Universal Function Approximation Through Over-the-Air Computing: A Deep Learning Approach

Over-the-air (OTA) computing has emerged as a promising technique that utilizes the superposition property of the wireless multiple access channel (MAC) as a means for computation. In this work, we propose a deep learning-based mechanism that approximates the pre- and post-processing functions of OTA computing, with the ultimate goal of approximating any desired target function. Specifically, we adopt a centralized training-decentralized execution approach that allows independent execution of deep neural networks (DNNs) on both devices and server to interpret the pre- and post-processing functions. The analysis is extended to the case of representing the pre- and post-processing functions to a higher dimensional space, further facilitating the reconstruction of the target function. To evaluate the effectiveness of the proposed method, we introduce a benchmark that serves as a lower bound on the computational distortion, i.e., the average mean square error (MSE) between the target function and the OTA computing estimation, which is described by closed-form solutions. It is noteworthy that the considered benchmark can serve as a reference point for any OTA computing-based application with any target function. Furthermore, the performance of the proposed decentralized DNN over-the-air computing execution (DOTACE) is evaluated through simulations, demonstrating its potential.