Abstract: Learning-based single image super-resolution (SISR) methods are continuously showing superior effectiveness and efficiency over traditional model-based methods, largely due to the end-to-end training. However, different from model-based methods that can handle the SISR problem with different scale factors, blur kernels and noise levels under a unified MAP (maximum a posteriori) framework, learning-based methods generally lack such flexibility. To address this issue, this paper proposes an end-to-end trainable unfolding network which leverages both learningbased methods and model-based methods. Specifically, by unfolding the MAP inference via a half-quadratic splitting algorithm, a fixed number of iterations consisting of alternately solving a data subproblem and a prior subproblem can be obtained. The two subproblems then can be solved with neural modules, resulting in an end-to-end trainable, iterative network. As a result, the proposed network inherits the flexibility of model-based methods to super-resolve blurry, noisy images for different scale factors via a single model, while maintaining the advantages of learning-based methods. Extensive experiments demonstrate the superiority of the proposed deep unfolding network in terms of flexibility, effectiveness and also generalizability.

Abstract: In this paper, we propose a general framework for incomplete multiview clustering. The proposed method is the first work that exploits the graph learning and spectral clustering techniques to learn the common representation for incomplete multiview clustering. First, owing to the good performance of low-rank representation in discovering the intrinsic subspace structure of data, we adopt it to adaptively construct the graph of each view. Second, a spectral constraint is used to achieve the low-dimensional representation of each view based on the spectral clustering. Third, we further introduce a co-regularization term to learn the common representation of samples for all views, and then use the ${k}$ -means to partition the data into their respective groups. An efficient iterative algorithm is provided to optimize the model. Experimental results conducted on seven incomplete multiview datasets show that the proposed method achieves the best performance in comparison with some state-of-the-art methods, which proves the effectiveness of the proposed method in incomplete multiview clustering.

Abstract: Enabling bi-directional retrieval of images and texts is important for understanding the correspondence between vision and language. Existing methods leverage the attention mechanism to explore such correspondence in a fine-grained manner. However, most of them consider all semantics equally and thus align them uniformly, regardless of their diverse complexities. In fact, semantics are diverse (i.e. involving different kinds of semantic concepts), and humans usually follow a latent structure to combine them into understandable languages. It may be difficult to optimally capture such sophisticated correspondences in existing methods. In this paper, to address such a deficiency, we propose an Iterative Matching with Recurrent Attention Memory (IMRAM) method, in which correspondences between images and texts are captured with multiple steps of alignments. Specifically, we introduce an iterative matching scheme to explore such fine-grained correspondence progressively. A memory distillation unit is used to refine alignment knowledge from early steps to later ones. Experiment results on three benchmark datasets, i.e. Flickr8K, Flickr30K, and MS COCO, show that our IMRAM achieves state-of-the-art performance, well demonstrating its effectiveness. Experiments on a practical business advertisement dataset, named KWAI-AD, further validates the applicability of our method in practical scenarios.

Abstract: In this paper, we propose an end-to-end graph learning framework, namely Iterative Deep Graph Learning (IDGL), for jointly and iteratively learning graph structure and graph embedding. The key rationale of IDGL is to learn a better graph structure based on better node embeddings, and vice versa (i.e., better node embeddings based on a better graph structure). Our iterative method dynamically stops when the learned graph structure approaches close enough to the graph optimized for the downstream prediction task. In addition, we cast the graph learning problem as a similarity metric learning problem and leverage adaptive graph regularization for controlling the quality of the learned graph. Finally, combining the anchor-based approximation technique, we further propose a scalable version of IDGL, namely IDGL-Anch, which significantly reduces the time and space complexity of IDGL without compromising the performance. Our extensive experiments on nine benchmarks show that our proposed IDGL models can consistently outperform or match the state-of-the-art baselines. Furthermore, IDGL can be more robust to adversarial graphs and cope with both transductive and inductive learning.

Abstract: This paper investigates the passive beamforming and information transfer (PBIT) technique for multiuser multiple-input multiple-output (Mu-MIMO) systems with the aid of reconfigurable intelligent surfaces (RISs), where the RISs enhance the primary communication via passive beamforming (P-BF) and at the same time deliver additional information by the on-off reflecting modulation (in which the RIS information is carried by the on/off state of each reflecting element). For the P-BF design, we propose to maximize the achievable user sum rate of the RIS-aided Mu-MIMO channel and formulate the problem as a two-step stochastic program. A sample average approximation (SAA) based iterative algorithm is developed for the efficient P-BF design of the considered scheme. To strike a balance between complexity and performance, we further propose a simplified P-BF algorithm by approximating the stochastic program as a deterministic alternating optimization problem. For the receiver design, the signal detection at the receiver is a bilinear estimation problem since the RIS information is multiplicatively modulated onto the reflected signals of the reflecting elements. To solve this bilinear estimation problem, we develop a turbo message passing (TMP) algorithm in which the factor graph associated with the problem is divided into two modules: one for the estimation of the user signals and the other for the estimation of the on-off state of each RIS element. The two modules are executed iteratively to yield a near-optimal low-complexity solution. Furthermore, we extend the design of the Mu-MIMO PBIT scheme from single-RIS to multi-RIS, by leveraging the similarity between the single-RIS and multi-RIS system models. Extensive simulation results are provided to demonstrate the advantages of our P-BF and receiver designs.

Abstract: The emerging orthogonal time frequency space (OTFS) modulation technique has shown its superiority to the current orthogonal frequency division multiplexing (OFDM) scheme, in terms of its capabilities of exploiting full time-frequency diversity and coping with channel dynamics. The optimal maximum a posteriori (MAP) detection is capable of eliminating the negative impacts of the inter-symbol interference in the delay-Doppler (DD) domain at the expense of a prohibitively high complexity. To reduce the receiver complexity for OTFS scheme, this paper proposes a variational Bayes (VB) approach as an approximation of the optimal MAP detection. Compared to the widely used message passing algorithm, we prove that the proposed iterative algorithm is guaranteed to converge to the global optimum of the approximated MAP detector regardless the resulting factor graph is loopy or not. Simulation results validate the fast convergence of the proposed VB receiver and also show its promising performance gain compared to the conventional message passing algorithm.

Abstract: The field of medical image reconstruction has seen roughly four types of methods. The first type tended to be analytical methods, such as filtered backprojection (FBP) for X-ray computed tomography (CT) and the inverse Fourier transform for magnetic resonance imaging (MRI), based on simple mathematical models for the imaging systems. These methods are typically fast, but have suboptimal properties such as poor resolution-noise tradeoff for CT. A second type is iterative reconstruction methods based on more complete models for the imaging system physics and, where appropriate, models for the sensor statistics. These iterative methods improved image quality by reducing noise and artifacts. The U.S. Food and Drug Administration (FDA)-approved methods among these have been based on relatively simple regularization models. A third type of methods has been designed to accommodate modified data acquisition methods, such as reduced sampling in MRI and CT to reduce scan time or radiation dose. These methods typically involve mathematical image models involving assumptions such as sparsity or low rank . A fourth type of methods replaces mathematically designed models of signals and systems with data-driven or adaptive models inspired by the field of machine learning . This article focuses on the two most recent trends in medical image reconstruction: methods based on sparsity or low-rank models and data-driven methods based on machine learning techniques.

Abstract: Many contemporary signal processing, machine learning and wireless communication applications can be formulated as nonconvex nonsmooth optimization problems. Often there is a lack of efficient algorithms for these problems, especially when the optimization variables are nonlinearly coupled in some nonconvex constraints. In this work, we propose an algorithm named penalty dual decomposition (PDD) for these difficult problems and discuss its various applications. The PDD is a double-loop iterative algorithm. Its inner iteration is used to inexactly solve a nonconvex nonsmooth augmented Lagrangian problem via block-coordinate-descent-type methods, while its outer iteration updates the dual variables and/or a penalty parameter. In Part I of this work, we describe the PDD algorithm and establish its convergence to KKT solutions. In Part II we evaluate the performance of PDD by customizing it to three applications arising from signal processing and wireless communications.

Abstract: Blind image deblurring remains a topic of enduring interest. Learning based approaches, especially those that employ neural networks have emerged to complement traditional model based methods and in many cases achieve vastly enhanced performance. That said, neural network approaches are generally empirically designed and the underlying structures are difficult to interpret. In recent years, a promising technique called algorithm unrolling has been developed that has helped connect iterative algorithms such as those for sparse coding to neural network architectures. In this article, we propose a neural network architecture based on this idea. We first present an iterative algorithm that may be considered as a generalization of the traditional total-variation regularization method in the gradient domain. We then unroll the algorithm to construct a neural network for image deblurring which we refer to as Deep Unrolling for Blind Deblurring (DUBLID). Key algorithm parameters are learned with the help of training images. Our proposed deep network DUBLID achieves significant practical performance gains while enjoying interpretability and efficiency at the same time. Extensive experimental results show that DUBLID outperforms many state-of-the-art methods and in addition is computationally faster.

Abstract: This paper is concerned with solving combinatorial optimization problems, in particular, the capacitated vehicle routing problems (CVRP). Classical Operations Research (OR) algorithms such as LKH3 (Helsgaun, 2017) are extremely inefficient (e.g., 13 hours on CVRP of only size 100) and difficult to scale to larger-size problems. Machine learning based approaches have recently shown to be promising, partly because of their efficiency (once trained, they can perform solving within minutes or even seconds). However, there is still a considerable gap between the quality of a machine learned solution and what OR methods can offer (e.g., on CVRP-100, the best result of learned solutions is between 16.10-16.80, significantly worse than LKH3's 15.65). In this paper, we present the first learning based approach for CVRP that is efficient in solving speed and at the same time outperforms OR methods. Starting with a random initial solution, our algorithm learns to iteratively refines the solution with an improvement operator, selected by a reinforcement learning based controller. The improvement operator is selected from a pool of powerful operators that are customized for routing problems. By combining the strengths of the two worlds, our approach achieves the new state-of-the-art results on CVRP, e.g., an average cost of 15.57 on CVRP-100.

Abstract: Maximum likelihood methods are based on the probability and statistics theory, and significant for parameter estimation and system modeling. This paper combines the maximum likelihood principle with the data filtering technique for parameter estimation of a class of bilinear systems. The input-output representation of a bilinear system is derived through eliminating the state variables in the model. Then, a filtering based maximum likelihood iterative least squares algorithm is proposed for identifying the parameters of bilinear systems with colored noises by filtering the input-output data with a filter. A least squares based iterative algorithm is given for comparison. The simulation results indicate that the proposed algorithm is effective for identifying bilinear systems. The filtering based maximum likelihood iterative least squares algorithm is more accurate under different noise variance, and has higher computational efficiency.

Abstract: In this paper, we consider the problem of learning control policies that optimize areward function while satisfying constraints due to considerations of safety, fairness, or other costs. We propose a new algorithm - Projection Based ConstrainedPolicy Optimization (PCPO), an iterative method for optimizing policies in a two-step process - the first step performs an unconstrained update while the secondstep reconciles the constraint violation by projection the policy back onto the constraint set. We theoretically analyze PCPO and provide a lower bound on rewardimprovement, as well as an upper bound on constraint violation for each policy update. We further characterize the convergence of PCPO with projection basedon two different metrics - L2 norm and Kullback-Leibler divergence. Our empirical results over several control tasks demonstrate that our algorithm achievessuperior performance, averaging more than 3.5 times less constraint violation andaround 15% higher reward compared to state-of-the-art methods.

Abstract: Many challenging image processing tasks can be described by an ill-posed linear inverse problem: deblurring, deconvolution, inpainting, compressed sensing, and superresolution all lie in this framework. Traditional inverse problem solvers minimize a cost function consisting of a data-fit term, which measures how well an image matches the observations, and a regularizer, which reflects prior knowledge and promotes images with desirable properties like smoothness. Recent advances in machine learning and image processing have illustrated that it is often possible to learn a regularizer from training data that can outperform more traditional regularizers. We present an end-to-end, data-driven method of solving inverse problems inspired by the Neumann series, which we call a Neumann network. Rather than unroll an iterative optimization algorithm, we truncate a Neumann series which directly solves the linear inverse problem with a data-driven nonlinear regularizer. The Neumann network architecture outperforms traditional inverse problem solution methods, model-free deep learning approaches, and state-of-the-art unrolled iterative methods on standard datasets. Finally, when the images belong to a union of subspaces and under appropriate assumptions on the forward model, we prove there exists a Neumann network configuration that well-approximates the optimal oracle estimator for the inverse problem and demonstrate empirically that the trained Neumann network has the form predicted by theory.

Abstract: Many problems in robotics involve multiple decision making agents. To operate efficiently in such settings, a robot must reason about the impact of its decisions on the behavior of other agents. Differential games offer an expressive theoretical framework for formulating these types of multi-agent problems. Unfortunately, most numerical solution techniques scale poorly with state dimension and are rarely used in real-time applications. For this reason, it is common to predict the future decisions of other agents and solve the resulting decoupled, i.e., single-agent, optimal control problem. This decoupling neglects the underlying interactive nature of the problem; however, efficient solution techniques do exist for broad classes of optimal control problems. We take inspiration from one such technique, the iterative linear-quadratic regulator (ILQR), which solves repeated approximations with linear dynamics and quadratic costs. Similarly, our proposed algorithm solves repeated linear-quadratic games. We experimentally benchmark our algorithm in several examples with a variety of initial conditions and show that the resulting strategies exhibit complex interactive behavior. Our results indicate that our algorithm converges reliably and runs in real-time. In a three-player, 14-state simulated intersection problem, our algorithm initially converges in < 0.25 s. Receding horizon invocations converge in < 50 ms in a hardware collision-avoidance test.

Abstract: Feature extraction and feature selection have been regarded as two independent dimensionality reduction methods in most of the existing literature. In this paper, we propose to integrate both approaches into a unified framework and design an unsupervised linear feature selective projection (FSP) for feature extraction with low-rank embedding and dual Laplacian regularization, with the aim to exploit the intrinsic relationship among data and suppress the impact of noise. Specifically, a projection matrix with an $l_{2,1}$ l 2 , 1 -norm regularization is introduced to project original high dimensional data points into a new subspace with lower dimension, where the $l_{2,1}$ l 2 , 1 -norm regularization can endow the projection with good interpretability. We deploy a coefficient matrix with low rank constraint to reconstruct the data points and the $l_{2,1}$ l 2 , 1 -norm is imposed to regularize the data reconstruction errors in the low-dimensional subspace and make FSP robust to noise. Furthermore, a dual graph Laplacian regularization term is imposed on the low dimensional data and data reconstruction matrix for preserving the local manifold geometrical structure of data. Finally, an alternatively iterative algorithm is carefully designed for solving the proposed optimization model. Theoretical convergence and computational complexity analysis of the algorithm are also provided. Comprehensive experiments on various benchmark datasets have been carried out to evaluate the performance of the proposed FSP. As indicated, our algorithm significantly outperforms other state-of-the-art methods for feature extraction.

Abstract: Multiaccess edge computing (MEC) is regarded as a promising solution to overcome the limit on the computation capacity of mobile devices. This article investigates an energy-efficient unmanned aerial vehicle (UAV)-enabled MEC framework incorporating nonorthogonal multiple access (NOMA), where multiple UAVs are deployed as edge servers to provide computation assistance to terrestrial users and NOMA is adopted to reduce the energy consumption of task offloading. A utility is formed to mathematically evaluate the weighted energy cost of the system. Due to the coupling of parameters, the minimization of utility is a highly nonconvex problem and therefore, the problem is decomposed into two more tractable subproblems, i.e., the optimal allocation of radio and computation resources given UAV trajectories, and the trajectory planning based on given resource allocation schemes. These two problems are converted to convex ones via successive convex approximation (SCA) and quadratic approximation, respectively. Then, an efficient iterative algorithm is proposed where these two subproblems are alternately solved to gradually approach the optimal resource management of the proposed system. Sufficient numerical results show that our proposed strategy has a remarkable advantage over existing systems in terms of energy efficiency.

Abstract: In this paper, we introduce a deep-learning-based framework to solve electromagnetic inverse scattering problems. This framework builds on and extends the capabilities of existing physics-based inversion algorithms. These algorithms, such as the contrast source inversion, subspace-optimization method, and their variants face a problem of getting trapped in false local minima when recovering objects with high permittivity. We propose a novel convolutional neural network architecture, termed the contrast source network, that learns the noise space components of the radiation operator. Together with the signal space components directly estimated from the data, we iteratively refine the solution and show convergence to the correct solution in cases where traditional techniques fail without any significant increase in computational time. We also propose a novel multiresolution strategy that helps in producing high resolution solutions without any significant increase in computational costs. Through extensive numerical experiments, we demonstrate the ability to recover high permittivity objects that include homogeneous, heterogeneous, and lossy scatterers.

Abstract: In this paper, we propose an end-to-end graph learning framework, namely Iterative Deep Graph Learning (IDGL), for jointly and iteratively learning graph structure and graph embedding. The key rationale of IDGL is to learn a better graph structure based on better node embeddings, and vice versa (i.e., better node embeddings based on a better graph structure). Our iterative method dynamically stops when the learned graph structure approaches close enough to the graph optimized for the downstream prediction task. In addition, we cast the graph learning problem as a similarity metric learning problem and leverage adaptive graph regularization for controlling the quality of the learned graph. Finally, combining the anchor-based approximation technique, we further propose a scalable version of IDGL, namely IDGL-Anch, which significantly reduces the time and space complexity of IDGL without compromising the performance. Our extensive experiments on nine benchmarks show that our proposed IDGL models can consistently outperform or match the state-of-the-art baselines. Furthermore, IDGL can be more robust to adversarial graphs and cope with both transductive and inductive learning.

Abstract: This paper considers the sum spectral efficiency (SE) optimization problem in multi-cell Massive MIMO systems with a varying number of active users. This is formulated as a joint pilot and data power control problem. Since the problem is non-convex, we first derive a novel iterative algorithm that obtains a stationary point in polynomial time. To enable real-time implementation, we also develop a deep learning solution. The proposed neural network, PowerNet, only uses the large-scale fading information to predict both the pilot and data powers. The main novelty is that we exploit the problem structure to design a single neural network that can handle a dynamically varying number of active users; hence, PowerNet is simultaneously approximating many different power control functions with varying number inputs and outputs. This is not the case in prior works and thus makes PowerNet an important step towards a practically useful solution. Numerical results demonstrate that PowerNet only loses 2% in sum SE, compared to the iterative algorithm, in a nine-cell system with up to 90 active users per in each coherence interval, and the runtime was only 0.03 ms on a graphics processing unit (GPU). When good data labels are selected for the training phase, PowerNet can yield better sum SE than by solving the optimization problem with one initial point.

Abstract: In this paper, we investigate the unmanned aerial vehicle (UAV)-aided simultaneous uplink and downlink transmission networks, where one UAV acting as a disseminator is connected to multiple access points (AP), and the other UAV acting as a base station (BS) collects data from numerous sensor nodes (SNs). The goal of this paper is to maximize the system throughput by jointly optimizing the 3D UAV trajectory, communication scheduling, and UAV-AP/SN transmit power. We first consider a special case where the UAV-BS and UAV-AP trajectories are pre-determined. Although the resulting problem is an integer and non-convex optimization problem, a globally optimal solution is obtained by applying the polyblock outer approximation (POA) method based on the problem’s hidden monotonic structure. Subsequently, for the general case considering the 3D UAV trajectory optimization, an efficient iterative algorithm is proposed to alternately optimize the divided sub-problems based on the successive convex approximation (SCA) technique. Numerical results demonstrate that the proposed design is able to achieve significant system throughput gain over the benchmarks. In addition, the SCA-based method can achieve nearly the same performance as the POA-based method with much lower computational complexity.

Abstract: It is well known that unmanned aerial vehicles (UAVs) can help terrestrial base stations (BSs) offload data traffic from crowded areas to improve coverage and boost throughput. However, the limited backhaul capacity cannot cope with the ever-increasing data demands, for which caching is introduced to relieve the backhaul bottleneck. In this paper, we focus on a multi-UAV assisted wireless network, and target to fully utilize the benefits of wireless caching and UAV mobility for multiuser content delivery. By taking into account the limited storage, our goal is to maximize the minimum throughput among UAV-served users by jointly optimizing cache placement, UAV trajectory, and transmission power in a finite period. The resultant problem is a mixed-integer non-convex optimization problem. To facilitate solving this problem, an alternating iterative algorithm is proposed by adopting the block alternating descent and successive convex approximation methods. Specifically, this problem is split into three subproblems, namely cache placement optimization, trajectory optimization, and power allocation optimization. Then these subproblems are solved alternately in an iterative manner. We show that the proposed algorithm can converge to the set of stationary solutions of this problem. Besides, we further analyze the computational complexity of this algorithm. Numerical results show that great throughput enhancement is achieved by applying our proposed joint design in comparison with other benchmarks without trajectory design and power control.

Abstract: This paper is devoted to investigation of the fractional order fuzzy dynamical system, in our case, modeling the recent pandemic due to corona virus (COVID-19). The considered model is analyzed for exactness and uniqueness of solution by using fixed point theory approach. We have also provided the numerical solution of the nonlinear dynamical system with the help of some iterative method applying Caputo as well as Attangana-Baleanu and Caputo fractional type derivative. Also, random COVID-19 model described by a system of random differential equations was presented. At the end we have given some numerical approximation to illustrate the proposed method by applying different fractional values corresponding to uncertainty.

Abstract: Parsimonious representations are ubiquitous in modeling and processing information. Motivated by the recent Multi-Layer Convolutional Sparse Coding (ML-CSC) model, we herein generalize the traditional Basis Pursuit problem to a multi-layer setting, introducing similar sparse enforcing penalties at different representation layers in a symbiotic relation between synthesis and analysis sparse priors. We explore different iterative methods to solve this new problem in practice, and we propose a new Multi-Layer Iterative Soft Thresholding Algorithm (ML-ISTA), as well as a fast version (ML-FISTA). We show that these nested first order algorithms converge, in the sense that the function value of near-fixed points can get arbitrarily close to the solution of the original problem. We further show how these algorithms effectively implement particular recurrent convolutional neural networks (CNNs) that generalize feed-forward ones without introducing any parameters. We present and analyze different architectures resulting from unfolding the iterations of the proposed pursuit algorithms, including a new Learned ML-ISTA, providing a principled way to construct deep recurrent CNNs. Unlike other similar constructions, these architectures unfold a global pursuit holistically for the entire network. We demonstrate the emerging constructions in a supervised learning setting, consistently improving the performance of classical CNNs while maintaining the number of parameters constant.

Abstract: This paper studies the extended state observer (ESO)-based distributed model predictive control (DMPC) approach to deal with multiple mobile robot formation with unknown disturbances. The distributed control problem with path parameters synchronization and disturbance rejection is formulated for formation system according to the tracking error dynamic model, where the reference paths are parameterized. A local distributed controller is designed by using DMPC strategy for each mobile robot in the absence of disturbance by including parameter synchronization constraints in the quadratic performance index as coupling terms. The DMPC optimization problem is solved by using Nash-optimization iteration strategy with the maximum number of iteration constraint. To improve the ability of anti-jamming, a feedforward compensation controller is designed by using ESO method, where the ESO is designed by pole assignment. The convergence of the proposed iterative algorithm is given. Furthermore, the input-to-state stability property of the proposed composite controller, combining a feedforward compensation controller and local distributed controller, is analyzed for the closed-loop system. Finally, the validity of the proposed algorithm is verified by two simulation examples.

Abstract: Iterative learning control (ILC) is a high-performance technique for repeated control tasks with design postulates on a fixed reference profile and identical initial conditions. However, the tracking performance is only critical at few points in point-to-point tasks, and their initial conditions are usually trial-varying within a certain range in practice, which essentially degrades the performance of conventional ILC algorithms. Therefore, this study reformulates the ILC problem setup for point-to-point tasks and considers the effort of trial-varying initial conditions in algorithm design. To reduce the tracking error, it proposes a worst-case norm-optimal problem and reformulates it into a convex optimisation problem using the Lagrange dual approach. In this sense, a robust ILC algorithm is derived based on iteratively solving this problem. The study also shows that the proposed robust ILC is equivalent to conventional norm-optimal ILC with trial-varying parameters. A numerical simulation case study is conducted to compare the performance of this algorithm with that of other control algorithms while performing a given point-to-point tracking task. The results reveal its efficiency for the specific task and robustness against trial-varying initial conditions.

Abstract: Mathematical modeling has always been one of the most potent tools in predicting the behavior of dynamic systems in biology. In this regard, we aim to study a three-species prey–predator model in the context of fractional operator. The model includes two competing species with logistic growing. It is considered that one of the competitors is being predated by the third group with Holling type II functional response. Moreover, one another competitor is in a commensal relationship with the third category acting as its host. In this model, the Atangana–Baleanu fractional derivative is used to describe the rate of evolution of functions in the model. Using a creative numerical trick, an iterative method for determining the numerical solution of fractional systems has been developed. This method provides an implicit form for determining solution approximations that can be solved by standard methods in solving nonlinear systems such as Newton’s method. Using this numerical technique, approximate answers for this system are provided, assuming several categories of possible choices for the model parameters. In the continuation of the simulations, the sensitivity analysis of the solutions to some parameters is examined. Some other theoretical features related to the model, such as expressing the necessary conditions on the stability of equilibrium points as well as the existence and uniqueness of solutions, are also examined in this article. It is found that utilizing the concept of fractional derivative order the flexibility of the model in justifying different situations for the system has increased. The use of fractional operators in the study of other models in computational biology is recommended.

Abstract: Optimization theory assisted algorithms have received great attention for precoding design in multiuser multiple-input multiple-output (MU-MIMO) systems. Although the resultant optimization algorithms are able to provide excellent performance, they generally require considerable computational complexity, which gets in the way of their practical application in real-time systems. In this work, in order to address this issue, we first propose a framework for deep-unfolding, where a general form of iterative algorithm induced deep-unfolding neural network (IAIDNN) is developed in matrix form to better solve the problems in communication systems. Then, we implement the proposed deepunfolding framework to solve the sum-rate maximization problem for precoding design in MU-MIMO systems. An efficient IAIDNN based on the structure of the classic weighted minimum mean-square error (WMMSE) iterative algorithm is developed. Specifically, the iterative WMMSE algorithm is unfolded into a layer-wise structure, where a number of trainable parameters are introduced to replace the highcomplexity operations in the forward propagation. To train the network, a generalized chain rule of the IAIDNN is proposed to depict the recurrence relation of gradients between two adjacent layers in the back propagation. Moreover, we discuss the computational complexity and generalization ability of the proposed scheme. Simulation results show that the proposed IAIDNN efficiently achieves the performance of the iterative WMMSE algorithm with reduced computational complexity.