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Efficient Graph Neural Networks for Multicarrier Wideband Hybrid Beamforming Optimization

Beier Li, Mai Vu

arXiv:2609.09708v1eess.SPcs.LGcs.NI

TL;DR

Wideband multicarrier systems face severe beam squinting, while per-antenna true-time-delay hardware is costly. The paper develops three GNN-based hybrid beamforming structures using a bipartite OFDM graph and reports superior performance, robustness, and generalization across the evaluated settings.

  • Problem

    Wideband 6G hybrid beamforming must address beam squinting caused by frequency-dependent beam shifts and a shared analog beamformer, but scalable and efficient solutions remain limited.

  • Method

    The paper represents a shared analog beamformer and multiple subcarriers with a bipartite graph and develops node-update, edge-update, and hybrid SVD-based GNN structures.

  • Results

    The proposed GNNs outperform traditional optimization and existing ML-based designs in spectral efficiency, beam-squinting resiliency, computational efficiency, imperfect-CSI robustness, and generalization.

  • Takeaways & Limitations

    The trained GNNs can adapt to changing subcarrier counts without retraining and extend directly to multi-user scenarios through graph expansion and problem reformulation.

Abstract

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6G wireless technology is poised to adopt higher and wider frequency bands, leveraging highly directional beamforming. However, the vast bandwidths amplify the impact of beam squinting. Traditional solutions, such as adding a true-time-delay filter to each antenna, are cost-prohibitive due to the required hardware scale. This paper proposes a signal processing alternative using Graph Neural Networks (GNNs) to optimize hybrid beamforming in multicarrier wideband systems. Using a bipartite graph to represent a shared analog beamformer among multiple subcarriers, we develop three GNN structures with distinct digital beamformer representations (i) at the subcarrier nodes, (ii) at the edges, or (iii) integrating traditional singular-value decomposition solutions. By designing an efficient message-passing mechanism, these structures offer insights into the impact of different GNN designs on communication system performance and computational complexity. Extensive analysis and ablation studies show that our proposed GNN structures outperform traditional optimization methods and existing ML-based solutions. Furthermore, the proposed GNNs exhibit strong resiliency to beam squinting and better robustness against imperfect CSI than even fully digital beamforming and all existing hybrid designs. These GNNs can also be extended to multi-user scenarios and demonstrate excellent generalization capabilities, allowing trained models to adapt to diverse multicarrier and multi-user settings without retraining.

I. INTRODUCTION

Wideband 6G systems intensify beam squinting, while existing hardware and optimization approaches remain costly or computationally demanding. This paper addresses the gap with GNN-based hybrid beamforming designs that model multicarrier structure directly and outperform prior methods across efficiency, robustness, and generalization.

  • Motivation: True-time-delay solutions can mitigate beam squinting but require many per-antenna filters, making signal-processing alternatives more attractive as antenna counts grow.Optimization methods such as AMO and ICD also require substantial computation after channel updates.
  • Motivation: The proposed GNNs target wideband multicarrier hybrid beamforming, where beam squinting becomes severe at sub-THz bandwidths and prior ML designs mainly address narrowband systems.Beam squinting is driven by the shared analog beamformer across subcarriers; the paper focuses primarily on single-user MIMO-OFDM to isolate this effect.
  • GNN Design: The graph uses one shared analog node connected to multiple subcarrier nodes, preserving OFDM structure while enabling scalable message passing across varying subcarrier configurations.The GNN architecture shares update functions, so trainable parameters do not depend on graph size.
  • GNN Design: The paper studies node-update and edge-update GNNs by placing each subcarrier’s digital beamformer representation at a node or edge, and extends both designs to multi-user settings without changing update rules.The edge-update formulation is presented as a first use of edge-update GNNs for hybrid beamforming.
  • GNN Design: The hybrid GNN learns the analog beamformer while computing digital beamformers through a closed-form singular-value decomposition solution.This combines machine learning with traditional signal processing rather than learning both beamformers directly.
  • Results: Across comparisons and ablations, the proposed GNNs outperform traditional optimization and existing ML methods in spectral efficiency, generalization, beam-squinting resiliency, and computational efficiency, including under imperfect CSI.The advantages are reported as particularly prominent when the number of antennas increases.

B. Channel Model

The wideband channel model captures frequency-dependent propagation and beam squinting, where a shared analog beamformer must serve all subcarriers while digital beamformers vary per subcarrier.

  • B. Channel Model: The clustered double-directional model represents each path using cluster and ray gains, propagation delays, and arrival/departure angles.The model uses α_il,k for complex path gain and β_il,k = e^−j2πτ_ilf_k for the path-delay component.
  • B. Channel Model: For uniform planar arrays, the array response depends on antenna spacing, wavelength, and two-dimensional antenna indices.The antenna indices span the planar array dimensions, with wavelength varying by subcarrier.
  • B. Channel Model: Beam squinting shifts the beam direction across subcarriers and degrades array gain as bandwidth expands.Fig. 2 uses f_c = 142 GHz and B = 20 GHz, showing the shift across four subcarriers and normalized gain versus fractional bandwidth for three AoAs.
  • B. Channel Model: The analog beamformer is shared across all subcarriers, whereas each subcarrier has an independent digital beamforming matrix.The joint formulation therefore evaluates beam-squint mitigation across all subcarriers under a common analog matrix W.

D. Problem Formulation

The paper formulates wideband hybrid beamforming as a shared-analog, subcarrier-specific digital design problem and uses permutation-aware bipartite GNNs to obtain scalable solutions without repeated optimization for each CSI realization.

  • D. Problem Formulation: The design optimizes transmit beamformers under perfect CSI using achievable spectral efficiency as the objective.The formulation considers the base-station transmit beamformers and imposes the hybrid beamforming constraints.
  • D. Problem Formulation: The hybrid beamforming problem is non-convex because of the constant-modulus constraint, making phase-only analog reconstruction potentially suboptimal.Traditional alternating optimization reconstructs the analog beamformer by extracting phases from complex-valued updates.
  • D. Problem Formulation: Traditional optimization requires re-optimization for every new CSI, whereas trained ML models can use forward computation on unseen CSI.This motivates learning-based methods that reduce the repeated computational burden.
  • A. Motivations of Selecting GNNs: GNNs are selected because shared update functions keep trainable parameters independent of graph size and support varying subcarrier configurations.This structure also preserves the symmetry between the shared analog beamformer and multicarrier digital beamformers.
  • B. Message-passing Graph Structure: The proposed bipartite graph contains one analog node shared by K subcarrier nodes, with representations placed at analog nodes, subcarrier nodes, and edges.The three representation strategies support node-update, edge-update, and hybrid GNN structures for digital beamformer modeling.
  • B. Message-passing Graph Structure: Each updating layer generates neighbor messages and updates representations through permutation-invariant aggregation implemented with shared functions across subcarriers.The layer stack progressively refines representations before the final beamformer reconstruction.
  • B. Message-passing Graph Structure: The original optimization problem and the proposed GNN outputs are permutation equivariant with respect to subcarrier ordering.This property supports adaptation across dynamic subcarrier-level configurations without changing the learned update functions.

C. Considerations in Designing GNN Structures

The paper models the shared analog beamformer as an analog node connected to subcarrier nodes, then compares node-update and edge-update representations for digital beamformers. These designs use message passing and permutation-equivariant aggregation to balance performance, flexibility, complexity, and scalability.

  • Graph and design choices: The bipartite graph represents one shared analog beamformer at node x connected to subcarrier nodes, with three digital-beamformer representation strategies explored.The strategies place digital representations at subcarrier nodes, at edges, or within a hybrid structure using traditional optimization.
  • Training: The models are trained unsupervised to maximize average data rate, with normalization applied to reconstructed outputs to satisfy the power constraint.Mini-batch stochastic gradient descent updates the trainable GNN parameters during offline training.
  • Scalability: Permutation equivariance and scalability allow training with fewer subcarriers while supporting variable subcarrier counts during inference.The approach requires CSI samples spanning relevant frequency ranges and channel conditions.
  • Node-update GNN: NU-GNN embeds each digital beamformer c_k in a subcarrier node while keeping vectorized CSI edge features fixed.The analog beamformer is represented at the analog node, and node representations are updated through message passing.
  • Edge-update GNN: EU-GNN embeds digital beamformers in edge representations initialized by CSI, rather than using subcarrier node representations.This structure updates the analog node and edge representations through separate MLP-based message-passing operations.
  • Message passing: Mean aggregation combines information from all neighboring subcarriers, capturing their collective beam-squint impact rather than only the dominant subcarrier.The message-generation and representation-update mechanisms directly affect GNN performance and computational complexity.

B. Edge Update GNN (EU-GNN) Structure

EU-GNN represents digital beamformers on edges initialized from CSI, while updating both the shared analog-node representation and edge representations through message passing. Direct phase learning produces an analog beamformer that inherently satisfies the constant-modulus constraint, while digital outputs are normalized for power compliance.

  • EU-GNN updates: EU-GNN updates the shared analog node and edge representation e_k, with CSI h_k serving only as the edge representation’s initial value.Two MLPs generate and update representations through messages associated with the graph edges.
  • Analog representation: Directly learning the analog beamformer phase inherently satisfies the constant-modulus constraint without an additional projection or approximation.This avoids a potentially harmful non-differentiable projection during learning.
  • Beamformer reconstruction: The edge-based digital representations are reconstructed into complex digital beamforming matrices and normalized to satisfy the transmit-power constraint.The same reconstruction process applies to EU-GNN edge representations in place of node representations.

V. ANALOG NODE GNN WITH ATTENTION STRUCTURE

AN-GNN learns the analog beamformer while deriving digital beamformers from a closed-form singular-value-decomposition solution at each layer. It then uses attention-based aggregation to combine subcarrier information when updating the analog representation.

  • AN-GNN design: AN-GNN learns the analog representation but obtains digital beamformer representations through closed-form expressions, keeping CSI edge features fixed.This combines machine learning for analog design with traditional optimization for digital beamforming.
  • Attention aggregation: AN-GNN uses an attention aggregation module to combine message information across subcarriers when updating the analog node.The structure includes separate MLPs for message generation and analog-representation updates.
  • Digital beamformer solution: For a given analog beamformer, the digital beamformer is solved from the effective channel using a singular-value-decomposition solution based on dominant right singular vectors.The truncated right singular-vector matrix contains columns associated with the largest nonzero singular values.
  • Layered update process: At each GNN layer, the previous analog representation is converted into a phase matrix, combined with each channel, and used to generate the next digital representations.These digital representations are then combined with edge features to generate messages for updating the analog node.

C. Aggregations Via an Attention Mechanism

Attention aggregation assigns learned, permutation-invariant weights to subcarrier information before updating the analog beamformer. The pure-learning NU-GNN and EU-GNN extend to multi-user transmission, whereas AN-GNN does not because its single-user digital closed-form solution is unavailable there.

  • Attention mechanism: Attention scores are computed for each subcarrier, converted to weights by softmax, and used in a weighted sum that emphasizes relatively important subcarriers.The scores are generated by a linear layer followed by LeakyReLU activation.
  • Permutation invariance: The attention aggregation is permutation invariant because its score computation and softmax preserve invariance under subcarrier reordering.The resulting attention weights and aggregated messages remain jointly invariant to the ordering of subcarriers.
  • Training procedure: The AN-GNN procedure reconstructs the analog phase, computes digital beamformers, propagates messages, and evaluates normalized beamformers through the training loss.Algorithm 1 repeats these operations across layers and epochs while updating GNN parameters.
  • Multi-user extension: The multi-user extension preserves the bipartite graph and embedded information for NU-GNN and EU-GNN while reformulating the objective for multi-user transmission.The received-signal model includes inter-user interference and uses the resulting objective to construct the training loss.
  • Multi-user limitation: AN-GNN cannot be directly extended to multiple users because the digital beamformers lack the single-user closed-form solution used by its design.This scope boundary applies specifically to the AN-GNN structure, not to the direct NU-GNN and EU-GNN extensions.

B. Extended Graph Model and Updating Rules

The multi-user extension adds user-specific subcarrier nodes while preserving permutation equivariance, allowing the proposed GNNs to generalize across subcarrier and user orders without retraining.

  • The multi-user graph groups each user’s subcarrier nodes into a user-specific set while retaining the proposed graph structure.
  • Permutation equivariance holds across both subcarrier and user orders in every GNN updating layer.
  • The equivariance property supports generalization across varying subcarrier and user orders without retraining.
  • NU-GNN and EU-GNN use the same message-generation and representation-update rules in the multi-user setting, with expanded node indexing and a different loss function.
  • Digital beamformers are reconstructed per user and subcarrier, concatenated, normalized, and paired with analog-beamformer reconstruction to satisfy the transmit-power constraint.
  • Inference complexity is analyzed for the three GNNs because offline training is completed before deployment.

A. Proposed GNN Structures

The proposed GNN structures use MLP-based message passing with complexity dominated by repeated updating layers, while AMO incurs higher-order optimization costs as antenna dimensions grow.

  • NU-GNN and EU-GNN use four MLPs with shared depth but different layer sizes, and their forward cost depends on hidden-layer depth and feature dimensions.
  • AN-GNN adds a fully connected aggregation layer and SVD-based subcarrier features, while its first complexity term remains dominant.
  • The total complexity of a GNN scales with the number of updating layers, with NU-GNN and EU-GNN differing in feature dimensions and aggregation operations.
  • AMO complexity includes Armijo line search, retraction, cost-function evaluation, Riemannian gradients, and repeated beamformer updates.
  • AMO also incurs matrix inversion and multiplication for subcarrier beamformers, although these operations remain minor when NRF is much smaller than Nt.
  • The proposed GNNs replace repeated optimization with feed-forward updating layers, whereas AMO repeats Riemannian updates until convergence.

C. Comparison Between GNNs and AMO

The proposed GNNs achieve strong spectral-efficiency performance while offering lower and more stable computational demands than AMO, particularly as antenna and subcarrier counts increase.

  • The GNN pass has second-order antenna complexity, whereas AMO reaches cubic order, so AMO’s cost grows faster with antenna-array size.
  • GNN and AMO complexity both grow linearly with subcarrier count, but AMO has the larger antenna-dependent multiplier.
  • The evaluation uses 64 antennas, four RF chains, eight receive antennas, and a 142 GHz carrier with 20 GHz bandwidth during training.
  • The channel model includes two clusters and three rays per cluster with large- and small-scale fading effects.
  • NU-GNN outperforms AMO and all other machine-learning benchmarks in convergence, while EU-GNN approaches AMO and AN-GNN performs similarly to ICD.
  • The comparison includes fully digital beamforming, AMO, ICD, AV-all, MCM, FNN, AN-FNN, and LCMLP-GNN baselines.

2) Ablation Study:

Ablations favor the simpler proposed message passing for NU-GNN and EU-GNN, while attention improves AN-GNN; broader evaluations show strong efficiency, scalability, and beam-squint resilience.

  • Ablation study: Adding extra messages to NU-GNN duplicates information already present, increasing complexity without improving convergence values.
  • Ablation study: EU-GNN variants with extra messages or CSI edge features improve convergence slightly, but comparable performance and longer training favor the simplest variant.
  • Ablation study: Attention modestly improves AN-GNN convergence at a small training-time cost and is selected for superior generalization ability.
  • Performance and scalability: Over 32 times: AMO is slower than the proposed GNNs per CSI update, while the GNNs also show much lower computation-time variability.
  • Performance and scalability: As Nt increases, the proposed models gain advantage over traditional methods while remaining competitive with fully digital beamforming.
  • Performance and scalability: 125%: the proposed GNNs outperform LCMLP-GNN in spectral efficiency despite LCMLP-GNN’s faster online inference.
  • Performance and scalability: The proposed GNNs grow quadratically with Nt, whereas AMO and ICD grow cubically, matching the observed inference-time trends.
  • Beam-squint resilience: NU-GNN and EU-GNN remain more resilient to increasing fractional bandwidth than all baselines, mitigating beam squinting across wider channels.

3) Beam Squinting Resiliency:

The proposed GNNs maintain beam-pattern alignment and spectral-efficiency robustness across changing subcarrier counts, imperfect CSI, and multi-user settings. NU-GNN leads spectral efficiency and inference speed, while EU-GNN is most stable; AN-GNN’s training savings disappear at inference.

  • Beam-pattern resiliency: NU-GNN and EU-GNN maintain near-perfect beam-pattern alignment across subcarriers, while AMO, MCM, and AV-all show larger shifts or distorted sidelobes.For one CSI sample, AMO has a 5° shift, AN-GNN stays within 2°, and NU-GNN/EU-GNN nearly align the first and last subcarriers.
  • Subcarrier generalization: The proposed GNNs generalize from K = 4 training subcarriers to K = 64 inference without retraining, with attention improving AN-GNN generalization.The evaluation varies the number of subcarriers and compares attention-based aggregation with element-wise mean aggregation.
  • Multi-user generalization: Across user configurations, the proposed GNNs outperform AMO, MU-LCMLP-GNN, BD, and MRT, but remain below the fully digital WMMSE benchmark.The models are trained with K = 4 subcarriers and evaluated with K = 64; the authors attribute the remaining gap to unmodeled inter-user interference.
  • Imperfect CSI: Under increasing location-error variance, NU-GNN and EU-GNN retain higher spectral efficiency than fully digital beamforming across the evaluated imperfect-CSI conditions.All methods degrade as location error increases, but fully digital designs are more sensitive because they rely on precise per-subcarrier CSI.
  • Model trade-offs: NU-GNN achieves the highest spectral efficiency and lowest average inference time, while EU-GNN is the most stable during training and inference.AN-GNN reduces training cost through fewer parameters, but digital-beamformer computation removes that advantage during inference.

APPENDIX A: PROOF OF PROPOSITION 1

The appendix establishes permutation equivariance for the beamforming optimization and GNN pipeline. Subcarrier permutations are preserved through shared updates, invariant aggregation, and final beamformer reconstruction, including the multi-user extension.

  • Optimization equivariance: The optimization mapping remains permutation equivariant when RF-chain, data-stream, and subcarrier indices are permuted consistently.The permuted beamforming variables continue to satisfy the power constraint while optimizing the correspondingly permuted channel objective.
  • GNN equivariance: Shared subcarrier updates and permutation-invariant aggregation make the learned representations equivariant to subcarrier reordering.The final reconstruction converts permuted representations into correspondingly permuted analog and digital beamforming matrices.
  • Multi-user extension: The multi-user extension preserves permutation equivariance by grouping subcarrier nodes into user-specific subsets and treating user permutations as block-wise subcarrier permutations.The existing GNN updating rules remain unchanged under this grouping.
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