# RISNet: a Dedicated Scalable Neural Network Architecture for Optimization of
Reconfigurable Intelligent Surfaces

Bile Peng, Finn Siegismund-Poschmann and Eduard A. Jorswieck
Institute for Communications Technology
Technische Universität Braunschweig, Germany
Email: {b.peng, f.siegismund-poschmann<EMAIL_ADDRESS>
###### Abstract

The reconfigurable intelligent surface (RIS) is a promising technology for
next-generation wireless communication. It comprises many passive antennas,
which reflect signals from the transmitter to the receiver with adjusted
phases without changing the amplitude. The large number of the antennas
enables a huge potential of signal processing despite the simple functionality
of a single antenna. However, it also makes the RIS configuration a high
dimensional problem, which might not have a closed-form solution and has a
high complexity and, as a result, severe difficulty in online real-time
application if we apply iterative numerical solutions. In this paper, we
introduce a machine learning approach to maximize the weighted sum-rate (WSR).
We propose a dedicated neural network architecture called _RISNet_. The RIS
optimization is designed according to the RIS property of product and direct
channel and homogeneous RIS antennas. The architecture is scalable due to the
fact that the number of trainable parameters is independent from the number of
RIS antennas (because all antennas share the same parameters). The weighted
minimum mean squared error (WMMSE) precoding is applied and an alternating
optimization (AO) training procedure is designed. Testing results show that
the proposed approach outperforms the state-of-the-art block coordinate
descent (BCD) algorithm. Moreover, although the training takes several hours,
online testing with trained model (application) is almost instant, which makes
it feasible for real-time application. Compared to it, the BCD algorithm
requires much more convergence time. Therefore, the proposed method
outperforms the state-of-the-art algorithm in both performance and complexity.

###### Index Terms:

reconfigurable intelligent surface, weighted sum-rate, machine learning,
weighted minimum mean-square error precoder, alternating optimization.

##  I Introduction

The reconfigurable intelligent surface (RIS) is a large antenna array composed
of many passive antennas. It receives the signal from the transmitter,
performs a simple signal processing without external power, and transmits it
to the receiver. It enables a new opportunity to optimize the wireless
channel. For decades, the channel properties were given and could not be
modified. Due to the simple structure, low cost, and high integrability with
other communication technologies, the RIS is widely considered as a key
enabling technology of the next-generation wireless communication systems [1,
2]. Among the various possible applications of the RIS, we consider the
weighted sum-rate (WSR) maximization problem in the RIS-assisted downlink
broadcast channel in this paper.

Two main challenges of the RIS optimization are the joint optimization of
precoding at the high dimensional base station (BS) and configuration of the
RIS due to the large number of antennas. For the first challenge, we can apply
the well-established weighted minimum mean squared error (WMMSE) precoder [3,
4, 5], which promises optimality of the WSR in presence of both interference
and thermal noise for multiple users. However, the joint optimization between
BS and RIS is still an open problem. For the second challenge, majorization-
maximization (MM) [6, 7], stochastic successive convex approximation [8],
alternating optimization (AO) [9], fully convolutional network (FCN) [10] and
alternating direction method of multipliers (ADMM) [11] are applied to
optimize the RIS configuration. These proposed algorithms have achieved
reasonable performance at the cost of high computational effort. In a broader
class of RIS optimization problem formulations, the Riemannian manifold
conjugate gradient (RMCG) and the Lagrangian method are applied to optimize
multiple RISs and BS to serve users on the cell edge [12]. Spatial
multiplexing in RIS-assisted uplink is considered [13]. A joint precoding
scheme with an AO method is proposed for cell-free RIS-aided communication
[14]. Downlink signal-to-noise ratio (SNR) is maximized for single user with
deep reinforcement learning (DRL) [15]. The spectrum and energy efficiency are
optimized with RIS [16, 17]. The interference caused by secondary networks is
mitigated with an RIS [18]. The BS precoding and phase shifts of the RIS are
iteratively optimized with the orthogonal frequency-division multiplexing
(OFDM) transmission scheme [19, 20]. The robust transmission scheme design
against imperfect channel state information (CSI) is addressed as well [21].
These methods are either suboptimal in performance and cannot realize the
potential of the RIS, or have high complexity and are not suitable for online
real-time application.

In order to further realize the potential of the RIS and reduce the
complexity, this paper presents a dedicated and scalable neural network
architecture _RISNet_ , which optimizes the RIS phase shifts for each RIS
antenna according to the channel features and realizes an improved WSR.

##  II Problem Formulation

The RIS-aided communication is considered with direct propagation path between
BS and users. The objective is to serve multiple users with orthogonal
multiple access (i.e., the spatial division multiple access (SDMA)) and to
maximize the WSR of the users. The BS is assumed to have multiple antennas and
can perform precoding subject to the transmit power constraint $E_{Tr}$. Each
RIS antenna has the ability to reflect signal from the transmitter with an
adjusted complex phase.

We denote the precoding matrix of size $M\times U$ as $\mathbf{V}$, where $M$
is the number of BS antennas and $U$ is the number of single-antenna users,
the channel from BS to RIS of size $N\times M$ as $\mathbf{H}$, where $N$ is
the number of RIS antennas. The RIS signal processing is denoted by the
diagonal matrix $\boldsymbol{\Phi}$, which has the size of $N\times N$, where
the diagonal element $\phi_{nn}$ in row $n$ and column $n$ is $e^{j\psi_{n}}$,
with $\psi_{n}\in[0,2\pi)$ being the phase shift of RIS antenna $n$. The
channel matrix from RIS to users is denoted as $\mathbf{G}$ of size $U\times
N$, where the element $g_{un}$ in row $u$ and column $n$ is the channel gain
from RIS antenna $n$ to user $u$. The direct channel from BS to users of size
$U\times M$ is denoted as $\mathbf{D}$, where the element $d_{um}$ in row $u$
and column $m$ is the channel gain from the $m$th BS antenna to the $u$th
user. The signal model of precoding and transmission is given as

$\mathbf{y}=\left(\mathbf{G}\boldsymbol{\Phi}\mathbf{H}+\mathbf{D}\right)\mathbf{V}\mathbf{x}+\mathbf{n},$
(1)

where $\mathbf{x}$ is the column vector of transmitted symbols, $\mathbf{y}$
is the column vector of received symbols and $\mathbf{n}$ is the column vector
of thermal noise. The system model is illustrated in Fig. 1.

Figure 1: System model of RIS-assisted downlink broadcast channel.

Let $\mathbf{C}$ be the matrix of precoding and transmission, i.e.,

$\mathbf{C}=(\mathbf{G}\boldsymbol{\Phi}\mathbf{H}+\mathbf{D})\mathbf{V}$ (2)

and $c_{uv}$ be the element of $\mathbf{C}$ in row $u$ and column $v$, our
objective is to maximize the WSR. Define $\alpha_{u}$ as the weight of user
$u$, the problem is formulated as

$\begin{array}[t]{ll}\max_{\mathbf{V},\boldsymbol{\Phi}}&\sum_{u=1}^{U}\alpha_{u}\log_{2}\left(1+\frac{c_{uu}^{2}}{\sum_{v\neq
u}c_{uv}^{2}+\frac{1}{\rho}}\right)\\\
\text{s.t.}&\text{tr}\left({\mathbf{V}\mathbf{V}^{H}}\right)\leq E_{Tr}\\\
&|\phi_{nn}|=1\\\ &|\phi_{nn^{\prime}}|=0\text{ for }n\neq
n^{\prime},\end{array}$ (3)

where $\rho$ is the transmit signal-to-noise ratio (TSNR), which is the ratio
between transmit power and noise power. In this work, we use the WMMSE
precoding matrix as $\mathbf{V}$ and optimize $\boldsymbol{\Phi}$ to maximize
the WSR.

##  III Channel Features

We propose to obtain $\boldsymbol{\Phi}$ with a neural network according to
the channel matrices, among which $\mathbf{H}$ is assumed to be constant
because BS and RIS are stationary and the environment is relatively invariant,
$\mathbf{G}$ and $\mathbf{D}$ vary with the user positions and are necessary
to compute $\boldsymbol{\Phi}$ and $\mathbf{V}$. We interpret $\mathbf{G}$ and
$\mathbf{D}$ as two feature maps. While each column of $\mathbf{G}$ can be
unambiguously mapped to each RIS antenna, $\mathbf{D}$ is the direct channel
and is irrelevant to the RIS. Therefore, we define
$\mathbf{J}=\mathbf{D}\mathbf{H}^{+}$ and (1) becomes

$\mathbf{y}=\left(\mathbf{G}\boldsymbol{\Phi}+\mathbf{J}\right)\mathbf{H}\mathbf{V}\mathbf{x}+\mathbf{n}.$
(4)

Equation (4) can be interpreted as follows: signal $\mathbf{x}$ is precoded
with $\mathbf{V}$ and then transmitted through channel $\mathbf{H}$ to the
RIS, then through channel $\mathbf{G}\boldsymbol{\Phi}+\mathbf{J}$ to the
users. Both $\mathbf{G}$ and $\mathbf{J}$ have $N$ columns and their columns
can be unambiguously mapped to the RIS antennas. Therefore, the features of
RIS antenna $n$ can be defined as

$\displaystyle\boldsymbol{\gamma}_{n}=($
$\displaystyle|g_{1n}|,\arg(g_{1n}),\dots,|g_{Un}|,\arg(g_{Un}),$ (5)
$\displaystyle|j_{1n}|,\arg(j_{1n}),\dots,|j_{Un}|,\arg(j_{Un})),$

where $g_{un}$ is the element in row $u$ and column $n$ of $\mathbf{G}$,
$j_{un}$ is defined similarly to matrix $\mathbf{J}$. In this way, we use 4$U$
features to characterize the wireless channels with $U$ users with respect to
each RIS antenna.

##  IV The RISNet Architecture

In this section, we present the RISNet architecture. The basic idea of the
RISNet is that an RIS antenna needs its local information as well as the
information of the whole RIS to make a good decision on its phase shift. The
local information is obtained based on the information of the considered
antenna exclusively whereas the global information is the mean of the
information of all RIS antennas. We present two versions of RISNet. The first
version accepts the concatenated channel features of all users as input.
Permutation of the users causes different output. This version is named
_permutation-variant_ RISNet. The second version accepts separated channel
features of each user and user permutation has no impact on the output. This
version is named _permutation-invariant_ RISNet. The permutation-variant
RISNet can work with asymmetric objective functions, such as WSR with
different weights of users or nonorthogonal multiple access (NOMA) in future
works. The permutation-invariant RISNet works only with symmetric objective
functions, but its permutation-invariance makes it more generalized to unseen
channel data.

###  IV-A Permutation-variant RISNet

Denote the input of layer $i$ as $\mathbf{F}_{i}$ of shape $B_{i}\times N$,
where $B_{i}$ is the feature dimension of layer $i$ and the $n$th column of
$\mathbf{F}_{i}$ is the feature vector of RIS antenna $n$. For $i=1$,
$\mathbf{F}_{1}^{u}$ (i.e., the input of the RISNet) is defined as the channel
feature

$\mathbf{F}_{1}=\boldsymbol{\Gamma},$ (6)

where the $n$th column of $\boldsymbol{\Gamma}$ is $\boldsymbol{\gamma}_{n}$
defined in (5).

For layer $i<L$, we compute the local feature as

$\mathbf{F}^{l}_{i+1}=\text{ReLU}(\mathbf{W}^{l}_{i}\mathbf{F}_{i}+\mathbf{b}_{i}^{l})$
(7)

where $\mathbf{W}^{l}_{i}$ of shape $B_{i+1}^{l}\times B_{i}$ and
$\mathbf{b}_{i}^{l}$ of shape $B_{i+1}^{l}\times 1$ are trainable weight and
bias for local feature in layer $i$, where $B_{i+1}^{l}$ is the local feature
dimension for layer $i+1$. Note that $\mathbf{b}_{i}^{l}$ is added to every
column to $\mathbf{W}^{l}_{i}\mathbf{F}_{i}$. The global feature is computed
as

$\mathbf{F}^{g}_{i+1}=\text{ReLU}(\mathbf{W}^{g}_{i}\mathbf{F}_{i}+\mathbf{b}_{i}^{g})\mathbf{1}_{N}/N$
(8)

for all RIS antennas, where $\mathbf{W}^{g}_{i}$ of shape $B_{i+1}^{g}\times
B_{i}$ and $b_{i}^{g}$ of shape $B_{i+1}^{g}\times 1$ are trainable weight and
bias for global feature in layer $i$, with $B_{i+1}^{g}$ being the global
feature dimension of layer $i+1$, and $\mathbf{1}_{N}$ is a matrix of all ones
of shape $N\times N$. Compared to (7), the global information is the same for
all RIS antennas, which is the mean of
$\text{ReLU}(\mathbf{W}^{g}_{i}\mathbf{F}_{i}+\mathbf{b}_{i}^{g})$ along the
columns. Note that $\mathbf{b}_{i}^{g}$ is added to every column of
$\mathbf{W}^{g}_{i}\mathbf{F}_{i}$ as in (7).

The output feature by layer $i$ is the concatenation of the channel features
and the two features defined above:

$\mathbf{F}_{i+1}^{u}=\left(\left(\boldsymbol{\Gamma}\right)^{T},\left(\mathbf{F}_{i+1}^{l}\right)^{T},\left(\mathbf{F}_{i+1}^{g}\right)^{T}\right)^{T}.$
(9)

Therefore, the feature dimension of the layer $i+1$ is
$B_{i+1}=4U+B_{i+1}^{l}+B_{i+1}^{g}$a since the channel feature dimension is
$4U$.

In the final layer ($i=L$), the output of the RISNet is computed as

$\mathbf{f}_{L+1}=\text{ReLU}\left(\mathbf{w}_{L}\sum_{u}\mathbf{F}_{L}^{u}+b_{L}\right)$
(10)

where $\mathbf{w}_{L}$ and $b_{L}$ are trainable weights and bias,
respectively.

The information processing of one layer of the permutation-invariant RISNet is
illustrated in Fig. 2.

$\boldsymbol{\Gamma}$$\mathbf{F}_{i}^{l}$$\mathbf{F}_{i}^{g}$AntennaFeatureLocal
layer $i$Global layer
$i$$\boldsymbol{\Gamma}$$\mathbf{F}_{i+1}^{l}$$\mathbf{F}_{i+1}^{g}$

Figure 2: Information processing of one layer in the RISNet.

###  IV-B Permutation-invariant RISNet

Denote the input of layer $i$ from user $u$ as $\mathbf{F}_{i}^{u}$ of shape
$B_{i}\times N$, where the $n$th column of $\mathbf{F}_{i}^{u}$ is the feature
vector of RIS antenna $n$. For $i=1$, $\mathbf{F}_{1}^{u}$ (i.e., the input of
the RISNet) is defined as

$\mathbf{F}_{1}^{u}=\boldsymbol{\Gamma}^{u},$ (11)

where $\boldsymbol{\Gamma}^{u}=1,2,\dots,U$ is the channel feature of user
$u$.

For layer $i<L$, we compute the ego local feature as

$\mathbf{F}^{u,e,l}_{i+1}=\text{ReLU}(\mathbf{W}^{e,l}_{i}\mathbf{F}^{u}_{i}+\mathbf{b}_{i}^{e,l}),$
(12)

where $\mathbf{W}^{e,l}_{i}$ of shape $B^{e,l}_{i+1}\times B_{i}$ and
$\mathbf{b}_{i}^{e,l}$ of shape $B^{e,l}_{i+1}\times 1$ are trainable weight
and bias for ego and local feature in layer $i$, where $B^{e,l}_{i+1}$ is the
ego and local feature dimension for layer $i+1$. The ego global feature is
computed as

$\mathbf{F}^{u,e,g}_{i+1}=\text{ReLU}(\mathbf{W}^{e,g}_{i}\mathbf{F}^{u}_{i}+\mathbf{b}_{i}^{e,g})\mathbf{1}_{N}/N$
(13)

for all RIS antennas and all $u$, where $\mathbf{W}^{e,g}_{i}$ of shape
$B^{e,g}_{i+1}\times B_{i}$ and $\mathbf{b}_{i}^{e,g}$ of shape
$B^{e,g}_{i+1}\times 1$ are trainable weight and bias for ego and global
feature in layer $i$, where $B_{i+1}^{e,g}$ is the ego and global feature
dimension of layer $i+1$. Compared to (12), the global information is the same
for all RIS antennas, which is the mean of
$\text{ReLU}(\mathbf{W}^{e,g}_{i}\mathbf{F}^{u}_{i}+\mathbf{b}_{i}^{e,g})$.

The opposite local and opposite global features of user $u$ are computed
similarly as

$\mathbf{F}^{u,o,l}_{i+1}=\sum_{v\neq
u}\text{ReLU}(\mathbf{W}^{o,l}_{i}\mathbf{F}^{v}_{i}+\mathbf{b}_{i}^{o,l})/(U-1)$
(14)

and

$\mathbf{F}^{u,o,g}_{i+1}=\sum_{v\neq
u}\text{ReLU}(\mathbf{W}^{o,g}_{i}\mathbf{F}^{v}_{i}+\mathbf{b}_{i}^{o,g})\mathbf{1}_{N}/(N(U-1))$
(15)

respectively, where $\mathbf{W}^{o,l}_{i}$ of shape $B^{o,l}_{i+1}\times
B_{i}$, $\mathbf{b}_{i}^{o,l}$ of shape $B^{o,l}_{i+1}\times 1$,
$\mathbf{W}^{o,g}_{i}$ of shape $B^{o,g}_{i+1}\times B_{i}$ and
$\mathbf{b}_{i}^{o,g}$ of shape $B^{o,g}_{i+1}\times 1$ are trainable weights
for opposite local feature, bias for opposite local feature, weights for
opposite global feature and bias for opposite global feature, respectively,
with $B^{o,l}_{i+1}$ and $B^{o,g}_{i+1}$ being the opposite local and opposite
global feature dimension of layer $i+1$, respectively. The sum operator in
(14) and (15) are regarding the tensors.

The output feature by layer $i$ of user $u$ is the concatenation of the
channel features and the four features defined above:

$\displaystyle\mathbf{F}_{i+1}^{u}=\Big{(}$
$\displaystyle\left(\boldsymbol{\Gamma}^{u}\right)^{T},\left(\mathbf{F}_{i+1}^{u,e,l}\right)^{T},\left(\mathbf{F}_{i+1}^{v,o,l}\right)^{T},$
(16)
$\displaystyle\left(\mathbf{F}_{i+1}^{u,e,g}\right)^{T},\left(\mathbf{F}_{i+1}^{v,o,g}\right)^{T}\Big{)}^{T}.$

Therefore, the feature dimension of layer $i+1$ is
$B_{i+1}=4+B^{e,l}_{i}+B^{e,g}_{i}+B^{o,l}_{i}+B^{o,g}_{i}$.

In the final layer ($i=L$), the output of the RISNet is computed as

$\mathbf{f}_{L+1}=\text{ReLU}\left(\mathbf{w}_{L}\sum_{u}\mathbf{F}_{L}^{u}+b_{L}\right)$
(17)

where $\mathbf{w}_{L}$ and $b_{L}$ are trainable weights and bias,
respectively.

The information processing of one layer of the permutation-invariant RISNet is
illustrated in Fig. 3. Please note that only two users are shown in the figure
for the illustration simplicity. If more than two users are served, opposite
information is averaged over the users according to (14) and (15).

$\boldsymbol{\Gamma}^{1}$$\mathbf{F}_{i}^{1,e,l}$$\mathbf{F}_{i}^{2,o,l}$$\mathbf{F}_{i}^{1,e,g}$$\mathbf{F}_{i}^{2,o,g}$Ego
local layer $i$Opposite local layer $i$Ego global layer $i$Opposite global
layer
$i$$\boldsymbol{\Gamma}^{2}$$\mathbf{F}_{i}^{2,e,l}$$\mathbf{F}_{i}^{1,o,l}$$\mathbf{F}_{i}^{2,e,g}$$\mathbf{F}_{i}^{1,o,g}$AntennaFeatureEgo
local layer $i$Opposite local layer $i$Ego global layer $i$Opposite global
layer
$i$$\boldsymbol{\Gamma}^{1}$$\mathbf{F}_{i+1}^{1,e,l}$$\mathbf{F}_{i+1}^{2,o,l}$$\mathbf{F}_{i+1}^{1,e,g}$$\mathbf{F}_{i+1}^{2,o,g}$$\boldsymbol{\Gamma}^{2}$$\mathbf{F}_{i+1}^{2,e,l}$$\mathbf{F}_{i+1}^{1,o,l}$$\mathbf{F}_{i+1}^{2,e,g}$$\mathbf{F}_{i+1}^{1,o,g}$

Figure 3: Information processing of one layer in the permutation-invariant
RISNet. Note that two users are assumed for simpliciticy of presentation. More
than two users are possible.

Training of the neural work is performed as Algorithm 1 describes.

Algorithm 1 RISNet training

repeat

Randomly select a batch of data samples.

Compute WMMSE precoding vectors $\mathbf{V}$ for every data sample in the
batch.

Compute the gradient of the objective with current $\mathbf{V}$ w.r.t. the
neural network parameters

Perform a gradient ascent step with the Adam optimizer

until Predefined number of iterations achieved

##  V Training and Testing Results

The training and testing results are presented in this section. Important
parameters of scenario and model are presented in Table I.

TABLE I: Setting and Parameter values Parameter |  Value
---|---
Number of BS antennas |  9
Number of RIS antennas |  1024
Number of users |  4
Channel models |  Rayleigh fading channels
Transmit SNR |  $10^{11}$, $5\times 10^{11}$, $10^{12}$
Weights of users |  (0.25, 0.25, 0.25, 0.25)
Number of layers |  8
Learning rate |  $8\times 10^{-4}$
Optimizer |  ADAM
Feature dimension for permutation-variant RISNet |  16
Feature dimension for permutation-invariant RISNet |  8
Iterations for permutation-variant version |  500
Iterations for permutation-invariant version |  1000
Batch size |  512
Optimizer |  ADAM
Number of data samples in training set |  10240
Number of data samples in testing set |  1024

The improvement of WSR is shown in Fig. 4, where one iteration is a gradient
ascent step with a batch of channel data. From the figure we can observe that
the training has improved the WSR considerably. Furthermore, the permutation-
variant RISNet quickly converges while the permutation-invariant RISNet takes
significantly longer time to train. However, the achieved WSR is higher with
the permutation-invariant RISNet. The difference is bigger with higher TSNR.
It is to note that the permutation-invariant RISNet is only applicable to
permutation-invariant objective functions. If the users have different weights
or if we apply NOMA instead of SDMA, we have to use the permutation-variant
RISNet because a user permutation changes the optimal RIS configuration.

$0$$100$$200$$300$$400$$500$$600$$700$$800$$900$$1{,}000$$0$$0.5$$1$$1.5$$2$IterationWSR
(bit/Hz/s)TSNR=$10^{11}$, PVTSNR=$10^{11}$, PITSNR=$5\times 10^{11}$,
PVTSNR=$5\times 10^{11}$, PITSNR=$10^{12}$, PVTSNR=$10^{12}$, PI Figure 4:
Improvement of WSR in training. PV stands for permutation-variant and PI
stands for permutation-invariant in the legend.

We choose random phase shifts of the RIS and the block coordinate descent
(BCD) algorithm proposed in [8] as two baselines to compare the proposed
approach. The testing data are different from the training data and are
identical to all four approaches shown in the figure. We can observe that both
versions of the RISNet outperform the two baselines significantly. In
addition, the testing with 1024 data samples takes only one minute on a laptop
with RISNet. On the other hand, the optimization with the BCD algorithm takes
more than 24 hours on the server with the same data samples. This indicates
that the proposed method is more advantageous in both performance and
complexity, therefore is closer to reality.

$0.1$$0.2$$0.3$$0.4$$0.5$$0.6$$0.7$$0.8$$0.9$$1$$\cdot
10^{12}$$0$$0.5$$1$$1.5$$2$TSNRWSR (bit/Hz/s)RISNet, PV (ours)RISNet, PI,
(ours)RandomBCD [8] Figure 5: Testing results of different approaches. PV
stands for permutation-variant and PI stands for permutation-invariant in the
legend.

##  VI Conclusion

We consider the WSR maximization problem in an RIS-aided wireless
communication network. Due to the complexity of the objective function, this
problem does not have a closed-form solution. The large number of RIS antennas
makes the optimization problem very high dimensional, which makes the
computational complexity of an iterative numerical solution very high. In this
work, we propose an unsupervised machine learning to solve this problem. A
dedicated and scalable neural network architecture _RISNet_ is introduced,
which achieves a high performance while retaining a low complexity. A
permutation-variant version of the RISNet is introduced for general use cases
and a permutation-invariant version is proposed for symmetric objective
function, where the permutation of users does not change the RISNet output.
This property makes it more stable to unseen channel data but limits its
application to symmetric objective functions. Training and testing results
show that the proposed solution achieves a better performance than the state-
of-the-art algorithm with a much lower complexity in testing (application),
thus making it not only more advantageous, but also closer to reality.

Source code and data set of this paper will be made publicly available if the
paper is accepted.

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# A note on combining chaotic dynamical systems using the fuzzy logic XOR
operator.

Rezki Chemlal  Laboratoire de Mathématiques Appliquées, Faculté des sciences
exactes.Université Abderahmane Mira Bejaia.06000 Bejaia Algeria

###### Abstract

In this paper we explore whatever combining two chaotic dynamical systems
using the fuzzy logic operator XOR can maintain or not the chaotic properties
of the resulting dynamical system. This study is motivated by techniques used
in applications to secure communications ,images encryption and cryptography.

Key words : Chaos, fuzzy logic, ergodic theory, full branch.

##  I Introduction

Chaotic dynamical systems are commonly used as models in a wide range of
applications, cryptography [6],[1],[9],[12] image encryption and retrieval
[7],[11],[2] and achievement of associative memory properties [4],[3],[5].

Extreme sensitivity to initial condition is an interesting property of chaotic
systems. This property makes chaotic systems a worthy choice for constructing
cryptosystems or for image encryption.

Another idea is to mix two or more chaotic dynamical systems to gain more
”unpredictably” or more ”confusion” in order to enhance encryption process
[8],[7].

This rise a natural question, whatever combining two chaotic dynamical systems
permits to maintain chaotic property of the resulting one ? In particular
whatever combining two chaotic dynamical systems by fuzzy logic operators,
mainly the operator xor, gives rise to a chaotic dynamical system.

In other words consider two chaotic dynamical systems $\left(I,F\right)$ and
$\left(I,G\right)$ is the dynamical system $\left(I,F\ast G\right)$ still
chaotic ? Where * stands for any fuzzy logic operator.

We studied some combination of known chaotic dynamical systems and checked
whatever the combination is still chaotic or not. This gave us some
preliminary remarks about how to combine chaotic dynamical systems in order to
maintain the chaotic properties of the resulting dynamical system.

##  II Dynamical systems

###  II-A Topological dynamics

A dynamical system $\left(X,F\right)$ consists of a compact metric space $X$
and a continuous self–map $F$ .

A point $x$ is said periodic if there exists $p>0$ with
$F^{p}\left(x\right)=x.$ The least $p$ with this property is called the period
of $x.$ A point $x$ is eventually or ultimately periodic if
$T^{m}\left(x\right)$ is periodic for some $m\geq 0.$

In the same way a dynamical system is said periodic if there exists $p>0$ with
$F^{p}\left(x\right)=x$ for every $x\in X$ and eventually periodic if $F^{m}$
is periodic for some $m\geq 0.$

A point $x\in X$ is said to be an equicontinuity point, or to be Lyapunov
stable, if for any $\epsilon>0$, there exists $\delta>0$ such that if
$d\left(x,y\right)<\delta$ one has
$d\left(F^{n}\left(y\right),F^{n}\left(x\right)\right)<\epsilon$ for any
integer $n\geq 0$.

We say that $\left(X,F\right)$ is sensitive if for any $x\in X$ we have :

$\exists\epsilon>0,\forall\delta>0,\exists y\in
B_{\delta}\left(x\right),\exists n\geq 0\text{ }\mathrm{such}\text{ {that
}}d\left(F^{n}\left(y\right),F^{n}\left(x\right)\right)\geq\epsilon.$

We say that $\left(X,T\right)$ is expansive if we have :

$\exists\epsilon>0,\forall x\neq y\in X,\exists n\geq
0,d\left(F^{n}\left(x\right),F^{n}\left(y\right)\right)\geq\epsilon.$

A dynamical system $\left(X,T\right)$__ is transitive if for any nonempty open
sets $U,V\subset A^{\mathbb{Z}}$ there exists $n>0$ with $U\cap
F^{-n}\left(V\right)\neq\emptyset.$ This is equivalent to the existence of a
point with a dense orbit.

A dynamical system is said topologically mixing if for any nonempty open sets
$U,V\subset A^{\mathbb{Z}},U\cap F^{-n}\left(V\right)\neq\emptyset$ for all
sufficiently large $n$.

Let $\left(X,F\right)$ be a dynamical system, endow the set $X$ with a sigma
algebra $\mathbb{B}$. The function $F$ preservers some measure $\mu$ on the
sigma algebra $\mathbb{B}$ iff for every $B\in\mathbb{B}$ we have
$\mu(F^{-1}(B))=\mu(B).$ We say then that $\left(X,\mathbb{B},F,\mu\right)$ is
a measurable dynamical system.

The topological support of a measure is defined as the set of all points $x\in
X$ for which every open neighborhood of $x$ has positive measure.

A dynamical system $\left(X,\mathbb{B},F,\mu\right)$ is ergodic if every
invariant subset of $X$ is either of measure 0 or of measure 1. Equivalently,
if for any measurable $U,V\subset X,$ there exists some $n\in\mathbb{N}$ such
that $\mu\left(U\cap F^{-n}\left(V\right)\right)>0.$

If a dynamical system is ergodic then it is transitive on the topological
support of the measure.

For dynamical systems on the real line a common used measure is the Lebesgue
measure, the topological support of the Lebesgue measure is $\mathbb{R}$.

###  II-B Chaotic dynamical systems on the interval

Dynamical systems defined on the real line have a particular behavior, a rich
literature is devoted for the subject, we will recall here some results about
their properties.

In matter of chaos The Devaney’s chaos is seen as a combination of
unpredictably (sensitivity) and regular behaviors (periodic points),
transitivity ensuring that the system is undecomposable.

###### Definition 1

A topological dynamical system $\left(X,f\right)$ is chaotic in the sense of
Devaney if :
(1) Is transitive.
(2) The set of periodic points is dense in X.
(3) Is sensitive to initial conditions.

It is know that for every dynamical system the conditions 1 and 2 implies the
condition 3 which lead to the so called Modified Devaney definition of chaos.

For interval maps transitivity is enough to imply the other two conditions.

###### Proposition 2

An interval map is chaotic in the sense of Devaney if and only if it is
transitive.

For the proof of this result you can look at [13].

##  III Results

###  III-A Preliminary remarks

One issue when using the fuzzy logic xor operator is preserving the invariance
of the resulting dynamical system if we want to combine two dynamical systems
they must be defined on the same interval but this is not enough to ensure
invariance of the resulting combined dynamical system.

If the two dynamical systems are defined on the interval $\left[0,1\right]$ it
is easy to show that the result will be invariant on the interval
$\left[0,1\right].$ One solution to overcome the problem of the invariance is
to rescale every dynamical system defined on a given interval to
$\left[0,1\right].$

Below we give two examples the first one of an xor combination of two chaotic
dynamical systems which is not chaotic and the second one where the result is
a chaotic dynamical system.

###### Example 3

Let us consider the two dynamical systems
$\left(\left[0,1\right],f_{r}\right)$ and $\left(\left[0,1\right],T\right)$
where $f_{r}$ and $T$ are the logistic map and the tent map respectively.

$f_{r}\left(x\right)=r.x.\left(1-x\right),T\left(x\right)=\left\\{\begin{array}[]{c}-2x+1,0\leq
x\leq 0.5\\\ 2x-1,0.5\leq x\leq 1\end{array}\right.$

These two dynamical systems are well known chaotic systems , let us consider
their fuzzy xor combination H defined by

$H\left(x\right)=\max\left(f_{r}\left(x\right),T\left(x\right)\right)-\min\left(f_{r}\left(x\right),T\left(x\right)\right)$

It possesses two fixed points, the fixed point 0 is instable while the fixed
point 0.23 is asymptotically stable. The basin of attraction of the point $0$
contains 4 isolated points while the basin of attraction of 0.23 contain the
hole interval except the basin of attraction of 0.

###### Example 4

Consider the two following dynamical systems the two dynamical systems
$\left(\left[0,1\right],T\right)$ and $\left(\left[0,1\right],ST\right)$ where
$T$ is the tent map and ST that is defined by

$ST\left(x\right)=\left\\{\begin{array}[]{c}-2x+1:0\leq x\leq\frac{1}{2}\\\
2x-1:\frac{1}{2}\leq x\leq\frac{1}{4}\end{array}\right.$

The graph of the function $ST$ is given below, you can see it as an inverted
Tent map graph. The map $ST$ is chaotic as the point $\frac{\pi}{3.5}$ has a
dense orbit. The map $S$ $xor$ $ST$ is chaotic .

###  III-B Numerical experiments summary

Along with the two examples shown before we have tested some other dynamical
systems, part of the results is shown in the following table.

xor | Doubling map | Cubic map | Logistic map | Tent map | Inverted Tent map
---|---|---|---|---|---
Doubling map | ….. | Non chaotic | Non chaotic | Non chaotic | Non chaotic
Cubic map | ….. | ….. | Non chaotic | Non chaotic | Non chaotic
Logistic map | ….. | ….. | ….. | Non chaotic | Chaotic
Tent map | ….. | ….. | ….. | ….. | Chaotic
Inverted Tent map | ….. | ….. | ….. | ….. | …..

The observation of the results of the table suggests that combining two
dynamical systems using the xor operator leads to the resulting dynamical
system to be chaotic if their graphs have some form of symmetry to the
horizontal line $y=\frac{1}{2}.$

###  III-C Mirror effect and number of full branches

The aime of this section is to come with some criterion choice. The optimal
situation is that the two dynamical systems have to be symmetrical to the
horizontal line $y=\frac{1}{2}$ this is what we will call a mirror effect. In
this situation we can show that the combination is a chaotic dynamical using
tools from ergodic theory.

In practical this result could be a tool to choose the dynamical systems to
combine, the closet to symmetry to $y=\frac{1}{2}$ they are the best chances
the combination to work we have.

###### Definition 5

Let $I\subset\mathbb{R}$ be an interval. A map $f:I\rightarrow I$ is a full
branch map if there exists a finite or countable partition $\mathcal{P}$ of
$I$ into subintervals such that for each $w\in\mathcal{P}$ the map
$\left.f\right|_{int\left(w\right)}:int\left(w\right)\rightarrow
int\left(I\right)$ is a bijection.
A map $f$ is a piecewise continuous (resp $C^{1},C^{2},affine$) full branch
map if for each $w\in\mathcal{P}$ the map
$\left|f\right._{int\left(w\right)}:int\left(w\right)\rightarrow
int\left(I\right)$ is a homeomorphism (resp $C^{1}$ diffeomorphism$,C^{2}$
diffeomorphism$,affine$).

###### Definition 6

A full branch map has bounded distortion if

$\underset{n\in\left\\{1,2\right\\}}{\sup}\underset{w^{\left(n\right)}\in\mathcal{P}^{\left(n\right)}}{\sup}\underset{x,y\in
w^{\left(n\right)}}{\sup}\log\left|Df^{\left(n\right)}\left(x\right)/Df^{\left(n\right)}\left(y\right)\right|<\infty$

here $\left(n\right)$ stands for the $nth$ derivative.

If the function is piecewise affine then the distortion is 0.

###### Example 7

The tent map, the doubling map and the logistic map have two full branches
with bounded distortion, the cubic map have three full branches with bounded
distortion.

###### Proposition 8

Consider $\left(\left[0,1\right],f\right)$ and
$\left(\left[0,1\right],g\right)$ two dynamical systems, suppose that the
graphs of $f$ and $g$ are symmetrical according to the horizontal line
$y=\frac{1}{2}$ and that the number of full branches of $f$ and $g$ are equal
to $k$.
The dynamical system $\left(\left[0,1\right],f\text{ }xor\text{ }g\right)$ has
$2k$ full branches.

###### Proof:

Suppose that we have a partition $\mathcal{P}$ of $\left[0,1\right]$ into
subintervals such that for each $w\in\mathcal{P}$ the map
$\left.f\right|_{int\left(w\right)}:int\left(w\right)\rightarrow\left]0,1\right[$
is a bijection.
As $g$ is a mirror of $f$ we obtain by symetry

$\left(f\text{ }xor\text{
}g\right)\left(x\right)=\left\\{\begin{array}[]{c}2\left|f\left(x\right)-0.5\right|\text{
}if\text{ }f\left(x\right)\leq 0.5\\\ 2\left|f\left(x\right)-0.5\right|\text{
}if\text{ }f\left(x\right)\geq 0.5\end{array}\right.$

As $f$ is a bijection there is a partition $w=w_{1}\cup w_{2}$ such that :

$\displaystyle\left(f\text{ }xor\text{ }g\right)\left(x\right)$
$\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{c}2\left|f\left(x\right)-0.5\right|\text{
}if\text{ }x\in w_{1}\\\ 2\left|f\left(x\right)-0.5\right|\text{ }if\text{
}x\in w_{2}\end{array}\right.$ $\displaystyle\Rightarrow$
$\displaystyle\left\\{\begin{array}[]{c}\left(f\text{ }xor\text{
}g\right)\left(w_{1}\right)=\left]0,1\right]\\\ \left(f\text{ }xor\text{
}g\right)\left(w_{1}\right)=\left]0,1\right[\end{array}\right.$

Thus $f$ $xor$ $g$ has two full branches on $w.$

###### Proposition 9

Consider $\left(\left[0,1\right],f\right)$ and
$\left(\left[0,1\right],g\right)$ two dynamical systems, suppose that the
graphs of $f$ and $g$ are symmetrical according to the horizontal line
$y=\frac{1}{2}$ and that $f$ and $g$ have the full branch property then $f$
$xor$ $g$ is chaotic .

###### Proof:

Suppose that $f$ has $k$ branches then $f$ $xor$ $g$ has $2k$ branches and a
relevant partition $w.$
As the graph of $g$ is symmetrical to the graph of $f$ then they have the same
distortion.

On each branch of the partition we have

$\underset{n\geq
1}{\sup}\underset{w^{\left(n\right)}\in\mathcal{P}^{\left(n\right)}}{\sup}\underset{x,y\in
w^{\left(n\right)}}{\sup}\log\left|D\left(f\text{ }xor\text{
}g\right)^{n}\left(x\right)/D\left(f\text{ }xor\text{
}g\right)^{n}\left(y\right)\right|\leq\\\
\max\left(\begin{array}[]{c}\underset{n\geq
1}{\sup}\underset{w^{\left(n\right)}\in\mathcal{P}^{\left(n\right)}}{\sup}\underset{x,y\in
w^{\left(n\right)}}{\sup}\log\left|Df\text{
}^{\left(n\right)}\left(x\right)/D\left(f\text{
}\right)^{n}\left(y\right)\right|\\\ ,\underset{n\geq
1}{\sup}\underset{w^{\left(n\right)}\in\mathcal{P}^{\left(n\right)}}{\sup}\underset{x,y\in
w^{\left(n\right)}}{\sup}\log\left|D\left(g\right)^{n}\left(x\right)/D\left(g\right)^{n}\left(y\right)\right|\end{array}\right)$

Hence $f$ $xor$ $g$ is a full branch map with bounded distortion. Then the
Lebesgue measure is ergodic [10].
As the Lebesgue measure is ergodic then $f$ $xor$ $g$ is transitive on the
topological support of the Lebesgue measure.Hence it is chaotic.

##  IV Conclusion

We investigated an ideal situation to combine chaotic one dimensional maps
using the fuzzy logic xor operator.

Using tools from ergodic theory we were able to establish a result using
slightly strong condition than Devaney’s chaos, it is worth noting that this
condition is satisfied by a majority of classical chaotic maps on the
interval.

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