# Goodput Maximization with Quantized Feedback in the Finite Blocklength
Regime for Quasi-Static Channels

Hasan Basri Celebi, and Mikael Skoglund Hasan Basri Celebi was with the KTH
Royal Institute of Technology, Stockholm, Sweden. He is now with ABB AB,
Västerås, Sweden (e-mail: hasan-basri.celebi@se.abb.com). Mikael Skoglund is
with the School of Electrical Engineering and Computer Science, KTH Royal
Institute of Technology, Stockholm, Sweden (e-mail: skoglund@kth.se).This work
was funded in part by the Swedish Foundation for Strategic Research (SSF)
under grant agreement RIT15-0091.

###### Abstract

In this paper, we study a quantized feedback scheme to maximize the goodput of
a finite blocklength communication scenario over a quasi-static fading
channel. It is assumed that the receiver has perfect channel state information
(CSI) and sends back the CSI to the transmitter over a resolution-limited
error-free feedback channel. With this partial CSI, the transmitter is
supposed to select the optimum transmission rate, such that it maximizes the
overall goodput of the communication system. This problem has been studied for
the asymptotic blocklength regime, however, no solution has so far been
presented for finite blocklength. Here, we study this problem in two cases:
with and without constraint on reliability. We first formulate the
optimization problems and analytically solve them. Iterative algorithms that
successfully exploit the system parameters for both cases are presented. It is
shown that although the achievable maximum goodput decreases with shorter
blocklengths and higher reliability requirements, significant improvement can
be achieved even with coarsely quantized feedback schemes.

###### Index Terms:

Channel coding, channel state information, goodput maximization, low-
complexity receivers, quantized feedback, URLLC.

##  I Introduction

The conventional mobile broadband communication systems are characterized to
make sure that they provide high goodput and capacity to the end users.
Although this is still expected in future generation communication systems,
new requirements are also introduced for new applications. Three main usage
scenarios are specified in the $5^{\text{th}}$ generation of mobile
communication systems [1, 2]: (1) enhanced Mobile Broadband (eMBB), which aims
at enabling communication with extremely high data rates across a wide
coverage area, (2) massive Machine-Type Communication (mMTC), where the goal
is to enable massive number of devices to communicate with the network
infrastructure and between each other without any human interaction, and (3)
Ultra-Reliable Low-Latency Communication (URLLC), which provides communication
support for novel traffic types, such as mission-critical applications, with
stringent constraints on reliability and end-to-end latency.

With the recent major releases of 3GPP Radio Access Network working group, to
support the URLLC applications, it is aimed at achieving reliability values
not lower than %$99.9999$ with end-to-end latency not higher than $1$ ms [3,
4, 5]. In order to meet such demands, it is required to increase the
efficiency of the communication system and decrease both the latency and the
overall number of packet errors. However, one major bottleneck in wireless
communication is the presence of channel fading, which is caused by the
multipath propagation and leads to undesired fluctuations in the received
signal power, which may result in loss of the received packet.

In this study, we consider finite blocklength communication over a frequency-
flat quasi-static fading wireless channel where the channel is assumed to stay
constant over some coherence time. One of the most important performance
criteria over quasi-static fading channel is the overall goodput of the
system, which can be measured by the achievable expected rate over a large
number of packet transmissions with variable transmission rates. While
studying the achievable rates over quasi-static fading channels, it is
important to select the right performance limit of the system [6]. For
applications that are latency tolerant, i.e. no constraint on latency is
introduced, it can be assumed that one packet can span over infinitely many
independent fading occurences. In this case, the valid performance limit of
the system can be computed with the ergodic capacity [7]. On the other hand,
for applications that are intolerant to latency, outage capacity is the valid
performance limit [8], which is the maximal transmission rate such that the
probability of the instantaneous mutual information falling below the selected
transmission rate does not exceed a desired outage threshold.

However, when stringent latency constraints are introduced, such as in URLLC
systems, performance limits that are achieved in the infinite blocklength
regime cannot be justified with finite blocklength [9]. Therefore, significant
amount of research on maximal achievable transmission rates for finite
blocklength has been conducted in the recent years. It is shown in [10] that
compared to the asymptotic limits, i.e. infinite blocklength regime, a rate
penalty needs to be paid when transmitting in finite blocklength regime.
Achievable bounds for various channels models with different fading
environments have been presented in [11] and [12].

While characterizing the achievable performance of the system over a fading
channel, accurate knowledge of the channel-state-information (CSI) is required
[13], which can be achieved by transmitting a separate training sequence that
is already known to the receiver [14]. However, CSI at the transmitter is also
crucial for reliable communication since, with this information, resources and
the transmission strategy can be adapted according to the channel state which
can greatly improve the performance [13]. In this study, it is assumed that
perfect CSI is available at the receiver but only partial knowledge is
available at the transmitter. This assumption is a realistic scenario for
URLLC applications where CSI is crucial at the transmitter to maintain the
high-reliability level however transmission of the full CSI cannot be afforded
due to the latency requirements. Thus, based on this partial information, we
develop an information-theoretic approach to investigate the achievable
goodput of the system over a large number of blocks transmitted at variable
rates.

###  I-A Related Work and Motivation

Fixed-transmission rate communication systems with perfect CSI over slowly
fading channels are studied in [15]. It is shown that the outage probability
can be minimized by adapting a power control system when perfect CSI at the
transmitter is available. A great amount of research has also studied the case
where the transmitter has partial CSI information. Notably, the benefit of
sending partial information over the feedback is to reduce the system overhead
[16], compared to the case that each receiver feeds all channel information
back to the transmitter. Minimization of outage probability with fixed
transmission rate, when partial CSI at the transmitter is available, is
studied in [17]. A more systematic feedback scheme is studied in [18] where
the transmitter can only achieve quantized CSI information. In [19],
performance of a wireless communication system with partial CSI information at
the transmitter, which is transmitted over a error-free quantized feedback
channel, is considered in asymptotic regime. An adaptive feedback scheme is
proposed that maximizes the expected rate.

Furthermore, a one-bit feedback scheme, where the user compares the received
signal strength to a threshold and sends back a binary signal informing the
transmitter that if the signal power is above or below the threshold, has been
studied in [20]. Partial CSI information at the transmitter for orthogonal and
non-orthogonal multiple access communication systems is studied in [21] and
the sufficient conditions are presented.

On the other hand, a well-studied sequential feedback system, named as hybrid
automatic repeat request, also considers a communication system with partial
feedback. The performance of hybrid automatic repeat request protocols in
infinite blocklength regime is studied in [22]. Performance comparison between
the communication systems with quantized feedback and hybrid automatic repeat
request protocol is studied in [23], where a hybrid system is proposed.
Recently, performance of communication systems with hybrid automatic repeat
request protocol in finite blocklength regime is studied in [24].

###  I-B Our Work and Contributions

In this work, we propose an adaptive quantized feedback scheme in the finite
blocklength regime that exploits the maximum goodput by searching the optimum
selections of quantization regions with corresponding transmission rates. We
first formulate the goodput maximization problem and then analytically solve
it. An iterative algorithm that achieves the optimal feedback scheme is also
proposed. A close approximation on the optimal selection of the transmission
rate on each quantization region is introduced.

Next, we study the second optimization problem being the quantized feedback
scheme design to maximize the goodput with reliability constraint. In that
case we first study the allocation of the maximum error allowance for each
quantization region. There, we propose a waterfilling-like algorithm to assign
the maximum error probability to each quantization region. We then
analytically solve the optimization problem with the augmented Lagrangian
method. An iterative algorithm that can achieve the maximum goodput while
guaranteeing the reliability constraint is proposed. The feasibility
conditions are also presented. Notably, to broaden the generality of the
study, the feedback design problem is explicitly formulated, where no specific
channel distribution is taken into account. Thus, results are valid for any
continuous channel distribution.

##  II System Model

We consider the discrete time complex baseband single-input single-output
wireless communication system illustrated in Fig. 1. A codeword of blocklength
$n$ is transmitted over a quasi-static fading channel, where the complex
valued channel coefficient $h$ is an independent and identically distributed
(i.i.d.) random variable according to some distribution but remains constant
over the codeword transmission. The received signal can be expressed as

$\boldsymbol{y}=h\boldsymbol{x}+\boldsymbol{z},$ (1)

where $\boldsymbol{x}$ and $\boldsymbol{z}$ represent the $n$-length
transmitted codeword and complex Gaussian noise vector where the samples are
i.i.d and $z_{i}\sim CN(0,1)$, where $z_{i}$ represents the $i$th component of
$\boldsymbol{z}$. Let $\gamma=|h|^{2}$ describe the i.i.d channel power.
Notably, $\gamma$ is a continuous random variable with its corresponding
probability density function (pdf), denoted as $f(\gamma)$, and cumulative
distribution function (cdf), $F(\gamma)$. It is assumed that $f(\gamma)$ is
positive over $0\leq\gamma\leq\infty$ and both $f(\gamma)$ and $F(\gamma)$ are
continuous.

It is assumed that the channel coefficient is known perfectly at the receiver.
We consider that the receiver divides the positive real line into $\Phi$
number of quantization regions and applies a deterministic index mapping on
the channel power, given as

$M(\gamma)=i,$ (2)

such that $\gamma\in[\phi_{i},\phi_{i+1})$, $i=1,2,\cdots,\Phi$, where
$\phi_{i}$ and $\phi_{i+1}$ represent the boundary of the $i^{\text{th}}$
quantization region. The selected index, $i$, is then transmitted back to the
transmitter over the error-free quantized feedback channel. Therefore, CSI is
partially known to the transmitter.

Next, a transmission rate $r_{i}$ is selected according to $i$. In this paper,
we study the optimum feedback scheme that maximizes of the goodput of the
communication system, which is the overall correctly received information rate
(in bits-per-second). For instance, for a fixed-rate communication system,
i.e. $\Phi=1$, with error probability $\epsilon$ the goodput of the
communication system can be computed as

$r(1-\epsilon)$ (3)

where $r$ is the transmission rate and $(1-\epsilon)$ represents the success
rate of the communication system. Thus, the goal of the study is to
investigate the optimum selections of $r_{i}$’s and $\phi_{i}$’s such that the
long-term goodput of the communication system is maximized.

A power budget constraint is allocated on the power of the codeword such that

$\lVert\boldsymbol{x}\rVert^{2}\leq P.$ (4)

It is clear that equality in Eq. (4) yields the maximum goodput since there is
no cost on transmission rate is incurred with increasing the power up to the
upper limit.

Next, previously obtained results under the assumption of infinite blocklength
are first introduced and then our contributions are presented.

Figure 1: System model

##  III Infinite Blocklength Analysis

It is shown by Shannon in [25] that there exists a codebook describing the
collection of codewords, with size $2^{nC}$ codewords, such that the codeword
error probability (CEP) vanishes as $n\rightarrow\infty$, where CEP is the
probability that the receiver decides in favor of a codeword that is different
from the one actually sent and $C$ is the channel capacity and represents the
maximum transmission rate for which error-free communication can be achieved.

The goodput of the communication system can be maximized by increasing the
transmission rate up to the channel capacity and the channel capacity can be
maximized by selecting the maximum allowed power $P$. Thus, for a given
channel gain realization $\gamma$, the instantaneous channel capacity can be
computed as

$C(\gamma)=\log\left(1+P\gamma\right).$ (5)

When perfect CSI is available at the transmitter, it is possible to match the
transmission rate according to the instantaneous channel capacity, i.e.
$C(\gamma)$. In this case, the maximum achievable goodput is the ergodic
capacity of the channel which is

$\int_{0}^{\infty}f(\gamma)C(\gamma)\mathrm{d}\gamma.$ (6)

On the other hand, when no CSI is present at the transmitter, the maximum
achievable goodput can be found by solving the following optimization problem

$\underset{r\geq 0}{\text{maximize}}\leavevmode\nobreak\ \leavevmode\nobreak\
\int_{\frac{1}{P}(2^{r}-1)}^{\infty}rf(\gamma)\mathrm{d}\gamma.$ (7)

Notice that Eq. (6) and Eq. (7) provide the best and the worst performance
bounds, i.e. perfect and no CSI at the transmitter, to the quantized feedback
scheme.

Let the transmitter select a capacity-achieving codebook according to the
feedback index $i$, such that111All logarithms in this paper are with base 2.

$r_{i}=\log\left(1+P\gamma_{i}\right),$ (8)

where $\gamma_{i}\in[\phi_{i},\phi_{i+1}]$ represents the selected
reconstruction point of the $i$th quantization region. Now, suppose that a
given channel realization $\gamma$ belongs to the $i$th quantization region,
i.e. $\gamma\in[\phi_{i},\phi_{i+1})$. We know that as long as $r_{i}\leq
C(\gamma)$, error-free transmission is possible. Thus, successful decoding is
only possible when $\gamma\geq\gamma_{i}$. If $\gamma<\gamma_{i}$, the system
is in outage. Once $\gamma_{i}$, $\forall i\in\\{1,2,\cdots,\Phi\\}$, are
selected, the outage probability of the system can be computed as

$\sum_{i=1}^{\Phi}F(\gamma_{i})-F(\phi_{i}).$ (9)

Notice that selecting $\gamma_{i}=\phi_{i}$ yields outage-free transmission.
However, goodput maximization is not guaranteed with this selection.

We, next, formulate the optimization problem on designing the feedback scheme
that maximizes the goodput in the infinite blocklength regime. The optimal
feedback scheme is the solution of the following optimization problem

$\displaystyle\underset{\phi_{i},\gamma_{i},r_{i}}{\text{maximize}}\leavevmode\nobreak\
$ $\displaystyle\leavevmode\nobreak\
\sum_{i=1}^{\Phi}r_{i}\big{(}F(\phi_{i+1})-F(\gamma_{i})\big{)}$ (10a) subject
to $\displaystyle\leavevmode\nobreak\ \phi_{i}\leq\gamma_{i},$ (10b)
$\displaystyle\leavevmode\nobreak\ \gamma_{i}\leq\phi_{i+1}.$ (10c)

This optimization problem has been solved in [19] and [23]. It is shown that
the optimum can be achieved by setting

$r_{i}=\log\left(1+P\phi_{i}\right)$ (11)

and the reconstruction points are found by an iterative algorithm that can
solve the following equation

$F(\phi_{i+1})=F(\phi_{i})+\frac{1}{P}f(\phi_{i})(1+\phi_{i}P)\log\left(\frac{1+\phi_{i}P}{1+\phi_{i-1}P}\right),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \text{for}\leavevmode\nobreak\
i=1,2,\cdots,\Phi,$ (12)

where the first quantization region does not necessarily start from zero and
therefore a new variable $\phi_{0}=0$ is introduced. Thus, the overall outage
probability of the optimum feedback scheme is $F(\phi_{1})$ since data
transmission with zero outage probability occurs in all quantization regions
except the region between $(\phi_{0},\phi_{1})$. When the channel power
coincides within this region, the communication system is in outage and data
is lost. Notably, the equation in (12) can be achieved by applying the Karish-
Kuhn-Tucker (KKT) conditions to the optimization problem. An iterative
algorithm is proposed in [19] to find the optimum reconstruction points.

##  IV Finite Blocklength Analysis

Now assume a blocklength constrained communication system where $n$ is not
allowed to take arbitrarily large values. In information theoretic studies,
this is called the finite blocklength regime and, in this regime, $C(\gamma)$
is not a correct measure to calculate the maximal allowed transmission rate.
In this case, the achievable transmission rate with codewords of length $n$
with non-zero CEP $\epsilon$ can be closely approximated by the normal
approximation, which can be computed as

$R(n,\gamma,\epsilon)=C(\gamma)-\sqrt{\frac{V(\gamma)}{n}}Q^{-1}(\epsilon)+O\left(\frac{\log
n}{n}\right).$ (13)

where the quantity $V(\gamma)$ is called the channel dispersion and, for AWGN
channel, it is obtained as

$V(\gamma)=\left(1-\frac{1}{(1+\gamma P)^{2}}\right)(\log\exp(1))^{2},$ (14)

and $Q^{-1}(\cdot)$ is the inverse of the Gaussian $Q-$function

$Q(t)=\frac{1}{\sqrt{2\pi}}\int_{t}^{\infty}\exp\left(-\frac{u^{2}}{2}\right)\mathrm{d}u.$
(15)

The expression in (13) is termed as the normal approximation to the maximal
transmission rate in the finite blocklength regime. Notice that
$R(n,\gamma,\epsilon)$ increases unboundedly as $\epsilon\rightarrow 1$.
However, with strict CEP requirements, i.e., small $\epsilon$, the second term
in Eq. (13) introduces a back-off from $C(\gamma)$ to ensure the transmission
achieves the reliability requirements with $n$ blocklength. Thus, the
achievable rate decreases substantially. On the other hand, the achievable
rate increases monotonically as the blocklength $n$ increases and converges to
Shannon’s capacity formula in the case of asymptotically long blocklength,
i.e., $n\rightarrow\infty$.

Although Eq. (13) gives a close approximation on the limit of achievable
rates, it does not tell how to achieve it [26, 27]. In the recent years,
several encoding and decoding schemes have been proposed that can perform
close to Eq. (13). Their performance comparisons are depicted in [2, 28, 29].
In this paper, it is assumed that the transmitter selects finite-length
codewords from a codebook that can achieve the achievable rates given by Eq.
(13).

For a given transmission rate $r$, the probability of decoding errors can then
be found by solving Eq. (13) for $\epsilon$

$\epsilon=\Omega(\gamma,r),$ (16)

where

$\Omega(\gamma,r)=Q\left(\big{(}C(\gamma)-r\big{)}\sqrt{\frac{n}{V(\gamma)}}\right)$
(17)

The behavior of $\Omega(\gamma,r)$ is depicted in Fig. 2 for several
blocklength values. As expected, in the asymptotic regime, i.e.,
$n\rightarrow\infty$, error-free transmission can be achieved when $r\leq C$.
On the other hand, if $r>C$, no information can be sent. Unlike the asymptotic
regime, in the finite blocklength regime, information transmission becomes
possible even for transmission rates higher than the capacity, albeit at the
expense of high CEP values. On the other hand, even for rates lower than the
capacity, CEP may not vanish. This result shows that the optimal feedback
scheme in the asymptotic regime may not correspond to the optimal selections
for finite blocklength.

Figure 2: Differences between the CEP characteristics in infinite and finite
blocklength regimes.

###  IV-A Finite Blocklength Analysis in Quasi-Static Channels

The instantaneous capacity of a channel, given a channel gain $\gamma$, can be
calculated as in Eq. (5). Assuming that a fixed transmission rate $r$ is
selected, there is a certain probability that the transmission is in outage
and reliable communcation is not possible. The outage probability of a
communication system can be calculated as

$\epsilon^{\text{out}}=\mathbb{P}\\{\log(1+P\gamma)<r\\},$ (18)

where the probability is over the channel gain $\gamma$. Thus, the outage
capacity or the $\epsilon$-capacity of a communication system can be found as
the largest $r$ that fulfills
$\mathbb{P}\\{\log(1+\gamma\mathcal{P})<r\\}\leq\epsilon$:

$C^{\epsilon^{\text{out}}}=\sup\\{r\geq 0:\leavevmode\nobreak\
\mathbb{P}\\{\log(1+P\gamma)<r\\}\leq\epsilon\\}$ (19)

Notice that the outage capacity $C^{\epsilon^{\text{out}}}$ reveals the
maximum transmission rate such that the CEP of the communication system is
$\epsilon$, assuming $n\rightarrow\infty$.

The performance of finite blocklength codes in quasi-static fading channels is
analyzed in [30] and [11], and it is shown that the channel dispersion
$V(\gamma)$ is zero for many classes of channel models such that the
achievable rate can be closely approximated by the outage capacity

$R\left(n,\gamma,\epsilon^{\text{out}}\right)=C^{\epsilon^{\text{out}}}+O\left(\frac{\log
n}{n}\right).$ (20)

Notice that since the dispersion is zero, no additional term on $V(\gamma)/n$
appears in Eq. (20). This shows that $R(n,\gamma,\epsilon)$ in Eq. (20)
converges to the outage capacity $C^{\epsilon^{\text{out}}}$ faster than
$R(n,\gamma,\epsilon)$ in Eq. (13) converges to the capacity $C(\gamma)$.
Thus, one can conclude that the blocklength in the finite blocklength regime
under quasi-static fading channels has smaller impact on the achievable rate.

However, this result does not hold for all types of channels. Consider, for
example, a Rician fading channel with $h\sim\mathcal{CN}(\mu,\sigma^{2})$. In
case the line-of-sight component $\mu$ remains constant and the variance
$\sigma^{2}$ of the non-LoS component converges to zero, the channel
approaches to an AWGN channel, where the maximum achievable rate is given by
Eq. (13). In that case, the convergence to the capacity occurs on the order of
$1/\sqrt{n}$, not $(\log n)/n$. Therefore, the expression in (20) does not
hold for such a channel.

To extend the normal approximation in (13) for different types of quasi-static
fading channels, the following expression is proposed in [11]

$\epsilon=\mathbb{E}\big{[}\Omega(\gamma,r)\big{]},$ (21)

where the expectation is taken with respect to the distribution of the random
channel gain $\gamma$. Hence, the overall success rate of the communication
system can be calculated as

$1-\epsilon=\int_{0}^{\infty}f(\gamma)\big{(}1-\Omega(\gamma,r)\big{)}\mathrm{d}\gamma.$
(22)

##  V Goodput Maximization with Quantized Feedback

The overall success rate given in Eq. (22) is achieved when the transmitter
selects a fixed rate. A better performance can be achieved if the transmitter
has knowledge on the channel gain. In this section, we investigate the goodput
in finite blocklength regime for the system model presented in Fig. 1.

Given a transmission rate $r_{i}$ and by using Eq. (22), the goodput of the
quantization region between $\phi_{i}$ and $\phi_{i+1}$ yields

$r_{i}\int_{\phi_{i}}^{\phi_{i+1}}f(\gamma)\big{(}1-\Omega(\gamma,r_{i})\big{)}\mathrm{d}\gamma.$
(23)

The total goodput of the communication system can be calculated by summing Eq.
(23) over all $i$’s. The integral in Eq. (23) can be decomposed as

$\int_{\phi_{i}}^{\phi_{i+1}}f(\gamma)\big{(}1-\Omega(\gamma,r_{i})\big{)}\mathrm{d}\gamma=F(\phi_{i+1})-F(\phi_{i})-\int_{\phi_{i}}^{\phi_{i+1}}f(\gamma)\Omega(\gamma,r_{i})\mathrm{d}\gamma$
(24)

Notably, the objective function defined for infinite blocklength regime is
equivalent to the expression above without the third term. The third term in
Eq. (24) introduces a backoff due to the finite blocklength. Notice that
setting $r_{i}=0$ yields the goodput in Eq. (23) to be zero. As $r_{i}$
increases, Eq. (23) increases up to some value but then, it starts to decrease
due to the third term since it can be shown that

$\lim\limits_{r_{i}\rightarrow\infty}\int_{\phi_{i}}^{\phi_{i+1}}f(\gamma)\Omega(\gamma,r_{i})\mathrm{d}\gamma=F(\phi_{i+1})-F(\phi_{i}),$
(25)

which reduces the goodput to zero.

To demonstrate the differences between the achievable goodput values with
infinite and finite blocklength, we show a case study where $h$ is Rician with
$K=10$, $K$ being the ratio between the power in the direct path and the power
in the scattered paths, and therefore $\gamma$ is a noncentral-$\chi^{2}$
distributed random variable with

$f_{\chi^{2}}(\gamma)=(K+1)\exp\big{(}-\gamma(K+1)+K\big{)}I_{0}\left(2\sqrt{\gamma
K(K+1)}\right),$ (26)

where $I_{0}(\cdot)$ is the zero-order modified Bessel function of the first
kind. We select $\Phi=3$ and the quantization regions are
$\phi_{i}=\\{0,0.9,1.2,\infty\\}$, not necessarily being the optimum
selections. The achievable goodput values for each quantization region with
respect to $r_{i}\in[0,5]$ are depicted in Fig. 3. More specifically, the
black solid lines and dashed red lines represent the results achieved with Eq.
(10a) and Eq. (23), respectively. Maximums are highlighted with square and
circle markers.

Figure 3: An illustration of the achievable goodput values with infinite and
finite blocklengths for $\Phi=3$ where $\phi_{i}=\\{0,0.9,1.2,\infty\\}$.

Fig. 3 reveals several important findings. It can be seen that there are
significant differences between the maximum achievable goodput values for
finite and infinite blocklength regimes at each quantization region. Due to
the backoff, maximum achievable goodput is lower in finite blocklength for all
quantization regions. In addition, one can end up with even lower goodput
values if the optimal scheme for asymptotic regime is applied for finite
blocklength. On the other hand, this difference is slightly smaller for $i=1$
but increases as $i$ increases. Similar empirical results are observed in
various different channel types. Thus, the impact of finite blocklength is
smaller for quantization regions with lower channel power, i.e. $i=1$, whereas
it increases for quantization regions with higher channel power, i.e. $i>1$.
This is due to the fact that the channel dispersion, $V(\gamma)$, is shown to
be zero when the channel experiences deep fading, which is the case when
$i=1$. However, when $i>1$, since the channel gain becomes higher, the channel
converges to an AWGN channel, i.e. $V(\gamma)\neq 0$, and the impact of finite
blocklength becomes extremely important.222A more thorough analysis is
presented in Fig. 6(b) in Sec. VII.

###  V-A Problem Formulation

The optimization problem on the maximization of the goodput in finite
blocklength regime can be formulated as

$\displaystyle\underset{r_{i},\phi_{i}}{\text{maximize}}\leavevmode\nobreak\
\leavevmode\nobreak\ $
$\displaystyle\sum_{i=1}^{\Phi}r_{i}\int_{\phi_{i}}^{\phi_{i+1}}f(\gamma)\big{(}1-\Omega(\gamma,r_{i})\big{)}\mathrm{d}\gamma$
(27a) s.t. $\displaystyle r_{i}\geq 0,$ (27b)
$\displaystyle\phi_{i+1}\geq\phi_{i},$ (27c)

where the maximization is done over the variables $r_{i}$ and $\phi_{i}$ with
the constraints that are listed in Eq. (27b) and Eq. (27c),
respectively.333Notice that no constraint on the CEP is introduced in this
optimization problem. Therefore this problem may not be suitable for URLLC
applications where stringent requirements on reliability is introduced.
Instead, this optimization problem is more appropriate for low-latency
applications. However, we first investigate this optimization problem and
present the results for the completeness of the study.

The proposed optimization problem is difficult to solve since $r_{i}$ and
$\phi_{i}$ are coupled together. To circumvent this difficulty, we introduce a
block coordinate descent based iterative solution where we optimize one set of
variables while keeping the others fixed, and vice versa [31, 32]. Then, each
sub-problem is solved in an iterative manner. Below, the sub-optimization
problems are introduced

Optimization problem 1:

$\displaystyle\underset{r_{i}}{\text{maximize}}\leavevmode\nobreak\
\leavevmode\nobreak\
\sum_{i=1}^{\Phi}r_{i}\int_{\phi_{i}}^{\phi_{i+1}}f(\gamma)\big{(}1-\Omega(\gamma,r_{i})\big{)}\mathrm{d}\gamma\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \text{s.t.}\leavevmode\nobreak\ \leavevmode\nobreak\
\eqref{eq_opt_prob_1_cons_1}.$ (28)

Optimization problem 2:

$\displaystyle\underset{\phi_{i}}{\text{maximize}}\leavevmode\nobreak\
\leavevmode\nobreak\
\sum_{i=1}^{\Phi}r_{i}\int_{\phi_{i}}^{\phi_{i+1}}f(\gamma)\big{(}1-\Omega(\gamma,r_{i})\big{)}\mathrm{d}\gamma\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \text{s.t.}\leavevmode\nobreak\ \leavevmode\nobreak\
\eqref{eq_opt_prob_1_cons_2}.$ (29)

Next, we investigate these two optimization problems.

###  V-B Problem 1: Optimization of Transmission Rate

######  Theorem 1.

For fixed $\phi_{i}$, the optimum $r_{i}$ can be found by solving the
following equation

$r_{i}=-\frac{\int_{\phi_{i}}^{\phi_{i+1}}f(\gamma)\big{(}1-\Omega(\gamma,r_{i})\big{)}\mathrm{d}\gamma}{\int_{\phi_{i}}^{\phi_{i+1}}H(\gamma,r_{i})\mathrm{d}\gamma},$
(30)

where

$H(\gamma,r_{i})=-\frac{1}{\sqrt{2\pi}}\exp\left(-\frac{n\big{(}C(\gamma)-r_{i}\big{)}^{2}}{2V(\gamma)}\right)f(\gamma)\sqrt{\frac{n}{V(\gamma)}}.$
(31)

###### Proof.

We first investigate the convexity of the objective function with respect to
$r_{i}$. The second-order derivative of the objective function follows

$\frac{\partial^{2}}{\partial
r_{i}^{2}}r_{i}\int_{\phi_{i}}^{\phi_{i+1}}f(\gamma)\big{(}1-\Omega(\gamma,r_{i})\big{)}\mathrm{d}\gamma=2\int_{\phi_{i}}^{\phi_{i+1}}H(\gamma,r_{i})\mathrm{d}\gamma+r_{i}\int_{\phi_{i}}^{\phi_{i+1}}H(\gamma,r_{i})\frac{n\big{(}C(\gamma)-r_{i}\big{)}}{V(\gamma)}\mathrm{d}\gamma$
(32)

where we used the following equality

$\frac{\partial}{\partial
t}Q(t)=-\frac{1}{\sqrt{2\pi}}\exp\left(-\frac{t^{2}}{2}\right).$ (33)

Since $n>0$, $C(\gamma)>0$, $V(\gamma)>0$, and, by definition, $f(\gamma)>0$,
and, however, $H(\gamma,r)$ is always negative. Therefore, the first term in
Eq. (32) is always negative. Similar to the first term, the second term is
also negative as long as $C(\gamma)>r_{i}$. However, depending on the interval
$\phi_{i}$ and $\phi_{i+1}$, the second integral becomes positive as $r_{i}$
increases. Thus, Eq. (32) is negative between $r_{i}\in[0,r_{i}^{\prime}]$ and
the objective function is concave and for higher values of $r_{i}$, Eq. (32)
is positive and the objective is convex, where $r_{i}^{\prime}$ is the point
of inflection and can be found by solving the following equation

$\bigg{[}2\int_{\phi_{i}}^{\phi_{i+1}}H(\gamma,r_{i})\mathrm{d}\gamma\\\
+r_{i}\int_{\phi_{i}}^{\phi_{i+1}}H(\gamma,r_{i})\frac{n\big{(}C(\gamma)-r_{i}\big{)}}{V(\gamma)}\mathrm{d}\gamma\bigg{]}_{r_{i}=r_{i}^{\prime}}=0.$
(34)

We call this function a concave-convex function [33].

Since the function is concave-convex and since it tends to zero as $r_{i}=0$
and $r_{i}\rightarrow\infty$, one can show that the maximum always lies within
the concave region. Therefore, the optimum value of $r_{i}$ that maximizes the
objective function can be found by taking the first derivative, setting it to
zero, and solving it for $r_{i}$, which yields Eq. (30). ∎

Although it is possible to numerically evaluate optimum $r_{i}$, no closed
form expression can be achieved due to the integrals. Next, we derive an
approximation on $r_{i}$.

######  Theorem 2.

A close appoximation on the optimal $r_{i}$ can be achieved with the following
equation

$\Big{[}F(\min\\{\Delta_{2},\phi_{i+1}\\})-F(\max\\{\Delta_{1},\phi_{i}\\})\Big{]}u(\phi_{i+1}-\Delta_{1})u(\Delta_{2}-\phi_{i})\\\
\times\left(\frac{1}{2}+b\Big{(}a-\mathbb{E}\big{[}\gamma\big{|}\max\\{\Delta_{1},\phi_{i}\\}\leq\gamma\leq\min\\{\Delta_{2},\phi_{i+1}\\}\big{]}\Big{)}\right)\\\
+r_{i}\Big{[}F(\phi_{i+1})-F(\max\\{\Delta_{2},\phi_{i}\\})\Big{]}u(\phi_{i+1}-\Delta_{2})\\\
=-r_{i}\int_{\phi_{i}}^{\phi_{i+1}}H(\gamma,r_{i})\mathrm{d}\gamma.$ (35)

where $u(\cdot)$ represents the unit step function and
$\Delta_{1}=\frac{1}{2b}+a$, $\Delta_{2}=-\frac{1}{2b}+a$ with
$a=\frac{1}{P}(\exp(r_{i})-1)$ and
$b=-P\sqrt{\frac{n}{2\pi(\exp(2r_{i})-1)}}$.

###### Proof.

The objective function for single $i$ can be approximated by linearizing the
$\Omega(\cdot)$ function at the point $\gamma=a$. The approximation of the
$\Omega(\cdot)$ function can be written as [34]

$\displaystyle\Omega(\gamma,r_{i})\approx\begin{cases}1&,\leavevmode\nobreak\
\gamma<\Delta_{1},\\\ 1/2+b(\gamma-a)&,\leavevmode\nobreak\
\gamma\in\left[\Delta_{1},\Delta_{2}\right],\\\ 0&,\leavevmode\nobreak\
\gamma>\Delta_{2},\end{cases}$ (36)

where $b$ is the derivative of the $Q(\cdot)$ function at $\gamma=a$, that is

$b=\frac{\partial}{\partial\gamma}\Omega(\gamma,r_{i})\leavevmode\nobreak\
\bigg{|}_{\gamma=a}=-P\sqrt{\frac{n}{2\pi(\exp(2r_{i})-1)}}.$ (37)

Implementing $a$ and $b$ into Eq. (36) and solving the integral yields

$\int_{\phi_{i}}^{\phi_{i+1}}f(\gamma)\big{(}1-\Omega(\gamma,r_{i})\big{)}\mathrm{d}\gamma\approx\begin{cases}0,\leavevmode\nobreak\
\text{if}\leavevmode\nobreak\ \phi_{i}\leq\phi_{i+1}\leq\Delta_{1},\\\
r_{i}\left(\frac{1}{2}+ba-b\mathbb{E}\left[\gamma|\Delta_{1}\leq\gamma\leq\phi_{i+1}\right]\right)\left(F(\phi_{i+1})-F(\Delta_{1})\right),\leavevmode\nobreak\
\\\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \text{if}\leavevmode\nobreak\
\phi_{i}\leq\Delta_{1}\leavevmode\nobreak\ \text{and}\leavevmode\nobreak\
\Delta_{1}\leq\phi_{i+1}\leq\Delta_{2},\\\
r_{i}\left(F(\phi_{i+1}-F(\Delta_{2}))\right)+r_{i}\left(\frac{1}{2}+ba-b\mathbb{E}\left[\gamma|\Delta_{1}\leq\gamma\leq\Delta_{2}\right]\right)\\\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
\times\left(F(\Delta_{2})-F(\Delta_{1})\right),\leavevmode\nobreak\
\text{if}\leavevmode\nobreak\ \phi_{i}\leq\Delta_{1}\leavevmode\nobreak\
\text{and}\leavevmode\nobreak\ \Delta_{2}\leq\phi_{i+1},\\\
r_{i}\left(\frac{1}{2}+ba-b\mathbb{E}\left[\gamma|\phi_{i}\leq\gamma\leq\phi_{i+1}\right]\right)\left(F(\phi_{i+1})-F(\phi_{i})\right),\leavevmode\nobreak\
\\\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \text{if}\leavevmode\nobreak\
\Delta_{1}\leq\phi_{i}\leq\phi_{i+1}\leq\Delta_{2},\\\
r_{i}\left(F(\phi_{i+1})-F(\Delta_{2})\right)+r_{i}\left(\frac{1}{2}+ba-b\mathbb{E}\left[\gamma|\phi_{i}\leq\gamma\leq\Delta_{2}\right]\right)\\\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
\times\left(F(\Delta_{2})-F(\phi_{i})\right),\leavevmode\nobreak\
\text{if}\leavevmode\nobreak\
\Delta_{1}\leq\phi_{i}\leq\Delta_{2}\leq\phi_{i+1},\\\
r_{i}\left(F(\phi_{i+1})-F(\phi_{i})\right),\leavevmode\nobreak\
\text{if}\leavevmode\nobreak\
\Delta_{2}\leq\phi_{i}\leq\phi_{i+1},\end{cases}$ (38)

where

$\mathbb{E}\left[\gamma|\gamma_{1}\leq\gamma\leq\gamma_{2}\right]=\frac{1}{F(\gamma_{2})-F(\gamma_{1})}\int_{\gamma_{1}}^{\gamma_{2}}\gamma
f(\gamma)\mathrm{d}\gamma.$ (39)

represents the expected value of the truncated random variable $\gamma$ such
that $\gamma$ can only have values between
$\gamma_{1}\leq\gamma\leq\gamma_{2}$. Merging all cases in Eq. (38) leads to

$\int_{\phi_{i}}^{\phi_{i+1}}f(\gamma)\big{(}1-\Omega(\gamma,r_{i})\big{)}\mathrm{d}\gamma\approx\Big{[}F(\phi_{i+1})-F(\max\\{\Delta_{2},\phi_{i}\\})\Big{]}u(\phi_{i+1}-\Delta_{2})\\\
+\Big{[}F(\min\\{\Delta_{2},\phi_{i+1}\\})-F(\max\\{\Delta_{1},\phi_{i}\\})\Big{]}u(\phi_{i+1}-\Delta_{1})u(\Delta_{2}-\phi_{i})\\\
\times\left(\frac{1}{2}+b\Big{(}a-\mathbb{E}\big{[}\gamma\big{|}\max\\{\Delta_{1},\phi_{i}\\}\leq\gamma\leq\min\\{\Delta_{2},\phi_{i+1}\\}\big{]}\Big{)}\right).$
(40)

Finally, setting the first order derivative of Eq. (40) to zero yields Eq.
(35). ∎

###  V-C Problem 2: Optimization of Quantization Regions

Now we investigate the second step of the optimization problem where $r_{i}$
is fixed but we let $\phi_{i}$ to vary. We then assume that a constraint
qualification holds at the maximizers of (13), so that the KKT conditions are
necessary optimality conditions.

######  Theorem 3.

For fixed $r_{i}$, the optimal quantization regions can be calculated by
solving the following equation for $\phi_{i}$

$\frac{r_{i-1}}{r_{i}}=\frac{1-\Omega(\phi_{i},r_{i})}{1-\Omega(\phi_{i},r_{i-1})}.$
(41)

###### Proof.

Let $\lambda_{i}\geq 0$ represents the Lagrangian multiplier associated with
the CEP constraints. The Lagrangian function of (29) is

$\mathcal{L}=\sum_{i=1}^{\Phi}r_{i}\int_{\phi_{i}}^{\phi_{i+1}}f(\gamma)\big{(}1-\Omega(\gamma,r_{i})\big{)}\mathrm{d}\gamma-\sum_{i=1}^{\Phi}\lambda_{i}(\phi_{i}-\phi_{i+1})$
(42)

where $\lambda_{i}\geq 0$ are the non-negative Lagrange multipliers that are
associated with the constraint in Eq. (27c).

By a direct investigation of the activeness of the linear constraint, we can
write the KKT conditions as

$\displaystyle\frac{\partial}{\partial\phi_{i}}\mathcal{L}=r_{i-1}\big{(}1-\Omega(\phi_{i},r_{i-1})\big{)}f(\phi_{i})-r_{i}\big{(}1-\Omega(\phi_{i},r_{i})\big{)}f(\phi_{i})-\lambda_{i}+\lambda_{i-1}=0,$
(43) $\displaystyle\lambda_{i}(\phi_{i}-\phi_{i+1})=0,$ (44)

where in Eq. (43) we used the Leibniz integral rule which states that the
derivative of a definite integral can be expressed as

$\frac{\partial}{\partial
v}\left(\int_{u_{1}(v)}^{u_{2}(v)}h(v,t)\mathrm{d}t\right)=\int_{u_{1}(v)}^{u_{2}(v)}\frac{\partial}{\partial
v}h(v,t)\mathrm{d}t+h(v,u_{2}(v))\frac{\partial}{\partial
v}u_{2}(v)-h(v,u_{1}(v))\frac{\partial}{\partial v}u_{1}(v).$ (45)

Solving the KKT conditions in Eq. (43) and (44) yields Eq. (41). ∎

Solving Eq. (41) for $\phi_{i}$ leads to the quantization scheme that
maximizes the overall goodput with the sub-optimum rate selections, which are
also updated with every step of the iteration of the algorithm. Here, one can
set $\phi_{1}=0$ and $\phi_{\Phi+1}=\infty$, and Eq. (41) needs to be solved
for all $\phi_{i}$, $i=2,\cdots,\Phi$ in every iteration. Notably, an
interesting outcome of Eq. (41) is that the optimum selection of $\phi_{i}$
given $r_{i}$ and $r_{i-1}$ does not depend on $f(\gamma)$.

Algorithm 1 Goodput maximization

1:Initialize $\phi_{i}$ for $i=1,2,\cdots,\Phi+1$

2:repeat

3: for $i=1,2,\cdots,\Phi$ do

4: Solve (35) for $r_{i}$

5: end for

6: for $i=\Phi,\Phi-1,\cdots,2$ do

7: Solve (41) for $\phi_{i}$

8: end for

9:until Convergence

The proposed algorithm is shown in Algorithm 1. It can be seen that the
iterative algorithm continues to run until the quantization regions achieve a
steady state in which the minimum difference between the values of the updated
quantization regions at two subsequent iterations is less than some previously
setted threshold value, denoted as $\tau$.

##  VI Goodput Maximization with Constraint on CEP

The maximization problem studied in Sec. V does not impose a constraint on the
CEP. However, reliability in a URLLC system is crucial and such a constraint
needs to be implemented. In this section, we jointly optimize the goodput and
CEP.

###  VI-A Problem Formulation

With a fixed rate $r_{i}$, the CEP at the $i^{\text{th}}$ quantization region
can be calculated as

$\epsilon_{i}=\int_{\phi_{i}}^{\phi_{i+1}}f(\gamma)\Omega(\gamma,r_{i})\mathrm{d}\gamma$
(46)

and the overall CEP of the system is the sum of Eq. (46) over all $i$’s. Thus,
with the reliability constraint, the optimization problem can now be
formulated as

$\displaystyle\underset{r_{i},\phi_{i},\epsilon_{i}}{\text{maximize}}\leavevmode\nobreak\
\leavevmode\nobreak\ $
$\displaystyle\sum_{i=1}^{\Phi}r_{i}\int_{\phi_{i}}^{\phi_{i+1}}f(\gamma)\big{(}1-\Omega(\gamma,r_{i})\big{)}\mathrm{d}\gamma$
(47a) s.t. $\displaystyle\sum_{i=1}^{\Phi}\epsilon_{i}\leq\varepsilon_{m}$
(47b)
$\displaystyle\eqref{eq_opt_prob_1_cons_1},\eqref{eq_opt_prob_1_cons_2},$

where Eq. (47b) specifies the maximum error probability. Notably, Eq. (47b)
introduces contraint on the sum of the CEPs of the quantization regions. This
constraint increases the complexity of the optimization problem such that the
optimum cannot be achieved due to the intractable complexity. However, in
order to relax the optimization problem, we introduce a new variable
$\varepsilon_{i}$, which denotes the CEP constraints on the $i^{\text{th}}$
quantization region such that the sum of $\varepsilon_{i}$’s is less than or
equal to the overall CEP constraint $\varepsilon_{m}$. Thus, the optimization
problem follows

$\displaystyle\underset{r_{i},\phi_{i},\epsilon_{i},\varepsilon_{i}}{\text{maximize}}\leavevmode\nobreak\
\leavevmode\nobreak\ $
$\displaystyle\sum_{i=1}^{\Phi}r_{i}\int_{\phi_{i}}^{\phi_{i+1}}f(\gamma)\big{(}1-\Omega(\gamma,r_{i})\big{)}\mathrm{d}\gamma$
(48a) s.t. $\displaystyle\epsilon_{i}\leq\varepsilon_{i},$ (48b)
$\displaystyle\sum_{i=1}^{\Phi}\varepsilon_{i}\leq\varepsilon_{m},$ (48c)
$\displaystyle\varepsilon_{i}>0,$ (48d)
$\displaystyle\eqref{eq_opt_prob_1_cons_1},\eqref{eq_opt_prob_1_cons_2}.$

Notice that $\varepsilon_{i}$, for $i=1,2,\cdots,\Phi$, are introduced as
variables and the optimum selection of $\varepsilon_{i}$ is also unknown.

(a)

(b)

Figure 4: An illustration of the proposed CEP assignment when $\Phi=4$, $h$ is
Rician with $K=10$, and $\phi_{i}=\\{0,0.5,1,2,\infty\\}$ for $i=1,\cdots,5$.
(a) Truncated mean values are depicted with respect to the quantization
regions. Here, the pdf of the channel power is also shown with the blue solid
line. (b) The assigned CEP values are shown with respect to the quantization
regions, where $\epsilon_{m}=10^{-5}$.

###  VI-B Problem 1: CEP Allocation

In this section, we evaluate the impact of the CEP allocation on the goodput
maximization problem with quantized feedback. Assuming fixed $\phi_{i}$, we
cast the following CEP allocation problem

$\displaystyle\underset{r_{i},\epsilon_{i},\varepsilon_{i}}{\text{minimize}}\leavevmode\nobreak\
\leavevmode\nobreak\ $
$\displaystyle\sum_{i=1}^{\Phi}r_{i}\int_{\phi_{i}}^{\phi_{i+1}}f(\gamma)\Omega(\gamma,r_{i})\mathrm{d}\gamma$
(49) s.t.
$\displaystyle\eqref{eq_opt_prob_5_const_1},\eqref{eq_opt_prob_5_const_2},\eqref{eq_opt_prob_5_const_3},\eqref{eq_opt_prob_1_cons_1}.$

Notice that this problem is exponentially complex to find the optimal
solution. However, we provide a suboptimal algorithm that can find a solution
to the problem.

Let us first assume that the optimum is achieved with equality for the
constraint in Eq. (48b) noting that this assumption is a valid assumption for
URLLC systems where $\varepsilon_{m}$ is a small value, i.e. in the order of
$10^{-6}$ or even lower. In this case, we relax the problem and introduce the
following

$\displaystyle\underset{\varepsilon_{i}}{\text{minimize}}\leavevmode\nobreak\
\leavevmode\nobreak\ \sum_{i=1}^{\Phi}r_{i}\varepsilon_{i}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \text{s.t.}\leavevmode\nobreak\ \leavevmode\nobreak\
\eqref{eq_opt_prob_5_const_2},\eqref{eq_opt_prob_5_const_3},\eqref{eq_opt_prob_1_cons_1}.$
(50)

Assuming that $r_{i}$ and $\epsilon_{i}$ are independent variables, optimum
can only be found if $r_{i}$’s are all equal. In this case, the optimum is
$\varepsilon_{i}=\varepsilon_{m}/\Phi$. However, for any other cases no
optimum solution can be found since the problem is unbounded and for any
target value there would be feasible points with objective values smaller than
the target.

Recall that $r_{i}$ and $\epsilon_{i}$ are connected according to Eq. (46).
Using this relationship a grid search algorithm can be proposed. However, a
significant drawback here is the accuracy of the normal approximation since,
as discussed in [35], the normal approximation becomes inaccurate for very
small blocklength, i.e., $n<100$, and for low CEP , i.e., $\epsilon<10^{-5}$.
Therefore, in this section, we propose a sub-optimal waterfilling-like
algorithm where we set each $\varepsilon_{i}$ inversely proportional to the
truncated mean of $\gamma$ within the quantized region.

For the system model presented in Eq. (1), the optimum power selection for the
$m^{\text{th}}$ codeword transmission with the waterfilling algorithm can be
found by solving [19, 20, 24]

$P_{m}=\left[\frac{1}{\delta}-\frac{1}{h_{m}^{2}}\right]^{+},$ (51)

where $[z]^{+}=\max\\{0,z\\}$ and $\delta$ satisfies

$\frac{1}{M}\sum_{m=1}^{M}\left[\frac{1}{\delta}-\frac{1}{h_{m}^{2}}\right]^{+}=nP,$
(52)

where $M$ represents the total number of transmissions. Similar to
waterfilling algorithm, here, we introduce the scaling coefficient $\alpha$
such that

$\alpha=\frac{\varepsilon_{m}}{\sum_{i=1}^{\Phi}\mathbb{E}\left[\gamma_{i}|\phi_{i}\leq\gamma_{i}\leq\phi_{i+1}\right]^{2}}$
(53)

and the CEP constraint on the $i^{\text{th}}$ quantization region is

$\varepsilon_{i}=\frac{\alpha}{\mathbb{E}\left[\gamma_{i}|\phi_{i}\leq\gamma_{i}\leq\phi_{i+1}\right]^{2}}.$
(54)

Thus, the CEP constraint on the $i^{\text{th}}$ quantization region is
inversely related with the truncated mean of the random variable $\gamma$.
With this method, it is also guaranteed that $\varepsilon_{i}>0$ for all $i$
values. An illustration of the proposed method is depicted in Fig. 4 for
Rician channel with $K=10$ when $\Phi=4$. Notice that the quantization regions
are separated at $\phi_{i}=\\{0,0.5,1,2,\infty\\}$ and the truncated mean
values are depicted with respect to the quantization regions. Based on the
truncated means, the assigned CEP values can be seen in Fig. 4(b) with respect
to the quantization regions, where $\epsilon_{m}=10^{-5}$.

Next, we introduce a block coordinate descent based iterative algorithm where
for each step the variables are updated respectively.

###  VI-C Problem 2: Optimization of Transmission Rate

Given a set of $\phi_{i}$’s, one can estimate the required CEP constraints on
every $i^{\text{th}}$ quantization region. Next, we search for the optimal
$r_{i}$ that maximizes the goodput while still maintaining the CEP
constraints. The optimization problem can be written as

$\displaystyle\underset{r_{i}}{\text{maximize}}\leavevmode\nobreak\
\leavevmode\nobreak\
r_{i}\int_{\phi_{i}}^{\phi_{i+1}}f(\gamma)\big{(}1-\Omega(\gamma,r_{i})\big{)}\mathrm{d}\gamma\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \text{s.t.}\leavevmode\nobreak\
\leavevmode\nobreak\
\eqref{eq_opt_prob_5_const_1},\eqref{eq_opt_prob_1_cons_1},$ (55)

where the maximization operation is done over a single $r_{i}$ since the
overall maximization of the goodput is equivalent to the maximization of every
$i^{\text{th}}$ goodput separately.

A two-step straightforward solution to this problem is: (_i_) Follow Theorem 1
and find $r_{i}^{\prime}$ that maximizes the objective function without any
constraint on CEP. (_ii_) Check if $r_{i}^{\prime}$ satisfies the CEP
constraint on the $i^{\text{th}}$ quantization region. If it does not,
gradually decrease $r_{i}^{\prime}$ until the constraint is met.

######  Remark 1.

If the CEP constraint cannot be achieved with an $r_{i}>0$, one can only
select $r_{i}=0$ which yields the CEP of the $i$th quantization region to be

$F(\phi_{i+1})-F(\phi_{i}).$ (56)

In this case, if $F(\phi_{i+1})-F(\phi_{i})>\varepsilon_{i}$, no feasible
solution can be achieved with this particular selections of
$\varepsilon_{i}$’s. A possible solution is to update $\phi_{i}$’s and
$\varepsilon_{i}$’s accordingly.

###  VI-D Problem 3: Optimization of Quantization Regions

Next, we search for the optimal $\phi_{i}$’s with the following optimization
problem

$\displaystyle\underset{\phi_{i}}{\text{maximize}}$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\sum_{i=1}^{\Phi}r_{i}\int_{\phi_{i}}^{\phi_{i+1}}f(\gamma)\big{(}1-\Omega(\gamma,r_{i})\big{)}\mathrm{d}\gamma$
(57a) s.t. $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\eqref{eq_opt_prob_5_const_1},\eqref{eq_opt_prob_1_cons_2}.$ (57b)

This optimization problem belongs to the class of non-convex optimization
problem, since the objective and the constraint in (57) are non-convex in
$\phi_{i}$. Although the Lagrangian duality theory has been proven to be an
effective tool for solving the convex optimization problems with the primal or
primal-dual computational methods, these methods are not powerful in a non-
convex problem due to the non-zero duality gap between the primal and dual
problems [36]. In order to reduce the duality gap for non-convex problems the
augmented Lagrangian method is introduced [37]. In this method, the duality
gap is reduced by adding a penalized function, which consists of a quadratic
term, to the Lagrangian.

The Lagrangian of the optimization problem in (57) can be defined by

$\mathcal{L}=\sum_{i=1}^{\Phi}r_{i}\left(F(\phi_{i+1})-F(\phi_{i})-\int_{\phi_{i}}^{\phi_{i+1}}f(\gamma)\Omega(\gamma,r_{i})\right)\\\
-\sum_{i=1}^{\Phi}\lambda_{i}\left(\int_{\phi_{i}}^{\phi_{i+1}}f(\gamma)\Omega(\gamma,r_{i})\mathrm{d}\gamma-\varepsilon_{i}\right)\\\
-\sum_{i=1}^{\Phi}\mu_{i}\left(\phi_{i}-\phi_{i+1}\right),$ (58)

for all $\lambda_{i}\in\mathbb{R}^{+}$ and $\mu_{i}\in\mathbb{R}^{+}$. The
augmented Lagrangian method eliminates the constraints and adds them into the
objective function. Thus, the augmented Lagrangian function can be written as

$\mathcal{L}^{a}=\sum_{i=1}^{\Phi}r_{i}\left(F(\phi_{i+1})-F(\phi_{i})-\int_{\phi_{i}}^{\phi_{i+1}}f(\gamma)\Omega(\gamma,r_{i})\right)\\\
-\frac{\rho}{2}\Bigg{(}\sum_{i=1}^{\Phi}\Bigg{(}\bigg{[}\beta\left(\int_{\phi_{i}}^{\phi_{i+1}}f(\gamma)\Omega(\gamma,r_{i})\mathrm{d}\gamma-\varepsilon_{i}\right)-\frac{\lambda_{i}}{\rho}\bigg{]}^{+}\Bigg{)}^{2}-\sum_{i=1}^{\Phi}\left(\left[\phi_{i}-\phi_{i+1}-\frac{\mu_{i}}{\rho}\right]^{+}\right)^{2}\Bigg{)},$
(59)

where $\rho>0$ is the penalty parameter and $\beta>0$ is the normalization
factor since the the CEP values can be very small compared to the other
penalty terms. $\lambda_{i}$ and $\mu_{i}$ are the Lagrangian dual variables
associated with the constraints in (57). It is shown in [38] that the
augmented Lagrangian $\mathcal{L}^{a}$ is locally convex when the penalty
terms are sufficiently large.

Algorithm 2 Goodput maximization with constraint on maximum CEP

1:Initialize $\phi_{i}$ for $i=1,2,\cdots,\Phi+1$

2:Set $\varepsilon_{i}$ for $i=1,2,\cdots,\Phi$ accordingly

3:repeat

4: for $i=1,2,\cdots,\Phi$ do

5: Update $\varepsilon_{i}$ according to (54)

6: end for

7: for $i=1,2,\cdots,\Phi$ do

8: Solve (35) for $r_{i}$

9: if $\epsilon_{i}>\varepsilon_{i}$ then

10: repeat

11: $r_{i}=r_{i}-\tau$

12: if $r_{i}==0$ then

13: Update $\phi_{i}$ and $\phi_{i+1}$, and go to 4

14: end if

15: until $\epsilon_{i}\leq\varepsilon_{i}$

16: end if

17: end for

18: for $i=\Phi,\Phi-1,\cdots,2$ do

19: Set $\lambda_{i}^{(1)},\mu_{i}^{(1)}$, and $\rho^{(1)}$

20: repeat

21: Solve (60) for $\phi_{i}$

22: Update $\lambda_{i}^{(l+1)},\mu_{i}^{(l+1)}$, and $\rho^{(l+1)}$ according
to (61), (62), and (63)

23: until Convergence

24: end for

25:until Convergence

The solution to the problem can now be approximated by solving

$\underset{\phi_{i}}{\text{maximize}}\leavevmode\nobreak\ \leavevmode\nobreak\
\mathcal{L}^{a}.$ (60)

The essential feature of this method is that after each unconstrained
maximization, which yields a maximization point $\phi_{i}$, the variables
$\rho$, $\lambda_{i},$ and $\mu_{i}$ are updated by means of the iteration.
Thus, the Lagrangian dual variables are updated according to

$\lambda_{i}^{(l+1)}=\Bigg{(}\bigg{[}\rho^{(l)}\beta\bigg{(}\int_{\phi_{i}}^{\phi_{i+1}}f(\gamma)\Omega(\gamma,r_{i})\mathrm{d}\gamma-\varepsilon_{i}\bigg{)}-\lambda_{i}^{(l)}\bigg{]}^{+}\Bigg{)}^{2},$
(61)
$\mu_{i}^{(l+1)}=\left(\left[\rho^{(l)}(\phi_{i}-\phi_{i+1})-\mu_{i}^{(l)}\right]^{+}\right)^{2}$
(62)

and the penalty parameter $\rho$ is updated as

$\rho^{(l+1)}=2\rho^{(l)},$ (63)

where the upper-script in $z^{(l)}$ represents the variable $z$ at the $l$th
iteration step.444A discussion on the initialization of the Lagrangian dual
variables can be found in [37, Ch. 4]. Notice that, with the augmented
Lagrangian method, constraint violations are increasingly penalized as the
penalty parameter $\rho$ increases in every iteration step. This imposes the
maximizer of Eq. (60) to be in the feasible region.

The proposed method is described in Algorithm 2. It can be seen that at each
iteration, as $\phi_{i}$’s and $r_{i}$’s change, $\varepsilon_{i}$ values are
also updated accordingly. The algorithm terminates as convergence is achieved.
Otherwise, no feasible solution is found. Notably, the optimal solution is
achieved by using some well-known optimization methods, which are the block
coordinate descent type method and the augmented Lagrangian method, and the
convergence of these methods are well documented in the literature, [37] and
[31].

##  VII Numerical Results and Discussions

(a)

(b)

Figure 5: The behavior of the algorithm for the feedback scheme with $\Phi=3$
and $h$ is Rician distributed with $K=10$. (a) The contour of the goodput for
all possible selections of $\phi_{2},\phi_{3}\in[0,5]$. The path of updated
$\phi_{2}$ and $\phi_{3}$ in every subsequent iteration until convergence is
shown with the blue line with circles. (b) (LHS) Achievable goodput and (RHS)
the corresponding CEP of the system versus iteration number are depicted.

(a)

(b)

Figure 6: (a) Tightness of the proposed finite blocklength approximation in
Theorem 2. Numerical comparisons for Rayleigh and Rician channels are depicted
for $P=10$ dB and $K=10$. (b) Comparison of the proposed method with the
benchmark. The goodput gain represents the difference between the achievable
goodput values of the proposed method and the benchmark in bpcu. Results are
computed for $n=64$, $P=10$ dB and $K=10$.

The behavior of the Algorithm 1 for a simple case where $\Phi=3$ is shown in
Fig. 5, where the horizontal axes represent the quantization variables
$\phi_{2}$ and $\phi_{3}$ and the vertical axis represents the
goodput.555Notice that for $\Phi=3$ only two quantization variables, which are
$\phi_{2}$ and $\phi_{3}$, need to be optimized. The contour surface in Fig.
5(a) depicts the maximum achievable goodput according to the selections of
$\phi_{2}$ and $\phi_{3}$. The solid line with circles shows the path of
updated $\phi_{2}$ and $\phi_{3}$ in every subsequent iteration until the
algorithm terminates. Notice that no values are shown in the upper left half
of Fig. 5(a) since the numerical selections of $\phi_{2}$ and $\phi_{3}$ in
this region violate the constraint in Eq. (27c) and therefore no feasible
result can be achieved. In this example, the initial values of $\phi_{2}$ and
$\phi_{3}$ are selected to be $\phi_{2}=5$ and $\phi_{3}=10$. The channel
coefficient $h$ is selected to be distributed according to the Rician
distribution with $K=10$. Selections of $\phi_{2}$ and $\phi_{3}$ in every
iteration are shown with a circle in Fig. 5(a). As can be seen, the algorithm
can efficiently track the path to the optimum and reach it.

(a)

(b)

Figure 7: Maximum goodput achieved without any CEP constraint over Rayleigh
and Rician channels with different feedback schemes, $\Phi=\\{1,2,4\\}$.
Values for finite $n$ are depicted with blue solid lines, where we show
results for $n=\\{32,64,128\\}$. Notice that, for all cases, goodput values
for $n=128$ are the highest ones and $n=32$ are the lowest. For comparison,
the ergodic capacity and achievable goodput in asymptotic regime are also
shown with dash-dotted and dashed lines, respectively. (a) Rayleigh channel.
(b) Rician channel with $K=10$ dB.777 We used the red color to show the
maximum achievable goodput values for $\Phi=4$ in Fig. 7(b) for finite
blocklength results since they are mixed with the results of $\Phi=2$.

The corresponding goodput values after each iteration are also shown in Fig.
5(b), where one can see that the algorithm is able to find better selections
of $\phi_{i}$’s and $r_{i}$’s in every iteration such that they lead to higher
goodput and finally reach the optimal values. Moreover, we also show the
overall CEP value of the system with the corresponding feedback scheme with
the blue solid line in Fig. 5(b). It can be seen that as the goodput
increases, the overall CEP decreases and reaches to a steady state after 50
iterations.

The tightness of the approximation that is presented in Theorem 2 is shown in
Fig. 6(a) for different Rayleigh and Rician channel conditions with different
blocklength values. One can identify that Eq. (35) provides accurate
approximation for the $Q$-function and the results of the algorithm are tight.

It is also significant to compare the proposed method with proposed scheme
with a benchmark scheme to show the superiority of the proposed method. Here,
we derive the benchmark scheme by setting $\phi_{i}$’s according to the
solution of the asymptotic regime that is presented in Eq. (11) and (12) but
transmitting short blocklength at rate $R(n,\phi_{i},\epsilon)$, from Eq.
(13), when the channel gain is $\gamma\in[\phi_{i},\phi_{i+1}]$. Goodput gain,
which is the difference between the goodput value calculated with the proposed
method and the benchmark, for Rayleigh and Rician channels are depicted in
Fig. 6(b) for $n=64$ and $\Phi=\\{2,4,8\\}$. It is important to observe that
the gain is positive for all cases which means that the proposed method
superior compared to the benchmark. Additionally, one can also see that the
gain increases with higher SNR and higher quantization regions. And the gain
is also higher in Rician channels compared to Rayleigh channel at the same SNR
and $\Phi$, which is an expected result due to the effect of the channel
dispersion.

Next, we show the maximum goodput values achieved by several feedback schemes
with different numbers of quantization regions and blocklengths in Fig. 7.
Results for Rayleigh and Rician channels are depicted in Fig. 7(a) and Fig.
7(b), respectively, for $\Phi=\\{1,2,4\\}$ and $n=\\{32,64,128\\}$. For
comparison purposes, the ergodic capacity and achievable goodput values in the
asymptotic regime [19] are also plotted.

Figure 8: The effect of $\Phi$ on the goodput. Results are computed for Rician
channel with $K=10$ and $P=10$ dB.

(a)

(b)

Figure 9: Maximum goodput values with and without constraint on CEP over
Rician channel with $K=10$ dB and (a) $P=10$ dB, (b) $P=20$ dB. Results for
different feedback schemes with $\Phi=\\{2,4\\}$ and $n=\\{32,64,128\\}$ are
depicted with respect to $\varepsilon_{m}$, ranging from $10^{-1}$ to
$10^{-7}$.

Notice that $\Phi=1$ represents the case where no CSI at the transmitter is
available. Thus, it is important to observe that even with coarsely quantized
systems, significant gains for both channel distributions are observed in both
finite and infinite blocklength regimes. For example, in infinite blocklength
regime, to achieve a target goodput of 1 bits per channel use (bpcu), feedback
schemes with $\Phi=2$ and $\Phi=4$ require a power of roughly 2 and 3 dB less
than a $\Phi=1$ system does, respectively. Similar advantages are also
observed for finite blocklength cases. It can be seen that with the quantized
feedback scheme, the goodput of the communication system can be significantly
increased. As expected, for all cases, i.e. fixed SNR and $\Phi$, goodput with
infinite blocklength are higher than finite blocklength. Thus, longer
blocklength results in higher goodput. Except the case with Rayleigh
distribution and $\Phi=1$. In this case, differences among all schemes are
negligible. This is due to the fact that the dispersion of the channel is
zero, which is explained in detail in Sec. V. Therefore, the blocklength of
the channel code does not have any impact on the achievable rate. Furthermore,
one can also observe that the gap between the finite and infinite blocklength
cases is higher in Rician channel compared to Rayleigh. This is because
channel dispersion is higher for Rician with high $K$ and therefore
blocklength has higher impact on the achievable rate. For instance, the
goodput gap between infinite blocklength and $n=128$ for Rayleigh channel with
$\Phi=4$ at $10$ dB SNR is $0.11$ bpcu. However, this gap increases to $0.16$
bpcu for Rician channel.

An interesting and relevant investigation would be on the effect of the number
of quantization regions, $\Phi$, on the goodput performance. This comparison
is shown in Fig. 8. Here, the achievable goodput values for Rician channel
with $K=10$ and $P=10$ dB are depicted for various $n$ and feedback schemes
with different $\Phi$ values. For comparison purposes the ergodic capacity,
which does not depend on $\Phi$, and asymptotic results that are achieved with
(11) and (12) are also added. The important outcome from Fig. 8 is that the
goodput significantly increases as $\Phi$ increases when $\Phi$ is small,
whereas this increment is not significant as $\Phi$ is big. Thus, based on the
outcome of this scenario, one can say that the optimal $\Phi$ is 16, after
which the performance does not significantly improve.

Figure 10: Convergence of Algorithm 2. $\Phi=4$ | $\Phi=8$
---|---
$\varepsilon=10^{-4}$ | $\varepsilon=10^{-5}$ | $\varepsilon=10^{-6}$ | $\varepsilon=10^{-4}$ | $\varepsilon=10^{-5}$ | $\varepsilon=10^{-6}$
0.0019 | 0.0021 | 0.0103 | 0.0037 | 0.0084 | 0.0213
Table I: Optimality gap between the proposed algorithm and the optimal
solutions.

Up to now we showed the empirical results without any constraint on CEP. Next,
we focus on the second optimization problem where restriction on CEP is
imposed. In this case, we particularly focus on Rician channel with $K=10$ dB
with $P=\\{10,20\\}$ dB SNR. Maximum goodput values achieved by different
schemes with CEP constraint are plotted in Fig. 9 for $n=\\{32,64,128\\}$
where the horizontal axis represents the maximum allowed CEP value
$\varepsilon_{m}\in[10^{-1},10^{-7}]$. Algorithm 2 is used to solve the
optimization problem with CEP constraint. Additionally, the ergodic capacity
and achievable goodput values with no CEP constraint, which are attained by
Algorithm 1, are also plotted for comparison purposes. Since these values do
not change with $\varepsilon_{m}$, they all have the same value for different
$\varepsilon_{m}$ and are represented with a straight horizontal line in Fig.
9.

Notice that, in Fig. 9, significant goodput gaps between the achievable values
with and without CEP constraint are observed. This gap increases as
$\varepsilon_{m}$ decreases. For instance, to achieve $\epsilon_{m}=10^{-3}$
with $n=128$ when $P=10$ dB, it is required to introduce $0.53$ and $0.26$
bpcu of rate backoffs when $\Phi=2$ and $\Phi=4$, respectively. However, if
$\epsilon_{m}=10^{-5}$, these gaps become $1$ and $0.53$ bpcu for $\Phi=2$ and
$\Phi=4$, respectively. Thus, it is shown that imposing a restriction on CEP
reduces the maximum goodput of the system. In other words, the feedback scheme
achieves the required reliability by decreasing the transmission rate. Notably
these gaps are even higher when $P=20$ dB. Furthermore, it can also be seen
that for very low reliability values, i.e. $\epsilon_{m}<10^{-2}$, goodput
results with CEP constraint converge to the results achieved without a CEP
constraint, since the overall CEP that is achieved with Algorithm 1 is already
lower than $\varepsilon_{m}$.

Recall that Algorithm 2 can only find a feedback scheme when a feasible set
that fulfills the constraints is achieved, otherwise no feasible set can be
constructed. This problem, for instance, arises for $n=128$ and $P=10$ dB when
$\varepsilon_{m}<10^{-6}$. In this case, no results are shown. Similarly, when
$n=64$ and $n=32$, no solution is achieved when $\varepsilon_{m}<2.5\times
10^{-6}$ and $\varepsilon_{m}<6\times 10^{-6}$, respectively. One possible
solution to this problem is to increase the SNR of the received codeword.
Thus, as seen in Fig. 9(b), it is possible to achieve a feedback scheme for
$n=128$ and $\varepsilon_{m}=10^{-7}$ when $P=20$ dB. Similar results are
observed for $n=64$ and $n=32$.

We finally discuss about the convergence of Algorithm 2. Goodput values
achieved at every iteration are shown in Fig. 10 for $\Phi=\\{4,8\\}$,
$n=128$, and Rician fading channel with $K=10$, $P=10$ dB under various
reliability constraints. It is observed that the proposed algorithm converges
and a steady state is achieved after 100 iterations in all cases. In addition,
to see the optimality gap between the optimal solution and the scheme found by
Algorithm 2, we performed an exhaustive grid search and list the gaps in Table
I.

##  VIII Conclusions

Goodput maximization in the finite blocklength regime for a communication
scenario with partial CSI knowledge at the transmitter over quasi-static
channel is studied in this paper. An adaptive quantized feedback scheme that
exploits the maximum goodput by searching the optimum selections of
quantization regions with their corresponding transmission rates is formulated
and investigated. This problem is then analytically solved and an iterative
algorithm is proposed. Next, optimal quantized feedback scheme design for
goodput maximization is studied under reliability constraint. A sub-optimum
CEP allocation technique is presented. Then, the problem is analytically
solved with the augmented Lagrangian method and an iterative algorithm that
can achieve the maximum goodput while guaranteeing the reliability constraint
is proposed. Lastly, numerical results of the optimization problem with and
without constraint on CEP are presented. It is shown that significant
improvement in goodput can be achieved even with coarsely quantized feedback
schemes. However, differences between the achievable goodput results in
asymptotic and non-asymptotic regimes are observed. It is shown that these
differences vary with different channel distributions. Furthermore, we noticed
that when constraint on reliability is imposed, the feedback scheme achieves
the required reliability by decreasing the transmission rate and therefore
maximum achievable goodput decreases with higher reliability constraint.
Finally, it is worth mentioning that results that are presented in this paper
are valid for any continuous channel distributions since the optimization
problems are explicitly formulated without any specific channel distribution.

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11institutetext:  Pattern Recognition Lab, Friedrich-Alexander-Universität
Erlangen-Nürnberg, Erlangen, Germany
Department of Dermatology, Universitätsklinikum Erlangen, Erlangen, Germany
Department Artificial Intelligence in Biomedical Engineering, FAU Erlangen-
Nürnberg, Erlangen, Germany

# Comparative Analysis of Radiomic Features and
Gene Expression Profiles in Histopathology Data
Using Graph Neural Networks

Luis Carlos Rivera Monroy 1122 Leonhard Rist 11 Martin Eberhardt 22 Christian
Ostalecki 22 Andreas Bauer 22 Julio Vera 22 Katharina Breininger 33 and
Andreas Maier 11 112233<EMAIL_ADDRESS>
###### Abstract

This study leverages graph neural networks to integrate MELC data with
Radiomic-extracted features for melanoma classification, focusing on cell-wise
analysis. It assesses the effectiveness of gene expression profiles and
Radiomic features, revealing that Radiomic features, particularly when
combined with UMAP for dimensionality reduction, significantly enhance
classification performance. Notably, using Radiomics contributes to increased
diagnostic accuracy and computational efficiency, as it allows for the
extraction of critical data from fewer stains, thereby reducing operational
costs. This methodology marks an advancement in computational dermatology for
melanoma cell classification, setting the stage for future research and
potential developments.

##  1 Introduction

In post-genomic research, a primary objective is elucidating the relationship
between molecular networks and their corresponding cells or tissues. These
networks, crucial for specific cellular functions, depend on proteins being
precisely positioned, temporally synchronized, and in the correct
concentration for effective interactions [1].

Understanding and constructing protein networks by measuring and describing
local interactions is vital for deeper understanding of diseases. A
significant advancement in this field has been the development of imaging
technologies, such as fluoroscopy, which marked a breakthrough by enabling the
affinity analysis of multiple proteins. However, the capacity of these
technologies to represent the complexity of cellular networks is limited [2].
Addressing this, methods like Multi-Epitope-Ligand Cartography (MELC) have
enhanced the depth of tissue characterization through iterative staining,
enabling more comprehensive analysis [3, 4].

MELC represents an improvement over traditional imaging techniques by
providing detailed tissue characterization at a cellular level. When combined
with machine learning, MELC enables detailed cell-wise classification based on
gene expression profiles, offering invaluable insights for medical research,
including treatment effects and prognosis evolution [5, 6].

Recent advancements in oncology have leveraged Radiomics combined with machine
learning for precision diagnostics. Studies by Liang et al., Pattarone et al.,
Gomez et al., and Mercaldo et al. highlight the significant role of Radiomics
in cancer research, providing new diagnostic and predictive tools through
machine-learning classifiers [7, 8, 9, 10].

In melanoma, the most lethal skin cancer, prognosis traditionally depends on
expert analysis of clinical history and histopathology images [11]. Albrecht
et al. emphasize the importance of feature standardization in mathematical
models for melanoma assessment, considering the subtype diversity and the need
for detailed lesion characterization [12]. Graph-based methods are emerging,
combining biological parameters with various domain features for a more
generalized approach.

This study investigates semi-supervised cell classification in MELC pathology
samples by focusing on two types of features: gene expression profiles derived
from the multiplex digital image and Radiomics. These features are utilized in
both traditional machine learning models and a graph neural network to
evaluate their comparative performance. Following the methodology outlined by
our previous work [5], this research also incorporates dimensionality
reduction techniques to assess their impact on classification accuracy, mainly
when applied in conjunction with Graph Neural Networks.

##  2 Materials and methods

###  2.1 Dataset

This investigation evaluated the proposed methodology using a dataset designed
for cell-wise classification, comprising tissue samples from suspected
melanoma cases. The dataset, included specimens from 27 cases, 20 of which
were confirmed melanoma instances, with the remainder being healthy tissue
samples [5]. Each specimen was processed following the MELC protocol, which
involved incubation with affinity reagents, application of fluoroscope-coupled
antibodies, capturing images through immunofluorescence microscopy, and gently
bleaching the staining agents, allowing for the application of up to 100
different staining agents. In this study, the samples were exposed to 80-85
antigens, and their fluorescent responses were digitized.

The resulting images were digitally captured for analysis, each with a high
resolution of 0.45 $\mu m/pixel$ and dimensions of $2018\times 2018$ pixels.
In cases where tumors were identified, expert segmentation was performed by a
medical professional, thus maintaining consistency with the approach
established in our previous work and facilitating a direct comparison and
follow-up of the findings.

###  2.2 Graph structure

In our 2023 study, we demonstrated the effective use of graph-based models to
encapsulate complex cellular data, specifically in melanoma [5]. This research
utilized Cellpose [13] for cell segmentation, employing the MELC technique to
analyze cellular structures. Two distinct graph representations were explored:
the first, grounded in feature similarity and spatial proximity, adheres to
the methodology established by Wolf et al. [14]; the second utilizes cell
spatial coordinates, aligning with the approach of Palla et al. [15], and
adopts a neighborhood size as defined in our previous publication [5].

In these models, individual cells are conceptualized as nodes within a graph,
with a typical sample containing approximately 2,080 cells. The relationships
between these cells, or edges, are delineated based on the aforementioned
criteria of similarity or proximity. For training, the graph is structured to
comprise 40,500 nodes and 202,500 edges, providing a comprehensive framework
for analyzing cellular interactions in melanoma.

###  2.3 Dimensionality Reduction

Integrating Radiomic features into neighborhood graph-based approaches
significantly enhances cell classification in histopathological analysis.
These features include first-order intensity statistics, shape-based
descriptors, and complex texture features. Notably, texture features from the
Gray Level Co-occurrence Matrix (GLCM) and the Gray Level Run Length Matrix
(GLRLM) provide a detailed characterization of cellular attributes [16]. GLCM
analyzes the spatial relationship of pixel intensities, while GLRLM focuses on
the length of consecutive pixels having the same intensity in an image. When
applied as node attributes in graphs, these features impart a nuanced
understanding of each cell, addressing challenges like initialization
sensitivity and suboptimal local minima in graph optimization [17].

Furthermore, advanced dimensionality reduction techniques such as uniform
manifold approximation and projection (UMAP) and t-distributed stochastic
neighbor edging (tSNE) enhance this process. These techniques effectively
compress high-dimensional data, facilitating the creation of detailed yet
computationally efficient neighborhood graphs [18, 19]. This combination of
image analysis and graph-based modeling improves cell classification accuracy
and provides deeper insights into biological structures in histopathological
images [20]. The resulting graph models, enriched with Radiomic data, become
potent tools for identifying patterns in cell populations, enhancing
diagnostic precision and research in histopathology [18, 19].

###  2.4 Graph Random Neural Network

In the domain of semi-supervised learning for graph-structured data, the Graph
Random Neural Networks (GRAND) architecture, as introduced by Feng et al.
(2020), has demonstrated notable efficacy, particularly in weakly supervised
tasks, robust performance, and efficient training [21]. This architecture
addresses critical limitations inherent in traditional graph neural networks
(GNNs), such as over-smoothing, non-robustness, and inadequate generalization
in scenarios with sparse labeled nodes. GRAND distinguishes itself by
strategically decoupling feature propagation from transformation, a design
choice that mitigates complexity and over-smoothing issues prevalent in
conventional GNNs. Central to its success in graph-structured data analysis,
GRAND’s methodology integrates a random propagation strategy using DropNode to
create perturbed feature matrices, enhancing robustness and employing
consistency regularization to maintain uniform classification confidence
across nodes and their neighborhoods.

Figure 1: Computational Pipeline for MELC Data-Based Cell Classification. A)
Input: Multiple MELC sample images. B) Semi-supervised cell instance
segmentation. C) Feature Extraction: Radiomic features and Gene expression
profiling. D) Features encoded in a Graph Neural Network (GRAND). E) Output:
Classification of cells as cancer positive or negative

###  2.5 Experimental Design

In this study, we adhered to the methodology for performance evaluation as
outlined by our previous work, employing a data split of 70% for training, 10%
for validation, and 20% for testing [5]. This distribution was designed to
ensure a balanced representation of healthy and melanoma tissues across each
dataset segment. The optimization of hyperparameters was methodically
conducted on the training set, utilizing a Bayesian search strategy.

We employed XGBoost and Random Forest for baseline comparisons, widely
recognized as standard machine-learning algorithms for handling tabular data.
The graph neural network (GNN) training followed the original authors’
implementation guidelines and a similar approach to hyperparameter
optimization [21]. Our analysis aims to compare the efficacy of two distinct
types of features: gene expression profiles derived from multi-array imaging
and Radiomic-generated features, which encompass first-order intensity
statistics, shape-based descriptors, and complex texture features. This
comparative study is designed to elucidate these feature sets’ relative
strengths and applications in the context of our machine learning models. An
overview of the proposed pipeline can be seen in Figure 1.

##  3 Results

As outlined in Table 1, a discernible trend indicates enhanced performance in
node classification for melanoma cells when utilizing Radiomic features
combined with UMAP for training. This observation is drawn from a comparative
evaluation of two feature types in the GNN framework and their interactions
with various dimensionality reduction techniques.

##  4 Discussion

The analysis of our results, as detailed in Table 1, demonstrates that graph
representations incorporating cell spatial locations outperform those based on
gene expression profiles alone. This finding supports the importance of
spatial context in cell classification. We observed that methods using tabular
profiles and feature graphs show similar performance, attributed to their
comparable information encoding strategies.

In dimensionality reduction, UMAP emerges as the superior technique, aligning
with previous findings on its ability to capture spatial patterns and cellular
neighborhoods effectively [5]. Conversely, tSNE focuses on maintaining feature
correlations, offering value but less impacting classification performance.

A pivotal discovery in our study is the consistent effectiveness of Radiomic
features over direct gene expression profiles from digital pathological
samples. This trend is consistent across various methods and types of
dimensionality reduction, and Radiomic features achieve enhanced results when
combined with these techniques. Addressing limitations in staining agent
selection, Radiomic-extracted features emerge as a viable solution, extracting
relevant biological features for melanoma cell classification. This research
marks an advancement in computational dermatology, suggesting new avenues for
more precise and efficient melanoma cell classification techniques.

Table 1:  Comparative Analysis of Gene Expression Profiles and Radiomic
Features in Training Graph Convolutional Neural Networks. This table
summarizes the results of training graph convolutional neural networks using
two distinct data sources: gene expression profiles derived from images and
features extracted via Radiomics.

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