Function Computation Over Multiple Access Channels via Hierarchical Constellations

2026-01-17

AI Summary

This work addresses the problem of efficiently and reliably computing functions of multiple transmitters’ signals over a Gaussian multiple-access channel by proposing an over-the-air computation (OAC) coding and modulation framework that integrates hierarchical constellations with a variable-length block coding masking mechanism. For the first time, this approach jointly leverages hierarchical signaling and masking to enable synchronous computation of multiple function outputs in a single channel use while effectively suppressing noise-induced cross-layer error propagation. Theoretical analysis shows that, with K independent sources, the gap between the achievable computation rate and the optimum scales as O(log₂(1/ε)/K), which vanishes as K grows. With the masking mechanism, this gap is further improved to O(log₂ln(1/ε)), independent of the source distribution, thereby achieving a double-logarithmic error performance gain.

Technology Category

• Machine Learning: Hardware-aware ML
• Search and Optimization: Mixed Discrete/Continuous Search
• Multiagent Systems: Agent Communication

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs

Abstract

We study function computation over a Gaussian multiple-access channel (MAC), where multiple transmitters aim at computing a function of their values at a common receiver. To this end, we propose a novel coded-modulation framework for over-the-air computation (OAC) based on hierarchical constellation design, which supports reliable computation of multiple function outputs using a single channel use. Moreover, we characterize the achievable computation rate and show that the proposed hierarchical constellations can compute R output functions with decoding error probability epsilon while the gap to the optimal computation rate scales as O(\log_2(1/\epsilon)/K) for independent source symbols, where K denotes the number of transmitters. Consequently, this gap vanishes as the network size grows, and the optimal rate is asymptotically attained. Furthermore, we introduce a shielding mechanism based on variable-length block coding that mitigates noise-induced error propagation across constellation levels while preserving the superposition structure of the MAC. We show that the shielding technique improves reliability, yielding a gap that scales optimally as O(\log_2\ln{(1/\epsilon)}), regardless of the source distribution. Together, these results identify the regimes in which uncoded or lightly coded OAC is information-theoretically optimal, providing a unified framework for low-latency, channel-agnostic function computation.

Problem

• Function Computation
• Multiple Access Channel
• Over-the-Air Computation
• Hierarchical Constellations
• Computation Rate

Innovation

• Hierarchical Constellations
• Over-the-Air Computation
• Multiple-Access Channel
• Function Computation
• Shielding Mechanism