# A New Noncoherent Gaussian Signaling Scheme for Low Probability of Detection
Communications

Yuma Katsuki, , Giuseppe Thadeu Freitas de Abreu, , Koji Ishibashi, , and
Naoki Ishikawa Y. Katsuki and N. Ishikawa are with the Faculty of Engineering,
Yokohama National University, 240-8501 Kanagawa, Japan (e-mail: ishikawa-
naoki-fr@ynu.ac.jp). G. Abreu is with the School of Computer Science and
Engineering, Jacobs University Bremen, 28759 Bremen, Germany. K. Ishibashi is
with the Advanced Wireless and Communication Research Center, The University
of Electro-Communications, 182-8585 Tokyo, Japana. This work was supported in
part by the Japan Science and Technology Agency, Strategic International
Collaborative Research Program (JST SICORP), Japan, under Grant JPMJSC20C1.

###### Abstract

We propose a novel, Gaussian signaling mechanism for low probability of
detection (LPD) communication systems with either single or multiple antennas.
The new scheme is designed to allow the noncoherent detection of Gaussian-
distributed signals, enabling LPD communications using signals that follow the
complex Gaussian distribution in the time and frequency domains. It is
demonstrated via simulations that the proposed scheme achieves better
performance than a comparable conventional scheme over the entire SNR region,
with the advantage becoming more significant in scenarios with lower overhead.

###### Index Terms:

Multiple-input multiple-output (MIMO), low probability of detection (LPD),
Gaussian signaling.

##  I Introduction

Low probability of detection (LPD) systems are a new type of covert wireless
communication technology [1] that is expected to play a key role in
applications such as stealth IoT and military networks [2], which require high
levels of security. In the LPD communication systems, a legitimate transmitter
attempts to communicate with a legitimate receiver without being detected by
an illegitimate adversary.

Low-power zero-mean complex-valued Gaussian signals are ideal for LPD, as it
has been recently shown [3] that such signals minimize the probability of
detection by illegitimate adversaries. This is unlike most current wireless
systems, many of which rely on orthogonal frequency division multiplexing
(OFDM) waveforms. This is because although time-domain OFDM signals also
follow Gaussian distributions, the discrete nature of the digital
constellations utilized in such systems is visible in the frequency domain, a
feature which can therefore be exploited to detect the presence of
communications. In contrast, in LPD systems, signals must be Gaussian in both
the time and the frequency domains.

The security of LPD communication has been analyzed from an information-
theoretic perspective [4, 5], but under the assumption that perfect channel
state information (CSI) is available at both transmitter and receiver [4, 5],
which in turn implies the exchange of reference signals, thus increasing the
risk of detection by an adversary. This fundamental weakness of existing LDP
methods calls for the design of noncoherent LPD schemes, which on the other
hand is challenging under the optimal LDP Gaussian signaling, and therefore
remains an open issue hindering the practicality of the technology [2].

Against this background, a noncoherent detection scheme for LDP systems
employing Gaussian signals is proposed in this letter, as an enabling
technology for LPD communications. The proposed scheme builds on principles of
differential encoding [6] to construct complex Gaussian reference and data
signals, such that thanks to the differential structure, no periodic reference
signals are required to track CSI. The proposed scheme has advantages over the
representative Gaussian signaling scheme of [7, 8] decoded via the semi-blind
detector [9], which can be considered the state-of-the-art in the area. The
contributions of the article can be summarized as follows.

  * •

We design a new noncoherent detection scheme for LPD communications. The
proposed scheme inserts a reference matrix at the beginning of the
transmission frame and generates differentially-encoded symbols, all of which
follow the ideal Gaussian distribution in the frequency domain, resulting in
Gaussian-distributed time-domain signals. Since Gaussian-distributed signals
make CSI estimation difficult due to their maximally-entropic feature, the
issue is circumvented by the design of a robust noncoherent detector.

  * •

We demonstrate that the proposed scheme achieves better BER performance than
state-of-the-art of LPD methods employing Gaussian signaling [7, 8] with semi-
blind detector [9]. In addition, our analysis indicates that the detection
complexity is reduced by a linear factor while maintaining the same security
level as the ideal Gaussian signaling.

##  II System Model

Consider a multiple-input multiple-output (MIMO) system in which a legitimate
transmitter, Alice, equipped with $M$ transmit antennas, communicates with a
legitimate receiver, Bob, equipped with $N$ receive antennas, in the presence
of an illegitimate adversary, Willie, which tries to detect Alice’s
communications. The narrow-band received signals at Bob is modeled as111We
remark that this system model is readily applicable to MIMO-OFDM scenarios
[6].

$\mathbf{Y}(i)=\mathbf{H}(i)\mathbf{S}(i)+\mathbf{V}(i)\in\mathbb{C}^{N\times
T},$ (1)

for $0<i\leq W$, where $i$ is the transmission index, $W$ is the frame length,
$T$ is the number of time slots, $\mathbf{S}(i)\in\mathbb{C}^{M\times T}$ is a
space-time codeword, and the elements of $\mathbf{H}(i)\in\mathbb{C}^{N\times
M}$ and $\mathbf{V}(i)\in\mathbb{C}^{N\times T}$ follow $\mathcal{CN}(0,1)$
and $\mathcal{CN}(0,\sigma_{v}^{2})$, respectively, with the signal-to-noise
ratio (SNR) is defined as $1/\sigma_{v}^{2}$.

Let $B$ be the number of information bits conveyed by the codeword
$\mathbf{S}(i)\in\mathcal{S}$, where
$\mathcal{S}\triangleq\\{\mathbf{S}_{1},\cdots,\mathbf{S}_{2^{B}}\\}$ is the
codebook. The bit error rate (BER) associated with signals as in (1) depends
on the so-called coding gain [10]

$\displaystyle G\triangleq\min_{p\neq
q\in\\{1,\cdots,2^{B}\\}}\left((\mathbf{S}_{p}-\mathbf{S}_{q})^{\mathrm{H}}(\mathbf{S}_{p}-\mathbf{S}_{q})\right)^{\frac{1}{N}},$
(2)

which represents the minimum Euclidean distance between different space-time
codewords in the codebook.

We emphasize that the main interest of LPD communication with $M\ll N$ is in
uplink scenario, since for massive MIMO systems, downlink physical layer
security schemes such as the one proposed in [11] can generate Gaussian
signals. As a consequence, LPD communications in downlink will be considered
out of the scope of this letter.

##  III Reference State-of-the-Art Schemes

Before we introduce our proposed method, let us briefly revise relevant state-
of-the-art techniques, in particular the conventional Gaussian signaling
scheme of [7, 8] and the conventional semi-blind detector of [9]. Combining
both schemes is a straightforward task, and results in an exclusive Gaussian
signaling scheme that can work for MIMO-OFDM scenarios. Hence, we regard the
combination as a performance baseline reference to our contribution.

###  III-A Conventional Gaussian Signaling Scheme of [7, 8]

Chaos MIMO (C-MIMO) [7] is a MIMO modulation scheme designed with basis on
chaos theory222In our simulations, C-MIMO achieved the best coding gain among
physical layer encryption schemes. Thus, we consider it to be a representative
Gaussian signaling scheme., in which $B=MT$ information bits
$\mathbf{b}=[b_{1}~{}b_{2}~{}\cdots~{}b_{B}]\in\mathbb{B}^{B}$ are mapped onto
complex-valued Gaussian symbols, such that the associated the $M\times T$
space-time codeword is in a form [8]

$\mathbf{S}(i)\triangleq\frac{1}{\sqrt{M}}\\!\left[\begin{array}[]{cccc}s_{1}&s_{M+1}&\cdots&s_{MT-M+1}\\\
s_{2}&s_{M+2}&\cdots&s_{MT-M+2}\\\ \vdots&\vdots&\ddots&\vdots\\\
s_{M}&s_{2M}&\cdots&s_{MT}\end{array}\right]\\!\in\mathbb{C}^{M\times T},$ (3)

where each symbol $s_{k}$ is generated by the Box-Muller’s transform [8]

$s_{k}=\sqrt{-\log\left(c_{k}^{(x)}\right)}\left(\cos\left(2\pi
c_{k}^{(y)}\right)+j\sin\left(2\pi c_{k}^{(y)}\right)\right)$ (4)

and $j$ denotes the imaginary number.

Here, $c_{k}^{(x)}$ and $c_{k}^{(y)}$ are uniform pseudo-random numbers
generated by a shared key, the input bit sequence, and the Bernoulli shift map
transition, which are given by [8]

$\displaystyle
c_{k}^{(x)}=\mathrm{arccos}\left(\mathrm{cos}\left(37\pi\left(\mathrm{Re}[c_{k}]+\mathrm{Im}[c_{k}]\right)\right)\right)/\pi,$
(5a) $\displaystyle
c_{k}^{(y)}=\mathrm{arcsin}\left(\mathrm{sin}\left(43\pi\left(\mathrm{Re}[c_{k}]\\!-\\!\mathrm{Im}[c_{k}]\right)\right)\right)/\pi\\!+\\!1/2,$
(5b)

where we have

$c_{k}=\mathrm{Re}[z_{N_{s}+b_{(k+B/2)~{}\mathrm{mod}~{}B}}]+\mathrm{Im}[z_{N_{s}+b_{(k+B/2+1)~{}\mathrm{mod}~{}B}}].$
(6)

In (6), the chaos sequences are defined by [8]

$\displaystyle\mathrm{Re}[z_{l}]=2\cdot\mathrm{Re}[z_{l-1}]~{}\mathrm{mod}~{}(1-10^{-16}),$
(7a)
$\displaystyle\mathrm{Im}[z_{l}]=2\cdot\mathrm{Im}[z_{l-1}]~{}\mathrm{mod}~{}(1-10^{-16}).$
(7b)

for $l=1,2,\cdots,N_{s},N_{s}+1$, where the transition factor is typically set
to a large number such as $N_{s}=100$ [8], and both sequences are initialized
by $\mathrm{Re}[z_{0}]=\Gamma(\mathrm{Re}[c_{k-1}],b_{k-1})$,
$\mathrm{Im}[z_{0}]=\Gamma(\mathrm{Im}[c_{k-1}],b_{k~{}\mathrm{mod}~{}B})$,
where [8]

$\displaystyle\Gamma(a,b)\triangleq\left\\{\begin{array}[]{ll}a&\text{if}\;(b=0),\\\
1-a&\text{if}\;(b=1~{}\mathrm{and}~{}a>1/2),\\\
a+1/2&\text{if}\;(b=1~{}\mathrm{and}~{}a\leq 1/2),\end{array}\right.$ (11)

where $c_{0}\in\mathbb{C}$ is a pre-shared key that satisfies
$0<\mathrm{Re}[c_{0}]<1$ and $0<\mathrm{Im}[c_{0}]<1$.

We remark that due to the Box-Muller transform described by (4), the resultant
symbol $s_{k}$ follows $\mathcal{CN}(0,1)$.333The open-source implementation
of C-MIMO is available at
https://github.com/ishikawalab/wiphy/blob/master/wiphy/examples/okamoto2012chaos.py

###  III-B Conventional Coherent and Semi-Blind Detection of [9]

Algorithm 1 Conventional Semi-blind Detection [9].

0: $\bar{\mathbf{Y}}=\left[\vbox{\hbox{\kern-1.15198pt\raise 0.0pt\hbox{\kern
74.16682pt}\kern 0.0pt\vbox{}\kern-1.15198pt}}\right]\in\mathbb{C}^{N\times
WT},~{}\hat{\mathbf{H}}^{(0)}$

0: $\hat{\mathbf{S}}^{(l)}$

1: Set the iteration index $l=0$.

2: repeat

3: Given $\hat{\mathbf{H}}^{(l)}$, perform ML detection for each sub-matrix of
$\bar{\mathbf{Y}}$ and obtain a set of estimates
$\hat{\mathbf{S}}^{(l)}=\left[\vbox{\hbox{\kern-1.15198pt\raise
0.0pt\hbox{\kern 68.3335pt}\kern
0.0pt\vbox{}\kern-1.15198pt}}\right]\in\mathbb{C}^{M\times WT}$

4: Update the channel matrix by
$\hat{\mathbf{H}}^{(l+1)}=\bar{\mathbf{Y}}\left[\hat{\mathbf{S}}^{(l)}\right]^{+}$.

5: Set $l=l+1$.

6: until $l<I_{\mathrm{max}}$.

Under the assumption that a perfect estimate of $\hat{\mathbf{H}}$ is
available at the receiver, maximum likelihood (ML) detection can be carried
out via

$\hat{\mathbf{S}}(i)=\mathop{\rm
arg~{}min}\limits_{\mathbf{S}}\|\mathbf{Y}(i)-\hat{\mathbf{H}}\mathbf{S}\|^{2}_{\mathrm{F}},$
(12)

where an optimal $\mathbf{S}$ is searched over the set of C-MIMO codewords.

In practice, however, frequent transmission of reference signals is required
to improve the accuracy of CSI estimation by Bob, which in turn also increases
the probability that transmissions are detected by Willie. To alleviate this
problem, the semi-blind detection scheme of [9] may be exploited. In such a
scheme, a reference signal $\mathbf{I}_{M}$ is transmited at the first
instance ($i.e.$, when $i=0$), which is used to obtain a rough initial channel
estimate $\hat{\mathbf{H}}^{(0)}$. Then, over the subsequent transmission of
$W$ blocks, the received signals are concatenated and more accurate CSI is
obtained iteratively, as summarized in Algorithm 1, where we use the pseudo-
inverse matrix
$\mathbf{A}^{+}=\mathbf{A}^{\mathrm{H}}\left(\mathbf{A}\mathbf{A}^{\mathrm{H}}\right)^{-1}$.

##  IV Proposed Noncoherent Gaussian Signaling (NGS) for LPD Communications

In this section, we introduce the proposed NGS scheme satisfying the
requirements of LPD communications.

###  IV-A Gaussian Signal Transmission

Based on a shared secret seed, Alice and Bob generate a Gaussian reference
matrix $\mathbf{G}\in\mathbb{C}^{M\times KM}$, where $K$ is a repetition
number, and each element of $\mathbf{G}$ follows $\mathcal{CN}(0,1/M)$,
$i.e.$, $\mathrm{E}[\|\mathbf{G}\|^{2}_{\mathrm{F}}]=KM$. Similarly, Alice and
Bob generate a Gaussian projection matrix $\mathbf{E}(i)\in\mathbb{C}^{M\times
T}$, where each element of $\mathbf{E}(i)$ follows $\mathcal{CN}(0,1/M)$,
$i.e.$, $\mathrm{E}[\|\mathbf{E}(i)\|^{2}_{\mathrm{F}}]=T$, where the index
$i$ indicates that the projection matrix varies over time.

After the above Gaussian matrices are prepared, at the index $i=0$, the
Gaussian reference matrix is transmitted. Then, for $0<i\leq W$, $B$ bits of
information are mapped onto a unitary matrix
$\mathbf{X}(i)\in\mathbb{C}^{M\times M}$, which is selected from a codebook of
$2^{B}$ matrices $\\{\mathbf{X}_{1},\cdots,\mathbf{X}_{2^{B}}\\}$. Here, we
use the classic differential encoding in the form

$\tilde{\mathbf{S}}(i)=\left\\{\begin{array}[]{ll}\mathbf{I}_{M}&\text{if
}i=0,\text{or}\\\ \tilde{\mathbf{S}}(i-1)\mathbf{X}(i)&\text{if
}i>0.\end{array}\right.$ (13)

Finally, the space-time codeword at $i$ is generated by

$\mathbf{S}(i)=\left\\{\begin{array}[]{ll}\mathbf{G}&\text{if
}i=0,\text{or}\\\ \tilde{\mathbf{S}}(i)\mathbf{E}(i)&\text{if
}i>0.\end{array}\right.$ (14)

The important question here is what type of unitary matrix $\mathbf{X}(i)$
would lead each element of (14) to follow an independent complex Gaussian
distribution. Since any point on the unit circle does not change the
statistical property, one can expect that any unitary matrix that has one
unit-norm nonzero element in each column results in the Gaussian distribution.
Differential schemes that satisfy such property include the diagonal unitary
coding (DUC) [12] and differential spatial modulation [13] methods, but since
the DUC approach maximizes the coding gain due to its optimized structure, it
will be adopted here in our NGS scheme.

In the DUC scheme [12], the $B$ input bits are mapped onto the integers
$b=0,1,\cdots,2^{B}-1$, and the corresponding unitary matrix is generated as

$\mathbf{X}_{b}=\mathrm{diag}\left[\mathrm{exp}\left(j\frac{2\pi
b}{2^{B}}u_{1}\right),\cdots,\mathrm{exp}\left(j\frac{2\pi
b}{2^{B}}u_{M}\right)\right],$ (15)

where the factors $0<u_{1}\leq\cdots u_{M}\leq 2^{B}/2\in\mathbb{Z}$ are
designed so as to maximize the diversity product given by444The optimized
factors are available online at
https://github.com/ishikawalab/wiphy/blob/master/wiphy/code/duc.py.

$\min_{b\in\\{1,\cdots,2^{B}-1\\}}\left|\prod_{m=1}^{M}\mathrm{sin}\left(\frac{\pi
bu_{m}}{2^{B}}\right)\right|^{\frac{1}{M}}\\!\\!\\!\\!\\!.$ (16)

Notice that the encoding scheme described by (15) becomes identical to the
classic differential phase-shift keying modulation if $M=1$ and $u_{1}=1$. In
addition, although the ideal transmission rate is $R=B/T$, since the Gaussian
reference matrix $\mathbf{G}^{M\times KM}$ is inserted, the effective
transmission rate is

$R_{\mathrm{eff}}=\frac{BW}{MK+WT}=\eta R,$ (17)

where we have implicitly defined the transmission efficiency
$\eta\triangleq(1+MK/(WT))^{-1}.$

As given, the efficiency decreases as $K$ increases. The reference insertion
ratio is calculated as $1-\eta$. In the performance comparisons, we will set
the ratio to 5%, but we remark that the performance of the proposed scheme
remains constant in high-speed mobile scenarios even with lower reference
insertion ratios such as $1\%$ and $0.1\%$.

###  IV-B Noncoherent Detection of Gaussian Signaling

The noncoherent detection of the Gaussian signaling scheme proposed in
Subsection IV-A is not a straightforward task. We extend the noncoherent
detection scheme proposed in [6] to support time-varying Gaussian reference
and projection matrices. Specifically, for the data blocks $0<i\leq W$, the
proposed noncoherent ML detector is described by

$\hat{\mathbf{X}}(i)=\mathop{\rm
arg~{}min}\limits_{\mathbf{X}}\|\mathbf{Y}(i)-\hat{\mathbf{Y}}(i-1)\mathbf{X}\mathbf{E}(i)\|^{2}_{\mathrm{F}},$
(18)

where the matrices $\hat{\mathbf{Y}}(i)$ are given by

$\hat{\mathbf{Y}}(0)\triangleq\mathbf{Y}(0)\mathbf{G}^{+}=\mathbf{H}(0)\tilde{\mathbf{S}}(0)+\mathbf{V}(0)\mathbf{G}^{+},$
(19)

at $i=0$, where each element of $\mathbf{V}(0)\in\mathbb{C}^{N\times KM}$
follows $\mathcal{CN}(0,\sigma_{v}^{2})$, and

$\hat{\mathbf{Y}}(i)\triangleq\beta\mathbf{Y}(i)\mathbf{E}^{\mathrm{H}}(i)\\!+\\!\hat{\mathbf{Y}}(i-1)\hat{\mathbf{X}}(i)(\mathbf{I}_{M}\\!-\\!\beta\mathbf{E}(i)\mathbf{E}^{\mathrm{H}}(i)),$
(20)

for $i>0$, respectively, with the parameter $\beta\triangleq 1-\alpha$
containing a forgetting factor $0<\alpha<1$ that determines the inter-
dependence between $\mathbf{Y}(i)$ and $\hat{\mathbf{Y}}(i-1)$.

Notice that due to (18), even if Willie successfully detects the LPD
communications, he cannot decrypt data symbols because the projection of
$\mathbf{E}(i)$ induces phase ambiguity, which implies that the proposed NGS
also works as a physical layer encryption scheme.

##  V Theoretical Analysis

In this section, we compare the proposed NGS LPD communication scheme against
the conventional C-MIMO in terms of security and complexity. System
configurations of the conventional C-MIMO and the proposed NGS schemes are
summarized in Table I.

TABLE I: System configurations of the conventional C-MIMO and the proposed NGS schemes. | Reference Signal ($i=0$) | Data Symbols ($0<i\leq W$) | ML Detector
---|---|---|---
Conventional C-MIMO | Gaussian reference matrix $\mathbf{G}\in\mathbb{C}^{M\times KM}$ | Okamoto’s C-MIMO encoding method [8] | Semi-blind detector [9]
| (see Section IV-A) | (see Section III-A) | (see Algorithm 1)
Proposed NGS | Gaussian reference matrix $\mathbf{G}\in\mathbb{C}^{M\times KM}$ | DUC & Gaussian projection matrix $\mathbf{E}(i)$ | Noncoherent detector
| (see Section IV-A) | (see Section IV-A) | (see Section IV-B)

###  V-A Security Analysis

In LPD communications, the achievable security level has been evaluated by the
Willie’s minimum detection error probability [4, 5, 3]. Let the case when
Alice does not transmit symbols to Bob be referred to as the null-hypothesis
$\mathcal{H}_{0}$, with likelihood function $p_{0}(y)$, and the alternative
case when Alice does transmit to Bob be referred to as the alternative
hypothesis $\mathcal{H}_{1}$, with likelihood function $p_{1}(y)$. The
received signal at Willie under both hypotheses can be expressed as [3]

$\left\\{\begin{array}[]{ll}\mathcal{H}_{0}:y=v,\;\text{or}\\\
\mathcal{H}_{1}:y=s+v,\end{array}\right.$ (21)

Figure 1: Lower bounds of Willie’s minimum detection error probability
$\xi^{*}$.

where $s$ and $v$ denote the transmit signal and additive white Gaussian
noise, respectively.

Assuming that all transmit signals are Gaussian, Willie’s detection error
probability can be expressed as [3]

$\xi^{*}\geq\xi^{*}_{\mathrm{min}}\triangleq
1-\sqrt{\mathcal{D}(p_{1}\|p_{0})/2},$ (22)

where the lower-bounding quantity $\xi^{*}_{\mathrm{min}}$ was derived in [4,
3] and $\mathcal{D}(p_{1}\|p_{0})$ denotes the Kullback-Leiber (KL) divergence
between the likelihoods of observing $y$ under the null and the alternative
hypotheses, respectively.

In Fig. 1, we evaluate the lower bounds of Willie’s minimum detection error
probability $\xi^{*}_{\mathrm{min}}$, where the conventional C-MIMO and the
proposed NGS were considered. As expected, both schemes exhibit the same lower
bound since the resulting constellation in both successfully follow the ideal
Gaussian distribution. Additionally, it is observed that as $M$ increases,
$\xi^{*}_{\mathrm{min}}$ becomes larger because less power is allocated to
each antenna, which is preferable in LPD communications. In short, it can be
said that the proposed scheme has the same security level of conventional
C-MIMO.

We remark that the repetition number $K$ has no effect on
$\xi^{*}_{\mathrm{min}}$ since it does not influence the statistical property
of transmit signals, which can be inferred from [4, 5, 3].

###  V-B Complexity Analysis

Next, the conventional C-MIMO and the proposed NGS are evaluated in terms of
encoding and decoding complexities.

#### Encoding complexity

In both the C-MIMO and NGS schemes a total of $BW$ [bits] are transmitted
within the frame length $W$. However, in order to generate a codebook, C-MIMO
requires $2^{B}W$ complex-valued random variables, since C-MIMO directly maps
information bits onto complex-valued Gaussian symbols.

In contrast, NGS requires only $BW$ complex-valued random variables since the
codebook is constructed as in (15), before being transformed by the Gaussian
projection matrix. Thus, the number of Gaussian variables to be generated can
be significantly reduced in the proposed NGS LPD scheme as the number of bits
$B$ increases.

#### Decoding complexity

The complexity orders of the conventional semi-blind detector $C_{\mathrm{c}}$
(see Algorithm 1) and of the proposed detector $C_{\mathrm{p}}$ (see
Subsection IV-B) are, respectively

$\displaystyle C_{\mathrm{c}}=$ $\displaystyle
2^{B}WI_{\mathrm{max}}(\underbrace{4MNT}_{\hat{\mathbf{H}}\mathbf{S}}+\underbrace{4NT}_{\|\cdot\|^{2}_{\mathrm{F}}})$
$\displaystyle+\underbrace{MW(I_{\mathrm{max}}-1)\left(8MT+\frac{4N}{W}+\frac{M^{2}}{W}\right)}_{\bar{\mathbf{Y}}\left[\hat{\mathbf{S}}^{(l)}\right]^{+}}$
$\displaystyle=$
$\displaystyle\mathcal{O}\big{(}2^{B}WI_{\mathrm{max}}(4MNT+4NT)\big{)},$ (23)
$\displaystyle C_{\mathrm{p}}=$ $\displaystyle
2^{B}W(\underbrace{4MNT}_{\hat{\mathbf{Y}}(i-1)\mathbf{X}}+\underbrace{4NT}_{\|\cdot\|^{2}_{\mathrm{F}}}+\underbrace{4MT}_{\mathbf{X}\mathbf{E}(i)})$
$\displaystyle+\underbrace{MW(8MN+4NT+4MT)}_{\beta\mathbf{Y}(i)\mathbf{E}^{\mathrm{H}}(i)\\!+\\!\hat{\mathbf{Y}}(i-1)\hat{\mathbf{X}}(i)(\mathbf{I}_{M}\\!-\\!\beta\mathbf{E}(i)\mathbf{E}^{\mathrm{H}}(i))}$
$\displaystyle=$ $\displaystyle\mathcal{O}\big{(}2^{B}W(4MNT+4NT+4MT)\big{)},$
(24)

where only the significant numbers of real-valued floating-point
multiplication operations are counted.

It follows that the ratio between both is given by

$\displaystyle\frac{C_{\mathrm{p}}}{C_{\mathrm{c}}}$
$\displaystyle=\mathcal{O}\left(\frac{1}{I_{\mathrm{max}}}\left(\frac{1+\frac{1}{M}+\frac{1}{N}}{1+\frac{1}{M}}\right)\right)\approx\mathcal{O}\left(\frac{1}{I_{\mathrm{max}}}\right),$
(25)

which means that the complexity order of the conventional semi-blind detector
is approximately $I_{\mathrm{max}}$ higher than that of the proposed detector.

##  VI Numerical Performance Comparisons

In this section, the conventional C-MIMO decoded via the semi-blind detector
and the proposed NGS LPD scheme are empirically evaluated in terms of their
bit error rates (BERs) and coding gains. Again, the system configurations of
both schemes are summarized in Table I. In our simulations, we set the minimum
SNR to $-20$ dB since transmission power in LPD communications systems is
typically small, and adopt the forgetting factor $\alpha=0.8$. In addition,
although the proposed NGS supports time-varying channels, we limit the
comparisons to quasi-static Rayleigh fading channels because the conventional
C-MIMO with the semi-blind detector only supports such channel condition.

Note that also in the conventional C-MIMO scheme, a Gaussian reference matrix
$\mathbf{G}\in\mathbb{C}^{M\times KM}$ is transmitted first for the purpose of
channel estimation, since all signals in LPD systems must be complex Gaussian.
The semi-blind detection algorithm is subsequently performed using the rough
initial channel estimate $\hat{\mathbf{H}}^{(0)}=\hat{\mathbf{Y}}(0)$ given by
(19).

First, a comparison between the BERs of the conventional C-MIMO [7] scheme
with semi-blind detection [9] and the proposed NGS LPD method is offered in
Fig. 3, for the case where $M=2$, $N=64$, $T=1$, $B=2$, and overhead factor
$K=1$ or $K=3$. As a lower bound, the BER of the NGS scheme with perfect CSI
is also included. The results show that the proposed NGS LPD system achieves
the best BER in all cases, with a gain of 16 dB over conventional C-MIMO
observed for $K=1$, and a performance rapidly approaching the lower bound for
larger $K$.

In order to better investigate the reason behind the large BER advantage of
NGS LPD seen in Fig. 3, we compare in Fig. 3 the coding gains of NGS LPD and
alternative schemes555Note that although spatial multiplexing does not
generate Gaussian signals, it was added to the comparison only as a reference
to coding gain., with $N=1$ and $T=1$, and $M$ varying from $1$ to $8$.

The results of Fig. 3 show that thanks to the optimized DUC constellation, the
proposed NGS LPD scheme achieves coding gains equal or close to those of
conventional spatial multiplexing, while generating Gaussian signals. In
comparison, the coding gains of conventional C-MIMO was found to be close or
equal to these of random Gaussian constellations.

Figure 2: Comparison of BERs achieved with Gaussian reference signals and
Gaussian projection matrix, with $M=2$, $N=64$, $T=1$ and $B=2$.

Figure 3: Comparison of coding gains achieved with Gaussian projection matrix,
with $B=M$, $N=1$ and $T=1$.

##  VII Conclusion

We proposed a solution to an open issue regarding the noncoherent detection of
Gaussian-distributed signals, which is especially important for LPD
communications. To solve the issue, we relied on newly proposed Gaussian
projection scheme, applied over optimized DUC constellations. Reference
signals were also designed so as to follow the Gaussian distribution. Thus,
the proposed scheme communicates only using Gaussian signals, which satisfies
the common requirement of LPD communication. With the use of OFDM, the
Gaussian signals in the frequency domain result in Gaussian-distributed time-
domain signals. An analysis was provided, which clarified that the proposed
NGS generates perfectly Gaussian symbols at a fraction of the complexity of
C-MIMO schemes. Simulation results also revealed large BER gains over the
latter alternative, due to the higher coding gains afforded by the new method.
Since the proposed NGS requires shorter reference signals, it is expected to
be suitable for low-overhead LPD communications in high-mobility scenarios.

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be composed from the triple reggeon vertices (2.42) with the propagators
$\sim|k|^{-2}$ for the internal lines as discussed in subsect. 2.3.

Consider the collinear triple vertex (5.6) which has been obtained in sect.
5.1 by the three-point YSC in helicity form for $n=2m=2$. We observe that the
reggeon vertices are obtained from two factors $M_{3}$ (5.6), one with the
momenta $k_{I}$ substituted by the complex variable $\kappa_{I}$ and the other
with the momenta $k_{I}$ substituted by the complex conjugated variable
$\kappa_{I}^{*}$. The helicities are to be substituted in accordance with the
weights of the reggeized gluon being $(\ell_{I},\bar{\ell}_{I})=(0,0)$. This
means in both factors $h_{1}=h_{2}=+\frac{1}{2},h_{3}=-\frac{1}{2}$.

$\mathcal{V}^{-++}(\kappa_{1},\kappa_{2})=\mathcal{V}^{+--}(\kappa_{1},\kappa_{2})=M_{3}(\kappa_{1},\kappa_{2},-\kappa_{1}-\kappa_{2};\frac{1}{2},\frac{1}{2},-\frac{1}{2})M_{3}(\kappa_{1}^{*},\kappa_{2}^{*},-\kappa_{1}^{*}-\kappa_{2}^{*};\frac{1}{2},\frac{1}{2},-\frac{1}{2}).$
(6.24)

###  6.2 The factorized $R$ operator and the BFKL hamiltonian

We consider the $L$ operator in the case $n=2$ after projection to the weight
$2\ell$ in normal coordinates $x=\frac{x_{1}}{x_{2}}$ (5.9). The matrix
$L(v^{(1)},v^{(2)})$ factorizes as

$L(v^{(1)},v^{(2)})=\begin{pmatrix}v^{(2)}+1+x\partial&\partial\\\
x(-x\partial+2\ell)&v^{(1)}-x\partial\end{pmatrix}=\ \
v^{(1)}\hat{V}^{-1}(v^{(1)})\hat{D}\hat{V}(v^{(2)}),$ (6.25)
$\hat{V}(v)=\begin{pmatrix}v&0\\\ -x&-1\end{pmatrix},\ \
\hat{D}=\begin{pmatrix}1&-\partial\\\ 0&1\end{pmatrix}.$

We summarize the parameter permutation method for constructing the Yang-Baxter
$R$ operator [76, 77]. Consider the monodromy of two such $L$ operators,

$L_{1}(v_{1}^{(1)},v_{1}^{(2)})L_{2}(v_{2}^{(1)},v_{2}^{(2)})$ (6.26)

and look for operators representing the elementary permutations of the
parameter sequence $v_{1}^{(1)},v_{1}^{(2)},$ $v_{2}^{(1)},v_{2}^{(2)}$ in
action on the monodromy, e.g.

$\mathcal{S}_{12}(v_{2}^{(1)}-v_{1}^{(2)})L_{1}(v_{1}^{(1)},v_{1}^{(2)})L_{2}(v_{2}^{(1)},v_{2}^{(2)})=L_{1}(v_{1}^{(1)},v_{2}^{(1)})L_{2}(v_{1}^{(2)},v_{2}^{(2)})\mathcal{S}_{12}(v_{2}^{(1)}-v_{1}^{(2)}).$

Using the factorized form (6.25) we find

$\mathcal{S}_{12}(v)=x_{12}^{v}.$

For the analogous permutation of the first two ( $I=1$) or the last two
($I=2$) entries in the sequence we have
$\mathcal{S}_{11}(v_{1}^{(1)}-v_{1}^{(2)})$ and
$\mathcal{S}_{22}(v_{2}^{(1)}-v_{2}^{(2)})$ with

$\mathcal{S}_{II}(v)=x_{I}^{-v}\frac{\Gamma(x_{I}\partial_{I}+1)}{\Gamma(x_{I}\partial_{I}+1-v)}.$

Consider the permutation of the sequence
$v_{1}^{(1)},v_{1}^{(2)},v_{2}^{(1)},v_{2}^{(2)}$ to
$v_{2}^{(1)},v_{2}^{(2)},v_{1}^{(1)},v_{1}^{(2)}$. It appears in the $RLL$
relation with the complete Yang-Baxter operator (3.21) modified by the the
point permutation, $\check{\hbox{\bbd R}}_{12}=P_{12}\hbox{\bbd R}_{12}$. To
construct it in terms of the $\mathcal{S}_{IJ}$ we write a corresponding
product of elementary permutations and its representation in action on the
monodromy (6.26).

$\mathcal{S}_{12}(v_{1}^{(1)}-v_{2}^{(2)})\mathcal{S}_{22}(v_{1}^{(2)}-v_{2}^{(2)})\mathcal{S}_{12}(v_{1}^{(2)}-v_{1}^{(1)})\cdot\mathcal{S}_{12}(v_{1}^{(1)}-v_{1}^{(2)})\mathcal{S}_{11}(v_{1}^{(1)}-v_{2}^{(1)})\mathcal{S}_{12}(v_{1}^{(2)}-v_{2}^{(1)})=$
$\frac{\Gamma(x_{12}\partial_{2}+1+v_{1}^{(2)}-v_{1}^{(1)})}{\Gamma(x_{12}\partial_{2}+1+v_{2}^{(2)}-v_{1}^{(1)})}\frac{\Gamma(x_{21}\partial_{1}+1+v_{1}^{(2)}-v_{2}^{(1)})}{\Gamma(x_{21}\partial_{1}+1+v_{1}^{(2)}-v_{1}^{(1)})}.$

In terms of the spectral parameters and the weights we obtain a form of the
complete $R$ operator for the case $g\ell(2)$ in terms of the normal
coordinate and derivative operators,

$\check{\hbox{\bbd
R}}(v_{1}-v_{2};\ell_{1},\ell_{2})=\frac{\Gamma(x_{21}\partial_{2}-2\ell_{1})}{\Gamma(x_{21}\partial_{2}-\ell_{1}-\ell_{2}+v_{2}-v_{1})}\frac{\Gamma(x_{12}\partial_{1}+v_{1}-v_{2}-\ell_{1}-\ell_{2})}{\Gamma(x_{12}\partial_{1}-2\ell_{1})}$
(6.27)

It is inconvenient, a form with the operator arguments commuting with the
generators $S_{1}^{a}+S_{2}^{a}$ would be preferable.

We notice that on the highest weight states in the tensor product of
representations by polynomials in $x_{1},x_{2}$ are represented as in (5.17)
by powers of the differences $x_{12}^{n}$. They are eigenvectors of the
operator (6.27). The convenient form with the same eigenvalues is obtained as

$\hbox{\bbd
R}(v_{1}-v_{2};\ell_{1},\ell_{2})=\frac{\Gamma(-\hat{m}+v_{1}-v_{2})}{\Gamma(-\hat{m}+v_{2}-v_{1})}$
(6.28)

in terms of the operator $\hat{m}$ related to the tensor Casimir (2.47) as
$\hat{m}(1-\hat{m})=x_{12}^{2}\partial_{1}\partial_{2}.$

We recall that in the position representation the solution of the BKFL
equation is expanded in the basis functions $E^{n,\nu}$ (2.52) factorising
into holomorphic and anti-holomorphic factors with the representation labels
$n$ taking all integer values and $\nu$ taking all real values. The point
$x_{0}$ labels the vectors of a particular representation.

These holomorphic basis elements $E_{m}$ do not obey the highest weight
condition for finite $x_{0}$. The argument leading to the convenient form
(6.28) does not apply in the straightforward way. However, for a fixed $x_{0}$
we can transform the canonical pairs $x_{1},\partial_{1},x_{2},\partial_{2}$
by a Möbius transformation of the coordinates,

$x_{1}^{\prime}=\frac{1}{x_{10}},x_{2}^{\prime}=\frac{1}{x_{20}}.$ (6.29)

Then the generators ${S^{\prime}}_{1}^{a},{S^{\prime}}_{2}^{a}$ constructed
according to (5.10) in terms of the new canonical pairs (depending on $x_{0}$)
are such that $E_{m}$ obey the highest weight condition,

$({S^{\prime}}_{1}^{-}+{S^{\prime}}_{2}^{-})E_{m}(x_{1},x_{2};x_{0})=0,\ \ \
({S^{\prime}}_{1}^{0}+{S^{\prime}}_{2}^{0})E_{m}(x_{1},x_{2};x_{0})=mE_{m}(x_{1},x_{2};x_{0}).$

Thus (6.28 ) with $\ell_{1}=\ell_{2}=0$ is a convenient form of the $R$
operator related to the BFKL pomeron. The BFKL hamiltonian operator is
obtained up to normalization as the first non-trivial term in its expansion in
$\varepsilon=v_{1}-v_{2}$.

$H=\psi(\hat{m})+\psi(1-\hat{m})-2\psi(1)$

directly related to the eigenvalues (2.51) and the operators forms $H_{g}$
(2.46), (2.50).

This hamiltonian describes the nearest-neighbour interaction in the multiple
exchange of gluonic reggeons. The decomposition of the R operator results in
the set of commuting local obserables of the corresponding closed spin chain
[83].

###  6.3 4-point YSC and the dipole form of the BFKL kernel

We consider the YSC at $N=4,K=2,n=2$ in the homgeneous coordinate form as in
sect. 5.3, perform the integrals over $c$ and change to the normal coordinates
separating the scale factor as in (5.15). As discussed in sect. 5.3 we obtain
from (3.45) the R operator kernel in the normal coordinate form (5.16).

In sect. 3.6 we have considered integral operators on functions of two
n-dimensional position variables. Here we consider functions of two points
$\mathbf{x}_{1},\mathbf{x}_{2}$ in the complex plane and integral operators

$Q\psi(\mathbf{x}_{1},\mathbf{x}_{2})=\int
d^{2}\mathbf{x}_{1}^{\prime}\psi(\mathbf{x}_{1}^{\prime},\mathbf{x}_{2}^{\prime})\
K(\mathbf{x}_{2}^{\prime},\mathbf{x}_{1}^{\prime},\mathbf{x}_{2},\mathbf{x}_{1}).$

with the kernels composed of a holomorphic and an anti-holomorphic factor.
Both factors are derived from YSC of the form (5.16). In both factors we set
$\ell_{1^{\prime}}=\ell_{2^{\prime}}=\ell$. The position arguments are
substituted as the complex $x_{I},I=2^{\prime},1^{\prime},2,1$ in the first
factor and as the complex conjugated $x_{I}^{*}$ in the second.

$K(\mathbf{x}_{2^{\prime}},\mathbf{x}_{1^{\prime}},\mathbf{x}_{2},\mathbf{x}_{1})=\phi_{X}(x_{1^{\prime}},x_{2^{\prime}},x_{1},x_{2})\phi_{X}(x^{*}_{1^{\prime}},x^{*}_{2^{\prime}},x^{*}_{1},x^{*}_{2})=\frac{|x_{1^{\prime}2^{\prime}}|^{2(-1-2\ell-\varepsilon)}|x_{12}|^{2(1+2\ell-\varepsilon)}}{|x_{1^{\prime}2}|^{2(1-\varepsilon)}|x_{12^{\prime}}|^{2(1-\varepsilon)}}.$

In the decomposition in $\varepsilon$ the leading $\varepsilon^{-1}$ term
corresponds to the kernel of the permutation operator $P_{12}$. The next term
in the $\varepsilon$ expansion is

$|x_{1^{\prime}2^{\prime}}|^{2(-1-2\ell)}|x_{12}|^{2(1+2\ell)}\left(|x_{1^{\prime}2}|^{2}\delta^{(2)}(x_{12^{\prime}})+|x_{12^{\prime}}|^{2}\delta^{(2)}(x_{1^{\prime}2})-\delta^{(2)}(x_{12^{\prime}})\delta^{(2)}(x_{1^{\prime}2})\int
d^{2}x_{3}|x_{13}|^{2}|x_{13}|^{2}\right).$

In the limiting case corresponding to the reggeized gluon exchange $\ell\to 0$
the action is defined on functions vanishing at coinciding arguments
$x_{1^{\prime}}=x_{2^{\prime}}$. The resulting kernel can be written as

$|x_{12}|^{2}\int\frac{d^{2}x_{3}}{|x_{13}|^{2}|x_{23}|^{2}}\left(\delta^{(2)}(x_{21^{\prime}})\delta(x_{2^{\prime}3})+\delta^{(2)}(x_{12^{\prime}})\delta(x_{1^{\prime}3})-\delta^{(2)}(x_{12^{\prime}})\delta(x_{21^{\prime}})\right)$

This is the dipole (or Möbius) form of the BFKL kernel (2.48).

##  7 Conclusions

The L operator matrices with the Jordan-Schwinger presentation of the
$g\ell(n)$ algebra as its matrix elements can be regarded as the basis of our
approach. The monodromies built as ordered matrix products of L operators
enter the definition of the YSC. The properties of the L operators result in
relations for YSC and in methods of their construction.

By elementary canonical transformations different forms of the L operators are
obtained leading to different representation forms of the YSC. The original JS
form and the helicity form appear in the applications.

The R operators obeying the Yang-Baxter RLL relation involving products of two
such L operators play a central role. A particular solution for such a R
operator has the form of a closed contour integral over a shift operator. It
serves as the tool of constructing YSC. Integral operators built by particular
4-point YSC turn out to be solutions of the RLL relation. In the case $n=2$
the spectrum of operators obeying the RLL relation and a further operator form
of R have been studied and applied.

The L operators projected to the action on homogeneous functions depend on two
parameters, the order N monodromy composed of them and the corresponding
N-point YSC depend on the set of 2N parameters. We choose these parameters in
a form, where the action of the R operators implies permutations of these
parameters.

In the applications the monodromy parameters are related to the particle types
and helicities. The YSC with generic parameter values turn to physically
relevant objects by specification of the parameters.

We have shown that the 3-point amplitudes, the building blocks of the
iterative BCFW construction of tree amplitudes, are specifications of 3-point
YSC. We have explained that this iterative construction is in fact the
convolution method of YSC construction. We have shown how specifications of
the parameters of 4-point YSC result in the $2\to 2$ tree amplitudes of gluons
and quarks.

We have seen that the parton splitting amplitudes, appearing in the collinear
limit from the 3-point amplitudes, are obtained from specifications of 3-point
YSC. The 4-point YSC define as kernels integral operators obeying relations of
the RLL type. The latter allow to derive the eigenvalues of those operators.
We have shown how by the specification of parameters in 4-point YSC and
integration contours the kernels of the Bjorken evolution equation are
obtained. This has been confirmed by showing that the corresponding
eigenvalues and the anomalous dimensions are proportional.

The reggeon triple vertex decomposes in a holomorphic and an anti-holomorphic
factor, each of which is obtained as a specification of 3-point YSC. The R
operator obeying the RLL relation with the L operators specified to the normal
coordinate form of the $g\ell(2)$ case can be obtained by the parameter
permutation method and contains the (holomorphic part of) BFKL hamiltonian as
the first non-trivial term in its decomposition. The integral operator form of
this R operator is obtained by specifying a 4-point YSC as its kernel. For the
action on functions of two points in the transverse plane the corresponding R
operator with two factors of such 4-point YSC as its kernel is constructed.
The dipole form of the BFKL hamiltonian appears in the decomposition of the
latter R operator.

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# Quantifying Political Bias in News Articles

Gizem Gezici  Huawei Turkey R&D CenterIstanbulTurkey<EMAIL_ADDRESS>
###### Abstract.

Search bias analysis is getting more attention in recent years since search
results could affect In this work, we aim to establish an automated model for
evaluating ideological bias in online news articles. The dataset is composed
of news articles in search results as well as the newspaper articles. The
current automated model results show that model capability is not sufficient
to be exploited for annotating the documents automatically, thereby computing
bias in search results.

Bias evaluation, Fair ranking, Political bias, Web search

##  1\. Introduction

Search engines are ubiquitous. As reported by SmartSights (Sma, [n.d.]), in
2017 46.8% of the world population accessed the internet and by 2021, the
number is expected to reach 53.7%. Currently, on average 3.5 billion Google
searches are done per day (Int, [n.d.]). These statistics advocate that search
systems are “gatekeepers to the Web” for many people (Diaz, 2008). As
information seekers search the Web more, they are also more influenced by
Search Engine Result Pages (SERPs) and their influence -negative included-
potentially become visible in a wide range of areas. However, as with all
software-based systems, search platforms do not lack of human influence,
thereby they may suffer from embedded bias, i.e., corpus or algorithmic bias.
Experimental studies suggest that particular information types or sources
might be retrieved more or less, or might not be well represented (Culpepper
et al., 2018). For instance, during the elections, it is known that people
issue repeated queries on the Web about political candidates and events such
as “democratic debate”, “Donald Trump” and “climate change” (Kulshrestha et
al., 2018). Epstein and Robertson (2015) claim that SERPs returned in response
to these queries may influence the voting decisions of the users and report
that manipulated search rankings can change the voting preferences of
undecided individuals at least by 20%. As individuals rely to a greater extent
on the SERPs for their decision making, there is a thriving demand for these
systems explainable.

Empirical evidence has shown that individuals trust more sources ranked at
higher positions in SERPs, but the ranking criteria may rather depend more on
user satisfaction than the factual information, which jeopardizes the
phenomenon of providing reliable information in exchange for satisfying users
(Culpepper et al., 2018). In spite of the given research findings, the
majority of online users tend to believe that search engines provide _neutral_
results, i.e., serving only as facilitators in accessing information on the
Web due to their automated operations (Goldman, 2008). However, this
romanticised view of search platforms does not reflect reality and there seems
to be a growing skepticism related to objectivity and credibility of these
platforms. To illustrate that, a recent dispute between the U.S. President
Donald Trump and Google can be given, where Mr. Trump accused Google of
presenting only negative news about him when his name is searched. Google
refuted this claim by saying that: “When users type queries into the Google
Search bar, our goal is to make sure they receive the most relevant answers in
a matter of seconds” and “Search is not used to set a political agenda and we
don’t bias our results toward any political
ideology”111https://www.reuters.com/article/us-usa-trump-tech-alphabet/google-
responds-to-trump-says-no-political-motive-in-search-results-idUSKCN1LD1QP. In
this work, we hope to shed some light on that debate, by not specifically
examining queries regarding Donald Trump but by fulfilling an in depth
analysis of retrieved search answers to a broad set of queries related to
controversial topics based on concrete evaluation measures.

_Bias_ implies undue emphasis. For a retrieval system, it can be defined as
the balance and representativeness of Web documents retrieved from a database
for a set of queries (Mowshowitz and Kawaguchi, 2002). When a user issues a
query to a search engine, documents from different sources are collected,
ranked, and presented to the user. Assume that a user searches for _2016
presidential election_ and the top-n ranked results are displayed. In such a
search scenario, the retrieved results may exaggerate or downplay particular
perspectives and thereby provide an unbalanced picture of the given query as
claimed by Mr. Trump, though without any scientific support. Hence, the
potential _undue inclusion or exclusion_ of specific perspectives in the
retrieved results lead to bias (Mowshowitz and Kawaguchi, 2002). Note that the
existence of bias is different from relevance. In the presented scenario, even
though the retrieved list of documents may all be judged as relevant with
respect to the given query, if the selection of the documents is skewed or
slanted, i.e., emphasizing one perspective over another, then the
corresponding search engine is _biased_ due to an imbalanced representation of
the perspectives towards the query’s topic. Bias is especially important if
the query topic is _controversial_ having opposing perspectives as described
in the given scenario. The bias in SERPs can be used by search engines to
inform their users about the bias by making themselves more accountable which
is one of the crucial attributes that a retrieval system should possess
(Culpepper et al., 2018).

In this work, we focus on the SERPs coming from the news sources and
investigate two major search engines (Bing and Google) in terms of political
bias. Our analysis has mainly three sides where we evaluate the political bias
of the search engines separately, then make a comparison among them as well as
track the source of this bias in the SERPs. The bias may come from the data,
which may contain biases (input bias) or the search algorithm, which contains
sophisticated features (algorithmic bias). In the scope of this work, we
concentrate on the input bias that is intrinsically embedded in the data
itself.

In short, we aim to answer the following research questions:

RQ1::

On a conservative-to-liberal scale, do search engines return _politically
biased_ SERPs in response to queries related to controversial topics?

RQ2::

Are search engines _significantly different_ from each other towards
controversial topics?

RQ3::

Does the _source of bias_ come from the input data?

We address these research questions for controversial topics representing a
broad range of issues in SERPs of Google and Bing through content analysis,
i.e. analysing the textual content of the retrieved documents. We focus on
_content bias_ and describe our SERP politically content bias quantification
framework in which we propose three different measures of bias based on common
Information Retrieval (IR) _utility-based_ evaluation measures: Precision at
cut-off (P@n), Rank Biased Precision (RBP), and Discounted Cumulative Gain at
cut-off (DCG@n). While the first measure quantifies bias considering only a
weak ranking criterion, i.e. the first $n$ returned documents as in SERPs, the
other two measures incorporate stronger ranking bias.

In order to answer RQ1, we measure the degree of deviation of the ranked SERPs
from an _ideal_ distribution where different political perspectives are
_equally_ likely to appear (Mowshowitz and Kawaguchi, 2002). To detect
political bias which results from the imbalanced representation of politically
different points of view, we label the documents’ political perspectives with
classification and use these labels for bias evaluation. To address RQ2, we
compare the political bias in the SERPs of the two search engines to see if
they show similar bias for the corresponding controversial topics. To answer
RQ3, we measure the political bias in a similar manner as we do to answer the
RQ1. The only difference is that this time we measure the bias in the whole
corpus, i.e., all retrieved SERPs from Google and Bing, instead of the partial
of it, as top $n$ documents. Then, we compare the bias of the whole corpus
with the partial bias that we compute for the first part of our analysis to
check if they are consistent, or not.

Our main contributions in this work can be summarized as follows:

  1. (1)

We propose a _novel generalizable search bias evaluation framework_ to measure
the bias in the SERPs.

  2. (2)

Our fairness-aware set of measures are explainable utility-based IR measures
which take into account of relevance as well, while evaluating bias in ranked
results.

  3. (3)

We apply our framework to _compare the relative bias_ in the SERPs content of
two search engines for controversial queries. To the best of our knowledge,
this is one of the first works to compare the two search engines in terms of
_content bias_.

  4. (4)

We utilize the framework to _track the source of bias_ in SERPs, i.e.,
checking if the existing bias comes from the corpus or not.

Lastly, a preliminary version of the present framework has been published
earlier (Gezici et al., 2021). This work expands and surpasses the previous
endeavor mainly in the following points: (i) We used a deep learning framework
that is widely-used in NLP tasks for classification, i.e., to predict the
political perspectives of the retrieved documents from the two popular search
engines, instead of obtaining these labels via crowdsourcing. (ii) We also
crawled a different dataset of articles from the newspapers’ websites to fine-
tune the deep learning model specifically for the news articles, thereby
achieve a better classification model in perspective detection. (iii) Finally,
we included a new investigation on the source of bias in our experiments
through examining if the measured bias in the top-n documents comes from the
data.

The remainder of the paper is structured as follows. In Section 2 we present
the related work. The search bias evaluation framework is developed in Section
3. In Section 4 we detail the experimental setup, and present the results.

##  2\. Background & Related Work

Studies indicate that online IR platforms have influenced public by causing an
increase in polarization; for instance it has been shown that social media
accounts with non-Western names are more likely to be indicated as fraudulent
since most probably the system has been trained on Western names. Therefore,
proposing new metrics can be useful in evaluating and understanding such
ethical issues in IR systems (Culpepper et al., 2018). Regarding this
assertion, there have been a growing interest in quantifying bias in the
recent years

##  3\. Political Bias Evaluation Framework

2In this section, to answer the research questions asserted in the
introduction, we propose our political bias evaluation framework. The first
question focuses on detecting political bias (if exists) in SERPs and the
second one examines if the search engines show the same level of bias. On the
other hand, the third question investigates if the source of bias comes from
the data. In this framework, we initially identify political perspectives of
the news articles automatically via classification. Then, we present the
measures of bias and the proposed protocol to identify political bias and
investigate the source of bias in SERPs.

###  3.1. Preliminaries

Our first goal is to detect political bias with respect to the distribution of
political perspectives expressed in the contents of the SERPs. Let
$\mathcal{S}$ be the set of search engines and $\mathcal{Q}$ be the set of
queries about controversial topics. When a query $q\in\mathcal{Q}$ is issued
to a search engine $s\in\mathcal{S}$, the search engine $s$ returns a SERP
$r$. We define the political perspective of the $i$-th retrieved document
$r_{i}$ with respect to $q$ as $j(r_{i})$. A political perspective can have
the following values: _conservative_ , _liberal_ , _both or neither_.

A document political perspective can be:

conservative (  ):

when the document content is in favour of the conservative political agenda.

liberal (  ):

when the document content is in favour of the liberal political agenda.

both or neither (  ):

when the document content is in favour of both or neither perspectives.

For our analyses, we deliberately use _controversial_ topics such as abortion,
medical marijuana, gay marriage, and Cuba embargo which contain opposing
viewpoints since complicated concepts concerning the identity, religion, or
political leaning are the actual points where search engines are more likely
to provide biased results (Noble, 2018). Our second goal is related to the
first one as we compare the search engines in terms of the political bias that
they show in their SERPs.

Our third goal is to find the source of political bias by checking if the bias
comes from the data (input bias). We do this by measuring the bias also in the
whole corpus of the SERPs in addition to the partial dataset, i.e. top 10 or
100 documents of the corpus, and comparing the corpus bias (if exists) with
the partial one to see if they are consistent or not.

###  3.2. Measures of Bias

On the basis of the previously definition of bias given in Section 1, it can
be quantified by measuring the degree of deviation of the distribution of
documents from the ideal one. To give a generic definition of an ideal list in
the field of bias evaluation poses problems; but in the scope of this work, we
can assess the existence of political bias in a ranked list retrieved by a
search engine if the presented information _significantly deviates_ from true
likelihoods (White, 2013). Using this definition reversely, we can adopt the
assumption that the ideal list is the one that minimizes the difference
between two opposing political perspectives, which we indicate here as  and  .

####  3.2.1. Bias Score of a SERP

Formally, we measure the political bias in a SERP $r$ as follows:

(1) $\beta_{f}(r)=f_{\,\text{ \leavevmode\hbox to4.82pt{\vbox
to4.82pt{\pgfpicture\makeatletter\hbox{\hskip 2.41112pt\lower-2.41112pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,1}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{2.41112pt}{0.0pt}\pgfsys@curveto{2.41112pt}{1.33163pt}{1.33163pt}{2.41112pt}{0.0pt}{2.41112pt}\pgfsys@curveto{-1.33163pt}{2.41112pt}{-2.41112pt}{1.33163pt}{-2.41112pt}{0.0pt}\pgfsys@curveto{-2.41112pt}{-1.33163pt}{-1.33163pt}{-2.41112pt}{0.0pt}{-2.41112pt}\pgfsys@curveto{1.33163pt}{-2.41112pt}{2.41112pt}{-1.33163pt}{2.41112pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\,}(r)-f_{\,\text{
\leavevmode\hbox to4.82pt{\vbox to4.82pt{\pgfpicture\makeatletter\hbox{\hskip
2.41112pt\lower-2.41112pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\pgfsys@color@rgb@stroke{1}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{1}{0}{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,0,0}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{2.41112pt}{0.0pt}\pgfsys@curveto{2.41112pt}{1.33163pt}{1.33163pt}{2.41112pt}{0.0pt}{2.41112pt}\pgfsys@curveto{-1.33163pt}{2.41112pt}{-2.41112pt}{1.33163pt}{-2.41112pt}{0.0pt}\pgfsys@curveto{-2.41112pt}{-1.33163pt}{-1.33163pt}{-2.41112pt}{0.0pt}{-2.41112pt}\pgfsys@curveto{1.33163pt}{-2.41112pt}{2.41112pt}{-1.33163pt}{2.41112pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\,}(r),$

where $f$ is a function that measures the likelihood of $r$ in satisfying the
information need of the user about the view  and the view  .

####  3.2.2. Mean Bias (MB)

Before defining $f$, from Eq. (1), we define the mean bias of a search engine
$s$ as:

(2)
$\text{MB}_{f}(s,\mathcal{Q})=\frac{1}{|\mathcal{Q}|}\sum_{q\in\mathcal{Q}}\beta_{f}(s(q)).$

An unbiased search engine would produce a mean bias of $0$. A limitation of MB
is that if a search engine is biased towards the  perspective on one topic and
bias towards the  perspective on another topic, these two contributions will
cancel each other out. In order to avoid this limitation we also define
another metric.

####  3.2.3. Mean Absolute Bias (MAB)

The mean absolute bias, which consists in taking the absolute value of the
bias for each $r$. Formally, this is defined as following:

(3)
$\text{MAB}_{f}(s,\mathcal{Q})=\frac{1}{|\mathcal{Q}|}\sum_{q\in\mathcal{Q}}|\beta_{f}(s(q))|.$

An unbiased search engine produces a mean absolute bias of $0$. Although this
measure solves the limitation of MB, MAB says nothing about towards which
perspective the search engine is biased, making these two measures of bias
complementary.

####  3.2.4. Retrieval Evaluation Measures

In IR the likelihood of $r$ in satisfying the information need of users is
measured via retrieval evaluation measures. Among these measures we selected 3
_utility-based_ evaluation measures. This class of evaluation measures
quantify $r$ in terms of its worth to the user and are normally computed as a
sum of the information gain summed over the relevant documents retrieved by
$r$.

The 3 IR evaluation measures used in the following experiments are: P$@n$,
RBP, and DCG$@n$.

P$@n$::

P$@n$ for the  view is formalized in (Gezici et al., 2021). However,
differently from the previous definition of $j$ where the only possible
outcomes are $g_{1}$ and $g_{2}$, here $j$ can return any of the label
associated to a political perspective (  ,  , and  ). Hence, only conservative
and liberal documents, that is relevant to the topic, are taken into account,
since $j(r_{i})$ returns _both or neither_ when otherwise.

Replacing P$@n$ in Eq. (1) we obtain the first measure of bias:

(4)
$\beta_{\text{P}@n}(r)=\frac{1}{n}\sum_{i=1}^{n}\left([j(r_{i})=\text{\faThumbsOUp}]-[j(r_{i})=\text{\raisebox{-2.0pt}{\faThumbsODown}}]\right).$

The main limitation of this measure of bias is that it has a weak concept of
ranking, i.e. the first $n$ documents contribute equally to the bias score.
The next two evaluation measures overcome this issue by defining discount
functions.

RBP::

RBP weights every document based on the coefficients of a normalized geometric
series with value $p\in]0,1[$, where $p$ is a parameter of RBP. Similarly to
what is done for P$@n$, we reformulate RBP to measure bias as follows:

(5)
$\text{RBP}_{\text{\faThumbsOUp}}=(1-p)\sum_{i=1}p^{i-1}[j(r_{i})=\text{\faThumbsOUp}].$

Substituting Eq. (5) to Eq. (1) we obtain:

(6)
$\beta_{\text{RBP}}(r)=(1-p)\sum_{i=1}p^{i-1}\left([j(r_{i})=\text{\faThumbsOUp}]-[j(r_{i})=\text{\raisebox{-2.0pt}{\faThumbsODown}}]\right).$

DCG$@n$:

DCG$@n$, instead, weights each document based on a logarithmic discount
function. Similarity to what is done for P$@n$ and RBP, we reformulate DCG$@n$
to measure bias as follows:

(7)
$\text{DCG}_{\text{\faThumbsOUp}}@n=\sum_{i=1}^{n}\frac{1}{\log(i+1)}[j(r_{i})=\text{\faThumbsOUp}].$

Substituting Eq. (7) to Eq. (1) we obtain:

(8)
$\beta_{\text{DCG}@n}(r)=\sum_{i=1}^{n}\frac{1}{\log(i+1)}\left([j(r_{i})=\text{\faThumbsOUp}]-[j(r_{i})=\text{\raisebox{-2.0pt}{\faThumbsODown}}]\right)$

Since we are evaluating web-users, for P$@n$ and DCG$@n$ we set $n=10$ and for
RBP we set $p=0.8$.

This last formulation (Eq. ), although it looks similar to the rND measure, it
does not suffer from the first four limitations introduced in Section 2. In
particular all these presented measures of bias: 1) do not focus on one group;
2) use a binary score associated to the document stance or political
perspective, similar to the way these measures are used in IR when considering
relevance; also like in IR 3) can be computed at each rank; 4) exclude non-
relevant documents from the measurement of bias, and; 5) provide various user
models associated to the 3 IR evaluation measures: P@n, DCG@n, and RBP.

Figure 1. Flow-chart of the quantifying political bias procedure.

###  3.3. Quantifying Political Bias

Using the previously defined measures of bias in Section 3.2 we quantify the
political bias of the two search engines, Bing and Google. Then we compare
them in terms of the measured bias. Now, we describe each step of the proposed
mechanism used to quantify political bias in SERPs.

Dataset Construction.:

After having crawled all the SERPs for both search engines and all queries
$\mathcal{Q}$, for each returned document we obtain the political perspective
of the document with respect to the corresponding query in the top 10
documents and the whole corpus. Both is done automatically via classification
since we already have a dataset labelled by crowdsourcing for training.

Bias Evaluation.:

We compute the bias measures of every SERP with all three IR-based measures of
bias: P$@n$, RBP, and DCG$@n$ for the first 10 documents and only P$@n$ for
the whole corpus, because the other measures of bias do not provide meaningful
results due to the rank information used in these measures. We then aggregate
the results using the two measures of bias, MB and MAB.

Statistical Analysis.:

To identify whether the bias measured is not rooted from randomness, for the
bias measure MAB, we compute a one-sample t-test: the null hypothesis is that
no difference exists and that the true mean is equal to zero. If this
hypothesis is rejected, this means that there is a significant difference; we
claim that the evaluated search engine is biased. Then, we make a comparison
in the bias difference measured across the two search engines using a two-
tailed paired t-test: the null hypothesis is that the difference between the
two true means is equal to zero. If this hypothesis is rejected, this means
that there is a significant difference so we claim that there is a difference
in bias between the two search engines.

##  4\. Experimental Setup

Figure 2. Top Online News Media Bias Ratings of AllSides

###  4.1. Dataset

In this work, we used two datasets for political bias evaluation. The first
dataset contains SERPs crawled from two search engines, Google and Bing. The
second dataset, on the other hand is composed of the articles published in the
known newspapers.

News Articles in SERPs.:

We obtained the controversial queries issued for searching from ProCong.org
(ProCon.org, 2018) and applied some filtering steps on the initial query set.
After filtering, the final query set size became 57. We submitted each query
in the final query set to the US News search engines of Google and Bing using
a US proxy. Then, we extracted the whole corpus returned by both engines in
response to all the queries in the set. Note that the data collection process
was done in a controlled environment such that the queries are sent to the
search engines at the same time. For more details about the selection of the
queries and crawling the SERPs, please refer to our previous work (Gezici et
al., 2021). After having crawled all the SERPs returned from both engines and
extracted their contents, we annotated the top 10 documents. We obtained the
stance label of each document with respect to the queries via crowdsourcing.
To label the political perspective of queries, we also used crowdsourcing. To
obtain the political perspective of documents, we transformed the stance
labels into political perspectives based on the perspective of their
corresponding queries. The details about our crowdsourcing campaigns as well
as the transformation process can also be found in (Gezici et al., 2021).

News Articles in Newspapers.:

To enrich the pre-trained BERT model on perspective detection task even more,
we crawled the article contents of the selected newspapers. To choose the
newspapers to be crawled, we prepared a list by utilizing the media bias chart
of AllSides.com All ([n.d.]). AllSides.com is a US-based online website that
shares information and ideas from all sides of the political spectrum with the
purpose of fighting filter bubbles and polarization. The organization has
researched since 2012, uses multiple methodologies to rate media bias and even
achieved a patent on this area. Furthermore, AllSides only focuses on American
sources and its media bias rating scale is based on American politics for the
online news articles. For all these reasons, we chose the media bias chart of
AllSides to determine the political perspective of newspapers which is
displayed in Figure 2. Subsequently, we crawled the textual contents of the
news articles of the corresponding newspapers in the chart by using a Python
library called feedparser 222https://pythonhosted.org/feedparser/ that parses
feeds in all known formats such as Atom, RSS, and RDF. Then, we labelled the
crawled news articles by weak supervision such that labelling the articles of
left/lean-left newspaper as liberal, right/lean-right as conservative, and
center as neutral. We further used this newspaper dataset to fine-tune the
BERT model after the first fine-tuning done with the training data by learning
the topic keywords and semantic relations in the articles, thereby detecting
the political perspectives more accurately.

###  4.2. Classification

To answer the first research question, we used the political perspective
labels of the top 10 documents in the first dataset previously obtained via
crowdsourcing. In the scope of this work, we used the labelled dataset to
obtain those perspectives automatically. For automatic perspective detection,
we used pre-trained Bidirectional Encoder Representations from Transformers
(BERT) model which is proposed by Google that presents state-of-the-art
results in a wide variety of NLP tasks (Devlin et al., 2018). BERT’s key
difference from other deep models is having an attention mechanism called
Transformer that applies bidirectional training as opposed to directional
models to language modelling. In this way, BERT learns contextual relations
between words from all of the surroundings (left and right) of a given word.

We splitted the labelled dataset into train and test sets and fine-tuned the
pre-trained BERT model on the train test. Moreover, for making the BERT learn
semantic clues better in the SERPs, in this work we crawled the newspapers’
contents as the second dataset which is similar to those news articles in the
SERPs, labelled with weak supervision and used this dataset to fine-tune the
BERT model again. To secondly fine-tune the already fine-tuned BERT model, we
used weak supervision labels of the news articles as depicted in Figure 1. As
the lower-quality labels were being updated by the existing BERT model, the
model was expected to learn linguistic clues such as semantic relations and
important keywords which belong to a specific category (conservative, liberal,
or neutral) simultaneously. The second question is highly related to the first
one, thus we directly used the results of the first question which showed the
levels of political bias in the SERPs of Bing and Google, then made a
comparison among them by using the statistical analyis described in Section
3.3.

To address the third question, we used the whole corpus in the first dataset
to investigate if the input bias exists. To track the source of bias, we made
use of the final (fine-tuned twice) BERT model to label all the SERPs in the
first dataset and check if the bias in the top 10 documents is consistent with
the whole corpus.

###  4.3. Results

Table 1. Perspective dataset query | document | Weak Supervision
---|---|---
title |  content | Without | With
|  ✓ |  | 0.647 | 0.692
|  ✓ |  ✓ | 0.692 | 0.692
✓ |  ✓ |  | 0.676 | 0.707
✓ |  ✓ |  ✓ | 0.707 | 0.723

Exp. 1: Fine-tuning BERT on only perspective data using the following inputs:

  1. a)

only document content with title

  2. b)

concatenating query and document content with title

  3. c)

only document title

  4. d)

concatenating query and document title

Exp. 2: Fine-tuning BERT on only news data using the following inputs:

  1. a)

only document content with title

  2. b)

only document title

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