VecComp: Vector Computing via MIMO Digital Over-the-Air Computation

TL;DR

VecComp generalizes ChannelComp to vector-function computation over fading MIMO MACs by using CSI-unaware receiver beamforming and random, isotropic transmit beamformers to suppress fading. It provides a non-asymptotic MSE bound showing the required number of receiver antennas scales as N_r = O(1/ε^2) for a target error ε, and introduces exact and inexact setups with SDP-based encoder design to achieve reliable vector computation. The framework supports practical digital modulations (PAM, QAM) and enables two concrete cases: affine transformations and convolutions, with demonstrated gains in robustness and scalability through extensive simulations. VecComp thus offers a scalable, hardware-friendly path to one-shot vector computations in distributed ML and IoT settings, leveraging MIMO diversity to mitigate fading while preserving compatibility with digital communications. The work highlights the practical impact of enabling high-dimensional, low-latency data processing directly in the wireless MAC layer for next-generation networks.

Abstract

Recently, the ChannelComp framework has proposed digital over-the-air computation by designing digital modulations that enable the computation of arbitrary functions. Unlike traditional analog over-the-air computation, which is restricted to nomographic functions, ChannelComp enables a broader range of computational tasks while maintaining compatibility with digital communication systems. This framework is intended for applications that favor local information processing over the mere acquisition of data. However, ChannelComp is currently designed for scalar function computation, while numerous data-centric applications necessitate vector-based computations, and it is susceptible to channel fading. In this work, we introduce a generalization of the ChannelComp framework, called VecComp, by integrating ChannelComp with multiple-antenna technology. This generalization not only enables vector function computation but also ensures scalability in the computational complexity, which increases only linearly with the vector dimension. As such, VecComp remains computationally efficient and robust against channel impairments, making it suitable for high-dimensional, data-centric applications. We establish a non-asymptotic upper bound on the mean squared error of VecComp, affirming its computation efficiency under fading channel conditions. Numerical experiments show the effectiveness of VecComp in improving the computation of vector functions and fading compensation over noisy and fading multiple-access channels.

Figures (8)

• Figure 1: MIMO Setup with $k$ transmitters, where there are $N_t$ transmitter antenna at nodes and $N_r$ receivers antenna at the CP. Node $k$ uses a beamforming matrix $\bm{V}_k \in \mathbb{C}^{N_t \times L}$ to transmit the signal $\bm{x}_k$ derived from the encoded source signal $\bm{s}_k$ via the encoder $\mathscr{E}_k(\cdot)$. The signal passes through the channel matrix $\bm{H}_k$ and is combined into the additive noise, $\bm{z}$, thus representing the aggregated received signal. The received signal is then processed by the beamforming matrix $\bm{U}$ to produce the final output $\bm{y}$.
• Figure 2: The overlaps of the reshaped constellation points of QPSK modulation do not allow us to compute the product function.
• Figure 3: NMSE performance of VecComp as a function of the number of receiver antennas, $N_r$, for varying computation dimensions $L$. The setup assumes $N_t = L$ and $\text{SNR} = 20, \text{dB}$, with simulation results averaged over $10^3$ Monte Carlo trials. The figure illustrates the improvement in computation robustness and fading resilience as $N_r$ increases from $10$ to $50$, highlighting the effect of antenna scaling on VecComp’s ability to accurately compute the sum function in a fading environment.
• Figure 4: NMSE performance of VecComp as a function of the number of transmitting nodes, $K$, for different receiver antenna configurations. The simulation is conducted with $N_t=4$, $L=4$, and $\text{SNR}=5$ dB, with results averaged over $10^4$ Monte Carlo trials.
• Figure 5: Performance comparison between VecComp, MIMO OAC zhu2018mimo, and wide-band MIMO in terms of NMSE error averaged over $N_s =100$, where input values are given by $x_k=\{0,1,\ldots,7\}$ and the desired functions are $f_1 = \prod_{k}x_k$, $f_2 = \sum_{k}x_k/K$, $f_3 = \max_{k}x_k$, and $f_4 = \sum_{k}x_k^2$.

Theorems & Definitions (27)

• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Remark 5
• Theorem 1
• proof
• Corollary 1
• Remark 6
• Proposition 1