Learning-based precoding has been shown able to be implemented in real-time, jointly optimized with channel acquisition, and robust to imperfect channels. Nonetheless, existing works rarely explain when and why a deep neural network (DNN) for learning precoding policy can perform well. In this paper, we strive to understand the learning performance by taking baseband precoding as an example, for which the optimal precoding matrices of several problems such as sum rate maximization have mathematical structure. Toward this goal, we design a graph neural network (GNN) with edge-update mechanism, whose inductive bias matches to the precoding policy, and analyze its connection to the commonly used convolutional neural networks (CNNs). Noticing that the learning performance can be decomposed into approximation and estimation errors, which depend on the smoothness of a policy and the inductive bias of a DNN, we analyze in which system settings the precoding policy is harder to be approximated by a DNN and how the inductive biases introduced by parameter sharing affect estimation errors. We proceed to derive the estimation error bounds of the DNNs. Simulations validate our analyses and verify the gain of GNN over CNNs in terms of reducing sample complexity.

Learning precoding policies with neural networks enables low complexity implementation, robustness to channel impairments, and joint optimization with channel acquisition. However, pure data-driven methods for learning precoding suffer from high complexity of training and poor generalizability to problem scales, while existing model-driven learning methods are either algorithm-specific or problem-specific. In this paper, we propose a model-based graph neural network (GNN) to learn precoding policies by harnessing their properties and relevant mathematical model. We first show that a vanilla GNN cannot learn zero-forcing precoding when the numbers of antennas and users are large, and is not generalizable to the numbers of users. Then, we conceive a new GNN structure by resorting to the iterative Taylor’s expansion of matrix pseudo-inverse, which can adapt to the interference strength among users. Simulation results show that the proposed GNN can well-learn different precoding policies (say spectral efficient and energy efficient precoding policies as well as coordinated beamforming) with low training complexity. Moreover, it can be generalized to the number of users, which is highly desirable in practice since the number of scheduled users may change in milliseconds.

Graph neural networks (GNNs) can improve the efficiency of learning wireless policies by leveraging their permutation properties and topology prior. While mismatched permutation property to a policy may degrade the learning performance and overlooked permutations incurs low sample efficiency, there is still lacking a systematical approach for modeling graph and designing structure of GNNs to harness all permutation properties. Moreover, the information of input feature may lose during updating hidden representations with GNNs, which leads to poor learning performance. In this paper, we propose a unified framework to learn permutable wireless policies with multidimensional GNNs, which update the hidden representations of hyper-edges to avoid the information loss. We provide a method to construct graph for a policy, over which a GNN with proper parameter sharing can exploit all possible permutations of the policy. We also investigate the permutability of wireless channels that affects the sample efficiency, and show how to trade off the training, inference, and design complexities of GNNs. To showcase how to design the GNNs within the framework, we consider precoding optimization in different systems. Simulation results validate the gain of the proposed GNNs over existing counterparts from exploiting the permutation prior and avoiding the information loss.

Multi-modal data collected from various sensors is instrumental in enhancing proactive handover management, beam directions and received powers prediction. However, what essential information to extract and how to effectively allocate wireless resources to transmit the information to a central processor (e.g., a base station (BS)) for decision making is a challenging task. In this work, we consider an uplink multi-user goal-oriented system, where images extracted from users’ depth cameras reflect blockage status between users and their serving BS, which are then used for future received power prediction. In the system, we employ a convolutional neural network to learn a joint semantic source and channel encoder such that essential information is extracted from images. Subsequently, we model the multi-user subcarrier communication system as a hypergraph and use hyper-edge graph neural networks to learn precoders at the user side and detector at the BS side. Simulation results demonstrate that by jointly training a deep neural network-based encoder, decoder, precoder and detector, the communication system can achieve lower prediction errors than traditional precoder and detector, especially in low signal-to-noise ratio scenarios. We also show a trade-off between prediction performance and the computational complexity.

Graph neural networks (GNNs) have been shown promising in optimizing power allocation and link scheduling with good size generalizability and low sample complexity, which are important for learning wireless policies under dynamic environments. This attributes to their matched permutation equivariance (PE) properties to the policies to be learned. Nonetheless, existing works have demonstrated that only satisfying the PE property cannot ensure a GNN for learning precoding policy to be generalizable to the unseen problem scales, say the number of users. Incorporating models with neural networks helps improve size generalizability, which however is only applicable to specific problems. In this paper, we strive to design a size generalizable GNN that does not depend on any mathematical model, such that the GNN can learn wireless policies including but not limited to baseband and hybrid precoding in multi-user multi-antenna systems. To this end, we first identify the key characteristics of the update equation of a GNN that affect its size generalization ability. Then, we design a size-generalizable GNN that is with these key characteristics and satisfies the PE property of a precoding policy in a recursive manner. Simulation results show that the proposed GNN can be well-generalized to the number of users for learning precoding policies.

Learning precoding with deep neural networks (DNNs) enables real-time implementation and robustness to imperfect channels. However, existing DNNs for learning precoding suffer from high training complexity and weak generalization ability to problem scales, which impedes their practical use in wireless systems with user scheduling. In this paper, we propose a graph neural network (GNN) to learn precoding policies efficiently by resorting to the model of Taylor’s expansion of matrix pseudo-inverse. The GNN can capture the importance of neighbored edges when aggregating their information, which is critical for improving the learning efficiency. We also interpret the role of the trainable parameters on learning the powers and directions of the precoding vectors. Simulation results show that the proposed model-driven GNN can well-learn spectral and energy efficient precoding policies with low training complexity, and is generalizable to the numbers of users.

Learning-based resource allocation can be implemented in real-time, but deep neural networks (DNNs) developed in other fields such as computer vision are with high training complexity and weak generalizability. Leveraging domain knowledge in communications is promising for learning wireless policies efficiently. In this paper, we propose a framework of integrating the Shannon formula with DNNs, and derive a data rate-based DNN (DRNN), for learning resource allocation by taking power allocation as an example. The DRNN is with an iterative structure with multiple update layers, each consisting of a pre-determined model function, an update network, and a dimension reduction network. To justify the iterative structure, we prove the existence of an iteration function that converges to the optimal policy for the update layer to learn. To justify the structure of each update layer, we provide the conditions for the iteration function to be a composite function of the model function. We further incorporate permutation equivariance properties into the DRNN. Simulation results show that the numbers of training samples and free parameters, and the training time to achieve a desired system performance can be reduced remarkably by harnessing the data rate model and PE prior.

Hybrid precoding in millimeter wave systems can support high spectral efficiency with affordable cost. With deep learning, fairly good solutions that are robust to imperfect chan-nels can be obtained with low complexity from the non-convex optimization problems. Yet previous works for hybrid precoding are with high cost for training neural networks because the mathe-matical properties of the optimization problems are not taken into account. In this paper, we show that hybrid precoding problems exhibit a multi-set permutation equivariance (PE) property and a phase invariance property. We propose a method to design a graph neural network (GNN) that can satisfy the PE property, and propose a processing method to harness the phase invariance property. Simulation results show that the proposed GNN is more efficient than the commonly used convolutional neural network, which requires much fewer trainable parameters and training samples to achieve the same sum-rate and achieves higher sum-rate with the same number of training samples.

Optimizing power allocation in cellular systems with deep learning enables real-time coordination of inter-cell interference. When channels are time-varying, the deep neural networks (DNNs) need to be re-trained frequently and the training samples need to be re-collected in a timely manner. To achieve higher sum rate with fewer training samples and lower training cost, domain knowledge should be resorted for designing DNNs. In this paper, we propose a DNN structure where the formula of data rate is used to facilitate the learning of power allocation policy, called data-rate based DNN (DRNN). Since such a model-based deep learning method does not exclude the use of prior knowledge for reducing the hypothesis space of a DNN, we further exploit a permutation equivariance prior by introducing parameter sharing into the DNN structure. By integrating the model and prior into DNN, simulations show that either sum rate is improved for given number of training samples or training complexity is reduced to achieve an expected performance.

Optimizing precoding with deep learning enables its real-time implementation. In addition to the learning perfor-mance such as sum rate, training complexity is also important since neural networks (NNs) have to be re-trained in time-varying channels. By leveraging the prior-known property for a policy to be learned, inductive biases can be introduced to the structure of NNs to balance the learning performance and training com-plexity. Most existing works use convolutional neural networks for learning precoding policy, without considering whether their inductive biases match the precoding task. In this paper, we first show that full-digital precoding policy exhibits permutation equivariance property and introduce graph NN (GNN) to learn the policy. We then analyze and show the connections between the structures and inductive biases of several NNs. Simulation results show that the inductive bias of the GNN is well-matched to the precoding policy, and hence achieves higher sum-rate with given number of training samples and needs lower training complexity to achieve the same sum-rate than other NNs.

A well-trained deep neural network (DNN) enables real-time resource allocation by learning the relationship between a policy and its impacting parameters. When wireless systems operate in dynamic environments, the DNN has to be re-trained frequently and hence training complexity should be low. A promising approach to deal with this issue is to construct DNNs with prior knowledge. In this paper, we show that the power allocation policy in multi-cell-multi-user systems exhibits a combination of permutation equivariance properties, which can be harnessed by graph neural networks (GNNs). In particular, we construct a heterogeneous graph and resort to heterogeneous GNN for learning the policy, whose outputs are only equivariant to some permutations of vertexes rather than arbitrary permutations as homogeneous GNNs. We prove that the properties of the functions learned by existing heterogeneous GNN for the formulated graph are inconsistent with the properties of the policy. To avoid the performance degradation by embedding wrong priors, we design a parameter sharing scheme for heterogeneous GNN such that the learned relationship satisfies the desired properties. Simulation results show that the sample and computational complexities for training the constructed GNN are much lower than existing DNNs to achieve the same sum rate.

Optimizing power control in multi-cell cellular networks with deep learning enables such a non-convex problem to be implemented in real-time. When channels are time-varying, the deep neural networks (DNNs) need to be re-trained frequently, which calls for low training complexity. To reduce the number of training samples and the size of DNN required to achieve good performance, a promising approach is to embed the DNNs with a priori knowledge. Since cellular networks can be modelled as a graph, it is natural to employ graph neural networks (GNNs) for learning, which exhibit permutation invariance (PI) and equivalence (PE) properties. Unlike the homogeneous GNNs that have been used for wireless problems, whose outputs are invariant or equivalent to arbitrary permutations of vertexes, heterogeneous GNNs (HetGNNs), which are more appropriate to model cellular networks, are only invariant or equivalent to some permutations. If the PI or PE properties of the HetGNN do not match the property of the task to be learned, the performance degrades dramatically. In this paper, we show that the power control policy has a combination of different PI and PE properties, and existing HetGNN does not satisfy these properties. We then design a parameter sharing scheme for HetGNN such that the learned relationship satisfies the desired properties. Simulation results show that the sample complexity and the size of designed GNN for learning the optimal power control policy in multi-user multi-cell networks are much lower than the existing DNNs, when achieving the same sum rate loss from the numerically obtained solutions.

Deep neural networks (DNNs) have been employed for designing wireless networks in many aspects, such as transceiver optimization, resource allocation, and information prediction. Existing works either use fully-connected DNN or the DNNs with specific structures that are designed in other domains. In this paper, we show that a priori knowledge widely existed in wireless tasks is permutation invariance. For these tasks, we propose a DNN with special structure, where the weight matrices between layers of the DNN only consist of two smaller sub-matrices. By such way of parameter sharing, the number of model parameters reduces, giving rise to low sample and computational complexity for training a DNN. We take predictive resource allocation as an example to show how the designed DNN can be applied for learning an optimal policy with unsupervised learning. Simulations results validate our analysis and show dramatic gain of the proposed structure in terms of reducing training complexity.

Predictive resource allocation can exploit residual resources in wireless networks to support high throughput, improve user experience, and enhance energy efficiency. Most priori works assume that fine-grained knowledge for user trajectory and/or traffic load is known, which is hard to predict in practice. In this paper, we investigate predictive resource allocation to achieve high throughput for mobile users requesting video-on-demand (VoD) services, which employs cell-level coarse grained information. In the start of a prediction window, we only need to predict the cells the users to be associated with, the sojourn time of each user in each cell, the loads of VoD traffic and realtime traffic at each base station (BS). These information is translated into two thresholds, which are introduced to help each BS to determine when and how much data to transmit. Two-threshold-based algorithms are provided. Simulation results show that the algorithms perform closely to the optimal predictive resource allocation with perfect fine-grained information in terms of supporting high request arrival rate and improving user experience, and one algorithm even outperforms the optimal method with prediction errors.

Assigning radio resources in advance to nonrealtime (NRT) service in a proactive manner can exploit residual resource after serving realtime service to boost the performance of wireless networks. By predicting future average data rate of each mobile user requesting NRT service in a time window, either directly or indirectly, a plan for assigning future resources to each user can be made. Most existing works make the plan by solving optimization problems, which require high computational complexity when the number of users is large and the prediction window in long. In this paper, we design a deep neural network (DNN), which contains an autoencoder and a fully-connected neural network, to learn the resource allocation pattern in a prediction window. With the help of the DNN trained offline, the plan can be made with low complexity. To increase the generalizability to time-varying traffic load for both NRT and realtime services, we resort to selective sampling in active learning. Simulation results show that the proposed method performs closely to the optimal solution in supporting high throughput with given quality of service requirement.