Multi-Cell Cooperative AirComp Framework

• Multi-cell cooperative AirComp frameworks are architectures enabling simultaneous data aggregation and computation over multi-cell networks using joint transceiver design and resource optimization.
• They mitigate intra- and inter-cell interference through coordinated power control, interference alignment, and polarization techniques to enhance MSE performance in federated learning.
• Joint optimization algorithms, including alternating optimization and deep graph learning, balance trade-offs between convergence speed and fronthaul overhead for scalable, robust deployments.

A multi-cell cooperative Air-Computation (AirComp) framework is an architectural, signal processing, and resource optimization paradigm that enables simultaneous data aggregation and computation over wireless multi-access channels spanning multiple cells. These frameworks are foundational for large-scale wireless distributed learning (e.g., federated learning) and networked sensing, where both intra-cell and inter-cell interference must be jointly managed to efficiently aggregate local updates, gradients, or statistics from devices distributed across a multi-cell infrastructure. Core components include joint transceiver design, power control, interference alignment, and cooperation protocols among access points (APs), base stations (BSs), or fusion centers (FCs). The following sections detail principles, methodologies, cooperative strategies, optimization algorithms, and empirical findings from state-of-the-art frameworks.

1. System Architecture and Signal Model

Multi-cell cooperative AirComp frameworks consist of several spatially distributed cells, each containing a set of edge devices and at least one multi-antenna AP or BS. Devices are partitioned into groups or FL tasks and transmit analog-modulated symbols (e.g., model updates, gradient vectors) over the shared spectrum. In advanced setups, APs are interconnected by high-rate fronthaul to a central processing unit (CPU), forming cell-free massive MIMO topologies (Chen et al., 17 Jan 2025, Chen et al., 2024).

Signal Model (uplink aggregation example for cell-free mMIMO):

• Device $k$ in group $g$ transmits:

$${x_{k,d}^{t} = b_{k,d}^{t} s_{k,d}^{t}}, \quad |b_{k,d}^{t}|^2 \leq P_k$$

where $s_{k,d}^{t}$ is a normalized local update, $b_{k,d}^{t}$ is the transmit coefficient.

• AP $l$ receives:

$$y_{l,d}^{t} = \sum_{k=1}^K h_{kl}^{t} b_{k,d}^{t} s_{k,d}^{t} + n_{l,d}^{t}$$

with $h_{kl}^{t}$ as the complex channel and $n_{l,d}^{t}$ as AWGN.

For multi-task federated learning, groups submit distinct updates, and over-the-air aggregation must contend with inter-group (and inter-cell) interference (Chen et al., 17 Jan 2025). In dual-polarized architectures, the channel incorporates polarization vectors and movable antenna arrays to further exploit spatial degrees of freedom for aggregation and interference management (Hu et al., 14 Jan 2026).

2. Levels of Cooperation and Interference Mitigation

AirComp frameworks support several cooperation levels among APs/BSs:

Level 3: Fully centralized—all APs forward raw pilot/data to the CPU, which globally estimates channels and designs receive combining vectors and transmit coefficients for all devices. This approach yields optimal aggregation error (MSE) but requires high fronthaul capacity.

Level 2: Centralized combining with local forwarding—CPU estimates channels and designs global combining vectors, but each AP performs local combining and forwards per-group estimates to the CPU. The CPU aggregates these to approximate the global function (Chen et al., 17 Jan 2025, Chen et al., 2024).

Level 1: Fully local—each AP estimates channels and forms local combiners; only local estimates sent to CPU. No transmit coefficient optimization beyond full-power, yielding suboptimal but fronthaul-efficient operation.

Interference mitigation requires tailored coordination:

• Simultaneous Signal-and-Interference Alignment (SIA): APs partition their spatial channels into equal-dimension signal and interference subspaces; the signal alignment enables AirComp while interference alignment nullifies inter-cell leakage (Lan et al., 2020).
• Power Control: Multi-cell power optimization (centralized or distributed via interference temperature caps) minimizes per-cell MSE on the Pareto frontier, balancing individual cell accuracy against overall network interference (Cao et al., 2020).
• Polarization and Antenna Positioning: Dual-polarized, movable antenna arrays enable joint spatial and polarization alignment, enhancing MSE performance and adapting to time-varying channel statistics (Hu et al., 14 Jan 2026).

3. Joint Optimization Algorithms

Aggregation error (MSE) minimization problems in these frameworks are inherently non-convex, often featuring bilinear or quadratic constraints. Block-coordinate alternating optimization is a standard approach, combining closed-form solutions for subproblems with iterative refinement.

Typical Alternating Optimization Steps:

a) Receive Combiner Update (v-step):

With transmit coefficients fixed, per-group receive combining vectors are updated via generalized Rayleigh quotient minimization:

$${v_{g,d}^{t} = \Big(\sum_{k=1}^K |b_k|^2 (\hat{h}_k \hat{h}_k^H + C_k) + \delta^2 I_{LN} \Big)^{-1} \sum_{j \in \mathcal{K}_g} \gamma_{jg} b_j \nu_j \hat{h}_j}$$

b) Transmit Coefficient Update (b-step):

Given receive combiners, optimal transmit scalars exploit KKT stationarity conditions and are given by:

$${b_{k,d}^{t} = \frac{\omega_g \gamma_{kg} \nu_k (\hat{h}_k^H v_g)}{ \sum_{p=1}^G \omega_p [ |v_p^H \hat{h}_k|^2 + v_p^H C_k v_p ] + \mu_k }}$$

c) Polarization and Position Update:

Dual-polarized and movable arrays employ SCA (successive convex approximation) and semidefinite relaxation for tractable updates of polarization/antenna position (Hu et al., 14 Jan 2026).

d) Distributed/Decentralized Schemes:

Interference temperature-based distributed power control allows each AP to iteratively adjust local transmit powers, exchanging minimal state information for constraint satisfaction and Pareto optimality (Cao et al., 2020).

e) Deep Graph Learning:

Recent frameworks reinterpret AO iterations as layers of a graph neural network (GNN), learning interference-aware update directions for joint transmitter/receiver optimization and fast adaptation to network changes (Tang et al., 16 May 2025).

4. Performance Analysis and Empirical Findings

Mean Squared Error (MSE): The central metric, reflecting aggregation fidelity and thus learning convergence, is rigorously quantified in all frameworks:

• Cell-free mMIMO with fully centralized processing provides up to 9 dB MSE reduction vs. cellular baselines (Chen et al., 2024).
• Transmit-coefficient optimization is critical for multi-task AirComp: Level 2/3 cell-free designs nearly match noiseless benchmarks; cellular variants often fail with non-uniform device placement (Chen et al., 17 Jan 2025).
• Dual-polarized movable antenna schemes lower MSE by 18–40% over single-polarized and fixed arrays, with gains increasing with antenna count and transmission power (Hu et al., 14 Jan 2026).

Learning Convergence: MSE performance directly maps to federated learning test accuracy and training loss. Pareto boundary and gap profiling methods efficiently balance trade-offs among multi-cell tasks, preventing divergence or plateauing of loss in weak cells (Wang et al., 2022, Zeng et al., 2023).

Trade-Offs: Fronthaul overhead, computational complexity, and macro-diversity gains have been tabulated for canonical levels of cooperation (Chen et al., 2024): | Level | AP→CPU Scalars | CPU→AP Scalars | MSE Reduction | |-------|----------------|---------------|--------------| | 3 | $\tau_c N L$ | $K$ | Maximized | | 2 | $< Level 3$ | – | Moderate | | 1 | $< Level 2$ | – | Lowest |

5. Advanced Features: Statistical CSI and Hardware Adaptation

Multi-cell AirComp frameworks increasingly address real-world constraints:

• Statistical Channel Optimization: In fast-fading or highly dynamic environments, antenna positions and other slow hardware settings are optimized for average performance rather than instantaneous CSI (Hu et al., 14 Jan 2026).
• STAR-RIS and IRS Assistance: Reconfigurable surfaces offer flexible beamforming and interference suppression, yielding substantial uplink/downlink MSE improvement and accelerated vertical FL convergence (Zeng et al., 2023).
• CSI-Free Non-Coherent Computation: FSK-based majority vote schemes remove CSI and synchronization requirements entirely, harnessing natural interference for scalable global aggregation in federated edge learning (Adeli et al., 2022).

6. Implications, Practical Guidelines, and Future Directions

Design Recommendations:

• Use centralized cooperation where fronthaul supports it; otherwise, exploit local combining and LSFD.
• Partition available spatial DoF into matched signal/interference spaces for SIA-based frameworks (Lan et al., 2020).
• Incorporate dynamic, adaptive frameworks (e.g., GNN unfolding) for scalable real-time operation (Tang et al., 16 May 2025).
• Consider hardware augmentation (dual-polarization, STAR-RIS) for further MSE mitigation.

Challenges:

• Accurate CSI acquisition is nontrivial; frameworks mitigate via statistical methods and robust optimization.
• Quantization and analog nonlinearities can impair AirComp aggregation; compensation techniques are required.

Observations:

• Cell-free and multi-cell cooperation sharply enhance both computation fidelity and distributed learning convergence, with gains amplified under channel diversity and dynamic device distributions.
• Distributed optimization methods (e.g., interference temperature updates) converge rapidly, requiring minimal backhaul (Cao et al., 2020).
• Deep learning approaches (GNN unfolding) generalize across user and cell counts, supporting highly dynamic and large-scale deployments (Tang et al., 16 May 2025).

Multi-cell cooperative AirComp frameworks represent a mature class of physical-layer computation schemes enabling robust, scalable, and low-latency wireless distributed learning in 6G and beyond. Ongoing research emphasizes further integration of hardware intelligence, non-coherent signaling, and multi-modal resource adaptation.