Abstract--This paper presents a robust precoder design for resilient cell-free massive MIMO (CF-mMIMO) systems that minimizes the weighted sum of desired signal mean square error (MSE) and residual interference leakage power under a total transmit power constraint. The proposed robust preco der incorporates channel state information (CSI) error statis tics to enhance resilience against CSI imperfections. We employ an alternating optimization algorithm initialized with a min imum MSE-type solution, which iteratively refines the precoder w hile maintaining low computational complexity and ensuring fas t convergence. Numerical results show that the proposed meth od significantly outperforms conventional linear precoders, providing an effective balance between performance and computati onal efficiency. Cell-free massive multiple-input multiple-output (CF-mMIMO) networks have emerged as an extension of massive multiple-input multiple-output (MIMO) systems [1], [2] an d cornerstone of next-generation wireless systems by deploy ing a large number of distributed access points (APs) to jointly serve users without cell boundaries [3], [4], [5].

ABSTRACT Robust precoding is efficiently feasible in frequency divis ion duplex (FDD) systems by incorporating the learnt statistic s of the propagation environment through a generative model. W e build on previous work that successfully designed site-specific precoders based on a combination of Gaussian mixture models (GMMs) and graph neural networks (GNNs). In this paper, by utilizing a vector quantized-variational au toen-coder (VQ-V AE), we circumvent one of the key drawbacks of GMMs, i.e., the number of GMM components scales exponentially to the feedback bits. In addition, the deep lear n-ing architecture of the VQ-V AE allows us to jointly train the GNN together with VQ-V AE along with pilot optimization forming an end-to-end (E2E) model, resulting in considerable performance gains in sum rate for multi-user wireless systems. Simulations demonstrate the superiority of the pr o-posed frameworks over the conventional methods involving the sub-discrete Fourier transform (DFT) pilot matrix and i t-erative precoder algorithms enabling the deployment of sys - tems characterized by fewer pilots or feedback bits.

Hybrid precoding is a key ingredient of cost-effective massive multiple-input multiple-output transceivers. However, setting jointly digital and analog precoders to optimally serve multiple users is a difficult optimization problem. Moreover, it relies heavily on precise knowledge of the channels, which is difficult to obtain, especially when considering realistic systems comprising hardware impairments. In this paper, a joint channel estimation and hybrid precoding method is proposed, which consists in an end-to-end architecture taking received pilots as inputs and outputting pre-coders. The resulting neural network is fully model-based, making it lightweight and interpretable with very few learnable parameters. The channel estimation step is performed using the unfolded matching pursuit algorithm, accounting for imperfect knowledge of the antenna system, while the precoding step is done via unfolded projected gradient ascent. The great potential of the proposed method is empirically demonstrated on realistic synthetic channels.

Deep learning (DL) has emerged as a solution for precoding in massive multiple-input multiple-output (mMIMO) systems due to its capacity to learn the characteristics of the propagation environment. However, training such a model requires high-quality, local datasets at the deployment site, which are often difficult to collect. We propose a transformer-based foundation model for mMIMO precoding that seeks to minimize the energy consumption of the transmitter while dynamically adapting to per-user rate requirements. At equal energy consumption, zero-shot deployment of the proposed foundation model significantly outperforms zero forcing, and approaches weighted minimum mean squared error performance with 8x less complexity. To address model adaptation in data-scarce settings, we introduce a data augmentation method that finds training samples similar to the target distribution by computing the cosine similarity between the outputs of the pre-trained feature extractor. Our work enables the implementation of DL-based solutions in practice by addressing challenges of data availability and training complexity. Moreover, the ability to dynamically configure per-user rate requirements can be leveraged by higher level resource allocation and scheduling algorithms for greater control over energy efficiency, spectral efficiency and fairness.

Massive MIMO systems are moving toward increased numbers of radio frequency chains, higher carrier frequencies and larger bandwidths. As such, digital-to-analog converters (DACs) are becoming a bottleneck in terms of hardware complexity and power consumption. In this work, non-linear precoding for coarsely quantized downlink massive MIMO is studied. Given the NP-hard nature of this problem, a graph neural network (GNN) is proposed that directly outputs the precoded quantized vector based on the channel matrix and the intended transmit symbols. The model is trained in a self-supervised manner, by directly maximizing the achievable rate. To overcome the non-differentiability of the objective function, introduced due to the non-differentiable DAC functions, a straight-through Gumbel-softmax estimation of the gradient is proposed. The proposed method achieves a significant increase in achievable sum rate under coarse quantization. For instance, in the single-user case, the proposed method can achieve the same sum rate as maximum ratio transmission (MRT) by using one-bit DAC's as compared to 3 bits for MRT. This reduces the DAC's power consumption by a factor 4-7 and 3 for baseband and RF DACs respectively. This, however, comes at the cost of increased digital signal processing power consumption. When accounting for this, the reduction in overall power consumption holds for a system bandwidth up to 3.5 MHz for baseband DACs, while the RF DACs can maintain a power reduction of 2.9 for higher bandwidths. Notably, indirect effects, which further reduce the power consumption, such as a reduced fronthaul consumption and reduction in other components, are not considered in this analysis.

We investigate the multiuser scheduling problem in multiple-input multiple-output (MIMO) systems using orthogonal frequency division multiplexing (OFDM) and hybrid beamforming in which a base station (BS) communicates with multiple users over millimeter wave (mmWave) channels in the downlink. Improved scheduling is critical for enhancing spectral efficiency and the long-term performance of the system from the perspective of proportional fairness (PF) metric in hybrid beamforming systems due to its limited multiplexing gain. Our objective is to maximize PF by properly designing the analog and digital precoders within the hybrid beamforming and selecting the users subject to the number of radio frequency (RF) chains. Leveraging the characteristics of mmWave channels, we apply a two-timescale protocol. On a long timescale, we assign an analog beam to each user. Scheduling the users and designing the digital precoder are done accordingly on a short timescale. To conduct scheduling, we propose combinatorial solutions, such as greedy and sorting algorithms, followed by a machine learning (ML) approach. Our numerical results highlight the trade-off between the performance and complexity of the proposed approaches. Consequently, we show that the choice of approach depends on the specific criteria within a given scenario.

Recent neural heuristics for the Vehicle Routing Problem (VRP) primarily rely on node coordinates as input, which may be less effective in practical scenarios where real cost metrics-such as edge-based distances-are more relevant. To address this limitation, we introduce EFormer, an Edge-based Transformer model that uses edge as the sole input for VRPs. Our approach employs a precoder module with a mixed-score attention mechanism to convert edge information into temporary node embeddings. We also present a parallel encoding strategy characterized by a graph encoder and a node encoder, each responsible for processing graph and node embeddings in distinct feature spaces, respectively. This design yields a more comprehensive representation of the global relationships among edges. In the decoding phase, parallel context embedding and multi-query integration are used to compute separate attention mechanisms over the two encoded embeddings, facilitating efficient path construction. We train EFormer using reinforcement learning in an autoregressive manner. Extensive experiments on the Traveling Salesman Problem (TSP) and Capacitated Vehicle Routing Problem (CVRP) reveal that EFormer outperforms established baselines on synthetic datasets, including large-scale and diverse distributions. Moreover, EFormer demonstrates strong generalization on real-world instances from TSPLib and CVRPLib. These findings confirm the effectiveness of EFormer's core design in solving VRPs.

Incorporating mathematical properties of a wireless policy to be learned into the design of deep neural networks (DNNs) is effective for enhancing learning efficiency. Multi-user precoding policy in multi-antenna system, which is the mapping from channel matrix to precoding matrix, possesses a permutation equivariance property, which has been harnessed to design the parameter sharing structure of the weight matrix of DNNs. In this paper, we study a stronger property than permutation equivariance, namely unitary equivariance, for precoder learning. We first show that a DNN with unitary equivariance designed by further introducing parameter sharing into a permutation equivariant DNN is unable to learn the optimal precoder. We proceed to develop a novel non-linear weighting process satisfying unitary equivariance and then construct a joint unitary and permutation equivariant DNN. Simulation results demonstrate that the proposed DNN not only outperforms existing learning methods in learning performance and generalizability but also reduces training complexity.

Weighted sum rate maximization (WSRM) for precoder optimization effectively balances performance and fairness among users. Recent studies have demonstrated the potential of deep learning in precoder optimization for sum rate maximization. However, the WSRM problem necessitates a redesign of neural network architectures to incorporate user weights into the input. In this paper, we propose a novel deep neural network (DNN) to learn the precoder for WSRM. Compared to existing DNNs, the proposed DNN leverage the joint unitary and permutation equivariant property inherent in the optimal precoding policy, effectively enhancing learning performance while reducing training complexity. Simulation results demonstrate that the proposed method significantly outperforms baseline learning methods in terms of both learning and generalization performance while maintaining low training and inference complexity.

The deployment of deep learning (DL) models for precoding in massive multiple-input multiple-output (mMIMO) systems is often constrained by high computational demands and energy consumption. In this paper, we investigate the compute energy efficiency of mMIMO precoders using DL-based approaches, comparing them to conventional methods such as zero forcing and weighted minimum mean square error (WMMSE). Our energy consumption model accounts for both memory access and calculation energy within DL accelerators. We propose a framework that incorporates mixed-precision quantization-aware training and neural architecture search to reduce energy usage without compromising accuracy. Using a ray-tracing dataset covering various base station sites, we analyze how site-specific conditions affect the energy efficiency of compressed models. Our results show that deep neural network compression generates precoders with up to 35 times higher energy efficiency than WMMSE at equal performance, depending on the scenario and the desired rate. These results establish a foundation and a benchmark for the development of energy-efficient DL-based mMIMO precoders.