# An Unsupervised Deep Unfolding Framework for Robust Symbol Level Precoding

Abdullahi Mohammad, Christos Masouros, and Yiannis Andreopoulos

###### Abstract

Symbol Level Precoding (SLP) has attracted significant research interest due
to its ability to exploit interference for energy-efficient transmission. This
paper proposes an unsupervised deep-neural network (DNN) based SLP framework.
Instead of naively training a DNN architecture for SLP without considering the
specifics of the optimization objective of the SLP domain, our proposal
unfolds a power minimization SLP formulation based on the interior point
method (IPM) proximal ‘log’ barrier function. Furthermore, we extend our
proposal to a robust precoding design under channel state information (CSI)
uncertainty. The results show that our proposed learning framework provides
near-optimal performance while reducing the computational cost from
$\mathbcal{O}(n^{7.5})$ to $\mathbcal{O}(n^{3})$ for the symmetrical system
case where $n=\text{number of transmit antennas}=\text{number of users}$. This
significant complexity reduction is also reflected in a proportional decrease
in the proposed approach’s execution time compared to the SLP optimization-
based solution.

###### Index Terms:

Symbol level precoding, Constructive Interference, downlink beamforming, power
minimization, Deep Neural Networks.

##  I Introduction

Interference has been known to yield a decrease in the throughput and
communication reliability of a downlink multi-user multiple-input single-
output (MU-MISO) wireless system. Traditionally, interference is regarded as
the limiting factor against the ever-increasing needs for transmission rates
and quality of service (QoS) in fifth-generation (5G) wireless communication
systems and beyond [1, 2, 3]. However, recent studies on interference
exploitation have transformed the traditional paradigm in which known
inferences are effectively managed [1, 2, 4, 3, 5]. Consequently, transmit
beamforming techniques for the downlink channels for power minimization
problems under specific QoS become imperative for high-throughput systems
under interference.

The idea of exploiting interference was first introduced by Masouros and
Alsusa [6], where instantaneous interference was classified into constructive
and destructive. Initial suboptimal approaches to exploit constructive
interference (CI) were first introduced by Masouros et al. [7][8]. The first
form of optimization-based CI precoding was introduced in the context of
vector perturbation precoding through a quadratic optimization approach [9]. A
convex optimization-based CI scheme termed symbol-level-precoding technique
was proposed first with strict phase constraints on the received constellation
point [10], and with a robust relaxed-angle formulation [2]. We refer to
recent work [9, 10, 11, 12, 13] for more details on the optimization-based CI
precoding techniques.

As a result of the performance gains over conventional block-level-precoding
(BLP) schemes, the idea of CI has been applied in many domains, such as vector
perturbation [14], wireless information and power transfer [15], mutual
coupling exploitation [16], multiuser MISO downlink channel [17], directional
modulation [18], relay and cognitive radio [1, 19]. Despite the superior
performance offered by CI-based precoding methods, their increased
computational complexity can hinder their practical application when performed
on a symbol-by-symbol basis. To address this, Li and Masouros [20] proposed an
iterative closed-form precoding design with optimal performance for CI
exploitation in the MISO downlink by driving the optimal precoder’s
mathematical Lagrangian expression and Karush–Kuhn–Tucker conditions for
optimization with both strict and relaxed phase rotations.

Lately, there is growing interest in using deep neural networks (DNNs) for
wireless physical layer design [21, 22, 23]. More relevant to this work are
the learning-based precoding schemes for MU-MISO downlink transmission [24,
25, 26, 27, 28]. The benefit of using DNNs is that the computational burden of
the learning algorithm can be controlled via online training, and a variety of
loss functions can be used for each optimization objective. One of the
earliest attempts of using DNNs models for beamforming design was the work of
Alkhateeb et al. [22], where a learning-based coordinated beamforming
technique was proposed for link reliability and frequent poor hand-off between
base stations (BSs) in millimeter-wave (mmWave) communications. Kerret and
Gesbert [29] introduced DNNs precoding scheme to address the “Team Decision
problems” for a decentralized decision making in multiple-input-multiple-
output (MIMO) settings. Huang et al. [24] proposed a fast beamforming design
based on unsupervised learning that yielded performance close to that of the
weighted minimum mean-squared error (WMMSE) algorithm. A DNN-based precoding
strategy that utilized a heuristic solution structure of the downlink
beamforming was proposed by Huang et al. [27]. Furthermore, Xia et al. [26]
developed deep convolutional neural networks (CNNs) framework for downlink
beamforming optimization. The framework exploits expert knowledge based on the
known structure of optimal iterative solutions for sum-rate maximization,
power minimization, and SINR balancing problems.

DNN methods are typically used for unconstrained optimization problems.
Therefore, most of the DNN-based strategies for wireless physical layer
designs are based on supervised learning to approximate the optimal solutions.
Using such approaches, the constraints are implicitly contained in the
training dataset obtained from conventional optimization solutions. However,
if obtaining optimal solutions via traditional optimization methods is very
computationally expensive (or infeasible), using supervised learning methods
for DNN-based method may not be practical.

Furthermore, the common approach for solving constrained optimization with DNN
for wireless physical layer design is via function approximation. It involves
solving the problem, first using iterative algorithms or convex optimization
techniques, and finally approximating the optimal solution with a DNN
architecture [25, 27, 26]. Accordingly, the major drawback of these proposals
is that the efficacy of supervised learning is bounded by the assumptions and
accuracy of the optimal solutions obtained from the structural optimization
algorithm.

This work proposes an unsupervised learning-based approach for precoding
design by exploiting known interference in MU-MISO systems for the power
minimization problem under SINR constraints. The learning framework is
designed by unfolding an interior point method (IPM) iterative algorithm via
‘log’ barrier function. The proposed learning-based precoding scheme does not
require generating the training dataset from the conventional optimization
solutions, thereby saving considerable computational effort and time. Our
contributions are summarized below:

  * •

We introduce an unsupervised DNN-based power minimization SLP scheme for MU-
MISO downlink transmission. The proposed framework is designed by unfolding an
IPM algorithm via a ‘log’ barrier function that exploits the convexity
associated with the SLP inequality constraints. The learning framework
utilizes the domain knowledge to derive the Lagrange function of the original
SLP optimization as a loss function. This is used to train the network in an
unsupervised mode to learn a set of Lagrangian multipliers that directly
minimize the objective function to satisfy the constraints. A regularization
parameter is added to the Lagrange function to aid the training convergence,
and we provide detailed formulations leading to the unfolded unsupervised
learning architecture for constrained optimization problems.

  * •

We extend the formulation to design a robust learning-based precoder where the
uncertainty in channel estimation is considered.

  * •

We derive analytic expressions for the computational complexity of various SLP
and the proposed unsupervised learning precoding schemes. Our analysis
demonstrates that the proposed deep unfolding (DU) framework offers a
theoretical, computational complexity reduction from $\mathbcal{O}(n^{7.5})$
to $\mathbcal{O}(n^{3})$ for the symmetrical system case where $n$ = number of
transmit antennas = number of users This is reflected in a commensurate
decrease in the execution time as compared to the SLP optimization-based
method.

The remainder of the paper is organized as follows: The system model and the
methods for traditional precoding and SLP optimization-based for downlink MU-
MISO system are presented in Section II. The proposed unsupervised DU-based
precoding designs under perfect channel condition for power minimization are
introduced in Section III and extension to a robust precoding design under
uncertainty channel condition is described in Section IV. Section V presents
detailed analytic computational complexity evaluation of the proposed
precoding schemes. Simulations and results are presented in Section VI.
Finally, Section VII summarizes and concludes the paper.

Notations: We use bold uppercase symbols for matrices, bold lowercase symbols
for vectors and lowercase symbols for scalars. The $\mathbcal{l}_{2}$-norm and
$\mathbcal{l}_{1}$-norm are denoted by $\left\lVert\cdot\right\rVert_{2}$ and
$\left\lVert\cdot\right\rVert_{1}$, respectively. The $|\cdot|$ represents the
absolute value and $\bm{\theta}_{i}$ is the i-th trainable parameter
associated with DNN layers. Operators Re$(\cdot)$ and Im$(\cdot)$ represent
real and imaginary parts of a complex vector, respectively. Finally, notations
$\mathbcal{L}(\cdot)$ and $\mathbcal{H}(\cdot)$ are reserved for the loss and
parameter update functions, respectively.

##  II System Model and Problem Description

###  II-A Conventional Block Level Precoding for Power Minimization

Consider a signle-cell downlink channel with $N_{t}$ transmit antennas at the
BS transmitting to $K$ single-antenna users. Assume a quasi-static flat-fading
channel between the BS and the users, denoted by
$\mathbf{h}_{i}\in\mathbb{C}^{{N}_{t}\times 1}$. The received signal at user
$i$ is given by

$\begin{split}y_{i}&=\mathbf{h}^{T}_{i}\sum\limits^{K}_{k=1}\mathbf{w}_{k}{s}_{k}+v_{i}\\\
&=\mathbf{h}^{T}_{i}\sum_{k=1}^{K}\mathbf{w}_{k}e^{j(\varphi_{k}-\varphi_{i})}{s}_{i}+v_{i}\end{split}$
(1)

where $\mathbf{h}_{i}$, $\mathbf{w}_{i}$, ${s}_{i}$, ${v}_{i}$ and
$\varphi_{i}$ represent the channel vector, precoding vector, data symbol,
received noise and phase rotation for the i-th user. Conventionally, the power
minimization problem seeks to minimize the average transmit power by treating
all interference as detrimental subject to QoS constraints as defined below
[30]

$\displaystyle\underset{\mathbf{\\{w_{i}\\}}}{\text{min}}$
$\displaystyle{\sum_{i=1}^{K}\left\lVert\mathbf{w}_{k}\right\rVert_{2}^{2}}$
(2) s.t.
$\displaystyle\frac{|\mathbf{h}_{i}^{T}\mathbf{w}_{i}|^{2}}{\sum_{k=1,k\neq
i}|\mathbf{h}_{i}^{T}\mathbf{w}_{k}|^{2}+{v}_{0}}\geq\Gamma_{i}\ ,\
\forall{i}.$

where $\Gamma_{i}$ is the SINR threshold of the i-th user. It has been proven
that problem (2) is suboptimal from an instantaneous point of view, as it does
not take into account the fact that interference can constructively enhance
the received signal power [8].

Figure 1: Generic geometrical optimization regions for interference
exploitation for Precoding design in $\mathbb{M}$-PSK [2].

###  II-B Power Minimization via Symbol-Level Precoding

With the aim of utilizing the instantaneous interference in a multi-user
downlink channel scenario, the interference can be categorized into
constructive and destructive based on the known standards described [31].
Based on this, CI is defined as the interference that nudges the received
symbols off the modulated-symbol constellation’s decision thresholds [2]. Fig.
1 shows the generic geometrical representation of the CI, where the received
signal is expressed as
$\bar{\mathbf{y}}_{i}\triangleq\mathbf{h}_{i}^{T}\sum_{k=1}^{K}\mathbf{w}_{k}{e}^{j(\varphi_{k}-\varphi_{1})}$.
From the received symbol expression, the real and imaginary parts are
respectively given by: $\bm{\omega}_{Re}=\text{Re}({\bar{\mathbf{y}}_{i}})$
and $\bm{\omega}_{Im}=\text{Im}({\bar{\mathbf{y}}_{i}})$. In Fig. 1, we show
an indicative example corresponding to the constellation point $1+j$ in the
QPSK constellation, where the green shaded area represents the constructive
region of the constellation, based on the minimum distance $(\tau)$ from the
decision boundaries. The value of $\tau$ is determined by the SNR constraints.
In line with the preceding description, problem (2) is adjusted to include CI
in the power minimization formulation. This allows the interfering signals to
align with the symbol of interest constructively through precoding,
contributing to the desired signal. Therefore, for $\mathbb{M}$-PSK, the power
minimization SLP can be reformulated based on the constructive/destructive
interference classification criteria [32]. The maximum phase shift in the CI
region is given by $\phi=\pm\frac{\pi}{\mathbb{M}}$, where $\mathbb{M}$ is the
modulation order. Therefore, the SLP optimization is given by [2]

$\begin{aligned}
&\underset{\mathbf{\\{w_{i}\\}}}{\text{min}}&&{\left\lVert\sum_{k=1}^{K}\mathbf{w}_{k}{e}^{j(\varphi_{k}-\varphi_{1})}\right\rVert_{2}^{2}}\\\
&\text{s.t.}&&\left|\text{Im}{\left(\mathbf{h}_{i}^{T}\sum_{k=1}^{K}\mathbf{w}_{k}e^{j(\varphi_{k}-\varphi_{i})}\right)}\right|\leq\\\
&&&\left(\text{Re}{\left(\mathbf{h}_{i}^{T}\sum_{k=1}^{K}\mathbf{w}_{k}e^{j(\varphi_{k}-\varphi_{i})}\right)-\sqrt{\Gamma_{i}v_{0}}}\right)\text{tan}\phi\
,\ \forall{i}.\end{aligned}$

(3)

##  III Learning-Based Power minimization for SLP

This section presents the formulation of a learning-based CI power
minimization problem for SLP. Throughout this section, we assume a perfect CSI
known at the BS.

Motivated by the recent adoption of an IPM for image restoration [33], we
propose an unsupervised learning framework that unfolds a constrained
optimization problem into a sequence of learning layers/iterations for a
multi-user MISO beamforming. We first convert (3) into a standard IPM
formulation containing a slack variable, where necessary. The measure of the
fidelity of the solution to (3) is determined by learning a set of penalty
parameters in the form of Lagrange multipliers associated with the
constraints. From (3), we define the following

$\hat{\mathbf{h}}_{i}=\mathbf{h}_{i}\sum_{k=1}^{K}e^{j(\varphi_{k}-\varphi_{i})}$
(4) ${\mathbf{w}}=\sum_{k=1}^{K}\mathbf{w}_{k}.$ (5)

Accordingly, to ease the analysis, we partition the complex rotations into the
real and imaginary parts as follows

$\displaystyle\hat{\mathbf{h}}_{i}$
$\displaystyle=\hat{\mathbf{h}}_{Ri}+j\hat{\mathbf{h}}_{Ii}$ (6a)
$\displaystyle\mathbf{w}$ $\displaystyle=\mathbf{w}_{R}+j\mathbf{w}_{I}$ (6b)

where $\hat{\mathbf{h}}_{Ri}=\text{Re}(\hat{\mathbf{h}}_{i})$,
$\hat{\mathbf{h}}_{Ii}=\text{Im}(\hat{\mathbf{h}}_{i})$,
$\mathbf{w}_{R}=\text{Re}(\mathbf{w})$ and
$\mathbf{w}_{I}=\text{Im}(\mathbf{w})$.

The product of complex rotations of (6a) and (6b) can be written as

$\hat{\mathbf{h}}_{i}\mathbf{w}=(\hat{\mathbf{h}}_{Ri}+j\hat{\mathbf{h}}_{Ii})(\mathbf{w}_{R}+j\mathbf{w}_{I}).$
(7)

Using (6), the real and imaginary parts of (7) can be written in vector forms
as follows

$\displaystyle\text{Re}(\hat{\mathbf{h}}_{i}\mathbf{w})$
$\displaystyle=\begin{bmatrix}\hat{\mathbf{h}}_{Ri}\ \ \hat{\mathbf{h}}_{Ii}\
\end{bmatrix}\begin{bmatrix}\mathbf{w}_{R}\\\ -\mathbf{w}_{I}\ \end{bmatrix}$
(8a) $\displaystyle\text{Im}(\hat{\mathbf{h}}_{i}\mathbf{w})$
$\displaystyle=\begin{bmatrix}\hat{\mathbf{h}}_{Ri}\ \ \hat{\mathbf{h}}_{Ii}\
\end{bmatrix}\begin{bmatrix}\mathbf{w}_{I}\\\ \mathbf{w}_{R}\ \end{bmatrix}$
(8b)

Let $\bm{\Lambda}=[\hat{\mathbf{h}}_{Ri}\ \ \hat{\mathbf{h}}_{Ii}]^{T}$,
$\mathbf{w}_{1}=[\mathbf{w}_{R}\ -\mathbf{w}_{I}]^{T}$ and
$\mathbf{w}_{2}=[\mathbf{w}_{I}\ \ \mathbf{w}_{R}]^{T}$

$\text{Re}(\hat{\mathbf{h}}_{i}^{T}\mathbf{w})=\bm{\Lambda}_{i}^{T}\mathbf{w}_{1}\
\text{and}\
\text{Im}(\hat{\mathbf{h}}_{i}^{T}\mathbf{w})=\bm{\Lambda}_{i}^{T}\bm{\Pi}\mathbf{w}_{1}$
(9)

where

${\mathbf{w}_{2}=\bm{\Pi}}\mathbf{w}_{1}\ \text{and}\
\bm{\Pi}=\begin{bmatrix}\mathbf{O}_{N_{t}}&-\mathbf{I}_{N_{t}}\\\
\mathbf{I}_{N_{t}}&\mathbf{O}_{N_{t}}\end{bmatrix};\
\in\mathbb{R}^{2N_{t}\times 2N_{t}},$ (10)

Note that $\mathbf{I}_{N_{t}}$ is the identity matrix and $\mathbf{O}_{N_{t}}$
the matrix of zeros, respectively. Using the above definitions, problem (3)
can be recast into its mutlicast formulation [2]

$\displaystyle\underset{\mathbf{\\{w_{1}\\}}}{\text{min}}$
$\displaystyle{\left\lVert\mathbf{w}_{1}\right\rVert_{2}^{2}}$ (11) s.t.
$\displaystyle\left|{\bm{\Lambda}_{i}^{T}\bm{\Pi}\mathbf{w}_{1}}\right|\leq\left(\bm{\Lambda}_{i}^{T}\mathbf{w}_{1}-\sqrt{\Gamma_{i}v_{0}}\right)\text{tan}\phi\
,\ \forall{i}$

###  III-A Interior Point Method

Consider a general form of a nonlinear constrained optimization of the form
[34]

$\displaystyle\underset{\mathbf{x\in{\mathbb{R}^{N}}}}{\text{min}}$
$\displaystyle{f(\mathbf{x})}$ (12) s.t. $\displaystyle g(\mathbf{x})\geq 0$
$\displaystyle\left.\begin{aligned} C(\mathbf{x})=0\end{aligned}\right.,$

The rationale of adopting IPM is to substitute the initial constrained
optimization problem by a chain of unconstrained sub-problems of the form

$\displaystyle\underset{\mathbf{x\in{\mathbb{R}^{N}}}}{\text{min}}f(\mathbf{x})+\lambda{{C(\mathbf{x})}}+\mu{{B(\mathbf{x})}}.$
(13)

where ${B}(\cdot)\triangleq-\sum\ln{(\cdot)}$ is the logarithmic barrier
function associated with inequality constraint with unbounded derivative at
the boundary of the feasible domain, ${C}(\cdot)$ is a function associated
with equality constraint, $\mu$ and $\lambda$ are the Lagrangian multipliers
for inequality and equality constraints, respectively. For $K$ users, we
define a vector $\bm{\mu}\triangleq[\mu_{1},\cdots,\mu_{K}]$.

Following the above line of argument, the unconstrained sequence of (11) per
user can be written as

$\displaystyle\underset{\mathbf{w\in{\mathbb{R}^{2N_{t}\times
1}}}}{\text{min}}f(\mathbf{w}_{1})+{\mu}{{B(\mathbf{w}_{1})}},$ (14)

To facilitate the solution of (11) , we introduce additional notations. For
every inequality constraint, $\gamma\in\\{0,+\infty\\}$ and
$\mathbf{w}_{1}\in\mathbb{R}^{2N_{t}\times 1}$, we define the proximity
function as in [34] with respect to (14), which we shall later use to compute
the projected gradient descent as

$\displaystyle\text{prox}_{\gamma{\mu}{B}}{(\mathbf{w}_{1})}=$
$\displaystyle\underset{\mathbf{w_{1}\in{\mathbb{R}^{2N_{t}\times{1}}}}}{\text{argmin}}$
$\displaystyle{\frac{1}{2}\left\lVert\mathbf{w}_{0}-\mathbf{w}_{1}\right\rVert}_{2}^{2}+\gamma{\mu}{B}({\mathbf{w}_{1}}),$
(15)

where $\gamma$ is the step-size for computing the gradients and
$\mathbf{w}_{0}$ is the initial value of the precoding vector. To convert (3)
into its equivalent barrier function problem, we integrate the inequality
constraint into the objective by translating it into a barrier term as follows
[35]

$\displaystyle\underset{\mathbf{w_{1}}}{\text{min}}$
$\displaystyle{f(\mathbf{w}_{1})}-{\mu}\sum_{i=1}^{p}{\ln{\left(g({\mathbf{w}}_{1i})\right)}}$
(16) s.t. $\displaystyle{{C(\mathbf{w}_{1})}=0}$

where
$g({\mathbf{w}}_{1})=\left(\bm{\Lambda}_{i}^{T}\mathbf{w}_{1}-\sqrt{\Gamma_{i}v_{0}}\right)\text{tan}\phi-\left|{\bm{\Lambda}_{i}^{T}\bm{\Pi}\mathbf{w}_{1}}\right|$
and $p$ is the number of the optimization variables.

Going back to our initial SPL optimization to apply this framework, first we
rewrite the constraint of (11) as

${a}\leq{\bm{\Lambda}_{i}^{T}\bm{\Pi}\mathbf{w}_{1}}\leq{b},$ (17)

where

$\displaystyle{a}$
$\displaystyle=-\left(\bm{\Lambda}_{i}^{T}\bm{\Pi}\mathbf{w}_{1}-\sqrt{\Gamma_{i}v_{0}}\right){\text{tan}{\phi}},$
(18a) $\displaystyle{b}$
$\displaystyle=\left(\bm{\Lambda}_{i}^{T}\bm{\Pi}\mathbf{w}_{1}-\sqrt{\Gamma_{i}v_{0}}\right){\text{tan}{\phi}}.$
(18b)

Therefore, the original problem (11) becomes

$\displaystyle\underset{\mathbf{\\{w_{1}\\}}}{\text{min}}$
$\displaystyle{\left\lVert\mathbf{w}_{1}\right\rVert_{2}^{2}}$ (19) s.t.
$\displaystyle{a}\leq{\bm{\Lambda}_{i}^{T}\bm{\Pi}\mathbf{w}_{1}}\leq{b}\ ,\
\forall{i}.$

It is apparent that the constraint of (19) is contained within a hyperslab
[36].

####  III-A1 Hyperslab Constraints

Given the constraint in (19), the precoding vector $\mathbf{w}_{1}$ is
contained within a set of hyperslab $\mathcal{C}$ and also bounded by
$\\{{a},\ {b}\\}$. Therefore, $\mathcal{C}$ is defined as follows

$\mathbcal{C}=\\{\mathbf{w}_{1}\in\mathbb{R}^{N_{t}\times
1}\\}|_{{a}\leq\bm{\Lambda}^{T}\bm{\Pi}\mathbf{w}_{1}\leq{b}}.$ (20)

For all $\gamma>0$ and ${\mu}>0$, a proximity barrier function related to (20)
is given by

$\begin{array}[]{ll}{B}({\mathbf{w}}_{1})=\left\\{\begin{aligned}
&-\ln{\left({b}-\hat{{w}}_{1}\right)}-\ln{\left(-{a}+\hat{{w}}_{1}\right)},&&\text{if}\
-{a}\leq\hat{{w}}_{1}\leq{b}\\\
&+\infty,&&\text{otherwise}\end{aligned}\right.\end{array}$

(21)

where for convenience, we let
$\hat{{w}}_{1}=\bm{\Lambda}_{i}^{T}{\bm{\Pi}}{\mathbf{w}_{1}}$.

###  III-B Proximity Operator for the SLP Formulation

To unfold (19) into learning framework using IPM, we use its equivalent
proximity ‘log’ barrier function (21) and the proximal operator of
$\gamma{\mu}B(\mathbf{w}_{1})$ for every $\mathbf{w}_{1}$ defined as

$\Phi(\mathbf{w}_{1},\gamma,{\mu})=\text{prox}_{\gamma\bm{\mu}{{B}}}{(\mathbf{w}_{1})}=\mathbf{w}_{1}+\frac{X(\mathbf{w}_{1},\gamma,{\mu})-\bm{\Lambda}_{i}^{T}\mathbf{w}_{1}}{\|\bm{\Lambda}_{i}\|_{2}^{2}}{\bm{\Lambda}_{i}}$
(22)

where ${X}$ is a typical solution of the following cubic equation of the form

${x}^{3}-\left({b}+{a}+\bm{\Lambda}^{T}\mathbf{w}_{1}\right){x}^{2}+\\\
\left({b}{a}+\bm{\Lambda}^{T}\mathbf{w}_{1}({b}+{a})-2\gamma{{\mu}}\|\bm{\Lambda}\|_{2}^{2}\right){x}\\\
+\left(-{b}{a}\bm{\Lambda}^{T}\mathbf{w}_{1}+\gamma{\mu}\left({b}+{a}\right)\|\bm{\Lambda}\|_{2}^{2}\right)=0.$
(23)

It is important to note that the solution to (23) is obtained using the
analytic solution of the cubic equation. To build the structure of the
learning framework, as detailed in [33], we need to obtain the Jacobian matrix
of $\Phi(\mathbf{w}_{1},\gamma,{\mu})$ with respect to $\mathbf{w}_{1}$ and
the derivatives with respect to $\gamma$ and $\mu$ as follows

$\mathcal{J}_{{\Phi}}\mid_{(\mathbf{w}_{1})}=\mathbf{I}_{2N_{t}}+\frac{1}{\left\lVert\bm{\Lambda}_{i}\right\rVert_{2}^{2}}\
\times\\\
\left(\frac{\left({b}-X(\mathbf{w}_{1},\gamma,{\mu})\right)\left({a}-X(\mathbf{w}_{1},\gamma,{\mu})\right)}{\Upsilon(\mathbf{w}_{1},\gamma,{\mu})}-1\right)\bm{\Lambda_{i}\Lambda_{i}^{T}}$
(24)

$\Delta_{{\Phi}}\mid_{({{\mu}})}=\frac{{-\gamma}\left({b}+{a}-2X(\mathbf{w}_{1},\gamma,{\mu})\right)}{\Upsilon(\mathbf{w}_{1},\gamma,{\mu})}\bm{\Lambda_{i}}$
(25)

$\Delta_{{\Phi}}\mid_{({\gamma})}=\frac{{-{\mu}}\left({b}+{a}-2X(\mathbf{w}_{1},\gamma,{\mu})\right)}{\Upsilon(\mathbf{w}_{1},\gamma,{\mu})}\bm{\Lambda_{i}},$
(26)

where $\mathbf{I}_{2N_{t}}\in\mathbb{R}^{2N_{t}\times 2N_{t}}$. For hyperslab
constraints, $\bm{\Upsilon}(\cdotp)$ is the derivative of (23) with respect to
${x}$. Finally, using similar abstraction as in subsection III-A, the SLP
formulation can be expressed as a succession of sub-problems with respect to
the inequality constraint

$\displaystyle\underset{\mathbf{w_{1}\in{\mathbb{R}}^{2N_{t}\times
1}}}{\text{min}}$
$\displaystyle{\left\lVert\mathbf{w}_{1}\right\rVert}_{2}^{2}+\lambda{\mathbf{w}_{1}}+{\mu}{{B}(\mathbf{w}_{1})}.$
(27)

It is important to note that the original problem (11) does not have an
equality constraint. However, the term $\lambda\mathbf{w}_{1}$ introduced in
(27) is to provide additional stability to the network. Using the proximity
operator of the barrier, the update rule for every iteration is given by

$\mathbf{w}_{1}^{[r+1]}=\text{prox}_{\gamma^{[r]}\bm{\mu}^{[r]}B}\left(\mathbf{w}_{1}^{[r]}-\gamma^{[r]}\Delta{{D}(\mathbf{w}_{1}^{[r]},\lambda^{[r]})}\right)$
(28)

where

${D}(\mathbf{w}_{1}^{[r]},\lambda^{[r]})={\left\lVert\mathbf{w}_{1}\right\rVert}_{2}^{2}+\lambda{\mathbf{w}_{1}},$
(29)

and
$\Delta=\frac{\partial{{D}(\mathbf{w}_{1}^{[r]},\lambda^{[r]})}}{\partial{\mathbf{w}_{1}^{[r]}}}$.

###  III-C Deep SLP Network (SLP-DNet)

To build the proposed learning-based SLP architecture, we combine an IPM with
a proximal forward-backward procedure as shown in Algorithm 1 and transform it
into an NN structure represented by the proximity barrier term (see Fig. 2).
The learning architecture strictly follows the formulation (28). We show a
striking similarity between our proposal and the feed-forward. Intuitively, we
form cascade layers of NN from (28) as follows

$\mathbf{w}_{1}^{[r+1]}=\text{prox}_{\gamma^{[r]}{\mu}^{[r]}{B}}\left[\left(\mathbf{I}_{N_{t}}-2\gamma^{[r]}\right)\mathbf{w}_{1}^{[r]}+\gamma^{[r]}\lambda^{[r]}\textbf{1}^{T}\right].$
(30)

where $\textbf{1}\in\mathbb{R}^{1\times 2N_{t}}$ is a vector of ones. By
letting $\mathbf{W}_{r}=\mathbf{I}_{2N_{t}}-2\gamma^{[r]}$,
$\mathbf{b}_{r}=\gamma^{[r]}\lambda^{[r]}\textbf{1}^{T}$ and
$\bm{\Theta}_{r}=\text{prox}_{\gamma^{[r]}{\mu}^{[r]}{B}}$, the r-layer
network $\mathcal{L}^{[r-1]}\cdots\mathcal{L}^{[0]}$ will correspond to the
following

$\bm{\Theta}_{0}\left(\mathbf{W}_{0}+\mathbf{b}_{0}\right),\cdots,\bm{\Theta}_{r}\left(\mathbf{W}_{r}+\mathbf{b}_{r}\right)$
(31)

where $\mathbf{W}_{r}$ and $\mathbf{b}_{r}$ are described as weight and bias
parameters respectively. The nonlinear activation functions are defined by
$\bm{\Theta}_{r}$.

In the SLP-DNet design, the Lagrange multiplier associated with the equality
constraint is wired across the network to provide additional flexibility. It
is important to note that the architectures are the same for both non-robust
and robust power minimization problems described in Sections III and IV but
differ in proximity barrier functions (PBFs). Therefore, to simplify our
exposition, we build the structure of the learning framework based on (28) and
the DU framework described in [33], which gives rise to Algorithm 1.

Algorithm 1 Feed-forward-Backward Proximal IPM

0: $\mathbf{w}_{1}^{[0]}$, ${\gamma}^{[0]}$, ${\lambda}^{[0]}$ and
${\mu}^{[0]}$

0: $\mathbf{w}_{1}$ Initialization :

1: randomly initialize $\mathbf{w}_{1}^{[0]}\in{\mathbb{R}^{2N_{t}\times 1}}$,
${\mu}^{[0]}>0$, $\lambda^{[0]}>0$ and $\gamma^{[0]}>0$ $\forall\ {i}=1,\
\cdots,\ K$

2: for $r=0$ to $L$ do

3:
$\mathbf{w}_{1}^{[r+1]}=\text{prox}_{\gamma^{[r]}{\mu}^{[r]}B}\left(\mathbf{w}_{1}^{[r]}-\gamma^{[r]}\Delta{{D}(\mathbf{w}_{1}^{[r]},\
\lambda^{[r]})}\right).$

4: end for

5: return $\mathbf{w}_{1}$

Algorithm 2 Proximity Barrier Operator of a Nonrobust SLP-DNet

0: $\mathbf{h}_{Ri}$, $\mathbf{h}_{Ii}$, $\Gamma_{i}$ and ${w}_{0}\
(\text{noise power})$

0: $\mathbf{w}_{1}$, $\gamma$, ${\mu}$ and $\lambda$ Initialization :

1: randomly initialize $\mathbf{w}_{0}\in{\mathbb{R}^{2N_{t}\times 1}}$,
${\mu}^{[0]}>0$, $\lambda^{[0]}>0$ and $\gamma^{[0]}>0$ $\forall\ {i}=1,\
\cdots,\ K$.

2: Find the solution to (23) using Cardano formula.

3: For every solution in step 2, compute its corresponding Barrier function
using (21).

4: Compute the Proximity Operator of the Barrier at $\mathbf{w}_{0}$ using
(15), where $\Phi(\mathbf{w}_{1},\
\gamma,{\mu})=\text{prox}_{\gamma{\mu}{{B}}}{(\mathbf{w}_{1})}.$

5: Compute the derivatives of the Proximity Operator w.r.t $\mathbf{w}_{1}$,
${\mu}$ and $\gamma$ using (24), (25) and (26).

6: Update the training variables as follows:

  1. (a)

${\mu}^{[r+1]}={\mu}^{[r]}-\eta\frac{\partial{\Phi}(\mathbf{w}_{1}^{[r]},\
\gamma^{[r]},\ {\mu}^{[r]})}{\partial{{\mu}^{[r]}}}$

  2. (b)

$\gamma^{[r+1]}=\gamma^{[r]}-\eta\frac{\partial{\Phi}(\mathbf{w}_{1}^{[r]},\
\gamma^{[r]},\ {\mu}^{[r]})}{\partial{\gamma^{[r]}}}$

  3. (c)

$\lambda^{[r+1]}=\lambda^{[r]}-\eta\frac{\partial{{D}(\mathbf{w}_{1}^{[r]},\
\lambda^{[r]})}}{\partial{\lambda^{[r]}}}$ using (29)

where $\eta$ is the learning rate.

7: Use the results in step 6 and the Algorithm 1 to obtain the optimal
precoding tensor.

As shown in Fig. 2, SLP-DNet has two main units; the parameter update module
(PUM) and the auxiliary processing block (APB). The PUM has three core
components associated with Lagrangian multipliers (equality and inequality
constraints) and the training step-size, which are updated according to the
following

$\mathcal{H}(\mathbf{w}_{1},{\mu},\gamma,\lambda)=\text{prox}_{\gamma^{[r]}{\mu}^{[r]}B}\left(\mathbf{w}_{1}^{[r]}-\gamma^{[r]}\Delta{{D}(\mathbf{w}_{1}^{[r]},\lambda^{[r]})}\right).$
(32)

Furthermore, the component that forms the barrier term is constructed with one
convolutional layer, an average pooling layer, a fully connected layer, and a
Softplus layer to curb the output to a positive real value to satisfy the
inequality constraint. The APB unit is connected to the last r-th block of the
PUM in the form of a deep CNN to convert the output of the last parameter
update block into a target transmit precoding vector. The APB architecture is
made up of 3 convolution layers and 2 activation layers. In addition, a Batch
Normalization layer is added between each convolutional layer and the
activation layer to stabilize the mismatch in the distribution of the inputs
caused by the internal covariate shift [37]. For every r block (r-th layer),
there are three core components; $\mathcal{L}_{{\mu}}^{[r]}$,
$\mathcal{L}_{\gamma}^{[r]}$ and $\mathcal{L}_{\lambda}^{[r]}$ associated with
the learnable parameters (${\mu}$, $\gamma$ and $\lambda$), respectively as
shown in Fig. 2. These components form a learning block for computing the
barrier parameter (${\mu}$) associated with the inequality constraint, the
step-size ($\gamma$) and the equality constraint ($\lambda$), if exists. To
ensure that the constraints remain positive, a Softplus-sign function [38],
$\text{Softplus}(z)=\ln{(1+\text{exp}(z))}$ is used.

Figure 2: Complete SLP-DNet Architecture, showing the parameter update mouule,
the auxiliary processing block.

The Softplus-sign function is a smooth approximation of the rectified linear
unit (ReLu) activation function; and unlike the ReLu its gradient is never
exactly equal to zero [38], which imposes an update on $\gamma$, ${\mu}$ and
$\lambda$ during the backward propagation. The PBF for nonrobust SLP
formulation is summarized in Algorithm 2 below. A similar algorithm can also
be adopted for a robust PBF using a robust formulation. Finally, the output
from the auxiliary processing block is the precoding vector in the real
domain. The relation: $\mathbf{w}_{1}=[\mathbf{w}_{R}\ -\mathbf{w}_{I}]^{T}$
is used to convert it to its equivalent complex domain for every SINR value of
the i-th user.

####  III-C1 Duality and Loss Function of the SLP Formulation

In order to ease the formulation of the dual-problem of the original problem
(11), the left-hand-side of the inequality constraint is split into its
equivalent positive and negative parts as follows

$\displaystyle\underset{\mathbf{\\{w_{1}\\}}}{\text{min}}$
$\displaystyle{\left\lVert\mathbf{w}_{1}\right\rVert_{2}^{2}}$ (33) s.t.
$\displaystyle\bm{\Lambda}_{i}^{T}\bm{\Pi}\mathbf{w}_{1}\leq\left(\bm{\Lambda}_{i}^{T}\bm{\Pi}\mathbf{w}_{1}-\sqrt{\Gamma_{i}v_{0}}\right){\text{tan}{\phi}},\
\forall{i}$
$\displaystyle-\bm{\Lambda}_{i}^{T}\bm{\Pi}\mathbf{w}_{1}\leq\left(\bm{\Lambda}_{i}^{T}\bm{\Pi}\mathbf{w}_{1}-\sqrt{\Gamma_{i}v_{0}}\right){\text{tan}{\phi}},\
\forall{i}.$

The Lagrangian of (33) is defined as

$\mathcal{L}_{\text{rl}}(\mathbf{w}_{1},\ \bm{\mu}_{1},\
\bm{\mu}_{2})=\left\lVert\mathbf{w}_{1}\right\rVert_{2}^{2}\\\
+\bm{\mu}_{1}\left(\bm{\Lambda}^{T}_{i}\bm{\Pi}\mathbf{w}_{1}-\bm{\Lambda}_{i}^{T}\mathbf{w}_{1}\text{tan}{\phi}+\sqrt{\Gamma_{i}v_{0}}\right)\\\
-\bm{\mu}_{2}\left(\bm{\Lambda}^{T}_{i}\bm{\Pi}\mathbf{w}_{1}+\bm{\Lambda}_{i}^{T}\mathbf{w}_{1}\text{tan}{\phi}-\sqrt{\Gamma_{i}v_{0}}\right),$
(34)

where $\bm{\mu}_{1}$ and $\bm{\mu}_{2}$ are the Lagrangian multipliers
associated with the constraints and are related to the proximity barrier. The
subscript ‘rl’ stands for relaxed phase rotation. It can be easily proven that
the lower bound (LB) of (34) is
$\mathbcal{L}_{\text{rl}}(\mathbf{w}_{1},\bm{\mu}_{1},\bm{\mu}_{2})\geq\bm{\mu}_{1}\Lambda_{i}\left(\Pi-\text{tan}{\phi}\right)-\bm{\mu}_{2}\Lambda_{i}\left(\Pi+\text{tan}{\phi}\right)$.
From (34), the optimal precoder is obtained by differentiating
$\mathcal{L}_{\text{rl}}(\cdotp)$ w.r.t $\mathbf{w}_{1}$ and equating to zero.
By doing so, the optimal precoder can be found as

$\mathbf{w}_{1}=\frac{\left(\bm{\mu_{1}}^{T}+\bm{\mu_{2}}^{T}\right)\cdotp\bm{\Lambda}_{i}\text{tan}{\phi}-\left(\bm{\mu_{1}}^{T}-\bm{\mu_{2}}^{T}\right)\cdotp\bm{\Pi^{T}}\bm{\Lambda}_{i}}{2}.$
(35)

In the sequel, we show that (35) is used to generate the training input
(precoding vector) by randomly initializing the Lagrangian multipliers
($\bm{\mu}_{1}$ and $\bm{\mu}_{2}$) and then train the network to learn their
values that minimize the loss function (Lagrangian function). The loss
function is modified by adding $\mathbcal{l}_{2}$-norm regularization over the
weights to calibrate the learning coefficients in order to adjust the learning
process. It should be noted that the regularization here is not aimed at
addressing the problem of overfitting as in the case of supervised learning.
However, regularization in an unsupervised learning is used to normalize and
moderate weights attached to a neuron to help stabilize the learning algorithm
[39]. The loss function (34) over $N$ training samples is thus expressed as

$\mathcal{L}_{\text{rl}}(\mathbf{w}_{1},\ \bm{\mu}_{1},\
\bm{\mu}_{2})=\frac{1}{N}\sum^{N}_{i=1}\|\mathbf{w}_{1}\|_{2}^{2}\\\
+\frac{1}{N}\sum^{N}_{i=1}\left(\bm{\mu}_{1}\left(\bm{\Lambda}^{T}_{i}\bm{\Pi}\mathbf{w}_{1}-\bm{\Lambda}_{i}^{T}\mathbf{w}_{1}\text{tan}{\phi}+\sqrt{\Gamma_{i}v_{0}}\right)\right)\\\
-\frac{1}{N}\sum^{N}_{i=1}\left(\bm{\mu}_{2}\left(\bm{\Lambda}^{T}_{i}\bm{\Pi}\mathbf{w}_{1}+\bm{\Lambda}_{i}^{T}\mathbf{w}_{1}\text{tan}{\phi}-\sqrt{\Gamma_{i}v_{0}}\right)\right)\\\
+\frac{\vartheta}{NL}\sum^{N}_{i=1}\sum_{i=1}^{L}\|\bm{\theta}_{i}\|_{2}^{2},$
(36)

where $\bm{\theta}_{i}$ are the trainable parameters of the i-th layers
associated with the weights and biases, and $\vartheta>0$ is the penalty
parameter that controls the bias and variance of the trainable coefficients,
$N$, $L$ is the number of training samples (batch size or the number of
channel realization) and the number of layers, respectively.

###  III-D Learning-Based SLP for Strict Angle Rotation

In the previous subsection, we have presented SLP-DNet based on relaxed angle
formulation. In this subsection, we provide a formulation for strict phase
angle rotation where all the interfering signals align exactly to the phase
the signal of interest (i.e. $\phi=0$ in Fig. 1), the optimization problem is
[2]

$\displaystyle\underset{\mathbf{\\{w_{1}\\}}}{\text{min}}$
$\displaystyle{\left\lVert\mathbf{w}_{1}\right\rVert^{2}}$ (37) s.t.
$\displaystyle{\bm{\Lambda}_{i}^{T}\bm{\Pi}\mathbf{w}_{1}=0}\ ,\ \forall{i}$
$\displaystyle\bm{\Lambda}_{i}^{T}\mathbf{w}_{1}\geq\sqrt{\Gamma_{i}v_{0}}\ ,\
\forall{i}.$

We observe that the inequality constraint in (37) is affine. Based on this,
the proximal barrier function for the strict phase rotation is

$B_{\text{st}}(\mathbf{w}_{1})=\left\\{\begin{aligned}
&-\ln{\left(\bm{\Lambda}_{i}^{T}\mathbf{w}_{1}-\sqrt{{\Gamma}_{i}v_{0}}\right)},&&\text{if}\
\bm{\Lambda}_{i}^{T}\mathbf{w}_{1}\geq\sqrt{{\Gamma}_{i}v_{0}}\\\
&+\infty,&&\text{otherwise.}\end{aligned}\right.$ (38)

The subscript ‘st’ represents strict phase rotation. Therefore, for every
precoding vector $\mathbf{w}_{1}\in\mathbb{R}^{2N_{t}\times 1}$, the proximity
operator of ${\mu}\gamma{B}_{\text{st}}$ at $\mathbf{w}_{1}$ is given by

$\Phi_{\text{st}}(\mathbf{w}_{1},{\mu},\gamma)=\mathbf{w}_{1}+\\\
\frac{\bm{\Lambda_{i}}^{T}\mathbf{w}_{1}-\sqrt{\Gamma_{i}v_{0}}-\sqrt{(\bm{\Lambda}_{i}^{T}\mathbf{w}_{1}-\sqrt{\Gamma_{i}v_{0}})^{2}+4\gamma{\mu}\left\lVert\bm{\Lambda}_{i}^{T}\right\rVert_{2}^{2}}}{{{2\left\lVert\bm{\Lambda}_{i}\right\rVert_{2}^{2}}}}\bm{\Lambda}_{i}.$
(39)

Similar to the steps in subsection III-B, the learning-based framework for SLP
strict phase rotation is designed by finding the Jacobian matrix of
$\Phi(\mathbf{w}_{1},{\mu},\gamma)$ with respect to $\mathbf{w}_{1}$, and the
derivatives of $\Phi(\mathbf{w}_{1},{\mu},\gamma)$ with respect to $\gamma$
and $\mu$ can be easily obtained from (39). The loss function over $N$
training batches is given by

$\mathcal{L}_{\text{st}}(\mathbf{w}_{1},\bm{\lambda},\
\bm{\mu})=\frac{1}{N}\sum^{N}_{i=1}\left(\|\mathbf{w}_{1}\|_{2}^{2}+\bm{\lambda}\bm{\Lambda}^{T}_{i}\bm{\Pi}\mathbf{v}_{1}\right)+\\\
\frac{1}{N}\sum^{N}_{i=1}\left(\bm{\mu}\left(\sqrt{\Gamma_{i}v_{0}}-\bm{\Lambda}^{T}_{i}\mathbf{w}_{1}\right)\right)+\frac{\vartheta}{NL}\sum^{N}_{i=1}\sum_{i=1}^{L}\|\bm{\theta}_{i}\|_{2}^{2},$
(40)

where $\bm{\mu}$ and $\bm{\lambda}$ are the Lagrangian multipliers for
inequality and equality constraints, respectively. Finally, minimizing (40)
with respect to $\mathbf{w}_{1}$ (differentiating
$\mathcal{L}_{\text{st}}(\cdotp)$ w.r.t $\mathbf{w}_{1}$), gives the optimal
precoder as

$\mathbf{w}_{1}=\frac{\bm{\mu}^{T}\cdotp\bm{\Lambda}_{i}-\bm{\lambda}^{T}\cdotp\bm{\Pi}\bm{\Lambda}_{i}}{2}.$
(41)

##  IV Learning-Based Robust Power Minimization SLP with Channel Uncertainty

###  IV-A Channel Uncertainty and Channel error Model

So far, we have derived the unsupervised learning scheme in which the
uncertainty in estimating the channel coefficients is not considered. The
exact CSI is often unobtainable in practice. To model the user’s actual
channel in the uncertainty region, we consider an ellipsoid $\xi$ such that

$\hat{\mathbf{h}}_{i}=\bar{\mathbf{h}}_{i}+\bar{\bm{e}}_{i}\ \forall{k},$ (42)

where $\bar{\mathbf{h}}_{i}$ is the known CSI estimates at the BS and
$\bar{\bm{e}}_{i}$ denotes the channel error within the uncertainty region of
the ellipsoid (i.e $\hat{\mathbf{h}}_{i}\in\xi$). The model of the uncertainty
ellipsoid with the center $\bar{\mathbf{h}}_{i}$ is expressed as

$\xi=\left\\{\bar{\mathbf{h}}_{i}+\bar{\mathbf{e}}_{i}|_{\|\bar{\mathbf{e}}_{i}\leq
1\|}\right\\}.$ (43)

As shown in [2], the channel error is given by
$\left\\{\bar{\mathbf{e}}_{i}:\left\lVert\bar{\mathbf{e}}_{i}\right\rVert_{2}^{2}\leq{\varsigma}^{2}_{i}\right\\}$.
It is important to note that the BS is assumed to have the knowledge about the
channel error, excluding its corresponding error bound ${\varsigma}^{2}_{i}$.
For details and formulation of the conventional robust BLP, we refer the
reader to [40].

###  IV-B Robust Optimization-Based SLP Formulation

The multi-cast constructive interference formulation of the power minimization
problem for the worst-case CSI error is given by [40]

$\displaystyle\underset{\mathbf{\\{w\\}}}{\text{min}}$
$\displaystyle{\left\lVert\mathbf{w}\right\rVert_{2}^{2}}$ (44) s.t.
$\displaystyle\Bigl{|}{\text{Im}\left(\hat{\mathbf{h}}_{i}^{T}\mathbf{w}\right)\Bigl{|}}-\left({\text{Re}\left(\hat{\mathbf{h}}_{i}^{T}\mathbf{w}\right)}-\sqrt{\Gamma_{i}v_{0}}\right)\text{tan}{\phi}\leq
0,$
$\displaystyle\forall\left\lVert\bar{\mathbf{e}}_{i}\right\rVert^{2}\leq{\varsigma}_{i}^{2},\
\forall{i}.$

The intractability of the constraint in (44) can be effectively handled using
convex optimization methods. Therefore, to guarantee that the robust
constraint in (44) is satisfied, it is modified as follows [2]

$\max_{\|\bar{e}_{i}\|^{2}\leq\varsigma^{2}_{i}}\left(\Bigl{|}\text{Im}\left(\hat{\mathbf{h}}^{T}\mathbf{w}\right)\Bigl{|}-\left(\text{Re}\left(\hat{\mathbf{h}}^{T}\mathbf{w}\right)-\sqrt{\Gamma
v_{0}}\right)\text{tan}\phi\right)\leq 0.$ (45)

It is worth noting that the subscripts in (45) are ignored in order to
simplify the problem formulation. By defining the equivalent real-valued
channel and channel error vectors, the real and imaginary parts in the
constraint can be decomposed into two separate constraints as explained in
Section III (see (8a) and (8b)). Thus the robust formulation of the constraint
is equivalent to two separate real-valued constraints as follows

$\bm{\Lambda}^{T}\mathbf{w}_{1}-\bm{\Lambda}^{T}\mathbf{w}_{2}\text{tan}{\phi}+\varsigma\left\lVert\mathbf{w}_{1}-\mathbf{w}_{2}\text{tan}{\phi}\right\rVert_{2}+\sqrt{\Gamma
v_{0}}\text{tan}\phi\leq 0,$ (46)
$-\bm{\Lambda}^{T}\mathbf{w}_{1}-\bm{\Lambda}^{T}\mathbf{w}_{2}\text{tan}{\phi}+\varsigma\left\lVert\mathbf{w}_{1}+\mathbf{w}_{2}\text{tan}{\phi}\right\rVert_{2}+\sqrt{\Gamma
v_{0}}\text{tan}\phi\leq 0,$ (47)

where
$\bm{\Lambda}=\begin{bmatrix}\bar{\mathbf{h}}_{R}&\bar{\mathbf{h}}_{I}\end{bmatrix}^{T}$,
$\mathbf{e}\overset{\Delta}{=}\begin{bmatrix}\bar{\mathbf{e}}_{R}&\bar{\mathbf{e}}_{I}\end{bmatrix}^{T}$
and
$\hat{\mathbf{h}}=\bar{\mathbf{h}}_{R}+j\bar{\mathbf{h}}_{I}+\bar{\mathbf{e}}_{R}+j\bar{\mathbf{e}}_{I}$.
Finally, the robust CI formulation for power minimization problem becomes

$\displaystyle\underset{\mathbf{\\{w_{1},w_{2}\\}}}{\text{min}}$
$\displaystyle{\left\lVert\mathbf{w}_{1}\right\rVert_{2}^{2}}$ (48) s.t.
$\displaystyle\text{Constraints}\ (\ref{robust_constraint1})\ \text{and}\
(\ref{robust_constraint2}),\ \forall{i}$ $\displaystyle\text{where}\ \
\mathbf{w}_{1}=\bm{\Pi}\mathbf{w}_{2}.$

###  IV-C Proposed Unsupervised Learning-Based Robust SLP

In this subsection, we extend our proposed unsupervised learning formulation
to a worst-case CSI-error to design a robust precoding scheme for the power
minimization problem. As an extension of the previous formulations in
subsection III-B, the focus here is to derive a PBF for the robust learning-
based precoding scheme. Substituting for $\mathbf{w}_{1}$ in (48), we have

$\left(\bm{\Lambda}^{T}\bm{\Pi}-\bm{\Lambda}^{T}\text{tan}{\phi}\right)\mathbf{w}_{2}+\varsigma\left\lVert\left(\bm{\Pi}-\text{tan}{\phi}\right)\mathbf{w}_{2}\right\rVert_{2}\\\
+\sqrt{\Gamma v_{0}}\text{tan}\phi\leq 0,$ (49)
$-\left(\bm{\Lambda}^{T}\bm{\Pi}+\bm{\Lambda}^{T}\text{tan}{\phi}\right)\mathbf{w}_{2}+\varsigma\left\lVert\left(\bm{\Pi}+\text{tan}{\phi}\right)\mathbf{w}_{2}\right\rVert_{2}\\\
+\sqrt{\Gamma v_{0}}\text{tan}\phi\leq 0.$ (50)

Apparently, the constraints (49) and (50) are bounded by the
$\mathcal{l}_{2}$-norm. Therefore, problem (48) is rewritten as

$\displaystyle\underset{\mathbf{\\{w_{2}\\}}}{\text{min}}$
$\displaystyle{\left\lVert\mathbf{w}_{2}\right\rVert_{2}^{2}}$ (51) s.t.
$\displaystyle\text{Constraints}\ (\ref{robust_constraint12})\ \text{and}\
(\ref{robust_constraint22}),\ \forall{i}.$

The resulting barrier function of the corresponding constraints of (51) is the
sum of the individual barrier functions associated with the two inequality
constraints. We begin by introducing the feasible set of solutions bounded by
the Euclidean ball.

####  IV-C1 Bounded Euclidean ball Constraint

Suppose a problem whose set of feasible solutions is bounded by the Euclidean
ball [41]

$\mathbcal{C}=\\{\mathbf{z}\in\mathbb{R}^{n}\big{|}\left\lVert\mathbf{z}-\mathbf{x}\right\rVert_{2}\leq\beta\\},$
(52)

where $\beta>0$ and $\mathbf{x}\in\mathbb{R}^{n}$. Let $\gamma>0$ and $\mu>0$
be the step-size and the Lagrange multiplier associated with the inequality
constraint, respectively. Then the barrier function is expressed as [41]

${B(\mathbf{z})}=\left\\{\begin{aligned}
&-\ln{\left(\beta-\left\lVert\mathbf{z}-\mathbf{x}\right\rVert_{2}\right)},&&\text{if}\
\left\lVert\mathbf{z}-\mathbf{x}\right\rVert_{2}<\beta,\\\
&+\infty,&&\text{otherwise}\end{aligned}\right.$ (53)

For simplicity, we let
$\mathbf{Q}_{1}=\left(\bm{\Pi}-\mathbf{I}_{2N_{t}}\text{tan}{\phi}\right)$ and
$\mathbf{Q}_{2}=\left(\bm{\Pi}+\mathbf{I}_{2N_{t}}\text{tan}{\phi}\right)$.
Based on (53), the barrier function corresponding to the constraint (49) is
written at the bottom of the page.

${B_{1}(\mathbf{w}_{2})}=\left\\{\begin{aligned} &-\ln{\left(-\sqrt{\Gamma
v_{0}}\text{tan}{\phi}-\left(\bm{\Lambda}^{T}\mathbf{Q}_{1}\mathbf{w}_{2}+\varsigma\left\lVert\mathbf{Q}_{1}\mathbf{w}_{2}\right\rVert_{2}\right)\right)},&&\text{if}\
\bm{\Lambda}^{T}\mathbf{Q}_{1}\mathbf{w}_{2}+\varsigma\left\lVert\mathbf{Q}_{1}\mathbf{w}_{2}\right\rVert_{2}<-\sqrt{\Gamma
v_{0}}\text{tan}{\phi}\\\ &+\infty&&\text{otherwise}\end{aligned}\right.$ (54)

In the case of constraint (50), similar expression is also written for
${B_{2}(\mathbf{w}_{2})}$ using $\mathbf{Q}_{2}$. The resulting barrier
function is thus

${B}_{\text{robust}}(\mathbf{w}_{2})={B_{1}(\mathbf{w}_{2})}+{B_{2}(\mathbf{w}_{2})}$
(55)

Without loss of generality, the constraints (49) and (50) can be further
written as

$\bm{\Lambda}^{T}\mathbf{Q}_{1}\mathbf{w}_{2}+\varsigma\left\lVert\mathbf{Q}_{1}\mathbf{w}_{2}\right\rVert_{2}+\sqrt{\Gamma
v_{0}}\text{tan}{\phi}\leq 0,$ (56)
$\bm{\Lambda}^{T}\mathbf{Q}_{2}\mathbf{w}_{2}+\varsigma\left\lVert\mathbf{Q}_{2}\mathbf{w}_{2}\right\rVert_{2}+\sqrt{\Gamma
v_{0}}\text{tan}{\phi}\leq 0.$ (57)

It can be seen that the upper bound of the two constraints (56) and (57) is
zero, Therefore, the effective proximity operator of (55) is obtained the by
squaring (56) and (57) and adding the results. Following similar steps
presented in subsection III-B, we obtain the proximity operator of the barrier
for the robust SLP-DNet (see Appendix A for details).

####  IV-C2 Loss Function of the Robust Power Minimization Problem

The training loss function is the Lagrangian of (51), and can be written as

$\begin{aligned}
&\underset{\mathbf{\\{w_{2}\\}}}{\text{min}}&&{\left\lVert\mathbf{w}_{2}\right\rVert_{2}^{2}}\\\
&\text{s.t.}&&\bm{\Lambda}^{T}\mathbf{Q}_{1}\mathbf{w}_{2}+\varsigma\left\lVert\mathbf{Q}_{1}\mathbf{w}_{2}\right\rVert_{2}+\sqrt{\Gamma
v_{0}}\text{tan}{\phi}\leq 0\ \forall{i}\\\
&&&\bm{\Lambda}^{T}\mathbf{Q}_{2}\mathbf{w}_{2}+\varsigma\left\lVert\mathbf{Q}_{2}\mathbf{w}_{2}\right\rVert_{2}+\sqrt{\Gamma
v_{0}}\text{tan}{\phi}\leq 0\ \forall{i}.\end{aligned}$

(58)

Therefore, the loss function of (58) is the regularized Lagrangian
parameterized by $\bm{\theta}_{i}$ over the entire layers

$\mathcal{L}_{\text{robust}}(\mathbf{w}_{2},\ \bm{\mu}_{1},\
\bm{\mu}_{2})=\frac{1}{N}\sum^{N}_{i=1}\|\mathbf{w}_{2}\|_{2}^{2}\\\
+\frac{\bm{\mu}_{1}}{N}\sum^{N}_{i=1}\left({\varsigma}^{2}\left\lVert\mathbf{Q}_{1}\mathbf{w}_{2}\right\rVert_{2}^{2}-\left(\sqrt{\Gamma
v_{0}}\text{tan}{\phi}-\bm{\Lambda}^{T}\mathbf{Q}_{1}\mathbf{w}_{2}\right)^{2}\right)\\\
+\frac{\bm{\mu}_{2}}{N}\sum^{N}_{i=1}\left({\varsigma}^{2}\left\lVert\mathbf{Q}_{2}\mathbf{w}_{2}\right\rVert_{2}^{2}-\left(\sqrt{\Gamma
v_{0}}\text{tan}{\phi}-\bm{\Lambda}^{T}\mathbf{Q}_{2}\mathbf{w}_{2}\right)^{2}\right)\\\
+\frac{\vartheta}{NL}\sum^{N}_{i=1}\sum_{i=1}^{L}\|\bm{\theta}_{i}\|_{2}^{2}.$
(59)

The minimum of (59) with respect to $\mathbf{w}_{2}$ by equating its
derivative to zero

$\left(1+\left(\bm{\mu}_{1}\left\lVert\mathbf{Q}_{1}\right\rVert_{2}^{2}+\bm{\mu}_{2}\left\lVert\mathbf{Q}_{2}\right\rVert_{2}^{2}\right)\left({\varsigma}^{2}-\bm{\Lambda}^{T}\bm{\Lambda}\right)\right){\mathbf{w}_{2}}=\\\
-\left(\bm{\mu}_{1}\mathbf{Q}_{1}+\bm{\mu}_{2}\mathbf{Q}_{2}\right)\bm{\Lambda}\sqrt{\Gamma{v}_{0}\text{tan}{\phi}}.$
(60)

For convenience, we redefine the real matrices and vectors as
$\begin{bmatrix}\|\mathbf{Q}_{1}\|_{2}^{2}&\|\mathbf{Q}_{2}\|_{2}^{2}\end{bmatrix}=\bar{\mathbf{q}}_{\text{norm}}$;
$\begin{bmatrix}\mathbf{Q}_{1}&\mathbf{Q}_{2}\end{bmatrix}=\bar{\mathbf{Q}}$
and $\begin{bmatrix}\bm{\mu}_{1}&\bm{\mu}_{2}\end{bmatrix}=\bar{\bm{\mu}}$.
With these new notations, (60) is simplified to

$\left(\mathbf{I}_{2N_{t}}+\mathbf{\bar{q}}_{\text{norm}}\bm{\bar{\mu}}^{T}\left({\varsigma}^{2}-\bm{\Lambda^{T}\Lambda}\right)\right){\mathbf{w}_{2}}=-\bm{\Lambda}\mathbf{\bar{Q}}{\bm{\bar{\mu}}}^{T}\sqrt{\Gamma{w}_{0}}{\text{tan}{\phi}}$
(61)

From (61), the optimal transmit precoder is thus

$\mathbf{w}_{2}=-\bm{\Lambda}\mathbf{\bar{Q}}{\bm{\bar{\mu}}}^{T}{\mathbf{A}}^{-1}\sqrt{\Gamma{v}_{0}}{\text{tan}{\phi}},$
(62)

where
${\mathbf{A}}=\left(\mathbf{I}_{2N_{t}}+\mathbf{\bar{q}}_{\text{norm}}\bm{\bar{\mu}}^{T}\left({\varsigma}^{2}-\bm{\Lambda^{T}\Lambda}\right)\right)$.
Note that the Lagrange multipliers $\bm{\mu}_{1}$ and $\bm{\mu}_{2}$ are
associated with the barrier term and are randomly initialized from a uniform
distribution.

##  V Data Generation, Training and Computational Complexity

###  V-A Dataset Generation

The channel coefficients are used to form a dataset and are generated randomly
from a normal distribution with zero mean and unit variance. The data input
tensor is obtained using (4). We summarize the entire dataset preprocessing
procedure in Fig. 3. It can be observed that the input dataset is normalized
by the transmit data symbol so that data entries are within the nominal range,
and this could potentially aid the training.

Figure 3: Dataset Generating Block.

###  V-B SLP-DNet Training and Testing

The training of DNNs generally involves three steps: forward propagation,
backward propagation, and parameter update [42]. Except where necessary, the
training SINR is drawn from a random uniform distribution to enable learning
over a wide range of SINR values. The PUM contains r blocks, which form a
learning layer. Therefore, each block contains three core components and is
trained block-wise for l number of iterations.

Similarly, the APB is trained for k iterations. It is important to note that
the number of training iterations of the parameter update module may not
necessarily be equal to that of the APB. We train the PUM for 15 iterations
and the APB for 10 iterations. To improve the training efficiency, we modify
the learning rate by a factor $\alpha\in\mathbb{R}^{+}$ for every training
step. All the training is done with a stochastic gradient descent algorithm
using Adam optimizer [42]. Since the learning is done in an unsupervised
fashion, the loss function is the Lagrangian function’s statistical mean over
the entire training batch samples. During the inference, a feed-forward pass
is performed over the entire architecture using the learned Lagrangian
multipliers to calculate the precoding vector using (35) and (62) for both SLP
and robust SLP formulations, respectively. Finally, at inference, the trained
model is run with different SINR values to obtain the required optimal
precoding matrix.

###  V-C Computational Complexity Evaluation

In this subsection, we analyze and compare the computational costs of the
conventional BLP, optimization-based SLP, and the proposed SLP-DNet schemes.
The complexities are evaluated in terms of the number of real arithmetic
operations involved. For ease of analysis, we convert the SOCP (11) into a
standard linear programming (LP)

$\displaystyle\underset{\mathbf{\\{z\\}}}{\text{min}}$
$\displaystyle\mathbf{c}^{T}\mathbf{z}$ (63) s.t.
$\displaystyle\mathbf{c}_{k}^{T}\mathbf{z}\leq{-\text{tan}\phi\sqrt{\Gamma_{i}v_{0}}}\
,\ \forall{i}$

where
$\displaystyle\mathbf{c=\begin{bmatrix}0&\mathbf{w}_{1}^{T}\end{bmatrix}}^{T}\in\mathbb{R}^{(2N_{t}+1)\times{1}}$,
$\mathbf{z}=\begin{bmatrix}1&\mathbf{w}_{1}\end{bmatrix}^{T}\in\mathbb{R}^{(2N_{t}+1)\times
1}$,
$\mathbf{c}_{k}=\left[\left|\bm{\Lambda}_{i}^{T}\bm{\Pi}\mathbf{w}_{1}\right|\
\bm{\Lambda}_{i}^{T}\tan\phi\right]^{T}\in\mathbb{R}^{(2N_{t}+1)\times 1}$ and
$\mathbf{W}=[\mathbf{w}_{11},\cdots,\mathbf{w}_{1K}];\ \
\forall{i=1,\cdots,K}$. The complexity per iteration for solving convex
optimization via IPM is dominated by the formation
($\mathtt{C}_{\text{form}}$) and factorization ($\mathtt{C}_{\text{fact}}$) of
the matrix coefficients of $m$ linear equations in $m$ unknowns [43]. For
generic IPMs, the complexity is expressed as [43]

$\mathtt{C}_{\text{total}}=\mathtt{C}_{\text{iter}}\cdot\left(\mathtt{C}_{\text{form}}+\mathtt{C}_{\text{fact}}\right)$
(64)

where $\mathtt{C}_{\text{iter}}$ is the iteration complexity required to
attain an optimal solution. For a given optimal target accuracy, $\epsilon>0$,
$\mathtt{C}_{\text{iter}}$ is given by

$\mathtt{C}_{\text{iter}}=\sqrt{\sum_{j=1}^{N_{\text{lc}}}{d}_{j}+2N_{\text{sc}}}\times{\ln\left(\frac{1}{\epsilon}\right)}$
(65)

where $d$ is the dimension of the constraints, $N_{\text{lc}}$ and
$N_{\text{sc}}$ are the numbers of linear inequality matrix and second order
cone (SOC) constraints, respectively. The costs of formation and factorization
of matrix are respectively given by [43]

$\mathtt{C}_{\text{form}}=\underbrace{m\sum_{j=1}^{N_{\text{lc}}}d_{j}^{3}+m^{2}\sum_{j=1}^{N_{\text{lc}}}d_{j}^{2}}_{\text{due
to
$N_{\text{lc}}$}}+\underbrace{m\sum_{j=1}^{N_{\text{sc}}}d_{j=1}^{2}}_{\text{due
to $N_{\text{sc}}$}};\ \mathtt{C}_{\text{fact}}=m^{3}.$ (66)

Specifically, we observe that problem (63) has $K$ constraints with dimension
$2N_{t}+1$. Therefore, using (65) and (66), the total computational complexity
is thus

$\mathtt{C}_{\text{total}}=\sqrt{2N_{t}+1}\left[m(2N_{t}+1)+m(2N_{t}+1)^{2}+m^{3}\right]\ln{\left(\frac{1}{\epsilon}\right)}$.

(67)

The complexity of BLP can be derived in a similar way and is shown directly in
Table I. Conversely, the complexity of the proposed SLP-DNet schemes is the
sum of PUM and the APB complexities. Moreover, the complexity of the PUM is
dominated by the costs of computing the ‘log barrier’ and the feed-forward
pass of the shallow CNN (see Table III) that makes up the barrier term
associated with the inequality constraint. Similarly, the complexity of the
APB is also obtained by computing the arithmetic operations involved during
the forward pass of the deep CNN (see Table IV). To derive the analytical
complexity of SLP-DNet, we assume a sliding window is used to perform the
dominant computation of the convolution operation in the CNN and ignore the
nonlinear computational overhead due to activations. Therefore, the total
computational complexity is expressed as

$C_{SLP-DNet}=C_{\text{log-br}}+\\\
2\sum^{L_{\text{conv}}}_{l=1}n_{\text{h}}^{[l-1]}n_{\text{w}}^{[l-1]}\left[C_{\text{in}}^{[l-1]}f^{[l]2}+1\right]C_{\text{out}}^{[l]}+\\\
\sum^{L_{\text{fc}}}_{j=1}\left(2M_{\text{in}}^{[j-1]}+1\right)M_{\text{out}}^{[i]}$
(68)

where $n_{\text{h}}$, $n_{\text{w}}$, $f$, $C_{\text{in}}$ and
$C_{\text{out}}$ are the height, width of the input tensor, kernel size,
number of input and output channels, respectively. Similarly,
$L_{\text{conv}}$, $L_{\text{fc}}$, $M_{\text{in}}$ and $M_{\text{out}}$ are
the number of convolution and fully connected (FC) layers, number of input and
output neurons in the FC layer, respectively. $C_{\text{log-br}}$ denotes the
complexity of the ‘log-barrier’ function. Table I shows the summary of the
computational complexities of our proposals and the benchmark precoding
schemes. As an illustration, we consider the case of a symmetrical system
$(N_{t}=K=n)$, and show that the proposed approach has substantially reduced
computational complexity of $\mathbcal{O}(n^{3})$, while the optimization-
based SLP approach of $\mathbcal{O}(n^{6.5})$ and the conventional BLP is
$\mathbcal{O}(n^{7.5})$.

Table I: Complexity analysis of proposed SLP-DNet and benchmark SLP schemes. Problem | Arithmetic Operations (term; $m=\mathcal{O}(2N_{t}K)$) | Complexity Order ($n=N_{t}=K$)
---|---|---
Conventional BLP | $\sqrt{(4N_{t}+K+2)}\left[m(2N_{t}+1)+m(2N_{t}+1)^{2}+m(K+1)^{2}+m^{3}\right]\ln{\left(\frac{1}{\epsilon}\right)}$ | $\mathcal{O}(n^{6.5})$
SLP Optimization-based | $\sqrt{2N_{t}+1}\left[m(2N_{t}+1)+m(2N_{t}+1)^{2}+m^{3}\right]\ln{\left(\frac{1}{\epsilon}\right)}$ | $\mathcal{O}(n^{6.5})$
SLP-DNet | $4K^{2}N_{t}+42K^{2}+48KN_{t}+512K+2$ | $\mathcal{O}(n^{3})$
SLP-DNet Strict | $4K^{2}N_{t}+39K^{2}+46KN_{t}+512K+2$ | $\mathcal{O}(n^{3})$
Robust Conventional BLP | $\sqrt{2K(2N_{t}+1)}\left[mK(2N_{t}+1)^{3}+m^{2}K(2N_{t}+1)^{2}+m^{3}\right]\ln{\left(\frac{1}{\epsilon}\right)}$ | $\mathcal{O}(n^{7.5})$
Robust SLP Optimization-based | $\sqrt{2(2N_{t}+1)}\left[2mK(2N_{t}+1)^{2}+m^{3}\right]\ln{\left(\frac{1}{\epsilon}\right)}$ | $\mathcal{O}(n^{6.5})$
Robust SLP-DNet | $16KN_{t}^{2}+42K^{2}+48KN_{t}+512K$ | $\mathcal{O}(n^{3})$

##  VI Simulation, Results and Discussion

###  VI-A Simulation Set-up

We consider a single-cell MISO downlink in which the BS is equipped with four
antennas ($N_{t}=4$) that serve $K=4$ single users. We generate 50,000
training and 2000 test samples of Rayleigh fading channel coefficients,
respectively drawn from the same statistical distribution. The transmit data
symbols are modulated using QPSK and 8PSK modulation schemes. The training
SINR is randomly drawn from uniform distribution
$\Gamma_{\text{train}}\sim\mathcal{U}(\Gamma_{\text{low}},\,\Gamma_{\text{high}})$.
Adam optimizer [42] is used for stochastic gradient descent algorithm with
Lagrangian function as a loss metric.

Furthermore, a parametric rectified linear unit (PReLu) activation function is
used for both convolutional and fully connected layers instead of the
traditional ReLu function. The reason for this is to address the problem of
dying gradient [42]. The learning rate is reduced by a factor $\alpha=0.65$
after every iteration to aid the learning algorithm to converge faster. The
learning models are implemented in Pytorch 1.7.1 and Python 3.7.8 on a
computer with the following specifications: Intel(R) Core (TM) i7-6700 CPU
Core, 32.0GB of RAM. Table II summarizes the simulation parameters, while
Tables III and IV depict the NN component settings of the SLP-DNet.

Table II: Simulation settings Parameters | Values
---|---
Training Samples | 50000
Batch Size (B) | 200
Test Samples | 2000
Training SINR range | 0.0dB - 45.0dB
Test SINR range (i-th user SINR) | 0.0dB - 35.0dB
Optimizer | SGD with Adam
Initial Learning Rate $\eta$ | 0.001
Learning Rate decay factor $\alpha$ | 0.65
Weight Initializer | Xavier Initializer
Number of blocks in the parameter |
update unit | $B_{r}=2$
Training Iterations for each block |
of the parameter update unit | 15
Training iterations for the auxiliary unit | 10
Table III: Proximity Barrier Function NN Layout Layer | Parameter, $\text{kernel size}=3\times 3$
---|---
Input Layer | Input size $(\text{B},\ 1,\ 2N_{t},\ K)$
Layer 1: Convolutional | Size $(\text{B},1,K,20)$; zero padding
Layer 2: Average Pooling | Size $((1,\ 1),\ \text{stride}=(1,\ 1))$
Layer 3: Activation | Soft-Plus
Layer 4: Flat | Size $(\text{B}\times 20\times K^{2})$
Layer : Fully-connected | Size$(\text{B}\times 20\times K^{2},\ 1)$
Layer 5: Activation | Soft-Plus function
Table IV: Auxiliary Processing Block (APB) NN Structure Layer | Parameter, $\text{kernel size}=3\times 3$
---|---
Input Layer | Input size $(\text{B},\ 1,\ 2N_{t},\ K)$
Layer 1: Convolutional | Size $(\text{B},\ 1,\ K,\ 64)$,
|  $\text{dilation}=1$ and unit padding
Layer 2: Batch Normalization |  $\text{eps}=10^{-6}$, $\text{momentum}=0.1$
Layer 3: Activation | PReLu
Layer 4: Convolutional | Size $(\text{B},\ 1,\ 64,\ 2N_{t}K)$,
|  $\text{dilation}=1$ and unit padding
Layer 5: Batch Normalization |  $\text{eps}=10^{-6}$, $\text{momentum}=0.1$
Layer 6: Activation | PReLu
Layer 7: Convolutional | Size $(\text{B},\ 1,\ 2N_{t}K,\ 1)$,
|  $\text{dilation}=1$ and unit padding

###  VI-B Performance Evaluation of Non-Robust SLP-DNet

In this subsection, we evaluate the performance of our proposed unsupervised
learning framework for nonrobust scenario against the benchmark algorithms
[30, 2, 40] for both strict and relaxed angle rotations.

Firstly, we compare the average transmit power of the conventional BLP (2),
the SLP optimization-based problems (3), (11) and the SLP-DNet precoding
scheme based on (28) and Algorithm 2. The performances of SLP-DNet and the
benchmark schemes (conventional BLP and SLP optimization-based) for strict
angle rotation are shown in Fig. 4. It can be observed that the transmit power
of the proposed SLP-DNet closely matches the optimisation based SLP, both with
significant gains against BLP.

Similarly, we discern the same trend in Fig. 5 for the relaxed angle scenario
as observed in Fig. 4. Accordingly, we find from Fig. 5 that the relaxed angle
formulation offers significant power savings over the strict angle formulation
and is therefore adopted in the subsequent experiments. Furthermore, at 30dB,
the performance of SLP-DNet is within 5% of the SLP optimization-based
solution. Thus, while the SLP optimization-based offers a slightly lower
transmit power at SINR above 30dB, the proposed learning-based model’s
performance is within $96\%-98\%$ of the optimization-based solution.

Figure 4: Transmit Power vs SINR averaged over 2000 test samples for
conventional BLP, SLP optimization-based and nonrobust SLP-DNet schemes for
$\mathbb{M}$-PSK modulation with $N_{t}=4$, $K=4$ under strict angle rotation.
Figure 5: Transmit Power vs SINR averaged over 2000 test samples for
conventional BLP, SLP optimization-based and nonrobust SLP-DNet schemes for
$\mathbb{M}$-PSK modulation with $N_{t}=4$, $K=4$ under relaxed angle
rotation.

###  VI-C Performance Evaluation of Robust SLP-DNet

In this subsection, we evaluate the performance of the robust SLP-DNet against
the robust SLP optimization-based and conventional precoding algorithms.

Figs. 6 and 7 compare the performance of the proposed robust SLP-DNet with the
traditional robust block-level precoder [40] and robust SLP precoder [2] for
the $4\times 4$ MISO system evaluated at $\varsigma^{2}={10}^{-4}$. For
simplicity, we use QPSK modulation scheme. Fig. 6 depicts how the average
transmit power increases with the SINR thresholds, for CSI error bounds
$\varsigma^{2}={10}^{-4}$. The SLP optimization-based precoding scheme is
observed to show a significant power savings of more than 60% compared to the
conventional optimization solution. Similarly, the proposed unsupervised
learning-based precoder portrays a similar transmit power reduction trend.
They show considerable power savings of $40\%-58\%$ against the conventional
BLP.

Furthermore, we investigate the effect of the CSI error bounds on the transmit
power at 30dB. Fig. 7 depicts the transmit power variation with increasing CSI
error bounds. Moreover, a significant increase in transmit power can be
observed where the channel uncertainty lies within the region of CSI error
bounds of $\varsigma^{2}=10^{-3}$. Interestingly, like the SLP optimization-
based algorithm, the proposed SLP-DNet also shows a descent or moderate
increase in transmit power by exploiting the constructive interference.

Figure 6: Transmit Power vs SINR averaged over 2000 test samples for robust
conventional, SLP optimization-based and SLP-DNet solutions with $N_{t}=4$,
$K=4$ and $\varsigma^{2}=0.0002$. Figure 7: Transmit Power vs Error-bound for
robust conventional BLP, SLP optimization-based and SLP-DNet solutions with
$N_{t}=4$, $K=4$.

Figs. 8(a) and 8(b) depict the execution times for nonrobust and robust
formulations. It can be seen that both SLP optimization-based algorithm and
the proposed learning schemes are feasible for all sets of $N_{t}$ BS antenna
and $K$ mobile users. However, for conventional BLP, the solution is only
feasible for $N_{t}\geq K$.

Fig. 8(a) shows the average execution time of the proposed unsupervised
learning solutions per symbol averaged over 2000 test samples for nonrobust
formulations. The SLP-DNet is observed to be significantly faster than the SLP
optimization-based. For example, the theoretical complexity is polynomial
order-3 and polynomial order-6.5 or order-7.5 for SLP-DNet and conventional
methods, respectively. This is shown in the execution times, where there is a
significantly steeper increase in run-time as the number of users increases.
The decrease in computational cost is because the dominant operations involved
in SLP-DNet at the inference are simple matrix-matrix or vector-matrix
convolution. The same trend is also observed in the case of a robust channel
scenario, as shown in Fig. 8(b). Therefore, the results in Figs. 8(a) and 8(b)
demonstrate that the proposed unsupervised learning-based precoding solutions
offer a good trade-off between the performance and computational complexity.
Moreover, as per the results obtained, SLP-DNet’s performance is within the
range of $89\%-99\%$ of the optimal SLP optimization-based precoding solution.
Thus, our proposals demonstrate a favorable tradeoff between the performance
and the computational complexity involved.

Figure 8: Comparison of average execution time per sample averaged over 200
test samples for conventional BLP, optimization-based and SLP-DNet solutions
with $N_{t}=4$ and $K$ users $(2,\cdots,8)$.

##  VII Conclusion

This paper proposes an unsupervised learning-based precoding framework for a
multi-user downlink MISO system. The proposed learning technique exploits the
constructive interference for the power minimization problem so that for given
QoS constraints, the transmit power available for transmission is minimized.
We use domain knowledge to design unsupervised learning architectures by
unfolding the proximal interior point method barrier ‘log’ function. The
proposed learning scheme is then extended to robust precoding designs with
imperfect CSI bounded by CSI errors. We demonstrate that our proposal is
computationally efficient and allows for feasible solutions to be obtained for
problems where traditional numerical optimization like IPM and brute-force
maximum likelihood solvers would not converge or would be prohibitively
costly.

##  Appendix A Proximity operator barrier for Robust SLP

For every transmit precoding vector $\mathbf{w}_{2}\in\mathbb{R}^{2N_{t}\times
1}$, the proximity operator of the barrier
$\gamma\mu{B}_{\text{robust}}(\mathbf{w}_{2})$ is given by

$\Phi_{\text{rb}}(\mathbf{w}_{2},\gamma,\mu)=\frac{2\Gamma{v}_{0}\text{tan}^{2}{\phi}-X(\mathbf{w}_{2},\gamma,\mu)^{2}}{2\Gamma{v}_{0}\text{tan}^{2}{\phi}-X(\mathbf{w}_{2},\gamma,\mu)^{2}+2\gamma\mu}{\mathbf{w}_{2}}$
(69)

where $X(\mathbf{w}_{2},\gamma,\mu)$ is the unique solution of the cubic
equation expressed as [33]

$\leavevmode\resizebox{446.2658pt}{}{$\begin{array}[]{ll}{x}^{3}-\left(\left(\varsigma^{2}-\bm{\Lambda}^{T}\bm{\Lambda}\right)\left\lVert\mathbf{w}_{2}\right\rVert_{2}+4\bm{\Lambda}^{T}\mathbf{w}_{2}\text{tan}{\phi}\sqrt{\Gamma{v}_{0}}\right){x}^{2}\\\
+\left(2\Gamma{v}_{0}\text{tan}^{2}{\phi}+2\gamma\mu\right){{x}}+\\\
2\Gamma{v}_{0}\text{tan}^{2}{\phi}\left(\left(\varsigma^{2}-\bm{\Lambda}^{T}\bm{\Lambda}\right)\left\lVert\mathbf{w}_{2}\right\rVert_{2}+4\bm{\Lambda}^{T}\mathbf{w}_{2}\text{tan}{\phi}\sqrt{\Gamma{v}_{0}}\right)=0.\end{array}$}$
(70)

It can be observed that (70) is a cubic equation and can be solved
analytically. In the final analysis, following similar steps as in (22)-(26),
the robust deep-unfolded model is obtained by finding the Jacobean matrix of
(69) with respect to the optimization variable $\mathbf{w}_{2}$, and the
derivatives with respect to the step-size $\gamma>0$ and the Lagrange
multiplier associated with the inequality constraint $\mu>0$. We use similar
concepts presented in subsection III-B to formulate the learning algorithm of
the robust SLP as a series of sub-problems with respect to the combined effect
of the two inequality constraints as follows

$\displaystyle\underset{\mathbf{w_{2}\in{\mathbb{R}}^{2N_{t}\times
1}}}{\text{min}}$
$\displaystyle{\left\lVert\mathbf{w}_{2}\right\rVert}_{2}^{2}+\lambda{\mathbf{w}_{2}}+{\mu}{{B}_{\text{robust}}(\mathbf{w}_{2})}.$
(71)

Similar to a nonrobust SLP-DNet, the update rule for every iteration is
expressed as

$\mathbf{w}_{2}^{[r+1]}=\text{prox}_{\gamma^{[r]}{\mu}^{[r]}B_{\text{robust}}}\left(\mathbf{w}_{2}^{[r]}-\gamma^{[r]}\Delta{{D}_{\text{robust}}(\mathbf{w}_{2}^{[r]},\lambda^{[r]})}\right)$
(72)

where

${D}_{\text{robust}}(\mathbf{w}_{2}^{[r]},\lambda^{[r]})={\left\lVert\mathbf{w}_{2}\right\rVert}_{2}^{2}+\lambda{\mathbf{w}_{2}}.$
(73)

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# CKM AND $C\\!P$ VIOLATION IN BEAUTY AND CHARM DECAYS IN LHCb

F. BETTI on behalf of the LHCb collaboration  School of Physics and Astronomy,
University of Edinburgh,
James Clerk Maxwell Building, Peter Guthrie Tait Road,
Edinburgh, EH9 3FD, United Kingdom

###### Abstract

Measurements of $C\\!P$ violation and Cabibbo-Kobayashi-Maskawa matrix in
beauty and charm hadron decays are the core business of the LHCb physics
programme. In this contribution, the most recent measurements performed by the
LHCb collaboration on this topic are reported. The most precise measurement of
time-dependent $C\\!P$ asymmetry parameters in ${{B}^{0}_{s}}\\!\to\phi\phi$
decays has been done, providing results fully compatible with Standard Model
expectations. The polarisation-dependent $C\\!P$-violation parameters of the
same decay are measured for the first time. The combination of beauty and
charm results gives $\gamma=\left(63.8^{+3.5}_{-3.7}\right)^{\circ}$. The
first search for local $C\\!P$ violation in
${{D}_{({s})}^{+}}\\!\to{{K}^{-}}{{K}^{+}}{{K}^{+}}$ has been performed,
resulting in no local $C\\!P$ violation observed. The measurement of time-
integrated $C\\!P$ violation in ${{D}^{0}}\\!\to{{K}^{-}}{{K}^{+}}$ decays,
combined with previous measurements performed by LHCb, provides evidence of
direct $C\\!P$ violation in charm in a single decay channel at the level of
$3.8$ standard deviations.

##  1 Introduction

In the Standard Model (SM) of particle physics, the combined symmetry of
charge conjugation ($C$) and parity ($P$), collectively known as $C\\!P$, is
violated by the weak interaction. The $C\\!P$ violation is one of the Sakharov
conditions necessary for early universe baryogenesis, which, when satisfied,
ensure a universe dominated by matter. [1] However, the amount of $C\\!P$
violation measured experimentally so far is not enough to explain the matter-
antimatter asymmetry observed in the universe. Hence, the search for sources
of $C\\!P$ violation beyond the SM is strongly motivated, as well as the
precise measurement of $C\\!P$ violation within the SM itself.

Within the SM, $C\\!P$ violation is quantified by the Cabibbo-Kobayashi-
Maskawa (CKM) matrix, [2, 3] a 3x3 unitary matrix that describes the weak
interaction between up-type and down-type quarks. The CKM matrix elements
represent the transition amplitudes for quarks to change flavour, and must be
determined through experimental observations. The unitarity of the CKM matrix
is a fundamental property within the SM framework. The CKM matrix can be
parameterised using three mixing angles and a single complex phase, where this
phase accounts for all $C\\!P$ violation in the quark sector within the SM.
Therefore, in order to accurately assess $C\\!P$ violation in the SM, it is
crucial to precisely measure the elements of the CKM matrix. Moreover,
experimental verification of the unitarity relations provides important tests
of the SM and can reveal potential deviations from the SM expectations.

In this contribution, recent measurements performed by the LHCb collaboration
concerning $C\\!P$ violation in ${{B}^{0}_{s}}\\!\to\phi\phi$ decay and
mixing, determination of the $\gamma$ angle and $C\\!P$ violation in charm
mesons are presented.

##  2 $C\\!P$ Violation in ${{B}^{0}_{s}}\\!\to\phi\phi$ decays

In ${{B}^{0}_{s}}\\!\to\phi\phi$ decays, the time-dependent $C\\!P$ violations
arises from the interference between decay and mixing, characterised by the
phase $\phi_{s}^{{s}{\overline{s}}{s}}$ and the $|\lambda|$ parameter, that
are expected to be very close to 0 and 1, respectively, in the SM. [4, 5] New
physics contributions in the penguin decay or the ${B}^{0}_{s}$ mixing could
significantly alter the values of $\phi_{s}^{{s}{\overline{s}}{s}}$ and
$|\lambda|$. [6, 7] A new measurement carried out by the LHCb collaboration
with the data collected during the Run 2 of the LHC has determined
$\phi_{s}^{{s}{\overline{s}}{s}}$ and $|\lambda|$. The strategy consists in
measuring the differential decay rate in terms of ${B}^{0}_{s}$ decay time and
the three helicity angles that describe the three polarisation states of the
final state. The differential decay rate is a function of the polarisation-
dependent parameters $\phi_{s,i}$ and $|\lambda_{i}|$ ($i=0,\perp,\parallel$),
so the measurement is able to search for new physics in different polarisation
states. The polarisation-independent measurement is performed by requiring
$\phi_{s,i}={\phi_{s}^{{s}{\overline{s}}{s}}}$ and $|\lambda_{i}|=|\lambda|$.
The signal candidates are statistically separated by the
$\Lambda_{b}^{0}\\!\to\phi Kp$ and the combinatorial backgrounds by means of a
fit to the $\phi\phi$ invariant mass, shown in Figure 1. The background-
subtracted multidimensional distribution of helicity angle and decay time is
then fit by the differential decay rate.

Figure 1: Mass distribution of the ${{B}^{0}_{s}}\\!\to\phi\phi$ candidates,
superimposed by the fit projections (top left). Background-subtracted
distributions of angular variables $\cos\theta_{1}$ (top right) and $\chi$
(bottom left), and decay time (bottom right), superimposed by the fit
projections.

The result of the polarisation-independent measurement is [8]

$\displaystyle{\phi_{s}^{{s}{\overline{s}}{s}}}$ $\displaystyle=(-0.042\pm
0.075\pm 0.009)\ \mathrm{rad},$ $\displaystyle|\lambda|$
$\displaystyle=1.004\pm 0.030\pm 0.009,$

where the first uncertainties are statistical and the second are systematic.
The combination with the Run 1 measurement [9] gives

$\displaystyle{\phi_{s}^{{s}{\overline{s}}{s}}}$ $\displaystyle=(-0.074\pm
0.069)\ \mathrm{rad},$ $\displaystyle|\lambda|$ $\displaystyle=1.009\pm
0.030,$

that is in agreement with the SM expectations. This is the most precise
measurement of time-dependent $C\\!P$ asymmetry parameters in
${{B}^{0}_{s}}\\!\to\phi\phi$ decays up to date. The polarisation-dependent
$C\\!P$-violation parameters of this decay are measured for the first time:

$\displaystyle|\lambda_{0}|$ $\displaystyle=1.02\pm 0.17,$
$\displaystyle|\lambda_{\perp}/\lambda_{0}|$ $\displaystyle=0.97\pm 0.22,$
$\displaystyle|\lambda_{\parallel}/\lambda_{0}|$ $\displaystyle=0.78\pm 0.21,$
$\displaystyle\phi_{s,0}$ $\displaystyle=(-0.18\pm 0.09)\ \mathrm{rad},$
$\displaystyle\phi_{s,\parallel}-\phi_{s,0}$ $\displaystyle=(0.12\pm 0.09)\
\mathrm{rad},$ $\displaystyle\phi_{s,\perp}-\phi_{s,0}$
$\displaystyle=(0.17\pm 0.09)\ \mathrm{rad},$

resulting in no difference observed between different polarisation states.

##  3 Direct measurements of $\gamma$

The angle $\gamma$ of the CKM matrix, defined as

$\gamma\equiv\arg{\left(-\frac{V_{ud}V_{ub}^{*}}{V_{cd}V_{cb}^{*}}\right)},$
(1)

can be measured in tree-level decays sensitive to the interference between
$b\to cW$ and $b\to uW$ transition amplitudes. In fact, $\gamma$ is determined
by measuring asymmetries and yield ratios in ${{B}^{\pm}}\\!\to D(\to
f_{D})h^{\pm}$ decays ($h^{\pm}=\pi^{\pm},K^{\pm}$), where $f_{D}$ is a final
state common to ${D}^{0}$ and ${\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$.

In multibody decays, the sensitivity is reduced by the differences between the
favoured and suppressed decays in the structure of the intermediate
resonances. This effect is parameterised by the so-called coherence factor,
that allows the maximum sensitivity to be reached when it is close to 1,
whereas it totally dilutes the sensitivity when it is closed to 0. In
$B^{\pm}\\!\to D(\to K^{\mp}\pi^{\pm}\pi^{\pm}\pi^{\mp})h^{\pm}$ decays,
performing the measurement in 4 bins of $D$ phase space allows a significant
improvement on the sensitivity to $\gamma$ to be reached, [10] because in some
bins the coherence factor can be larger than the integrated one.

The measurement of $\gamma$ with $B^{\pm}\\!\to D(\to
K^{\mp}\pi^{\pm}\pi^{\pm}\pi^{\mp})h^{\pm}$ decays performed by the LHCb
collaboration with the full dataset gives: [11]

$\gamma=\left(54.8^{+6.0+0.6+6.7}_{-5.8-0.6-4.3}\right)^{\circ},$

where the first uncertainty is statistical, the second is systematic and the
third is due to the external inputs of ${D}^{0}$ mixing and hadronic decay
parameters. [12, 13] This is the second most precise determination of $\gamma$
from a single $D$ decay mode. A large improvement on the uncertainty related
to the external inputs is expected by including future LHCb determinations of
${D}^{0}$ mixing parameters and future measurements of hadronic $D$-decay
parameters performed with incoming BESIII data.

Since ${{B}^{\pm}}\\!\to D(\to K^{\pm}\pi^{\mp})h^{\pm}$ are sensitive also to
the ${D}^{0}$ mixing and hadronic decay parameters, the LHCb collaboration
started recently to perform the combination of all the measurements sensitive
to $\gamma$ and charm parameters. The most recent $\gamma$-charm combination
of LHCb measurements has been carried out including the latest determinations
of $\gamma$ with the $B^{\pm}\\!\to D(\to
K^{\mp}\pi^{\pm}\pi^{\pm}\pi^{\mp})h^{\pm}$ and $B^{\pm}\\!\to D(\to
h^{\pm}h^{\prime\mp}\pi^{0})h^{\pm}$ decay modes, [11, 14] the measurement of
${D}^{0}$ mixing parameters and $C\\!P$ violation parameters in mixing using
${{D}^{0}}\\!\to h^{-}h^{+}$ [15] and ${\kern
1.79993pt\overline{\kern-1.79993ptB}{}}\to{{D}^{0}}(\to
K_{s}^{0}{{\pi}^{+}}{{\pi}^{-}})\mu^{-}{{\overline{\nu}}_{\mu}}X$ decays, [16]
and the measurement of time-integrated $C\\!P$ asymmetry in
${{D}^{0}}\\!\to{{K}^{-}}{{K}^{+}}$ decays [17] with the data collected during
the Run 2 of the LHC. The resulting value of $\gamma$ is [18]

$\gamma=\left(63.8^{+3.5}_{-3.7}\right)^{\circ},$

fully compatible with indirect determinations. [19, 20]. The ${D}^{0}$ mixing
parameters are determined to be

$x=(0.398^{+0.050}_{-0.049})\%,\quad y=(0.636^{+0.020}_{-0.019})\%.$

The latest determination of $\gamma$, not included in the $\gamma$-charm
combination, has been done by the LHCb collaboration by using the $B^{\pm}\to
D(\to{{K}^{+}}{{K}^{-}}\pi^{\pm}\pi^{\mp})h^{\pm}$ and $B^{\pm}\\!\to
D(\to{{\pi}^{+}}{{\pi}^{-}}\pi^{\pm}\pi^{\mp})h^{\pm}$ decay modes with the
data collected during the Run 1 and Run 2 of the LHC. A phase-space integrated
measurement is done in both decay channels, while a binned analysis is
performed for the $D\\!\to{{K}^{+}}{{K}^{-}}\pi^{\pm}\pi^{\mp}$ mode,
resulting in [21]

$\gamma=\left(116^{+12}_{-14}\right)^{\circ}.$

The precision of this measurement will be improved once new charm model-
independent measurements of the ${D}^{0}$ decay parameters will be available.

##  4 $C\\!P$ violation in charm decays

Charm decay are an unique laboratory to study $C\\!P$ violation in up-type
quark decays. Due to the smallness of involved CKM elements and GIM mechanism,
$C\\!P$ violation in charm decays is predicted to be small. Moreover, the long
distance contributions are not negligible in the corresponding SM
calculations, making very difficult to provide precise expectations for the
related observables. The measured value of the difference between the time-
integrated $C\\!P$ asymmetries in ${{D}^{0}}\\!\to{{K}^{-}}{{K}^{+}}$ and
${{D}^{0}}\\!\to{{\pi}^{-}}{{\pi}^{+}}$ decays by the LHCb collaboration [22]
challenges the calculations based on first-principles QCD. Further
measurements are therefore needed in charm sector in order to clarify the
picture.

Multi-body decays are particularly interesting, because phase-space local
$C\\!P$ asymmetries can be larger than integrated ones. The LHCb collaboration
has recently performed the first search for local $C\\!P$ violation in
${{D}_{({s})}^{+}}\\!\to{{K}^{-}}{{K}^{+}}{{K}^{+}}$ decays using the data
collected between 2016 and 2018. The analysis makes use of the Miranda
technique, [23, 24] that consists in dividing the Dalitz plot in bins and
computing, for each bin $i$, the significance $S_{C\\!P}^{i}$ of the
difference in the numbers of ${{D}_{({s})}^{+}}$ candidates
$N^{i}({{D}_{({s})}^{+}})$ and ${{D}{}_{({s})}^{-}}$ candidates
$N^{i}({{D}{}_{({s})}^{-}})$, where the latter is corrected for global charge
asymmetry
$\alpha\equiv\sum_{i}N^{i}({{D}_{({s})}^{+}})/\sum_{i}N^{i}({{D}{}_{({s})}^{-}})$:

$S_{C\\!P}^{i}\equiv\frac{N^{i}({{D}_{({s})}^{+}})-\alpha\cdot
N^{i}({{D}{}_{({s})}^{-}})}{\sqrt{\alpha\left(\delta^{2}_{N^{i}({{D}_{({s})}^{+}})}+\delta^{2}_{N^{i}({{D}{}_{({s})}^{-}})}\right)}},$
(2)

where $\delta_{N^{i}({{D}_{({s})}^{+}})}$ and
$\delta_{N^{i}({{D}{}_{({s})}^{-}})}$ are the statistical uncertainties of the
signal candidate yields. The yields are obtained by fitting the
${D}_{({s})}^{+}$ invariant mass distributions in each bin. In the absence of
local $C\\!P$ violation, the values of $S_{C\\!P}^{i}$ follow a normal
distribution. A $\chi^{2}$ test is performed, with
$\chi^{2}\equiv\sum_{i}(S_{C\\!P}^{i})^{2}$, and the corresponding $p$-value
is calculated as the probability of obtaining a $\chi^{2}$ that is at least as
high as the value observed, under the assumption of $C\\!P$ conservation (null
hypothesis). If $p$-values are less than $3\times 10^{-7}$, $C\\!P$ violation
is observed. The analysis, validated on Cabibbo-favoured
${{D}^{+}}\\!\to{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}$ and
${{D}^{+}_{s}}\\!\to{{K}^{-}}{{K}^{+}}{{\pi}^{+}}$ control modes, where no
direct $C\\!P$ violation is expected, results in a $p$-value of $13.3\%$ for
the ${D}^{+}_{s}$ mode and $31.6\%$ for the ${D}^{+}$ mode, corresponding to
no evidence of local $C\\!P$ violation. [25]

The most recent measurement of $C\\!P$ violation in two-body charm decays has
been carried out by the LHCb collaboration with the data collected during the
Run 2 of the LHC. The flavor of the ${D}^{0}$ meson at the production is
obtained by looking at the charge of the pion in the
${{D}^{*+}}\\!\to{{D}^{0}}{{\pi}^{+}}$ decay. Since the raw asymmetry

$A_{\rm
raw}(\mbox{${{D}^{0}}\\!\to{{K}^{-}}{{K}^{+}}$})\equiv\frac{N\left({{D}^{*+}}\to{{D}^{0}}(\to{{K}^{-}}{{K}^{+}}){{\pi}^{+}}\right)-N\left({{D}^{*-}}\to{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}(\to{{K}^{-}}{{K}^{+}}){{\pi}^{-}}\right)}{N\left({{D}^{*+}}\to{{D}^{0}}(\to{{K}^{-}}{{K}^{+}}){{\pi}^{+}}\right)+N\left({{D}^{*-}}\to{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}(\to{{K}^{-}}{{K}^{+}}){{\pi}^{-}}\right)}$
(3)

is affected by the detection asymmetry of the tagging pion and the production
asymmetry of the ${D}^{0}$ meson, Cabibbo-favoured control modes are used to
cancel the nuisance asymmetries and obtain the physical $C\\!P$ asymmetry
$A_{C\\!P}({{D}^{0}}\to{{K}^{-}}{{K}^{+}})$. Two almost statistically
independent calibration procedures are used: one, already used in the Run 1
measurements, [26, 27] makes use of
${{D}^{+}}\\!\to{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}$ and ${{D}^{+}}\\!\to{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{0}}{{\pi}^{+}}$ decays, whereas the
other, which is used for the first time, exploits the
${{D}^{+}_{s}}\\!\to\phi{{\pi}^{+}}$ and ${{D}^{+}_{s}}\\!\to{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{0}}{{K}^{+}}$ decays. The
${{D}^{*+}}\\!\to{{D}^{0}}(\to{{K}^{-}}{{\pi}^{+}}){{\pi}^{+}}$ decay is
needed by both calibration procedures. The resulting time-integrated $C\\!P$
asymmetries are [17]

$\displaystyle{A_{C\\!P}({{D}^{0}}\to{{K}^{-}}{{K}^{+}})}|{{D}^{+}}$
$\displaystyle=(13.6\pm 8.8\pm 1.6)\times 10^{-4},$
$\displaystyle{A_{C\\!P}({{D}^{0}}\to{{K}^{-}}{{K}^{+}})}|{{D}^{+}_{s}}$
$\displaystyle=(2.8\pm 6.7\pm 2.0)\times 10^{-4},$

for the first and second calibration procedure respectively, where the first
uncertainties are statistical and the second are systematic, with statistical
(systematic) correlation equal to 0.05 (0.28). The time-integrated $C\\!P$
asymmetry in the ${{D}^{0}}\\!\to h^{-}h^{+}$ decay is equal to

${A_{C\\!P}({{D}^{0}}\to
h^{-}h^{+})}=a_{h^{-}h^{+}}^{\mathrm{d}}+\frac{\langle
t\rangle_{h^{-}h^{+}}}{\tau_{{D}^{0}}}\Delta Y,$ (4)

where $a_{h^{-}h^{+}}^{\mathrm{d}}$ is the direct $C\\!P$ asymmetry in the
decay, $\Delta Y$ parameterises the time-dependent $C\\!P$ asymmetry, $\langle
t\rangle_{h^{-}h^{+}}$ is the observed average ${D}^{0}$ decay time and
$\tau_{{D}^{0}}$ is the nominal ${D}^{0}$ lifetime. [28] By combining all the
LHCb measurements of $A_{C\\!P}({{D}^{0}}\to{{K}^{-}}{{K}^{+}})$, $\Delta
A_{{C\\!P}}\equiv{A_{C\\!P}({{D}^{0}}\to{{K}^{-}}{{K}^{+}})}-{A_{C\\!P}({{D}^{0}}\to{{\pi}^{-}}{{\pi}^{+}})}$,
$\Delta Y$ and $\langle t\rangle_{h^{-}h^{+}}$, the direct $C\\!P$ asymmetries
are obtained:

$\displaystyle a_{{{K}^{-}}{{K}^{+}}}^{\mathrm{d}}$ $\displaystyle=(7.7\pm
5.7)\times 10^{-4},$ $\displaystyle a_{{{\pi}^{-}}{{\pi}^{+}}}^{\mathrm{d}}$
$\displaystyle=(23.2\pm 6.1)\times 10^{-4},$

with a correlation of 0.88. This result corresponds to an evidence of direct
$C\\!P$ violation in ${{D}^{0}}\\!\to{{\pi}^{-}}{{\pi}^{+}}$ decays at the
level of $3.8$ standard deviations, and exceeds the SM expectations of
$U$-spin symmetry breaking at the level of 2 standard deviations.

##  5 Conclusions

LHCb continues to produce world-leading results on $C\\!P$ violation in beauty
and charm decays. A new precise test of the SM in ${B}^{0}_{s}$ mixing has
been made. The CKM angle $\gamma$ is now known with an uncertainty smaller
than $4^{\circ}$. The first evidence of $C\\!P$ violation in charm in a single
decay channel at the level of $3.8$ standard deviations has been reported, and
a new search of local $C\\!P$ violation in charm hadron multi-body decay has
been performed. The LHCb Upgrade I will improve such measurements with the
data collected during the Run 3 of the LHC, thanks to the higher integrated
luminosity and a better trigger efficiency.

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