EML-AirComp: Layered Over-the-Air Computation from a Single Nomographic Gate

2026-07-17

AI Summary

This work addresses the high communication overhead in computing multivariate functions over distributed wireless systems by proposing an exp-minus-log (EML) operational framework based on a unified two-input over-the-air computation (AirComp) gate. By constructing reusable EML tree structures, the framework enables multi-layer function evaluation over a compact domain without requiring node-specific nonlinear gates, while preserving the positive definiteness of intermediate logarithmic parameters and ensuring controllable error propagation. Under peak power constraints in AWGN and coherent flat-fading channels, deterministic error dynamics are analyzed through a quantization interface. Theoretical analysis yields bounds on the number of gate invocations, dependency depth, latency, and high-probability AWGN error margins. End-to-end positive definiteness and error conditions are validated via a four-node, two-hop example.

Abstract

Over-the-air computation (AirComp) exploits multiple-access superposition to compute functions of distributed data without separately decoding all terminal messages. We study a reusable two-input AirComp gate for the exp-minus-log (EML) operation $\eml(u,v)=\exp(u)-\log(v)$, $v>0$. Thus, all internal nodes of a prescribed real-admissible EML tree reuse one gate type, avoiding node-specific nonlinear gate designs. Given an explicit EML tree whose intermediate logarithm arguments remain positive on a given compact domain, we derive additive white Gaussian noise (AWGN) and coherent flat fading implementations under peak-power constraints. We then characterize the number of gate evaluations, the dependency depth, evaluation latency, node-wise feasibility, deterministic error propagation, positivity preservation, and a high-probability AWGN error bound for the complete tree. A four-terminal two-hop example gives explicit positivity and end-to-end error conditions, and a digital interface propagates quantization and gate errors across the tree.

Problem

• Over-the-air computation
• exp-minus-log
• nomographic function
• error propagation
• positivity preservation

Innovation

• Over-the-air computation
• EML gate
• Nomographic function
• Error propagation
• Positivity preservation