Optimal Structure of Receive Beamforming for Over-the-Air Computation

TL;DR

This work tackles fast data aggregation via over-the-air computation (AirComp) in multi-antenna AP settings. It jointly optimizes transmit scalars, denoising, and receive beamforming to minimize mean-squared error, deriving closed-form solutions for the transmitter side and formulating a non-convex QCQP in the receive beamformer. By exploiting an optimal beamforming structure through successive convex approximation and Lagrangian duality, the authors reduce the problem dimension from the number of antennas to the number of devices and propose two efficient algorithms, SDR-Opt and SCA-Opt, that achieve comparable MSE to baseline methods with significantly lower computation. Simulations in a realistic three-dimensional setting validate the approach, showing faster wireless aggregation with negligible performance gap and favorable scaling for large antenna arrays.

Abstract

We investigate fast data aggregation via over-the-air computation (AirComp) over wireless networks. In this scenario, an access point (AP) with multiple antennas aims to recover the arithmetic mean of sensory data from multiple wireless devices. To minimize estimation distortion, we formulate a mean-squared-error (MSE) minimization problem that considers joint optimization of transmit scalars at wireless devices, denoising factor, and receive beamforming vector at the AP. We derive closed-form expressions for the transmit scalars and denoising factor, resulting in a non-convex quadratic constrained quadratic programming (QCQP) problem concerning the receive beamforming vector. To tackle the computational complexity of the beamforming design, particularly relevant in massive multiple-input multiple-output (MIMO) AirComp systems, we explore the optimal structure of receive beamforming using successive convex approximation (SCA) and Lagrange duality. By leveraging the proposed optimal beamforming structure, we develop two efficient algorithms based on SCA and semi-definite relaxation (SDR). These algorithms enable fast wireless aggregation with low computational complexity and yield almost identical mean square error (MSE) performance compared to baseline algorithms. Simulation results validate the effectiveness of our proposed methods.

Figures (4)

• Figure 1: MSE versus $N$ at AP when $K = 10$.
• Figure 2: Average computation time versus $N$ when $K = 10$.
• Figure 3: MSE versus $K$ when $N = 120$.
• Figure 4: Average computation time versus $K$ when $N = 120$.

Theorems & Definitions (2)

• proof : Proof
• proof