Konstantinos G Rallis

Vasilis K Papanikolaou

Panagiotis D Diamantoulakis

Mohammad-Ali Khalighi
email: ali.khalighi@fresnel.fr

George K Karagiannidis

Energy Efficiency Maximization in Cooperative Hybrid VLC/RF Networks with NOMA

In this paper, a cooperative hybrid visible light communications (VLC)/radio frequency (RF) network that employs non-orthogonal multiple access (NOMA) is investigated. More specifically, the resource allocation and the operating mode of the system is optimized to maximize a weighted energy efficiency metric, taking into accounts the particularities of the hybrid network. The resulting optimization problem is efficiently tackled with the use of Dinkelbach's algorithm and difference of convex programming. Using this solution, a deep neural network (DNN) is trained to find the operation mode and alleviate the computational cost of the overall problem leading to a close solution to the optimal. Finally, the effectiveness of the proposed setup is presented via Monte Carlo simulation results.

I. INTRODUCTION

A major limitation of conventional radio frequency (RF) communications has been the ever increasing crowding of the RF spectrum, due to the growing number of connected devices and amount of data they require. To address these concerns, academia and industry have shifted their attention to emerging technologies, such as optical wireless communications (OWC), that utilize a different region of the electromagnetic spectrum, meaning they would be able to serve a plethora of devices with no interference to the RF network [START_REF] Uysal | Optical Wireless Communications: An Emerging Technology[END_REF]. More specifically, optical wireless access indoors has been mostly facilitated via visible light communications (VLC) technology through the use of light emitting diodes (LED) bulbs, which can offer high data rate communications at a low cost through the illumination infrastructure. VLC also offers high inherent physical layer security and a high frequency reuse factor since light is blocked by opaque structures such as walls. Due to these advantages, VLC has been recognized as a potential candidate technology for the 5G Public-Private-Partnership (PPP) project, while there are global efforts on VLC standardization such as IEEE P802.11bb. On the other hand, line-of-sight (LoS) can easily be blocked by the movement or rotation of the receiver and also VLC is in general dependent on the room illumination levels. The above reasons have led to the proposed heterogeneous networking of RF and VLC networks, in order to maintain coverage via RF while boasting higher data rates through VLC. Hybrid VLC/RF networks have been extensively studied during the recent years [START_REF] Chowdhury | Optical Wireless Hybrid Networks: Trends, Opportunities, Challenges, and Research Directions[END_REF] as a way to capitalize on the unused electromagnetic (EM) spectrum for wireless access.

Apart from the growing interest in previously unexploited regions of the EM spectrum, the efficient use of the available bandwidth is of paramount importance. To this end, nonorthogonal multiple access (NOMA) has attracted significant attention from the research community due to its increased spectral efficiency. NOMA with VLC has been examined in [START_REF] Marshoud | Non-Orthogonal Multiple Access for Visible Light Communications[END_REF]- [START_REF] Yang | Fair Non-Orthogonal Multiple Access for Visible Light Communication Downlinks[END_REF] and it was also shown experimentally in [START_REF] Lin | Experimental demonstration of bidirectional NOMA-OFDMA visible light communications[END_REF], while hybrid VLC/RF networks with NOMA were studied in [START_REF] Papanikolaou | User Grouping for Hybrid VLC/RF Networks With NOMA: A Coalitional Game Approach[END_REF]- [START_REF] Obeed | Power Allocation and Link Selection for Multicell Cooperative NOMA Hybrid VLC/RF Systems[END_REF]. Cooperative hybrid VLC/RF networks with NOMA were examined in terms of outage probability in [START_REF] Xiao | Hybrid Lightwave/RF Cooperative NOMA Networks[END_REF]. In that paper, a cross-band selection diversity combining scheme is proposed to improve the system's performance.

Moreover, the increasing number of access points (APs) is concerning given the resulting energy consumption. However, VLC APs which are usually already on for illumination coverage, can offer wireless access with minimal additional energy consumption. As such, VLC has been recognized as a promising indoor wireless network to reduce the energy requirements and improve energy efficiency. In this regard, in [START_REF] Kafafy | Power Efficient Downlink Resource Allocation for Hybrid RF/VLC Wireless Networks[END_REF], the authors examined the resource allocation in a hybrid VLC/RF network to maximize the energy efficiency of the system, while in [START_REF] Khreishah | A Hybrid RF-VLC System for Energy Efficient Wireless Access[END_REF], the authors minimized the energy consumption of a hybrid VLC/RF network while satisfying the users' quality of service (QoS) requirements. Additionally, in [START_REF] Aboagye | Design of Energy Efficient Hybrid VLC/RF/PLC Communication System for Indoor Networks[END_REF] the authors designed a hybrid VLC/RF network with a power line communications (PLC) backhaul and maximize the energy efficiency via optimizing the power allocation of the VLC/RF network and the backhaul flow of the PLC network. Moreover, in [START_REF] Al-Khori | Secrecy performance of decode-and-forward based hybrid rf/vlc relaying systems[END_REF], hybrid VLC/RF networks with a decode and forward relaying setup were investigated in terms of physical layer security. More recently, in [START_REF] Obeed | User Pairing, Link Selection, and Power Allocation for Cooperative NOMA Hybrid VLC/RF Systems[END_REF], [START_REF] Obeed | Power Allocation and Link Selection for Multicell Cooperative NOMA Hybrid VLC/RF Systems[END_REF], the authors investigated the power allocation and user pairing in a cooperative setup like the one [START_REF] Xiao | Hybrid Lightwave/RF Cooperative NOMA Networks[END_REF], with the aim to maximize the sum rate while satisfying QoS constraints of the users.

Motivated by the former results, in this paper, we investigate a cooperative hybrid VLC/RF network to maximize a weighted energy efficiency metric. The proposed metric takes into account the particularities imposed by the illumination function of the VLC APs. The cooperative network makes use of NOMA to serve its users via the VLC AP, while an RF link is present between the users for possible cooperation. The operation mode of the network is chosen to optimize energy efficiency and the resulting optimization problem is tackled with the use of a Dinkelbach based algorithm and difference of convex programming. Finally, in order to reduce the computational complexity of the proposed solution, a deep neural network (DNN) is trained to chose the operation mode of the network given the users' channel state information and QoS constraints. After training, the DNN can offer close to optimal performance at a fraction of the cost.

II. SYSTEM MODEL

We consider an indoor downlink system, consisting of one VLC AP and two types of users, namely a near user, denoted as U 1 , and a far user, denoted as U 2 . The users' locations are uniformly distributed in a cyclic disk with radius R 0 and an annular area bounded by radii R 0 and R v , respectively. Polar coordinates (r i , θ i ) are utilized to describe the exact position of user i from the center. Moreover, the VLC AP is placed on the ceiling with a vertical distance L from the ground plane. Both U 1 and U 2 are assumed to belong in the coverage area of the VLC AP. The VLC AP utilized NOMA for downlink transmission to U 1 and U 2 simultaneously, while U 1 can also act as a full-duplex (FD) decode-and-forward (DF) RF relay to assist the far user U 2 .

A. VLC Transmission

VLC transmission employs intensity modulation-direct detection (IM-DD) NOMA. The non-negative optical transmitted superposed signal is defined as x = √ P 1 x 1 + √ P 2 x 2 , with P 1 and P 2 being the transmitted power allocated for user U 1 and U 2 , respectively. To maintain illumination levels as well as to guarantee safety and hardware constraints, the maximum transmitted optical power is limited to P VLC max . Therefore, it holds that P 1 + P 2 ≤ P VLC max .

(1)

RF VLC LED U 1 U 2 R v R 0 Fig. 1. System Model.
Accordingly, the received message at user U i can be expressed as

y i = ηh i x + n i , ( 2
)
where η is the photo-detector responsivity in A/W, h i is the VLC channel gain between the VLC AP and U i , and n i is the additive white Gaussian noise (AWGN) at U i . Assuming LoS, the VLC channel gain is given by [START_REF] Komine | Fundamental Analysis for Visible-Light Communication System Using LED Lights[END_REF], [START_REF] Ma | Coordinated Broadcasting for Multiuser Indoor Visible Light Communication Systems[END_REF]

h i = m + 1 2πd 2 i A r cos m (φ i )T (ψ i )g(ψ i )cos(ψ i ), (3)
where A r is the detector area of the photodiode at user U i , φ i and ψ i denote the irradiance and the incidence angles of U i , respectively, while m is the Lambertian emission order, which is obtained as

m = -ln2/ln(cos(Φ 1/2 )), (4)
where Φ 1/2 is the transmitter semi-angle at half-power. On top of that, the distance d i is expressed as d i = r 2 i + L 2 regarding the polar coordinates of user U i and the ceiling height L. Moreover, T (ψ i ) and g(ψ i ) denote the gains of the optical filter and the optical concentrator, respectively, with the latter being expressed as

g(ψ i ) = n 2 sin 2 ΨFOV , 0 ≤ ψ i ≤ Ψ FOV 0, ψ i > Ψ FOV , ( 5
)
where Ψ FOV denotes the field-of-view (FOV) of the transmitter and n is the refractive index. Without loss of generality, we assume that the optical detector of each user is pointing upwards so that it holds that φ i ≈ ψ i . Then, taking into account the polar coordinates system, the channel gain from the VLC AP to the VLC user is given by

h i = C(m + 1)L (m+1) (r 2 i + L 2 ) m+3 2 , ( 6
)
where C = ArT (ψi)g(ψi) 2πd 2 i . According to the power domain NOMA principle, the weak user U 2 decodes its own message with an achievable rate R 2,V by treating U 1 's message as interference. At the strong user's detector, i.e., at U 1 , the message meant for U 2 is first decoded to be removed via successive interference cancellation (SIC). The rate at which U 1 can decode the message of U 2 is R 2→1,V . Following this, U 1 decodes its own message without interference with an achievable rate equal to R 1,V . Due to the intensity modulation-direct detection (IM/DD) utilized by the VLC system, the common Shannon formula cannot be used to evaluate the capacity since there are a number of additional constraints for the transmitted signal, including the non-negativity limitation of the input signal that modulates the optical intensity of the emitted light and the constraint on the total optical transmit power transmitted. Instead, by utilizing a lower bound of the corresponding capacity region, the achievable rates for each user can be described as [START_REF] Chaaban | On the Capacity of the Intensity-Modulation Direct-Detection Optical Broadcast Channel[END_REF]

R 2→1,V = B v log 2 1 + (ηh 1 P 2 ) 2 ((ηh 1 P 1 ) 2 + 9σ 2 )(1 + μ ) 2 -φ , (7) R 1,V = B v log 2 1 + (ηh 1 P 1 ) 2 9σ 2 (1 + μ ) 2 -φ , (8) R 2,V = B v log 2 1 + (ηh 2 P 2 ) 2 ((ηh 2 P 1 ) 2 + 9σ 2 )(1 + μ ) 2 -φ , (9)
where B v is the bandwidth of the VLC network, σ 2 is the noise variance and φ = 0.016, μ = 0.0015.

B. RF Transmission

During RF transmission, the baseband equivalent received signal at the terminal of user U 2 is expressed as follows

y R = P RF x t h RF + n R , ( 10
)
where n R is the AWGN noise at the RF receiver with variance

σ 2 R , h RF ∼ Rician(K r , d -ζ RF )
is the Rician RF channel. Without loss of generality, Rician fading is assumed since a strong LoS component is almost certainly present between the two indoor users as they are described in this scenario. Moreover P RF denotes the available power for retransmission at U 1 and ζ is the path-loss exponent. Given the topology of the scenario and by utilizing the polar coordinates of U 1 and U 2 , d RF , which is the Euclidean distance between the two users, can be calculated by

d RF = r 2 1 + r 2 2 -2r 1 r 2 cos (θ 2 -θ 1 ) (11)
Taking into account the path-loss formula

L dB RF = L d0 + 10ζ log 10 ( dR d0 ), L RF = 10
L dB RF 10 , where L d0 = 68dB and d 0 = 1 m, the achievable data rate from U 1 to U 2 via the RF link, according to Shannon capacity, is given by:

R 2,R = B R log 2 1 + L RF |h RF | 2 P RF σ 2 R , ( 12
)
where B R denotes the bandwidth of the RF system.

III. ENERGY EFFICIENCY OPTIMIZATION

In this section, the resource allocation of the hybrid VLC/RF network is investigated in order to maximize an energy efficient metric, while the users of the network fulfill their QoS requirements. Energy efficiency is defined as the ratio of achievable data rate per consumed power. In order to better capture the particularities of the heterogeneity of this type of network, such as the dependency of VLC transmission power to the illumination levels, a weighted energy efficiency metric is examined. More specifically, a weight α with 0 ≤ α ≤ 1 is used to tune the users' throughput, while a weight β, 0 ≤ β ≤ 1, adjusting the focus of the metric on the consumed power of each network. As such, the weighted energy efficiency metric that is studied in this paper is expressed as

EE = αR 1 + (1 -α)R 2 βP VLC + (1 -β)P RF , ( 13
)
where P VLC = P 1 + P 2 and R 2 is the achievable data rate of the second user, depending on the mode q of the cooperative network, i.e.,

R 2 = qR 2,V + (1 -q)R 2,R , q ∈ {0, 1}, (14)
where q = 1 is the pure VLC mode, while q = 0 is the cooperative RF mode.

Considering the achievable data rates of each user, their QoS constraints, and the required power for consumption, the following optimization problem to maximize energy efficiency is defined

max q,P1,P2,PRF αR1+(1-α)R2 βPVLC+(1-β)PRF s.t. C 1 : R 1,V ≥ R thr 1 , C 2 : R 2 ≥ R thr 2 , C 3 : R 2→1,V ≥ R thr 2 , C 4 : i∈VLC P VLC i ≤ P VLC max , C 5 : P RF 2 ≤ P RF max , (15)
where C 1 and C 2 are the QoS requirements of the users U 1 and U 2 , respectively. U 2 can achieve their requirement with either the VLC network or through the cooperative strategy. Since NOMA is utilized in this scenario, U 1 is required to be able to decode U 2 's message for two reasons; first, it is required for U 1 to perform SIC and decode their own message, and second in order to be able to transmit via RF the message to user U 2 . Therefore, constraint C 3 ensures that the rate at which U 1 decodes U 2 's message fulfills their QoS constraint. Finally, as mentioned in the previous section, the power consumption levels of the AP and U 1 are bounded by illumination and hardware constraints leading to C 4 and C 5 .

The optimization problem in ( 15) is a mixed integer nonlinear optimization problem. In order to solve this problem we perform a full search on the integer value q, since it can only take values of either zero or one. Following that, problem [START_REF] Komine | Fundamental Analysis for Visible-Light Communication System Using LED Lights[END_REF] reduces to a nonlinear problem. However, it is noted that the problem is non-convex, mainly due to the fractional objective function and constraints C 1 , C 2 , and C 3 , which contain the logarithms with squared power terms. In order to efficiently solve this problem in polynomial time, we need to transform it into an equivalent convex than can be efficiently solved by efficient fractional programming techniques, including convex optimization tools. To do so, first, we introduce two auxiliary variables r 1 and r 2 , such that

R thr 1 ≤ r 1 ≤ R 1,V and R thr 1 ≤ r 2 ≤ R 2 , ( 16
)
which change the constraints and introduce C 6 and C 7 . As such, the problem ( 15) is formulated as:

max q,r1,r2,P1,P2,PRF αr1+(1-α)r2 βPVLC+(1-β)PRF s.t. C 1 : R 2→1,V ≥ R thr 2 , C 2 : i∈VLC P VLC i ≤ P VLC max C 3 : P RF 2 ≤ P RF max , C 4 : r 1 ≤ R 1,V , C 5 : r 2 ≤ R 2 , C 6 : r 1 ≥ R thr 1 , C 7 : r 2 ≥ R thr 2 . ( 17
)
The objective function's form makes problem (17) a fractional programming problem. To solve this, Dinkelbach's algorithm is applied [START_REF] Dinkelbach | On Nonlinear Fractional Programming[END_REF]. Dinkelbach's algorithm is an iterative process that solves an equivalent parametric problem given as the difference between the numerator and the denominator multiplied by a parameter. As such, we set F = αr 1 +(1-α)r 2 and G = βP VLC + (1β)P RF . To maximize an objective function with Dinkelbach's algorithm, F should be concave and G should be convex. In this case, both functions are affine, so Dinkelbach's algorithm can be utilized. The fractional problem max{U (z) = F (z)/G(z)} can then be related to

H(u) = max{F (z) -uG(z)}, u ∈ R, (18)
with z = [r 1 , r 2 , P 1 , P 2 , P RF ]. Following that, our aim is to find an optimal solution z i of H(u i ) in each iteration i of the algorithm.

Ĥ is continuous, convex and strictly decreasing in R. It is assumed that z+ is optimal, if and only if it is optimal for H(u + ), where u + is the only zero of H. The Dinkelbach's Algorithm applied is explained in Algorithm 1.

Algorithm 1: Dinkelbach's Algorithm Initialization: Set the initial point u 0 < u + , for example u 0 = U (z 0 ) > 0 for some z 0 . Also set iteration index i = 0 and the convergence accuracy ; while H(u i ) > (for some given) do Calculate an optimal solution z i of

H(u i ) s.t. ( 17
).C 1 -(17).C 7 ; Let u i+1 = U (z i ); i ← i + 1; end Result: optimal u + ,z +
In Algorithm 1, constraints [START_REF] Chaaban | On the Capacity of the Intensity-Modulation Direct-Detection Optical Broadcast Channel[END_REF].C 1 -(17).C 7 still hold. However the problem remains non-convex and in order to be able to find a tractable solution in polynomial time, it is of essence to transform the problem to an equivalent convex one. We start by employing the geometric programming transformation, i.e., r 1 = e r1 , r 2 = e r2 , P 1 = e p1 , P 2 = e p2 , P RF = e pRF , [START_REF] Vucic | DC programming approach for resource allocation in wireless networks[END_REF] leading to z = [r 1 , r2 , p1 , p2 , pRF ] and e p1 + e p2 = e pVLC . After the transformation, the problem is expressed as follows

max q,z αe r1 + (1 -α)e r2 -u (βe pVLC + (1 -β)e pRF ) s.t. C 1 : log e 2p1 + 9σ 2 (ηh1) 2 -2p 2 + log (1 + μ ) 2 (2 R thr 2 + φ Bv -1) ≤ 0 C 2 : e p1 + e p2 ≤ P VLC max C 3 : p RF ≤ log(P RF max ) C 4 :log 2 e r1 + φ Bv -1 -2p 1 -log (ηh1) 2 9σ 2 (1+ μ) 2 ≤ 0 C (q=1) 5 : log 2 e r2 + φ Bv -1 + log e 2p1 + 9σ 2 (ηh2) 2 -2p 2 + log((1 + μ ) 2 ) ≤ 0 C (q=0) 5 : log 2 e r2 B R -1 -p RF -log LRFhRF σR ≤0 C 6 : r1 ≥ log(R thr 1 ) C 7 : r2 ≥ log(R thr 2 ). ( 20
)
Constraints C 2 , C 3 , C 6 , and C 7 are easily shown to be convex. More specifically, the first term of C 1 and the second term of C (q=1) 5

are convex as log-sum-exp terms. The remaining of terms that are needed to be convex are in the form of f (t) = log C 0 2 e t /C1 -1 . The second derivative of f (t) with respect to t can be written as d 2 f dt 2 = 2 q C0q(C02 q -1-q log(2)) (C02 q -1) 2

, where q = exp(t)/C 1 and C 0 , C 1 > 0. Considering ξ = C 0 2 qq log(2) -1 is an increasing function with respect to q and when q → 0, ξ → 0, it is shown that d 2 f dt 2 ≥ 0. Following that, it is proven that f (t) is convex with respect to t.

We define the objective function of (20) as Ξ(z, u) = αe r1 + (1α)e r2u (βe pVLC + (1β)e pRF ) = Φ(z) -uΘ(z). We observe that Ξ is convex with respect to u, but it is a difference of convex (DC) function with respect to z. Therefore, to calculate the optimal solution of (20) to plug into Algorithm 1, we have to apply a DC Algorithm (DCA) [START_REF] Vucic | DC programming approach for resource allocation in wireless networks[END_REF]. DCA is an iterative algorithm based on local optimality conditions and duality. It is noted that we choose to minimize the -Ξ(z, u i ) problem which is equivalent to maximize the Ξ(z, u i ) one. The idea of DCA is that at each iteration, the second component in the primal DC problem is replaced by its affine minorization in order to generate the convex problem

min z Ξ(z, u) s.t. (20).C 1 -(20).C 7 , (21)
where

Ξ(z, u) = uΘ(z) -(Φ(z k ) + ∇Φ(z k ) T (z -zk )).
The DCA scheme is described in Algorithm 2.

Algorithm 2: DCA Initialization: Choose an initial point z0 , z-1 ∈ R N and Let k = 0; Set iteration index i = 0 and the convergence accuracy

; while Ξ(z k , u i ) -Ξ(z k-1 , u i ) > do
Compute ∇Φ(z k ); Calculate an optimal solution zk+1 for convex problem [START_REF] Sun | Learning to Optimize: Training Deep Neural Networks for Interference Management[END_REF];

k ← k + 1; end Result: optimal z+
The resulting convex optimization problem can be efficiently solved with convex optimization methods in polynomial time, such as the interior-point method [START_REF] Boyd | Convex optimization[END_REF]. Following that, optimal z is transformed to z through [START_REF] Vucic | DC programming approach for resource allocation in wireless networks[END_REF] and it is plugged into Algorithm 1, in order to find the optimal u. At the end of this iterative process the optimal resource allocation is obtained. The operation mode q is selected from the solution that maximizes the total energy efficiency.

IV. MULTI-LAYER PERCEPTRON FOR MODE CLASSIFICATION

In order to reduce the complexity of obtaining the optimum as was described in the previous section, we design More specifically, our DNN model is a multi-layer perceptron (MLP), which is a feedforward network making it much easier to train. To train the MLP, training sets are generated via the optimal algorithm described in section III.

The training data are denoted as (x, y), where the input x refers to the channel vector h = [h 1 , h 2 , h RF ] and the QoS requirements of the users

R thr = R thr 1 = R thr 2 . So x = [h, R thr ]. The output y of the MLP is continuous in [0, 1]
an as such it cannot directly set the value of q. Therefore, for any fractional value 0.5 ≤ y ≤ 1, we set q = 1, otherwise if 0 ≤ y < 0.5, the value q = 0 is chosen. Apart from the input and output layers, the MLP consists of a 3 hidden layers. Each hidden layer consists of a total of 20 nodes. The activation function for the three hidden layers is ReLU, while for the output layer the sigmoid function was used. In order to increase the accuracy of the MLP, a pre-processing of the data took place through a min-max scaler pre-processor, which translates the input values to a range of [0,1] given their minimum and maximum values in the set.

V. NUMERICAL RESULTS AND DISCUSSION

In this section, Monte Carlo simulation results are presented for the proposed system for 10 4 iterations. The simulation parameters can be found in Table I. More specifically, in Figs 3 and4, the effect of the weighting factors α and β on the energy efficiency versus the threshold rates of the users can be observed. Without loss of generality, the two users are presented with identical QoS constraints in these results. In this regard, in Fig. 3, β = 0.5 meaning that both networks are equally accountable for the energy consumption of the network, while the different values of α highlight whether priority is given to one user over the other in terms of achievable data rate. The optimal system configuration in terms of energy efficiency is then seen to be the cooperative hybrid mode over the pure VLC mode when the rate requirements of the users are low. However, when users require higher throughput, the RF link requires a very high amount of energy to comply, while VLC can achieve the constraints at a lower energy cost. Interestingly, for lower rates, as the rate threshold increases, energy efficiency is also increasing, which means that a little amount of additional power can provide a higher increase in data rate, which can be justified through [START_REF] Xiao | Hybrid Lightwave/RF Cooperative NOMA Networks[END_REF], [START_REF] Obeed | User Pairing, Link Selection, and Power Allocation for Cooperative NOMA Hybrid VLC/RF Systems[END_REF]. Moreover, in Fig. 4, α = 0.5 is set, so that neither U 1 nor U 2 has a higher priority than the other and the effect of different values of β are examined. When β is low, the power consumption of the VLC AP does not contribute a lot to the energy efficiency calculation. This is an interesting and practical case, since VLC APs can most of the time provide illumination coverage and are already on, so β can be lower than 0.5 or even zero. On the other hand, when illumination is not required, β can be higher than 0.5 to highlight the increased cost of the VLC AP in this scenario. It can be observed from Fig. 4, that when β takes lower values, the optimal selection is the VLC for a lower required threshold rate. When the weight β is increased, despite its lower energy efficiency is general, the cooperative hybrid VLC/RF mode is selected, instead, for up to 6 bits/Hz threshold rate.

Finally, the results of the implemented MLP are shown in Fig. 5. The dataset contained 10 4 cases, 7000 of which were used for training, 1500 for validation, and 1500 for testing. Stochastic Gradient Descent (SGD) was utilized to optimize the MLP, while the binary cross-entropy was used as a loss function. The proposed MLP has an accuracy of 95%. The average energy efficiency achieved through the use of the MLP is very tight to the optimal bound. As such, the DNN implementation can indeed help alleviate the computational cost burden of the full optimization method described in section III.

VI. CONCLUSIONS

In this paper, we have presented a VLC network with NOMA that can operate in a cooperative hybrid VLC/RF mode to increase the energy efficiency of the system. A weighted energy efficiency metric was proposed to capture the particularities of the network and the optimal operation mode and power allocation was found via Dinkelbach's algorithm, DC programming and convex optimization methods. In order to reduce the computational complexity of the proposed solution a DNN was trained on optimal samples recorded from the optimal solution and reached an close to optimal performance. Future work of this project encompasses the training of the DNN to completely replace the optimization method after training and provide a close to optimal solution.

Fig. 2 .

 2 Fig. 2. Network graph of the proposed MLP

Fig. 3 .

 3 Fig. 3. Energy efficiency for different values of R thr and α with β = 0.5.

Fig. 4 .

 4 Fig. 4. Energy efficiency for different values of R thr and β with α = 0.5.

Fig. 5 .

 5 Fig. 5. Energy efficiency achieved through MLP selection and optimal selection for α = β = 0.5.

ACKNOWLEDGMENT

This publication is based upon work from COST Action CA19111 (European Network on Future Generation Optical Wireless Communication Technologies, NEWFOCUS), supported by COST (European Cooperation in Science and Technology).


Optimization of Receivers Field-of-Views in Multi-User VLC Networks: A Bio-inspired Approach M. W. Eltokhey, Member, IEEE, M. A. Khalighi, Senior Member, IEEE, Z. Ghassemlooy, Senior Member, IEEE

Abstract-Visible-light communication is a well known emerging technology, which enables high data rate wireless access in indoor environments, by making dual-use of light-emitting diode luminaires for providing lighting and communication. Managing the multiple access and addressing the users' mobility are among the current challenges of this technology. To manage the multipleaccess interference arising from non-intended received signals of the other users, different approaches have been proposed so far, including zero-forcing pre-coding and broadcasting of the users' signals. However, by this approach, the network performance is sensitive to the users' locations, since they affect the correlation properties of the network channel matrix; a higher correlation (e.g., for the case of users getting closer together) results in a performance degradation. To mitigate this, we propose a novel approach of optimizing the receivers' field-of-views based on the bio-inspired particle swarm optimization, in order to adapt to the users' locations, and hence improve the network robustness. The efficiency of the proposed method is demonstrated through numerical simulations.

Index Terms-Visible light communications; multiple-access interference; particle swarm optimization; ZF pre-coding; fieldof-view optimization.

I. MULTI-USER VISIBLE-LIGHT COMMUNICATION NETWORKS

The ever-increasing bandwidth requirements to support the growing demand for high-speed data transmission in indoor scenarios, is overwhelming the radio frequency (RF) spectrum, which is already under a huge pressure. Subsequently, there has been a growing interest in the visible-light communication (VLC) technology during the past two decades, as a solution for broadband indoor communications. This has been driven by numerous advantages of this technology, such as utilizing the light-emitting-diode (LED) -based lighting infrastructure, operating at an unregulated frequency band, being unaffected by RF interference, and providing inherent security due to light confinement in most of the indoor environments [START_REF] Ghassemlooy | Visible Light Communications: Theory and Applications[END_REF].

One important requirement in VLC systems is ensuring coverage in large-space indoor environments. This can be realized through the use of multi-cell architectures, in the same way it is done in wireless RF cellular networks. This M. W. Eltokhey and M. A. Khalighi are with Aix-Marseille University, CNRS, Centrale Marseille, Institut Fresnel, Marseille, France (e-mail: Mahmoud.Eltokhey@centrale-marseille.fr, Ali.Khalighi@fresnel.fr) Z. way, several LED luminaires will serve as access points (APs), which handle users within their respective illumination areas (cells). In such multi-cell VLC networks, which potentially should handle multiple users, each equipped with a receiver (Rx) that uses a photo-detector (PD), an important issue is to manage the multiple-access interference (MAI). In fact, MAI can arise from inter-user interference (IUI) or/and inter-cell interference (ICI), i.e., the received non-intended signals of the users in the same cell or/and in the other cells, respectively [START_REF] Eltokhey | Hybrid NOMA and ZF pre-coding transmission for multi-cell VLC networks[END_REF]. There has been a significant amount of research on the development of efficient multiple-access (MA) techniques for VLC networks so far [START_REF] Al-Ahmadi | Multi-user visible light communications: State-of-the-art and future directions[END_REF]. The most popular solutions proposed include orthogonal frequency-division MA (OFDMA), nonorthogonal MA (NOMA), and pre-coded multi-user multipleinput single-output (MU-MISO) [START_REF] Eltokhey | Multiple access techniques for large space indoor scenarios: A comparative study[END_REF], [START_REF] Abdalla | Interference in multi-user optical wireless communications systems[END_REF]. Here, we focus on the third technique, i.e., pre-coded MU-MISO due to its implementation simplicity, suggesting it as a practical solution. There, a common approach is to use linear zero-forcing precoding (ZFPC) due to its simplicity and its good performance at relatively high signal-to-noise ratios (SNRs), which is mostly the case in indoor VLC networks, because of the short link distance in the case of unblocked line-of-sight (LOS) [START_REF] Pham | Multi-cell VLC: Multi-user downlink capacity with coordinated precoding[END_REF], [START_REF] Feng | The spatial dimming scheme for the MU-MIMO-OFDM VLC system[END_REF].

The performance of ZFPC is largely affected by the locations of the users, as these determine the correlation between the users' channel gains [START_REF] Yu | Multi-user MISO broadcasting for indoor visible light communication[END_REF]. In fact, a high correlation (e.g., due to nearly-located users) results in a degradation of the overall network performance. Although different Rxs' parameters can be optimized to improve the performance, one efficient solution for reducing this correlation is to appropriately tune the Rxs' field-of-views (FOVs), as proposed in [START_REF] Hong | Performance of a precoding MIMO system for decentralized multiuser indoor visible light communications[END_REF], [START_REF] Marshoud | MU-MIMO precoding for VLC with imperfect CSI[END_REF].

This article considers tuning of the Rxs' FOVs in MU-MISO ZFPC VLC systems to improve the network performance by decreasing the correlation in the network channel matrix. The novelty of the proposed method is that it relies on the "bioinspired" algorithm of particle swarm optimization (PSO), which has been used in solving diverse problems in the area of optical wireless communications [START_REF] Eltokhey | Optimization of intensities and locations of diffuse spots in indoor optical wireless communications[END_REF], [START_REF] Demir | Unified resource allocation and mobility management technique using particle swarm optimization for VLC networks[END_REF]. We illustrate the advantages of this swarm intelligence technique in terms of efficiency and computational simplicity, and investigate the impact of varying the optimization parameters on the system performance. Also, we propose a solution to further reduce the computational complexity of the proposed optimization approach, and show its advantage in the case of user mobility.

In the sequel, we first describe the principles of a ZFPC VLC network. Next, we provide an overview of PSO and the proposed FOV tuning algorithm, before presenting the performance study of the proposed solution and discussing its practical implementation issues.

II. GENERAL DESCRIPTION OF ZFPC VLC NETWORKS

For VLC networks, the transmission link from the AP to the Rx is characterized mainly by the channel gain. We only account for the LOS path here. In fact, in the case of existing LOS in VLC networks, the contribution of non-LOS components is practically negligible [START_REF] Long | Investigating channel frequency selectivity in indoor visible light communication systems[END_REF]. The APs, i.e., the LED luminaires, are modeled by a Lambertian pattern of order m [START_REF] Long | Investigating channel frequency selectivity in indoor visible light communication systems[END_REF]. No lens is considered at the Rx, thus, the channel gain does not depend on the Rx's FOV.

Figure 1 illustrates the considered VLC network, with the main system blocks shown in Fig. 1(a). A central control unit coordinates between the APs for exchanging information (e.g., the channel state information, CSI), handling the ZFPC, as well as FOV optimization. To estimate the CSI at each Rx, pilot symbols are transmitted by the APs in the downlink. In the uplink, the estimated CSI are sent back to the APs using infrared links, for calculating the ZFPC matrix, and for FOV optimization.

To adjust the Rx's FOV, a mechanical iris is considered to be mounted on top of the plan of the PD's surface, to vary the maximum angle of reception at the Rx [START_REF] Abdalla | Dynamic FOV visible light communications receiver for dense optical networks[END_REF]. An illustration for the considered FOV tuning is presented in Figs. 1(b), 1(c), and 1(d).

We consider a MU-MISO broadcast VLC system, where N t APs cooperate to serve N r users, assuming N t ≥ N r [START_REF] Yu | Multi-user MISO broadcasting for indoor visible light communication[END_REF], in order to comply with the ZFPC constraints. To manage the MAI, for instance, at AP i , the desired signal of Rx j , d j , is multiplied by the pre-coding weight w ij . We consider the maxmin fairness criterion in [START_REF] Yu | Multi-user MISO broadcasting for indoor visible light communication[END_REF] for calculating the ZFPC weights, due to it simplicity.

III. PARTICLE SWARM OPTIMIZATION

For optimization of the FOVs of the Rxs in the ZFPC system, we propose to use PSO, which mimics swarms movement [START_REF] Kennedy | Particle swarm optimization[END_REF]. Compared with other metaheuristic optimization techniques, the advantages of the PSO approach include its memory effect, in the sense that all particles acquire the knowledge of the good solutions in the solution space, because every particle shares its information with the others. In addition, PSO offers the advantages of a simple implementation and a relatively quick convergence, which are the reasons why it has received a great deal of attention in a wide range of applications. Consider a D-dimensional search space, which involves all possible combinations of the values of the D variables to be optimized. By PSO, within this search space, each point is a potential solution: each particle moves between points to find the optimal solution. The direction and the speed of the movements in the solution space are decided by the best solution found by the particle itself (personal best position, P best), and that found by the ensemble of particles (global best position, Gbest). The quality of these solutions is assessed using a fitness function designed according to the optimization goals.

PSO can be well described by an example of a swarm of bees (particles) searching in a garden (solution space) for location with most flowers (optimal solution), over a certain time (iterations). Bees communicate their experiences in terms of flowers' density at their locations to each other (P best and Gbest), to decide the speed and direction of movement (particles' velocities) in their next move. The flowers' density corresponds to how good the location is (solution's quality), where higher flowers' densities correspond to better solutions. Figure 2 illustrates the considered PSO algorithm, where as shown in Figs. 2(a) and 2(b), particles (represented by bees) wander the 2-variable solution space formed by the axes of Variable 1 and Variable 2 to search for the optimal solution. The axis of fitness value represents the evaluation of each possible solution using the fitness function, where larger fitness values represent better solutions. In Fig. 2(a) the particles are distributed in the solution space and exchange information about the P best and Gbest, which illustrates the exploration behaviour by the particles at the start of the optimization. Figure 2(b) shows the behavior of the particles at the last iterations, where they are about to converge to the optimal solution. A flow chart for the proposed FOV tuning algorithm using PSO is also shown in Fig. 2(c), where the dashed box at the right side depicts an example of two particles in a 3dimensional solution space, illustrating the influence of the original velocity, P best, and Gbest on the direction and the velocity of the particles' movements. Figure 2(d) illustrates the idea of reducing the correlation in the network channel matrix by optimization of the FOVs, highlighting the impact of varying the Rx FOV using PSO on the channel gains at the Rxs, and subsequently the ZF pre-coding network performance.

The main parameters of the PSO algorithm include the maximum number of iterations N it , the number of particles N p , and the number of variables D. For updating the velocities and the positions of the particles, the approach in [START_REF] Eltokhey | Optimization of intensities and locations of diffuse spots in indoor optical wireless communications[END_REF] is considered. Without loss of generality, for preventing particles from moving to locations outside the solution space, hard boundary conditions are considered here, where new positions (represented by new variables' values) exceeding the solution space are clipped, in order to remain at the boundaries.

IV. FOV TUNING USING PSO

Here, we describe the proposed FOV tuning algorithm. As shown in Fig. 2(c), the algorithm starts with generating for each particle (i.e., possible solution, represented by a set of FOVs for all Rxs) random positions (sets of FOVs) and velocities (rates of varying of the tested sets of FOVs in the solution space), prior to evaluating in each step (iteration) the performance of each particle and updating them. To evaluate the performance of a particle (the quality of the solution formed by a particular set of FOVs), the corresponding solution is converted from integer values to FOVs, which are forwarded to the Rxs in the downlink to apply them. Note that, to generalize the proposed algorithm to the cases where only discrete non-equally-separated FOV values are possible (e.g., FOVs ∈ [10 converted to integer values, and the problem is solved as an integer programming problem. In particular, the indices of the FOVs (in ascending order of FOVs) are used as the values in the solution space, to ensure equal spacing between the possible FOVs. The proposed approach is illustrated in Fig. 2(d), where in the downlink, the central control unit sends to the Rxs (via the APs) the calculated FOVs, as shown in Fig. 1(a). Next, the estimated channel gains at the Rxs (i.e., the elements of the network channel matrix) using these FOVs are sent back in the uplink to the central control unit to update the FOVs using PSO. Based on these evaluations, and after updating the positions and velocities of the particles and the system experience, i.e., P best and Gbest, PSO continues searching new FOVs that achieve a better performance by exploring possible better solutions. Once the maximum number of iterations is achieved, the Gbest is considered as the solution corresponding to the optimized FOVs. As shown in Fig. 1(a), the additional overhead only concerns the optimized FOVs in the downlink, and the SNR performance in the uplink, which remain very limited. Note that, given the limited users' mobility in indoor scenarios, the variations in the network channel matrix are relatively slow, which relaxes the adaptation requirements, and the constraints on the delay from the feedback channel.

A mechanical iris is used at the top of each PD to tune the corresponding Rx's FOV. To consider a practical design, we assume that the FOVs can be only tuned to discrete values, ranging between 10 • and 80 • , with a step of 5 • . Note that the speed of variation of the FOV highly depends on the parameters of the mechanical iris, as well as on the tools used for controlling it, which is beyond the scope of this optimization study. Note that, the relatively slow changes of the channel in indoor scenarios would justify the neglected delay in the adjustment of the parameters of the mechanical iris. To evaluate the solution of PSO for FOV tuning, we propose to consider the following fitness function:

Fitness = F 1 × Nr j=1 R j -F 2 × cond(H) -(F 3 × N 0 ), (1)
where cond(H) refers to the 2-norm condition number of the network channel matrix H, R j denotes the data-rate of the user j, and N 0 is defined as the number of users with no existing LOS link with any AP. Also, F 1 , F 2 , and F 3 are the weights given to the 1 st , 2 nd , and 3 rd terms of the fitness function, respectively. In fact, the 1 st term ensures promoting solutions with higher sum-rates (i.e., the total achievable network throughput), the 2 nd term downgrades solutions resulting in high condition number for network channel matrix, and the 3 rd term guarantees that the solutions resulting in users with LOS blockage are excluded. This is done by giving F 3 a large value, so that for any value of the sum-rate and condition number achieved by a solution, having a user with no LOS results in a bad evaluation of the solution. This point is explained in more detail in the next section. Note that, due to the non-linear nature of the considered problem, any variation in the Rxs' locations requires re-adapting the model to have FOVs that satisfy LOS coverage. This makes PSO more suitable compared to other techniques which require training data to update the model adaptively depending on the actual LOS coverage conditions.

V. PERFORMANCE ANALYSIS

This section presents numerical results, for comparing the network performance for the cases of non-optimized FOVs and optimized FOVs using PSO. Consider four Rxs in the VLC network, handled by ZFPC. All Rxs are assumed to point up to the ceiling. Four LED luminaires (APs) are assumed, connected to a central control unit. The indoor environment is considered of dimension (7 × 7 × 3) m 3 , while the APs are located at (x, y, z) positions (2.25, 2.25, 2.5), (2.25, 4.75, 2.5), (4.75, 2.25, 2.5), and (4.75, 4.75, 2.5). The number of LED chips per luminaire is 36, the current per luminaire is 3.6 A, the Lambertian order m of each LED is 1, and the LED conversion efficiency S is 0.44 W/A [START_REF] Eltokhey | Hybrid NOMA and ZF pre-coding transmission for multi-cell VLC networks[END_REF]. The noise power spectral density is fixed to 10 -21 A 2 /Hz, and the considered PD responsivity and area are 0.4 A/W and 1 cm 2 , respectively [START_REF] Eltokhey | Hybrid NOMA and ZF pre-coding transmission for multi-cell VLC networks[END_REF].

The network sum-rate (corresponding to the 1 st term in (1)) is calculated for a system bandwidth of 10 MHz, assuming DC-biased optical-orthogonal frequency-division multiplexing (DCO-OFDM) signal modulation. Also, the 2-norm condition numbers in the considered random user positions (corresponding to the 2 nd term in (1)) reached orders of 10 3 . Following the discussions at the end of the previous section, we hence set the constants F 1 , F 2 , and F 3 in (1) to 1, 10 5 , and 10 10 , respectively, to guarantee that the 3 rd term in (1) is always dominating, and to further ensure that the solutions with very large 2-norm condition numbers are downgraded. In addition, to further downgrade the solutions with LOS blockage, we set the corresponding network sum-rates to zero.

A. Performance over different optimization parameters

To choose the best combination of particles and iterations for a given total number of evaluations of N p × N it , Table I compares the average network sum-rate and the number of LOS blockage cases for different values of N p and N it , for 1000 random user positions with random Rxs' heights Z Rx between 0.85 and 1.35 m. The 1000 random positions are picked from 1250 random positions, with random elevation and azimuth angles for the Rxs' within the range of (0 • -45 • ) and (0 • -180 • ), respectively, such that they guarantee LOS coverage at the Rxs' FOVs of 80 • . This ensures the feasibility of using the proposed optimization in finding optimized solutions, given that each of the optimized FOVs has a maximum value of 80 • . The sum-rate is averaged over all users' positions corresponding to LOS coverage occurrence. Note that, LOS blockage refers to the cases where one or more Rxs have no As can be seen, increasing the number of evaluations from 50 to 100, and from 100 to 200, results in an improved performance in terms of average sum-rate and LOS blockage, as expected. In addition, it is noted from the cases with 50 or 200 evaluations that, having a larger number of particles than iterations (for the same number of evaluations) results in a better average sum-rate, and a lower LOS blockage (in case of 50 evaluations). This can be explained by the better exploration of the solution space before converging to the optimal solution.

B. Case study of optimized and non-optimized FOVs

To show in more detail how the FOVs and the sum-rate performance vary over iterations throughout the proposed optimization, we consider in Fig. 3 the performance of a certain scenario, where 4 Rxs with height of 0.85 m are located at (x, y) of (4.58, 0.32), (5.08, 4.03), (3.63, 5.47), and (1.49, 1.58). Figure 3(b) shows the fitness function calculated using (1) at each iteration. In the blue boxes, are indicated the FOVs, the sum-rates, and the condition numbers of H at every change of the fitness value, representing the most optimal performance achieved at that iteration. Also, the red boxes indicate the sumrates achieved in cases of fixed FOVs of 50 • , 70 • , and 90 • . The plot shows an improvement in the quality of the solution due to the further exploration of the solution space, where PSO converges to the optimal solution at iteration 6.

We note from the data provided in the red boxes that decreasing the FOV in general should improve the performance, however, this remains dependent on the correlation of the network channel matrix. This can be verified from the slight decrease in the 2-norm condition number and the small increase in the network sum-rate by increasing the FOV from 70 • to 90 • . This elucidates the complexity of the problem, and the fact that the sum-rate performance is not always improved by simply decreasing the FOVs.

C. Performance over random Rx locations

Using the best combination of N p and N it from Table I (i.e., N p = 20 and N it = 10), we compare in Table II It is observed that increasing the FOV decreases the risk of LOS blockage. However, in general, this comes at the expense of increased homogeneity between users' channel gains, thus a performance degradation due to higher probability of increased correlation in the network channel matrix (which is apparent only for the highly constrained case of random Rxs' orientations for the large FOVs of 80 • and 90 • ). Also, considering random Rxs' orientations in addition to random Rxs' heights led to a degraded sum-rate performance, due to lower number of solutions that satisfy LOS coverage. Moreover, increasing the Rx's height (i.e., decreasing the path length between the Tx and the Rx) increases the probability of LOS blockage, due to the decrease in the area covered on the AP plane by the Rx's FOV. On the other hand, irrespective of the considered Rxs' heights, the proposed FOV optimization shows a robust performance, with very limited LOS blockage occurrence. It is worth mentioning that, even by decreasing the number of evaluations to 50 (see Table I), the achieved performance with PSO was still better than the cases with non-optimized FOVs.

Similar observations can be made by considering the network average sum-rate, where the proposed approach outperforms the case of non-optimized FOVs. This advantage results from the adaptation of every Rx to users' positions to decrease the correlation in the network channel matrix. Increasing the Rxs' heights results in an average sum-rate degradation, because of higher correlation between users' channel gains, as a result of shorter transmission path lengths.

D. Performance in the case of user mobility

To further decrease the number of evaluations in the case of changing users' positions, we propose to forward the most recent optimized FOVs as initial positions for PSO, for a faster convergence. To prevent particles from converging to sub-optimal solutions because of insufficient exploration of the solution space, we propose to vary these initial values by a random factor between 0 and 50% for each particle, in order to increase the spread of the particles in the solution space. We refer to this approach as "memory-assisted" optimization. The efficiency of this approach was investigated on 1000 scenarios generated by the random way-point model, simulating the mobility of 4 Rxs with velocities varying between 0.1 and 0.5 m/sec, and a time interval of 5 sec between two successive Table III shows a comparison of the average sum-rate and the LOS blockage performance between the cases of conventional and memory-assisted PSO-based FOVs' optimization, as well as the case of fixed FOV of 80 • . This latter corresponds to the best performance for non-optimized FOVs, as seen in Table II. The results show a quite robust LOS blockage performance for the memory-assisted approach, compared with the conventional optimization. As before, for the case of conventional optimization, LOS blockage occurrence improves with increasing the number of evaluations from 50 to 100 or by decreasing the Rxs' heights. For the average sum-rate, both conventional and memory-assisted optimizations outperform the non-optimized case, while decreasing the Rxs' heights or increasing the number of evaluations results in an improved performance. However, we notice that the conventional optimization provides a better average sum-rate performance, which can be explained by a more constrained exploration of the solution space in the case of memory-assisted optimization. Indeed, the initial values are not distributed over the entire solution space as it is the case for conventional optimization. Lastly, concerning the computational complexity, for each particle and at each iteration, proposed PSO-based optimization needs to (i) update the velocities, calculate the fitness function, and enforce the boundary conditions; and (ii) calculate the ZFPC solution.

VI. CONCLUSIONS

We proposed in this article the optimization of the Rxs' FOVs based on PSO, for ZFPC MU-MISO VLC networks. Firstly, we highlighted the trade-off between the solution quality and the number of evaluations, and showed that using a sufficient number of particles is essential for an adequate exploration of the solution space. In general, decreasing the FOV improves the average sum-rate but it also increases the risk of LOS blockage. The proposed optimized FOV tuning approach offers the best trade-off between a good throughput performance and a robust LOS coverage. The performance of optimized and non-optimized FOVs were also compared in the case of user mobility, where it was shown that by using the last optimized FOVs as initial values for the PSO, a better LOS blockage performance is achieved using a smaller number of evaluations. However, this comes at the expense of a slightly lower average sum-rate, due to the constrained exploration of the solution space. Nevertheless, the robust performance achieved with a limited number of evaluations, makes this approach suitable for the cases of limited computational resources or relatively high user mobility. For scenarios with still more constrained computational resources, the optimization of FOVs can be carried out only in cases where the channel correlation exceeds a certain threshold.

Given the promises of the proposed scheme in improving the network performance, future research could investigate the merits of this approach while taking into account the practical constraints.

This includes, for instance: investigation of other pre-coding schemes (i.e., other than ZF pre-coding); consideration of the user mobility, random tilting of the Rxs, and parameters of the mechanical iris used for controlling the FOVs including the required delay for FOV tuning; optimization of the FOV with the lowest possible number of changes to ensure minimum delay; and improvement of the proposed memory-assisted optimization to increase the solution quality by controlling the impact of the most recent optimized FOVs.

Note that the proposed approach for FOV optimization can be applied to other areas in the field of wireless communications that require real-time adaptation to changing network configuration, where the presented study, in particular, concerning the consideration of the constraints on the possible values in the solution space, the trade-off between the PSO parameters, and the design of the fitness function, can be quite insightful.

Fig. 1 :

 1 Fig. 1: FOV tuning in the VLC network. (a): main blocks in the VLC system. (b), (c), (d): Examples of FOV tuning using mechanical iris for three Rxs, where FOV1 > FOV2 > FOV3.

Fig. 2 :

 2 Fig. 2: PSO algorithm description. (a), (b): Illustrations of particles at the first (exploring the solution space while exchanging information) and last (about to converge to the optimal solution) iterations, respectively. (c): Flow chart of the algorithm for an example of optimizing the FOVs of 3 Rxs, indices and q refer to the iteration number and the particle number, respectively, "Int." stands for integer. Dashed box at the left side shows how the variable values (FOVs) are extracted from integers during the optimization to solve the problem as integer programming problem, for a generalized solution in cases of having non-uniform steps between the possible FOVs. Dashed box at the right side illustrates an example of behavior of 2 particles, under the influence of original velocity, P best, and Gbest. (d): Illustration of the proposed FOV optimization for the case of two APs and two Rxs: By tuning the FOVs, the entries of the network channel matrix are controlled; hij denotes the channel gain of the communication link between APi and Rxj.

Fig. 3 :

 3 Fig. 3: Comparison between optimized and non-optimized FOVs for a 4 Rx -4 AP scenario. (a): Top view for the locations of the Rxs and the APs with heights of 0.85 m and 2.5 m, respectively. (b): Fitness value versus optimization's iterations. The dashed blue boxes represent the optimized FOVs over changes in fitness and their corresponding sum-rates. The dashed red boxes correspond to the non-optimized FOVs (50 • , 70 • , and 90 • ).

TABLE I :

 I Comparison between different combinations of number of particles (Np) and iterations (Nit) over average sum-rate and LOS blockage performance, for cases of random Rx heights in 4 Rx scenarios. indicates that there has been no LOS blockage over the considered random scenarios, which is generally the case due to a better exploration of the solution space.Given the considered 15 possible FOVs for the Rxs, the number of evaluations required for a "parameter sweep" (i.e., trying all possible combinations of FOVs for finding optimum values, without using an optimization algorithm) for scenarios with 2, 3, and 4 Rxs, are 225, 3375, and 50625, respectively. Obviously, the considered numbers of evaluations N p × N it in Table I are much smaller than those needed for a parameter sweep.

	Combination	Av. sum-rate (bps) ×10 7	LOS blockage
	Np = 5, N it = 10	4.19	25
	Np = 10, N it = 5	4.27	3
	Np = 10, N it = 10	4.42	2
	Np = 10, N it = 20	4.50	0
	Np = 20, N it = 10	4.60	0
	LOS path with any AP. Obviously, a lower LOS blockage
	indicates a better LOS coverage probability. A LOS blockage
	of 0

  the LOS blockage occurrence and the average sum-rate for the cases of non-optimized and optimized FOVs for 4 Rxs. Comparison is carried out over 1000 random scenarios. Non-optimized FOV values of 50 • , 60 • , 70 • , 80 • , and 90 • are considered, however, the results of LOS blockage for FOVs of 80 • and 90 • were excluded from the table as all Rxs have LOS coverage over all scenarios. For the same reason of LOS blockage, the average sum-rate results for FOVs of 50 • , 60 • , and 70 • were excluded.

TABLE II :

 II Number of scenarios with at least 1 Rx with no LOS coverage and average sum-rate performance for optimized (Opt.) and non-optimized fixed FOVs, for fixed and random ZRx between 0.85 and 1.35 m, and random Rxs' orientations.

			LOS blockage
	FOV	Z Rx = 0.85 m Z Rx = 1.15 m Random	Random
	50 •	668	873	Z Rx 840	orientations 786
	60 •	30	268	257	491
	70 •	0	0	0	194
	Opt.	0	0	0	15
		Average sum-rate (bps) ×10 7
	80 •	3.24	3.06	3.10	3.06
	90 •	3.24	3.06	3.10	2.88
	Opt.	5.18	4.50	4.60	4.46

TABLE III :

 III LOS blockage and average sum-rate performance for different cases of optimized and non-optimized FOVs for 1000 generated positions of 4 Rxs reflecting user mobility according to random way-point model.

		Non-optimized Memory-assisted PSO (Np = 10) Conventional PSO (Np = 10)
	Z Rx	FOV = 80 •	N it = 5	N it = 10	N it = 5	N it = 10
			LOS blockage
	0.85 m	0	0	0	1	0
	1.15 m	0	0	0	3	2
			Average sum-rate (bps) ×10 7
	0.85 m	2.92	4.20	4.36	4.43	4.58
	1.15 m	2.76	3.74	3.86	3.84	3.97
	positions. The comparison is carried out over Rxs' heights of
	0.85 m and 1.15 m, and for different numbers of particles and
	iterations.


Vincent Guillemot

Laurent Le Brusquet

Arthur Tenenhaus

Vincent Frouin

Graph-Constrained Discriminant Analysis of functional genomics data

Classification studies from microarray data have proved useful in tasks like predicting patient class. At the same time, more and more biological information about gene regulation networks has been gathered mainly in the form of graph. Incorporating the a priori biological information encoded by graphs turns out to be a very important issue to increase classification performance. We present a method to integrate information from a network topology into a classification algorithm: the graph-Constrained Discriminant Analysis (gCDA). We applied our algorithm to simulated and real data and show that it performs better than a linear Support Vector Machines classifier.

Introduction

In functional genomics, the issue of integrating information about gene-coregulation arises more and more frequently in the analysis of microarray data: this information may consist in a priori or results from other experiments. For example, some differential analysis procedures, incorporating gene networks, were recently proposed [START_REF] Wei | Incorporating gene networks into statistical tests for genomic data via a spatially correlated mixture model[END_REF]. Several methods can be found in the literature to achieve the integration of the a priori information one have on the relations between genes into the classification task [START_REF] Li | Network-constrained regularization and variable selection for analysis of genomic data[END_REF], [START_REF] Rapaport | Classification of microarray data using gene networks[END_REF].

In [START_REF] Li | Network-constrained regularization and variable selection for analysis of genomic data[END_REF], the authors proposed to fit a linear model y = βX on the dataset X to predict the binary variable y. Their approach follows regularization ideas that consist in optimizing the following criterion

J(β, c 1 , c 2 ) = (y -βX) (y -βX) + c 1 |β| + c 2 β L G β
with respect to β, where L G is the Laplacian of the a priori graph G (see paragraph 2.1). The solution of this optimization problem is achieved through a Lasso procedure [START_REF] Efron | Least angle regression[END_REF] and leads to a sparse model ; it is used to characterize the subnetworks from the reference network that are "expressed" in the dataset.

In [START_REF] Rapaport | Classification of microarray data using gene networks[END_REF], a spectral transformation is applied to the graph G and then a kernel Support Vector Machines classification is performed. The new metric between two expression profiles f and g is:

d φ ( f ; g) = f K φ g (1)
, where K φ is the positive semidefinite matrix obtained after the spectral transformation of graph G. These procedures lead to a better graphinterpretability of the resulting classifiers. But the goal of these integration algorithms is not to improve classification performance.

We propose a method, based on the Discriminant Analysis introduced by Fisher [START_REF] Fisher | The use of multiple measurements in taxonomic problems[END_REF], that takes into account an a priori and improves classification performance when compared to competitive classification algorithm (here Support vector Machines [START_REF] Cortes | Support-vector networks[END_REF]). We considered first simulated data to demonstrate our point; the parameters of the simulated dataset generator were tuned in order to mimic any given real dataset while following some interaction constraints. Then, we applied our method on data from the literature [START_REF] Alon | Broad patterns of gene expression revealed by clustering analysis of tumor and normal colon tissues probed by oligonucleotide arrays[END_REF]. We focused on binary classification problems, and left multiclass problems as future work.

The remainder of the paper is structured as follows: in the second part gCDA is presented and the last part is devoted to the validation of gCDA and its comparison with the state-of-the-art methods on simulated and real datasets.

Graph-Constrained Discriminant Analysis

A constrained version of Discriminant Analysis is developed to compute, from a learning dataset, a classifier able to predict the class of a new sample.

Interaction between genes

A graph G is defined by the set of its edges E and the set of its vertices V. Let w be the function w : V × V → {0, 1} such that w(i, j) is 1 if there is an edge between vertices i and j and 0 otherwise. For each vertex i of V, the connectivity degree d i is defined as the cardinality of the set of the vertices in V connected to i. In the following, G is a finite graph with p vertices (genes) and m edges.

We take a graph of independence as equivalent to a graph of gene regulations (which is only a partial view of the biological concept of a regulation network). Each vertex represents a variable and the absence of an edge be-tween two vertices means that the two variables are independent knowing the remaining variables. For example, there is no edge between variables 4 and 5 (denoted respectively by x 4 and x 5 ) in the graph depicted on figure 1: x 4 and x 5 are independent knowing the remaining variables. The precision matrix of the multivariate Gaussian variable X = [x 1 , ..., x n ], which is also its inverse covariance matrix Σ -1 , has the following property [START_REF] Whittaker | Graphical Models in Applied Multivariate Statistics[END_REF]:

x 4 ⊥ ⊥ x 5 | x j , j ∈ {1, ..., p} \ {4, 5} (2)

⇔

Σ -1 4,5 = 0 In this paper, we propose to use property [START_REF] Barabasi | Emergence of scaling in random networks[END_REF] to derive an estimation of the covariance matrix Σ G from the Laplacian matrix L G of an a priori graph G. Indeed L G , is a semi definite positive p × p matrix which coefficients are:

[L G ] i,j = -w(i, j) , if i = j d i , if i = j
and in which each null term corresponds to an absence of edge in G. Thus, after the addition of a δ > 0 on the diagonal of L G , we have a good positive definite candidate for the precision matrix of a multivariate Gaussian variable of which G is an independence graph:

Σ -1 G = L G + δI (3)
with I the identity matrix of size p. In the gCDA algorithm, the property 3 is used to compute covariance matrices from both microarray data and an a priori graph.

Discriminant Analysis

Let

X 1 = (x (i) 1 ) i∈{1,...,n 1 } (resp. X 2 = (x (i)
1 ) i∈{1,...,n 2 } ) be an n 1 -sample (resp. n 2 ) of a Gaussian multivariate variable with mean µ 1 (resp. µ 2 ) and covariance matrix Σ, and let X = [X 1 , X 2 ] be the corresponding 2 classes dataset. With these notations, we can define the between and within classes covariances as

S b = n 1 (x 1 -x)(x 1 -x) + n 2 (x 2 -x)(x 2 -x) (4)
and

S w = 1 n -2 2 ∑ k=1 1 n k n k ∑ j=1 x (j) k -x k x (j) k -x k ,
(5) where x k is the empirical mean of X k and x the empirical mean of X.

The goal of the Discriminant Analysis is to determine a linear transformation W of dataset X that minimizes the within class variance to between class variance ratio. Provided that S -1 w exists, this problem is equivalent to perform the eigenvalue decomposition of the matrix S b S -1 w :

S b S -1 w = VDV
and to choose W = V as the linear transformation.

Once W is computed, the prediction of the class of a new individual x new is based on the estimation of the posterior probability of the class ω k knowing the sample

x new : log Pr(ω k |z new = Wx new ) = - 1 2 (z new -µ k ) Σ -1 k (z new -µ k ) -log det Σ k + log π k + a
where π k is the prior probability of each class, Σ k is the estimation of the covariance matrix of class k after the transformation by W and a ∈ R is a constant which does not depend on the class k. x new will be attributed the class k new maximizing the posterior probabilities according to two different strategies. The socalled Linear Discriminant Analysis assumes that Σ k = Σ does not depend on the class k. On the contrary, the Quadratic Discriminant Analysis assumes that the estimation of Σ k has to be different from one class to another.

Taking into account the a priori graph

In the n p case, the Maximum Likelihood Estimators (MLE) presented in equations 4 and 5 are unbiased but show poor performances as regards their variance. In order to overcome this problem several approaches are proposed in the literature. In [START_REF] Guo | Regularized linear discriminant analysis and its application in microarrays[END_REF], the authors present a method that consists in inferring a regularized version of the covariance matrix:

Σ = α Σ + (1 -α)I (6)
and in [START_REF] Schäfer | A shrinkage approach to large-scale covariance matrix estimation and implications for functional genomics[END_REF], a similar strategy of regularization is used to shrink the estimator of the covariance matrix by replacing I in equation 6 by sparse positive definite matrices, and the parameter α is analytically computed. Our method is inspired by the previously reported ideas: in the graph-Constrained Discriminant Analysis, I is replaced by the covariance matrix obtained from the Laplacian of the a priori graph:

Σ = α Σ + (1 -α) (L G + δI) -1
The α parameter is determined with a crossvalidation procedure.

In this context, the within class variability estimation can be redefined:

Σ w (α) = n 1 n Σ 1 (α) + n 2 n Σ 2 (α) (7)
to perform a Linear or a Quadratic graph-Constrained Discriminant Analysis.

In the Linear gCDA, each class is supposed to have the same covariance and there is only one a priori graph:

Σ w (α) = αS w + (1 -α) (L G + δI) -1 (8)
In the Quadratic gCDA, each class is characterized by a different a priori graph:

Σ w (α) = αS w + (1 -α) L G 1 + δI -1 + L G 2 + δI -1 (9)

Results

In this section the two datasets we used to challenge gCDA are presented. The first one is a simulated dataset and the second one is a microarray dataset [START_REF] Alon | Broad patterns of gene expression revealed by clustering analysis of tumor and normal colon tissues probed by oligonucleotide arrays[END_REF]. In what follows, n 1 and n 2 are the number of individuals in classes 1 and 2, p is the number variables, G the graph of independence, µ the mean and Σ the covariance matrix. We compare gCDA to Support Vector Machines [START_REF] Cortes | Support-vector networks[END_REF].

The characteristics of the two datasets are represented in table 1, the number of iterations used to compute the test error rate with a Monte Carlo Cross Validation algorithm [START_REF] Boulesteix | Evaluating microarraybased classifiers: an overview[END_REF]

Data simulation

Since we want to evaluate the integration of a graph into a classification, we have to design a dataset with a given independence graph structure. Classically, microarray data are simulated by sampling the covariance matrix of a real dataset ; the underlying structure of interactions remains unknown in those simulations. Other simulation strategies consider the generation of a random graph, followed by the solution of a system of Ordinary Differential Equations [START_REF] Mendes | Artificial gene networks for objective comparison of analysis algorithms[END_REF], which implies the choice of numerous parameters for a graph with a realistic size. We formulate the graph constraint as follows: the dataset is a multivariate Gaussian sample with known mean, and with a covariance matrix built from the given graph of independence between the variables.

We make the assumption that the graph of independence underlying each class is the same and is characterized by a scale free distribution of the node degrees [START_REF] Barabasi | Emergence of scaling in random networks[END_REF]. Figure 1 shows a graphical representation of a random graph according to the algorithm we used. We no-tice that no loop is generated in such graphs, which is probably unrealistic from a biological point of view, but does not impact the classification procedure. For each class k, our simulation algorithm can be summed up as follows: a) random generation of network G b) the mean vector µ k of the expression profiles mimics the distribution of mean expressions in real datasets [START_REF] Golub | Molecular classification of cancer: class discovery and class prediction by gene expression monitoring[END_REF] c) the covariance matrix Σ G is computed from the graph G with respect to equation 3 d) an expression profile is a multivariate Gaussian random vector of mean µ k and covariance matrix Σ G For a given graph, we compute a corresponding covariance matrix and generate two samples of Multivariate Gaussian Variables X 1 and X 2 . We tested gCDA with the "real" graph and with another random graph. The results are presented on figure 2.

It can be seen that constraining the classification with a random graph is not better than

Real microarray data

100 genes were randomly chosen among the 2000 genes of the Alon dataset [START_REF] Alon | Broad patterns of gene expression revealed by clustering analysis of tumor and normal colon tissues probed by oligonucleotide arrays[END_REF]. Since we do not have any a priori graph of genetic regulation for this dataset, we applied a new algorithm using Partial Least Squares regressions [START_REF] Tenenhaus | Gene association networks from microarray data using a regularized estimation of partial correlation based on pls regression[END_REF] to infer the independence graph on an independent set of microarrays. The Alon dataset is then split into two independent datasets: the first one is used to infer a regulation graph for each class and on the second one is applied the Quadratic procedure described in section 2: the Linear procedure does not work, hypothetically because the independence graphs are different from one class to another (which is confirmed by the graph inference step: there are only 7 % common edges between the two graphs). Results are given in table 2. To compare the performance, we also report the results on the simulated data data gCDA SVM simulated 13.75% 17.75% Alon 3.75% 4.5%

Table 2. Results on real data

On figure 3, we see that there is an optimal α = 0.21 for which the test error rate is significantly inferior to the test error rate obtained with SVM (a Wilcoxon test applied on the two sets of B error rates gave a p-value of 1 %).

Conclusion

We show a significant improvement in classification performance when the underlying graph of regulation is known in the case of simulated data or when the underlying graph of regulation is inferred in the case of real microarray data.

The graph-constrained estimator of the covariance matrix will be studied in the spirit of [START_REF] Schäfer | A shrinkage approach to large-scale covariance matrix estimation and implications for functional genomics[END_REF] to characterize its bias and variance. We will also study the differences on simulated data between the Linear and the Quadratic versions of gCDA. And finally we will work on an implementation of gCDA able to cope with thousands of variables (for the moment it is possible to run gCDA with only hundreds of variables).

Figure 1 .

 1 Figure 1. An example of a random graph.

Figure 2 .

 2 Figure 2. Test error rate against α.

Figure 3 .

 3 Figure 3. Test error rate against α.

Table 1 . Characteristics of the datasets

 1 is fixed to B = 50.

		p	n 1 n 2
	Simulated 100 20 20
	Alon	100 26 14