized as follows. Section II presents the system
and power model. In Section III, the network power consumption minimization
problem is formulated as a MI-SOCP, followed by some analisis. Section IV
presents the low complexity inflation algorithm to obtain the suboptimal
solution. Simulation results and discussions are given in section V. Finally,
we conclude this paper in section VI.

##  II System and Power Model

###  II-A System Model

We consider a downlink Cloud-RAN with $L$ multiple-antenna RRHs and $K$
single-antenna MUs, where the $l$-th RRH with $N_{l}$ antennas is connected to
a BBU Pool by a limited fronthaul link, as shown in Fig.1. It is assumed that
all user data and the perfect CSI is available at the BBU pool. We define
$\mathcal{L}=\\{1,2,\ldots,L\\}$ and $\mathcal{K}=\\{1,2,\ldots,K\\}$ as the
set of RRH and MU indices, respectively. In a beamformer design problem, the
baseband signals of RRH $l$ can be expressed as

$\displaystyle x_{l}=\sum\limits_{k=1}^{K}\mathbf{w}_{l,k}s_{k},\forall
l\in\mathcal{L}$ (1)

where $\mathbf{w}_{l,k}\in\mathbb{C}^{N_{l}\times 1},\forall
l\in\mathcal{L},\forall k\in\mathcal{K}$ is the beamforming vector at RRH $l$
for MU $k$ and $s_{k}\in\mathbb{C}$ is the data symbol for MU $k$ with unit
power, i.e. $E[|s_{k}|^{2}]=1$. Specifically, we assume that there is a
maximum transmit power constraint for RRH $l$ which is given by:

$\displaystyle\sum\limits_{k=1}^{K}||\mathbf{w}_{l,k}||_{2}^{2}\leq
P_{l}^{MAX},\forall l\in\mathcal{L}$ (2)

Moreover, we consider a quasi-static fading environment and denote the channel
vector from RRH $l$ to MU $k$ as $\mathbf{h}_{l,k}\in\mathbb{C}^{N_{l}\times
1},\forall l\in\mathcal{L},\forall k\in\mathcal{K}$. Then, the received
baseband signal at MU $k$ is given by

$\displaystyle
y_{k}=\underbrace{\sum\limits_{l=1}^{L}\mathbf{h}_{l,k}^{H}\mathbf{w}_{l,k}s_{k}}_{\text{Desired
Signal}}+\underbrace{\sum\limits_{i\in\mathcal{K},i\neq
k}\sum\limits_{l=1}^{L}\mathbf{h}_{l,k}^{H}\mathbf{w}_{l,i}s_{i}}_{\text{Interference}}+z_{k},\forall
k\in\mathcal{K}$ (3)

where $z_{k}\sim\mathcal{CN}(0,\sigma_{k}^{2})$ is the additive Gaussian
noise. Treating the interference in Eqs.(3) as noise, the received signal-to-
interference-plus-noise ratio (SINR) at MU $k$ is given by

$\displaystyle
SINR_{k}=\frac{|\sum\limits_{l=1}^{L}\mathbf{h}_{l,k}^{H}\mathbf{w}_{l,k}|^{2}}{\sum\limits_{i\in\mathcal{K},i\neq
k}|\sum\limits_{l=1}^{L}\mathbf{h}_{l,k}^{H}\mathbf{w}_{l,i}|^{2}+\sigma_{k}^{2}},\forall
k\in\mathcal{K}$ (4)

###  II-B Power Model

Figure 1: An example of downlink C-RAN with limited-capacity fronthaul links.

We assume that each RRH in C-RAN may be in one of the two states, which are
sleep (SLP) and transmitting (TRA). Then, we define the binary variable tuple
$\\{a_{l},b_{l,k},\forall l\in\mathcal{L},\forall k\in\mathcal{K}\\}$ as the
network state, where $a_{l}\in\\{0,1\\}$ and $b_{l,k}\in\\{0,1\\}$. $a_{l}$
denotes RRH power indicator: $a_{l}=1$ indicates that the RRH $l$ is in
transmitting state, and $a_{l}=0$ otherwise, $\forall l\in L$. Moreover,
$b_{l,k}$ denotes user association indicator: $b_{l,k}=1$ means that the MU
$k$ is served by the RRH $l$, and $b_{l,k}=0$ otherwise, $\forall
k\in\mathcal{K},\forall l\in\mathcal{L}$. Clearly, according to the
relationships among $a_{l}$, $b_{l,k}$ and $\mathbf{w}_{l,k}$, we define the
user association constraints as follows:

$\begin{cases}\\{b_{l,k}=0,\forall k\in\mathcal{K}\\}\Leftrightarrow
a_{l}=0,\quad\quad\forall l\in\mathcal{L}\\\
b_{l,k}=0\Leftrightarrow\mathbf{w}_{l,k}=\mathbf{0},\quad\quad\forall
l\in\mathcal{L},\forall k\in\mathcal{K}\end{cases}$ (5)

The equivalent form of Eqs.(5) can be written as:

$\begin{cases}b_{l,k}\neq 0,\exists k\in\mathcal{K}\Leftrightarrow
a_{l}=1,\quad\quad\forall l\in\mathcal{L}\\\
b_{l,k}=1\Leftrightarrow\mathbf{w}_{l,k}\neq\mathbf{0},\quad\quad\forall
l\in\mathcal{L},\forall k\in\mathcal{K}\end{cases}$ (6)

According to [12], the power consumption of RRH $l$ in transmitting, denoted
by $P_{l}^{TRA}$, consists of both circuit and transmit power consumption.
Considering the Eqs.(5), $P_{l}^{TRA}$ can be expressed as:

$\displaystyle P_{l}^{TRA}$
$\displaystyle=P_{l}^{CIR}+\frac{1}{\eta_{l}}\sum\limits_{k=1}^{K}b_{l,k}\|\mathbf{w}_{l,k}\|_{2}^{2}$
(7)
$\displaystyle=P_{l}^{CIR}+\frac{1}{\eta_{l}}\sum\limits_{k=1}^{K}\|\mathbf{w}_{l,k}\|_{2}^{2}$

where $P_{l}^{CIR}$ and $\eta_{l}$ denote the circuit power consumption and
the efficiency of the radio frequency power amplifier, respectively. In
addition, let the constant $P_{l}^{SLP}$ denote the power consumption when the
RRH $l$ is in sleep. For Pico base station, the typical values are
$P_{l}^{CIR}=6.8W$, $P_{l}^{SLP}=4.3W$, and $\eta_{l}=0.25$ [12].

With the Eqs.(5) and Eqs.(7), the power consumption of RRH $l$, denoted by
$P_{l}$, can be expressed as

$\displaystyle P_{l}$ $\displaystyle=a_{l}P_{l}^{TRA}+(1-a_{l})P_{l}^{SLP}$
(8)
$\displaystyle=a_{l}P_{l}^{CIR}+\frac{a_{l}}{\eta_{l}}\sum\limits_{k=1}^{K}b_{l,k}\|\mathbf{w}_{l,k}\|_{2}^{2}+(1-a_{l})P_{l}^{SLP}$
$\displaystyle=a_{l}P_{l}^{CIR}+\frac{1}{\eta_{l}}\sum\limits_{k=1}^{K}\|\mathbf{w}_{l,k}\|_{2}^{2}+(1-a_{l})P_{l}^{SLP}$
$\displaystyle=P_{l}^{SLP}+a_{l}P_{l}^{CMS}+\frac{1}{\eta_{l}}\sum\limits_{k=1}^{K}\|\mathbf{w}_{l,k}\|_{2}^{2},\forall
l\in\mathcal{L}$

where $P_{l}^{CMS}=P_{l}^{CIR}-P_{l}^{SLP}$. We denote $\Phi$ as the parameter
tuple $\\{a_{l},b_{l,k},\mathbf{w}_{l,k},\forall l\in\mathcal{L},\forall
k\in\mathcal{K}\\}$. Then, by omitting the constant term $P_{l}^{SLP}$, we
define the network power consumption function $F(\Phi)$ as

$\displaystyle
F(\Phi)=\sum\limits_{l=1}^{L}(a_{l}P_{l}^{CMS}+\frac{1}{\eta_{l}}\sum\limits_{k=1}^{K}\|\mathbf{w}_{l,k}\|_{2}^{2})$
(9)

##  III Problem Formulation and Transformation

Based on the power consumption model, we formulate here the network power
minimization problem in C-RANs as a MINLP, which jointly consider the QoS
requirements of MUs, maximum transmit power and the fronthaul limitation of
each RRH.

###  III-A Problem Formulation

Similar to [13, 14, 15, 16], we employ the QoS constraints for MUs as follows:

$\displaystyle SINR_{k}\geq\gamma_{k},\forall k\in\mathcal{K},$ (10)

where $\gamma_{k}>0$ denotes the target SINR of MU $k$.

To quantify the fronthaul cost, one obvious metric is the average bits/sec.
However, considering that the information exchanged in the fronthaul includes
not only user data but also signaling overhead, this metric reveals too much
detail, and the associated problem is highly combinatorial [17]. Hence,
similar to [10] and [18], we adopt the number of active connection links which
gives a first order measurement of fronthaul load as the metric of fronthaul
cost instead. Then, we have the following constraints:

$\displaystyle\sum\limits_{k=1}^{K}b_{l,k}\leq C_{l},\forall l\in\mathcal{L}$
(11)

where $C_{l}$ denotes the maximum number of active connection links that RRH
$l$ can serve.

With Eqs.(5), the QoS constraints and the fronthaul capacity constraints, the
network power consumption minimization problem can be formulated as follows:

$\displaystyle\mathscr{P}^{(pri)}$ $\displaystyle:\min F(\Phi)$ (12a)
$\displaystyle s.t.\quad$ $\displaystyle SINR_{k}\geq\gamma_{k},\forall
k\in\mathcal{K}$ (12b)
$\displaystyle\sum\limits_{k=1}^{K}||\mathbf{w}_{l,k}||_{2}^{2}\leq
P_{l}^{MAX},\forall l\in\mathcal{L}$ (12c)
$\displaystyle\sum\limits_{k=1}^{K}b_{l,k}\leq C_{l},\forall l\in\mathcal{L}$
(12d) $\displaystyle a_{l}=0\Leftrightarrow\\{b_{l,k}=0,\forall
k\in\mathcal{K}\\},\forall l\in\mathcal{L}$ (12e) $\displaystyle
b_{l,k}=0\Leftrightarrow\mathbf{w}_{l,k}=\mathbf{0},\forall
l\in\mathcal{L},\forall k\in\mathcal{K}$ (12f) $\displaystyle
a_{l}\in\\{0,1\\},b_{l,k}\in\\{0,1\\},\forall l\in\mathcal{L},\forall
k\in\mathcal{K}$ (12g)

For the rest of this paper, we assume that problem $\mathscr{P}^{(pri)}$ is
always feasible when all RRHs are active, i.e. the QoS requirement of each MU
will be satisfied if $a_{l}=1,\forall l\in\mathcal{L}$.

In the following subsection, we will transform problem $\mathscr{P}^{(pri)}$
into a Mixed-Integer Second-Order Cone Program(MI-SOCP)[19], since the problem
with the user association constraints ( i.e. Eqs.(12e) and Eqs.(12f)) is hard
to be solved.

###  III-B Problem Transformation

We first handle Eqs. (12b). Since the phases of $\mathbf{w}_{l,k}$ do not
affect the objective function and Eqs. (12b), the QoS constraints could be
rewritten as the following second order cone (SOC) constraints[13]:

$\displaystyle\sqrt{\sum\limits_{i\neq
k}|\sum\limits_{l=1}^{L}\mathbf{h}_{l,k}^{H}\mathbf{w}_{l,i}|^{2}+\sigma_{k}^{2}}\leq\frac{1}{\sqrt{\gamma_{k}}}\mathrm{Re}\\{\sum\limits_{l=1}^{L}\mathbf{h}_{l,k}^{H}\mathbf{w}_{l,k}\\},\forall
k\in\mathcal{K}$ (13a)
$\displaystyle\qquad\qquad\qquad\mathrm{Im}\\{\sum\limits_{l=1}^{L}\mathbf{h}_{l,k}^{H}\mathbf{w}_{l,k}\\}=0,\forall
k\in\mathcal{K}$ (13b)

Then, we define problem $\mathscr{P}^{(pri)}$ which is a MI-SOCP:

$\displaystyle\mathscr{P}^{(ref)}:$
$\displaystyle\min\hat{F}(\Phi)=F(\Phi)+\frac{\zeta}{L\cdot
K}\sum\limits_{l=1}^{L}\sum\limits_{k=1}^{K}b_{l,k}$ (14a) $\displaystyle
s.t.\quad$ $\displaystyle(\ref{5}),(\ref{6})$ (14b)
$\displaystyle\sum\limits_{k=1}^{K}||\mathbf{w}_{l,k}||_{2}^{2}\leq
P_{l}^{MAX},\forall l\in\mathcal{L}$ (14c)
$\displaystyle\sum\limits_{k=1}^{K}b_{l,k}\leq a_{l}C_{l},\forall
l\in\mathcal{L}$ (14d) $\displaystyle||\mathbf{w}_{l,k}||_{2}\leq
b_{l,k}\sqrt{P_{l}^{MAX}},\forall l\in\mathcal{L},\forall k\in\mathcal{K}$
(14e) $\displaystyle a_{l}\in\\{0,1\\},b_{l,k}\in\\{0,1\\},\forall
l\in\mathcal{L},\forall k\in\mathcal{K}$ (14f)

where $\zeta$ is a constant with $\zeta\rightarrow 0$.

Let the parameter tuples
$\Phi^{(pri)}=\\{a_{l}^{(pri)},b_{l,k}^{(pri)},\mathbf{w}_{l,k}^{(pri)},\linebreak\forall
l\in\mathcal{L},\forall k\in\mathcal{K}\\}$ and
$\Phi^{(ref)}=\\{a_{l}^{(ref)},b_{l,k}^{(ref)},\mathbf{w}_{l,k}^{(ref)},\forall
l\in\mathcal{L},\linebreak\forall k\in\mathcal{K}\\}$ denote optimal solutions
of problem $\mathscr{P}^{(pri)}$ and problem $\mathscr{P}^{(ref)}$,
respectively. In what follows, we will reveals the relationship between
problem $\mathscr{P}^{(pri)}$ and problem $\mathscr{P}^{(ref)}$.

$\mathbf{Lemma1:}$ $\Phi^{(ref)}$ is a feasible solution of problem
$\mathscr{P}^{(pri)}$. In addition, $\Phi^{(pri)}$ is feasible in problem
$\mathscr{P}^{(ref)}$.

$\mathbf{Proof:}$ Please refer to the Appendix A.

$\mathbf{Theorem1:}$ $\Phi^{(ref)}$ is a good approximate solution of problem
$\mathscr{P}^{(pri)}$. More specifically, we have

$\displaystyle 0\leq F(\Phi^{(ref)})-F(\Phi^{(pri)})\leq\zeta.$ (15)

$\mathbf{Proof:}$ Please refer to the Appendix B.

##  IV The Low Complexity Inflation Algorithm

In recent years, some commercial software packages such as CPLEX[20] and
MOSEK[21] have been adopted to determine the optimal solution of MI-SOCP via
the branch-and-cut method[22]. However, the computational complexity may be
prohibitive for densely deployed scenarios in practice. Therefore, in this
section, we adopt a low complexity inflation algorithm introduced by [23] to
obtain the suboptimal solution of problem $\mathscr{P}^{(ref)}$ based on the
continuous relaxation of this problem.

By relaxing the integer constraints such as Eqs.(14f), we could transform a
MI-SOCP into a SOCP, which provides a local lower bound on the optimal
objective value of the original problem[19]. Then, the continuous relaxation
of problem $\mathscr{P}^{(ref)}$ can be expressed as:

$\displaystyle\mathscr{P}^{(con)}:$ $\displaystyle\min\hat{F}(\Phi)$ (16a)
$\displaystyle s.t.\quad$
$\displaystyle(\ref{5}),(\ref{6}),(\ref{ref1})-(\ref{ref2})$ (16b)
$\displaystyle a_{l}\in[0,1],b_{l,k}\in[0,1],\forall l\in\mathcal{L},\forall
k\in\mathcal{K}$ (16c)

We denote
$\Phi^{(con)}=\\{a_{l}^{(con)},b_{l,k}^{(con)},\mathbf{w}_{l,k}^{(con)},\forall
l\in\mathcal{L},\forall k\in\mathcal{K}\\}$ as the optimal solution of problem
$\mathscr{P}^{(con)}$.

In the following, we will describe the low complexity inflation algorithm. Let
the parameter tuple $\\{a_{l}^{(n)},b_{l,k}^{(n)},\forall
l\in\mathcal{L},\forall k\in\mathcal{K}\\}$ and $\hat{F}^{(n)}$ denote the
network state and optimal objective value in the $n$th iteration,
respectively. The initialization of the algorithm is configured as
$a_{l}^{(0)}=0,b_{l,k}^{(0)}=0,\forall l\in\mathcal{L},\forall
k\in\mathcal{K}$, and
$\hat{F}^{(0)}=\sum\limits_{l=1}^{L}(P_{l}^{CMS}+\frac{1}{\eta_{l}}P_{l}^{MAX})+\zeta$,
which means that the network is shut down at the beginning of iteration and
there is no $\hat{F}^{(n)},n=1,2,\ldots$ cloud be larger than $\hat{F}^{(0)}$.
In addition, $\mathcal{U}^{(n)}$ denotes the indeterminate RRH-MU pair set
with $\mathcal{U}^{(0)}=\\{(l,k)|\forall l\in\mathcal{L},\forall
k\in\mathcal{K}\\}$. We gradually set one of zero-valued variables in
$\\{b_{l,k}^{(n-1)},\forall(l,k)\in\mathcal{U}^{(n-1)}\\}$ to one in the $n$th
iteration.

It is obvious that how to select the proper RRH-MU pair in the set
$\mathcal{U}^{(n-1)}$ is critical for the performance of the inflation
algorithm. Jointly considering the contribution to the network performance and
the fronthaul limitation of each RRH, the priority level of each pair, denoted
as $\alpha_{l,k}$, can be defined as:

$\displaystyle\alpha_{l,k}=\frac{|\mathbf{h}_{l,k}^{H}\mathbf{w}_{l,k}^{(con)}|^{2}}{\sum\limits_{i\neq
k}|\mathbf{h}_{l,i}^{H}\mathbf{w}_{l,k}^{(con)}|^{2}}\cdot\frac{C_{l}}{\sum\limits_{j=1}^{L}C_{j}},\forall(l,k)\in\mathcal{U}^{(0)}$
(17)

where $|\mathbf{h}_{l,k}^{H}\mathbf{w}_{l,k}^{(con)}|^{2}$ represents the
useful signal power, and $\sum\limits_{i\neq
k}|\mathbf{h}_{l,i}^{H}\mathbf{w}_{l,k}^{(con)}|^{2}$ is the interference to
other MUs. According to the previous assumption, problem $\mathscr{P}^{(con)}$
is feasible with various QoS requirements, therefore, the priority level of
each pair can be always obtained by calculating Eq.(17). In $n$th iteration,
the $(l^{*},k^{*})$ denotes the RRH-MU pair with largest priority level in
$\mathcal{U}^{(n-1)}$, and we will set $b_{l^{*},j^{*}}^{(n)}=1$ and
$a_{l^{*}}^{(n)}=1$. Then, we will remove $(l^{*},j^{*})$ from
$\mathcal{U}^{(n-1)}$. Moreover, if the number of MUs served by RRH $l^{*}$
reaches the capacity, all the RRH-MU pairs except $(l^{*},k^{*})$ will be
removed from $\mathcal{U}^{(n-1)}$.

When parameter tuple $\\{a_{l}^{(n)},b_{l,k}^{(n)},\forall
k\in\mathcal{K},\forall l\in\mathcal{L}\\}$ in the $n$th iteration is fixed,
the MI-SOCP $\mathscr{P}^{(ref)}$ is transformed into a SOCP, which can be
expressed as:

$\displaystyle\mathscr{P}^{(n)}:$ $\displaystyle\min\hat{F}(\Phi)$ (18a)
$\displaystyle s.t.\quad$ $\displaystyle(\ref{5}),(\ref{6})$ (18b)
$\displaystyle\sum\limits_{k=1}^{K}||\mathbf{w}_{l,k}||_{2}^{2}\leq
a_{l}^{n}P_{l}^{MAX},\forall l\in\mathcal{L}$ (18c)
$\displaystyle\mathbf{w}_{l,k}=b_{l,k}^{n}\mathbf{w}_{l,k},\forall
l\in\mathcal{L},\forall k\in\mathcal{K}$ (18d)

If problem $\mathscr{P}^{(n)}$ is infeasible, we will set
$\hat{F}^{(n)}=\hat{F}^{(0)}$ and proceed to the next iteration. If problem
$\mathscr{P}^{(n)}$ is feasible and $\hat{F}^{(n)}>\hat{F}^{(n-1)}$, we will
stop and go back to the previous iteration, i.e., we set
$\hat{F}^{(n)}=\hat{F}^{(n-1)}$, $b_{l^{*},k^{*}}^{(n)}=0$ and
$a_{l^{*}}^{(n)}=\max\\{b_{l^{*},k},\forall k\in\mathcal{K}\\}$. Otherwise, we
proceed to the next iteration. The pseudo-code of the low complexity inflation
algorithm is presented in Algorithm $1$.

Algorithm 1 Low Complexity Inflation Algorithm

1: Initialization: $\mathcal{U}^{(0)}=\\{(l,k)|,\forall
l\in\mathcal{L},\forall k\in\mathcal{K}\\}$;
$a_{l}^{(0)}=0,b_{l,k}^{(0)}=0,\alpha_{l,k}\forall(l,k)\in\mathcal{U}^{(0)}$;
a sufficiently large $\hat{F}^{(0)}$; $n=1$.

2: while $\mathcal{U}^{n-1}$ is non-empty set do

3: Compute
$(l^{*},k^{*})=\operatorname*{argmax}\limits_{(l,k)\in\mathcal{U}^{(n-1)}}\alpha_{l,k}$.

4: Set $b_{l,k}^{(n)}=b_{l,k}^{(n-1)},a_{l}^{(n)}=a_{l}^{n-1}$.

5: Set $b_{l^{*},k^{*}}^{(n)}=1,a_{l^{*}}^{(n)}=1$.

6: Update $\mathcal{U}^{(n)}=\mathcal{U}^{(n-1)}\setminus\\{(l^{*},k^{*})\\}$.

7: if $\sum\limits_{j=1}^{K}b_{l^{*},k}^{(n)}=C_{l^{*}}$ then

8:
$\mathcal{U}^{(n)}=\mathcal{U}^{(n)}\setminus\\{(l^{*},k)|(l^{*},k)\in\mathcal{U}^{(n)},\forall
k\in\mathcal{K},k\neq k^{*}\\}$.

9: end if

10: if Problem $\mathscr{P}^{(n)}$ is infeasible. then

11: Set $\hat{F}^{(n)}=\hat{F}^{(0)}$.

12: else

13: if $\hat{F}^{(n)}>\hat{F}^{(n-1)}$ then

14: Set $\hat{F}^{(n)}=\hat{F}^{(n-1)}$.

15: Set
$b_{l^{*},k^{*}}^{(n)}=0,a_{l^{*}}^{(n)}=\max\limits_{k\in\mathcal{K}}b_{l^{*},k}$.

16: end if

17: end if

18: The iteration number $n=n+1$.

19: end while

The computational complexity of the inflation algorithm in Algorithm $1$
mainly consists in solving $(\sum\limits_{l\in\mathcal{L}}C_{l}+1)$ SOCP
problems, since a RRH can not serve any MU if the fronthaul reaches its
limitation. For each of the SOCP problem $\mathscr{P}^{(n)}$, the
computational complexity is
$\mathcal{O}((K\sum\limits_{l\in\mathcal{L}}N_{l})^{3.5})$ by using the
interior-point method[24]. Therefore, Algorithm $1$ is a polynomial time
algorithm and it converges in finite iterations.

##  V Simulation Results and Discussions

TABLE I: Simulation Parameters | Parameter
---
| Value
---
Path-loss at distance $d$ (km) |  $148.1+37.6log_{10}(d)$ dB
Standard deviation of log-norm shadowing |  $8$ dB
Small-scale fading distribution | $\mathcal{CN}(\mathbf{0,I})$
Noise power density |  $-174$ dBm/Hz
System bandwidth |  $10$MHz
Maximum transmit power of RRH $P_{l}^{MAX}$ |  $10$W
Transmitting antenna power gain |  $9$ dBi
Constant $\zeta$ | $10^{-3}$

In this section, we simulate the performance of Algorithm $1$. For comparison,
the scheme adopted by LTE-A is used as a benchmark, where the network is
divided into non-overlapping clusters according to the locations of RRHs and
MUs, and the RRHs in each cluster coordinately serve all the users within the
coverage area [25]. We consider a network comprising $L=10$ $2$-antenna RRHs
and $K=10$ or $K=15$ single-antenna MUs uniformly distributed in the square
region $[-1500\;1500]\times[-1500\;1500]$ meters. The channel model is shown
in Table I. The numerical results are averaged over $100$ randomly generated
network realization. According to [12], we set $P_{l}^{CIR}=6.8W$,
$P_{l}^{SLP}=4.3W$ and $\eta_{l}=0.25$.

Figure 2: Network power consumption versus Target SINR. Figure 3: Network
power consumption versus Fronthaul capacity. Figure 4: Number of active RRHs
versus Target SINR. Figure 5: Number of active RRHs versus Fronthaul capacity.

Fig.2 shows the variation in network power consumption versus target SINR when
fronthaul capacity equals $6$. We observe from this figure that the
performance of Algorithm $1$ is better than that of LTE-A beamforming
algorithm and the gap between the two algorithms increases with the target
SINR, since the RRHs serving each MU in Algorithm $1$ are overlapping and
adopt CoMP to jointly transmit data, which converts the ICI into useful
signals. In addition, it is easy to understand that the network consumption
with $K=10$ is lower than that with $K=15$, since higher transmitpower and
more active RRHs are required to serve the excess MUs.

Fig.3 depicts the variation in network power consumption versus fronthaul
capacity when target SINR is $6dB$. As we can see, the network power
consumption decreases with the fronthaul capacity. This is because the RRH can
serve more MUS and more diversity gain can be obtained. Moreover, the curves
descends slowly when each RRH can serve more MUs, as the fronthaul capacity is
not the primary limitation on the network performance.

Fig.4 and Fig.5 show the variation in number of active RRHs versus target SINR
and fronthaul capacity, respectively. The fronthaul capacity and target SINR
are $6$ and $6dB$ in the two figures, respectively. These two figures
illustrate that more RRHs have to be active with high SINR requirement or
small fronthaul capacity, and more RRHs can be in sleep if there are fewer MUs
when we adopt Algorithm $1$. In addition, Fig.4 shows that all RRHs are active
in LTE-A beamforming algorithm, even if the number of MUs is small and the
target SINR is low.

##  VI Conclusion

In this paper, we proposed a joint user association and downlink beamforming
scheme for C-RANs with limited fronthaul to minimize the network power
consumption. To be more specific, the design problem is formulated as a mixed
integer nonlinear program at first. We then transform the problem into a MI-
SOCP which is a SOCP when the integer variables are fixed. At last, we adopt a
low complexity inflation algorithm to obtain the suboptimal solution.
According to the simulation results, it has been observed that the adopted
algorithm can effectively lower the network consumption compared with the
scheme used by LTE-A. Since too many users served by the network will incurs
problem $\mathscr{P}^{(pri)}$ is infeasible, our future work is in progress to
consider the user admission control in the proposed scheme.

## APPENDIX A

## Proof OF Lemma 1

We can easily observe that the feasible region of problem
$\mathscr{P}^{(pri)}$ is a subset of that of problem $\mathscr{P}^{(ref)}$.
Hence, $\Phi^{(pri)}$ is a feasible solution of problem $\mathscr{P}^{(ref)}$.
In addition, $\Phi^{(ref)}$ satisfies the constraints (12b)-(12d) and (12g).
Accordingly, in the following, we will prove $\Phi^{(ref)}$ satisfies the
constraints (12e) and (12f).

In case that $a_{l}^{(ref)}=0$, the equalities $\\{b_{l,k}^{(ref)}=0,\forall
k\in\mathcal{K}\\}$ hold according to constraints (14d). In turn, if
$\\{b_{l,k}^{(ref)}=0,\forall k\in\mathcal{K}\\}$, we can easily know that
both $a_{l}=0$ and $a_{l}=1$ are feasible. If $a_{l}^{(ref)}=1$, we have
$\hat{F}(\Phi^{(ref)})|_{a_{l}^{(ref)}=1}>\hat{F}(\Phi^{(ref)})|_{a_{l}=0}$,
which is in contradiction to the assumption that $\Phi^{(ref)}$ is the optimal
solution of problem $\mathscr{P}^{(ref)}$. Hence, we have $a_{l}^{(ref)}=0$.
Moreover, through Eqs.(14e), we have $\mathbf{w}_{l,k}^{(ref)}=\mathbf{0}$ if
$b_{l,k}^{(ref)}=0$. We adopt similar contradicting argument to prove that
$b_{l,k}^{(ref)}$ equals $0$ if $\mathbf{w}_{l,k}^{(ref)}=\mathbf{0}$ as well.
Therefore, $\Phi^{(ref)}$ is feasible in problem $\mathscr{P}^{(pri)}$.

The proof is completed.

## APPENDIX B

## Proof OF Theorem 1

With the $\mathbf{Lemma1}$, we have

$\displaystyle\hat{F}(\Phi^{(ref)})-\hat{F}(\Phi^{(pri)})\leq 0$ (19)
$\displaystyle(F(\Phi^{(ref)})-F(\Phi^{(pri)}))+\frac{\zeta}{L\cdot
K}\sum\limits_{l=1}^{L}\sum\limits_{k=1}^{K}(b_{l,k}^{(ref)}-b_{l,k}^{(pri)})\leq
0$ $\displaystyle F(\Phi^{(ref)})-F(\Phi^{(pri)})\leq\frac{\zeta}{L\cdot
K}\sum\limits_{l=1}^{L}\sum\limits_{k=1}^{K}(b_{l,k}^{(pri)}-b_{l,k}^{(ref)})$

$\Phi^{(pri)}$ is a optimal solution of problem $\mathscr{P}^{(pri)}$, so

$\displaystyle 0\leq F(\Phi^{(ref)})-F(\Phi^{(pri)})\leq\frac{\zeta}{L\cdot
K}\sum\limits_{l=1}^{L}\sum\limits_{k=1}^{K}(b_{l,k}^{(pri)}-b_{l,k}^{(ref)})$
(20) $\displaystyle\leq\frac{\zeta}{L\cdot K}\cdot L\cdot K=\zeta$

The proof is completed.

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Derek Chauvin |  police neg. actor |  N/A |  blacks pos. target, police neg. actor |  blacks neg. actor
Thugs |  police neg. actor |  antifa neg. actor |  racism neg. actor, police neg. actor |  antifa neg. actor, BLM neg. actor
David Dorn |  blacks pos. target |  police pos. target |  blacks pos. target |  police pos. target
Lives |  blacks pos. target |  community pos. target, police pos. target |  blacks pos. target |  police pos. target
Donald Trump |  government neg. actor |  republicans pos. actor |  republicans neg. actor |  republicans pos. actor
They |  blacks pos. target |  democrats neg. actor, antifa neg. actor |  whites neg. actor, racism neg. actor |  democrats neg. actor, antifa neg. actor

Table 18: Examples of literal entity to abstract entity mapping. Sometimes
pro-BlackLM and pro-BlueLM use the same phrase to address different entities
and/or perspectives. The lexicon of the literal entities in this table can be
found in Table 21.

###  E.3 Discourse of the movements

Table 19 shows the discourse of the pro-BlackLM and pro-BlueLM camps using
perspectives and moral foundations.

To analyze the discourse of the movements we identify the high PMI
perspectives associated with each campaign using the same formula as described
in Appendix A.

For the identification of moral foundations in the tweets, we train a RoBERTa-
based classifier on out-of-domain data. We take the Twitter moral foundation
dataset proposed by Johnson and Goldwasser, 2018. In this dataset, 2k tweets
are annotated for one or more moral foundations. We encode the tweets using
our pre-trained RoBERTa-tapt ([CLS] token of the last hidden layer) and use a
fully connected layer to identify the moral foundation labels. This is a
multi-label classification task. Hence, we use BCE loss and AdamW optimizer
for optimization. We stop training the model if the validation accuracy does
not improve for three consecutive epochs. The moral foundation classification
F1 score on a held-out validation set was $75.19\%$. We infer the moral
foundation labels in the #BLM corpus using this classifier.

###  E.4 Temporal analysis

We calculate the percentage of tweets where a specific perspective is used on
a given day. For one type of perspective such as negative actors we calculate
the percentage taking tweets only mentioning negative actors in a particular
stance. The temporal trends are shown in Figure 8.

(a) Portrayal of entities as positive targets in pro-BlueLM over time.

(b) Portrayal of entities as positive actors in pro-BlueLM over time.

(c) Portrayal of entities as positive targets in pro-BlackLM over time.

(d) Portrayal of entities as negative actors in pro-BlackLM over time.

Figure 8: Temporal trends in all camps for different types of entity roles
identified using our model in the direct supervision setup. Note that, mostly
BLM is portrayed as positive actors in pro-BlackLM and the portrayal of other
entities as positive actors is significantly lower in this camp. Hence, the
trend for positive actors in pro-BlackLM is not shown in this analysis.

Discourse in Pro-#BlackLivesmatter | Discourse in Pro-#BlueLivesmatter
---|---
High PMI Perspectives |  MFs in Context |  Other Pers. in Context |  High PMI Perspectives |  MFs in Context |  Other Pers. in Context
police neg. actor |  fairness/cheating, authority/subversion |  blacks pos. target, blm movement pos. actor |  police pos. actor |  authority/subversion, loyalty/betrayal |  police pos. target, antifa neg. actor
blm movement pos. actor |  loyalty/betrayal, fairness/cheating |  blacks pos. target, racism neg. actor |  blm movement neg. actor |  authority/subversion, loyalty/betrayal |  antifa neg. actor, democrats neg. actor
racism/racists neg. actor |  fairness/cheating, loyalty/betrayal |  blacks pos. target, whites neg. actor |  antifa neg. actor |  authority/subversion, loyalty/betrayal |  blm movement neg. actor, democrats neg. actor
blacks pos. target |  fairness/cheating, loyalty/betrayal |  police neg. actor, blm movement pos. actor |  democrats neg. actor |  authority/subversion, loyalty/betrayal |  community pos. target, antifa neg. actor
whites neg. actor |  fairness/cheating, auth./subversion |  blacks pos. target, racism neg. actor |  community pos. target |  authority/subversion, loyalty/betrayal |  democrats neg. actor, antifa neg. actor
blm movement pos. target |  loyalty/betrayal, authority/subversion |  blacks pos. target, blm pos. actor |  police pos. target |  loyalty/betrayal, authority/subversion |  police pos. actor, antifa neg. actor
government neg. actor |  authority/subversion, loyalty/betrayal |  blacks pos. target, racism neg. actor |  republicans pos. actor |  loyalty/betrayal, fairness/cheating |  democrats neg. actor, antifa neg. actor
petition pos. target |  fairness/cheating, loyalty/betrayal |  blacks pos. target, police neg. actor |  |  |

Table 19: Discourse of movements explained with messaging choices and Moral
Foundations (MFs). Moral Foundation care/harm was used in all of the cases by
both sides. Hence, it is removed from the table. This table is created using
the predictions by our model in the direct supervision setup.

Literal Entities |  Example tweets in pro-#BlackLivesMatter |  Example tweets in pro-#BlueLivesMatter
---|---|---
Black American Victims |  Arrest the cops that killed #BreonnaTaylor! #BlackLivesMatter |  Thanks for the body cam. It is obvious to anyone, but #Liberals that #RayshardBrooks was violently resisting arrest. #Atlanta #AllLivesMatter
Derek Chauvin |  Derek Chauvin, the murderer policeman who killed George Floyd, would not have been arrested had it not been for the uprising in Minnesota. |  N/A
Thugs |  The behaviour of the American police is absolutely revolting. Racist yobs with military grade weapons empowered by a fascist president. Why should we automatically be expected to "respect" these thugs? Respect should be earned. #BlackLivesMatter |  BlackLivesMatter is the black version of the white trash Antifa thugs. They could care less about black lives, they’re thugs and trash. #RememberDavidDorn #AllLivesMatter
David Dorn |  This isn’t political, all #BlackLivesMatter. David Dorn, #AhmaudArbery, #GeorgeFloyd #BreonnaTaylor. Stop turning this into something else for a reaction. |  Where are you all when #DavidDorn was shot in the head by #BlackLivesMatter rioters and #AntifaTerrorist ? Disgusting silence.
Lives |  The countless black lives that have been lost in the U.S. due to acts of #policebrutality have been tolerated & condoned by the government enough to be considered Crimes against humanity imo. I think America should be held accountable for this on an Int. level. #BlackLivesMatter |  The Police should leave. It’s not worth their lives for this BS anymore. #BlueLivesMatter
Donald Trump |  @realDonaldTrump should get accustomed to being surrounded by fencing & prison guards. #protests #BLM #BunkerBoyTrump #Bunkerbaby #TrumpIsAnIdiot #TrumpIsACoward #DCProtests #ArrestTrump #BlackLivesMatter |  @realDonaldTrump has asked for Unity from Day 1 unlike the DemoKKKrats And @TheDemocrats and their leaders are the one calling for violence, not Trump
They |  HOW ARE THESE CASES NOT MAKING HEADLINES?! They are being murdered and nothing is being done about it. #BlackLivesMatter |  And this is what they want to reform!??! Morons cannot win!

Table 20: Examples of tweets where the pro-BlackLM and pro-BlueLM use the same
phrases to address different entities and/or perspectives. The literal
entities in the example tweets are bolded and underlined. The lexicon of the
literal entities in this table can be found in Table 21. “N/A” means the
entity is not mentioned frequently in a particular campaign.

Literal Entities |  Lexicon |  Description
---|---|---
Black American Victims |  # georgefloyd, george floyd, georgefloyd, # breonnataylor, breonna taylor, breonna, # georgeflyod, george, # georgefloyd #, # rayshardbrooks, rayshard brooks, rayshardbrooks, # ahmaudarbery, ahmaud arbery, robert fuller, samuel dubose, sandra bland, walter scott |  Names of the Black American persons who were killed.
Derek Chauvin |  derek chauvin, chauvin |  The police officer who killed George Floyd.
Thugs |  thugs, these thugs, the thugs |  A negative term to address a person or a group of persons.
David Dorn |  # daviddorn, david dorn |  Black Police officer who was killed during the George Floyd protests.
Lives |  their lives, lives, the lives, our lives, life, my life, his life, her life, their life, the life, a life |  Self-explanatory.
Donald Trump |  @ realdonaldtrump, @realdonaldtrump, realdonaldtrump, trump supporters, trump, # trump, president trump, donald trump, trump2020 |  45th U.S. President who is from the Republican Party.
They |  they |  The Pronoun they.

Table 21: Description of the literal entities studied in the qualitative
evaluations.

##  Appendix F Literature Review

Understanding the discourse of the #BLM movement: The discourse of the #BLM
movement is understudied in NLP. One of the early works Keith et al. (2017)
proposed a distantly supervised EM-based approach for Identifying the names of
civilians in news articles killed by police. Later Nguyen and Nguyen, 2018
incorporated Deep Learning methods in this task. Both of the works study only
named entities and lack the study of how entities are addressed using other
words (e.g., “Thugs”, “Heroes”) or simply using Pronouns. Recently, Ziems and
Yang, 2021 introduced the Police Violence Frame Corpus of 82k news articles
covering 7k police killings and they studied entity-centric framing where the
frame towards the victim of police violence is defined as the age, gender,
race, criminality, mental illness, and attacking/fleeing/unarmed status of the
victim. All of these works are done on more formal texts such as news articles
and do not address how different entities are addressed and what the sentiment
and perspectives expressed towards them are.

Recently, a shared task of identifying BLM-centric events from large
unstructured data sources has been proposed Giorgi et al. (2021b) and Giorgi
et al., 2022 introduced a large tweet corpus on #BLM paving the way for more
studies including ours in this area. In this paper, we unify the
identification of co-referenced entities, perspectives toward them in terms of
the moral roles assigned to them, and stance prediction on # the BLM movement
on highly noisy social media texts.

The discourse of the #BlackLivesMatter movement is mostly studied in social
science and computational social science literature. For example, one line of
research studied the social dynamics, ties, and in-group commitments that
influence the mobilization and formation of the movement. For example,
Williamson et al., 2018 found that #BLM protests are more likely to occur in
localities where more Black people have previously been killed by police, Hong
and Peoples, 2021 studied how social ties influence participation in the
movement and Peng et al., 2019 discovered that people who join the movement in
response to a real-life event are more committed to it compared to the other
types. Another line of research studied the power of social media in
mobilizing the movement (Mundt et al., 2018; Freelon et al., 2018; Grill,
2021). Some works have studied what properties accelerate mobilization. For
example, Casas and Williams, 2019 studied the power of images in mobilizing
the movement and Keib et al., 2018 studied what types of tweets are retweeted
more related to the movement. De Choudhury et al., 2016 studied temporal
analysis of the #BLM movement based on social media, engagement of people from
different demographics, and their correlation with textual features. One major
drawback of these studies is that many of them often involve expensive human
studies.

There have been several attempts to understand and analyze the narrative in
online text in response to the murders of Black individuals in social science.
For example, Eichstaedt et al., 2021 and Field et al., 2022 studied the
emotions expressed in the online text in response to murders of Black persons.
Another line of research studied the framing (Stewart et al., 2017; Ince et
al., 2017), rhetoric functions Wilkins et al. (2019), participation and
attention to topics Twyman et al. (2017), and narrative agency Yang (2016)
related to the movement. Some other works studied hashtag-based analysis of
the #BlackLivesMatter movement Blevins et al. (2019) and the divergence of the
counter-movements (e.g., the #AllLivesMatter movement) Gallagher et al.
(2018).

The existing studies in computational social science on narrative
understanding mostly rely on hashtag or lexicon-based analysis of the
movements, hence, they fail to capture the nuances in the messaging choices
and often cannot differentiate between movements and counter-movements when
representative hashtag from one movement is “hijacked” Gallagher et al. (2018)
by the supporters of the counter-movement.

In contrast to these existing studies in CSS and NLP on #BLM, in this paper,
we propose a holistic technical framework for characterizing such social
movements on online media by explicitly modeling the perspectives of different
camps.

Perspective Analysis and Stance Detection: Revealing perspectives in complex
and deceptive discussions is an important part of discourse analysis and it
has been studied in different settings and variations in recent studies. For
example, Thomas et al., 2006 and Burfoot et al., 2011 attempted to identify
stances in congressional floor-debate transcripts (against or in support of
proposed legislation). Another line of research studied stances in online
debate forums where the stance (pro vs. con) of a speaker on a specific issue
is predicted (Somasundaran and Wiebe, 2010; Walker et al., 2012; Hasan and Ng,
2013, 2014; Sridhar et al., 2015; Sun et al., 2018; Xu et al., 2018; Li et
al., 2018). Mohammad et al., 2016 introduced a shared task of predicting
stances in microblogs such as tweets where the task is to identify the stance
in a tweet with respect to a given target (e.g. entity, issue, and so on).
Consequently, more recent works have studied political stances in politically
controversial issues (Johnson and Goldwasser, 2016b, a; Ebrahimi et al., 2016;
Augenstein et al., 2016). Recently, another shared task has been proposed
Pomerleau and Rao (2017) where the task of fake-news detection is studied from
a stance detection perspective. In this task, a headline and a body text are
given - either from the same news article or from two different articles. The
task is to determine if the stance of the body text relative to the claim made
in the headline is in agreement, disagreement, discussion, or irrelevant.

Identification of stance is studied using various approaches Cignarella et al.
(2020). For example, Burfoot et al., 2011 and Sridhar et al. (2015) studied
stance detection using a collective classification approach. Sridhar et al.,
2015 and Hasan and Ng, 2014 studied not only stances but also the reasoning
behind stances and the disagreements among them. In a related work in this
line, Somasundaran and Wiebe, 2010 used lexicon-based features for detecting
“arguing” opinions, and supervised systems using sentiment and arguing
opinions were developed for stance classification. Sun et al., 2018 proposed a
neural model to learn mutual attention between the document and other
linguistic factors. Thomas et al., 2006 leveraged inter-document relationships
and Walker et al., 2012 leveraged the dialogic structure of the debates in
terms of agreement relations between speakers for stance detection. Another
line of research is built on relational learning-based methods for stance
detection where various types of contextualizing and relational properties are
explicitly modeled and incorporated in stance detection. For example, Johnson
and Goldwasser, 2016b proposed a relational learning framework incorporating
framing and temporal activity patterns, and Ebrahimi et al., 2016 proposed a
relational model incorporating the friendship networks of the authors. The
main drawback of such relational learning approaches is that all possible
solutions to the problem are explored and the one with the best gain is
returned, as a result, the inference tree becomes really large and
computationally expensive for a large amount of data.

For the analysis of perspectives in polarized topics, different socio-
linguistic theories have been used in the literature. For example, the Moral
Foundation Theory (MFT) (Haidt and Joseph, 2004; Haidt and Graham, 2007) and
framing analysis (Entman, 1993; Chong and Druckman, 2007; Boydstun et al.,
2014). Framing is referred to the approach of communication by focusing on
certain aspects of a story in order to bias the readers toward certain
stances. Previous studies used framing to understand political perspectives
and communication strategies by biased news sources and social media users
(Tsur et al., 2015; Baumer et al., 2015; Card et al., 2015; Field et al.,
2018; Demszky et al., 2019; Fan et al., 2019; Roy and Goldwasser, 2020). The
Moral Foundation Theory is also widely used for analyzing perspectives
(Dehghani et al., 2014; Fulgoni et al., 2016; Brady et al., 2017; Hoover et
al., 2020; Roy and Goldwasser, 2021). However, Moral Foundations are widely
used to understand sentence level perspectives. Roy et al. (2021) introduced
Morality Frames which is a knowledge representation framework for capturing
entity-centric moral sentiments. Because of the expressivity of entity-centric
moral foundations and their strong correlation with stances, in this paper, we
use morality frames for modeling perspectives in the #BlackLivesMatter and
#BlueLivesMatter movements.

Data Augmentation Approaches in NLP: Data augmentation refers to the technique
where additional and ideally diverse data are generated without explicitly
collecting new data. Data augmentation techniques can be useful in a low-
resource setting where obtaining dataset for training models are difficult.
Annotated data for nuanced perspectives are difficult to get as annotating
them pertains to specialized knowledge. Hence, automatic data augmentation is
desirable.

With the recent advances in deep learning-based models, different data
augmentation techniques have been proposed. For example, Kobayashi, 2018
proposed RNN-based, Kumar et al., 2019 trained sequence to sequence models,
and Yang et al., 2020 and Ng et al., 2020 proposed pretrained transformer-
based approaches for data augmentation. In most of these approaches, existing
text data is modified to augment new data. In contrast to these studies,
Anaby-Tavor et al., 2020 and Quteineh et al., 2020 directly estimated the text
generation process using GPT-2 Radford et al. (2019) rather than modifying an
existing example to generate new data.

With the recent advances of transformer-based pretrained auto-regressive Large
Language Models (LLMs) such as GPT-3 Brown et al. (2020), a new direction for
data augmentation has opened up where controlled text generation (Iyyer et
al., 2018; Kumar et al., 2020; Liu et al., 2020) is feasible by prompting
these pretrained LLMs. These LLMs are trained on a huge corpus of web crawls,
hence, have the capability to generate human-like text. One recent work by Liu
et al., 2022 combined the generative power of these LLMs and the evaluation
power of humans and proposed a human-LLM interaction loop to generate datasets
for Natural Language Inference (NLI) tasks. Inspired by these recent advances
in data augmentation using LLMs, in this paper, we propose to augment few-shot
training data using LLMs by prompting them in a structured way. This augmented
dataset is later used to bootstrap our proposed model.