Over-the-Air (OTA) computation is the problem of computing functions of distributed data without transmitting the entirety of the data to a central point. By avoiding such costly transmissions, OTA computation schemes can achieve a better-than-linear (depending on the function, often logarithmic or even constant) scaling of the communication cost as the number of transmitters grows. Among the most common functions computed OTA are linear functions such as weighted sums. In this work, we propose and analyze an analog OTA computation scheme for a class of functions that contains linear functions as well as some nonlinear functions such as $p$-norms of vectors. We prove error bound guarantees that are valid for fast-fading channels and all distributions of fading and noise contained in the class of sub-Gaussian distributions. This class includes Gaussian distributions, but also many other practically relevant cases such as Class A Middleton noise and fading with dominant line-of-sight components. In addition, there can be correlations in the fading and noise so that the presented results also apply to, for example, block fading channels and channels with bursty interference. We do not rely on any stochastic characterization of the distributed arguments of the OTA computed function; in particular, there is no assumption that these arguments are drawn from identical or independent probability distributions. Our analysis is nonasymptotic and therefore provides error bounds that are valid for a finite number of channel uses. OTA computation has a huge potential for reducing communication cost in applications such as Machine Learning (ML)-based distributed anomaly detection in large wireless sensor networks. We illustrate this potential through extensive numerical simulations.
翻译:在Air (OTA) 的计算中, 在不将全部数据传输到一个中心点的情况下计算分布式数据的计算功能存在问题。 通过避免如此昂贵的传输, OTA 计算方法可以实现比线性( 往往对数甚至恒定) 更好的通信成本缩放( 取决于函数, 通常是对数, 甚至是恒定) 。 在计算 OTA 的最常见函数中, 包括加权总和等线性函数。 在这项工作中, 我们提议和分析一个模拟 OTA 计算方法, 用于包含线性功能的某类函数以及一些非线性函数, 如矢量的矢量的 $p$- 调调值。 通过我们证明对快速淡化的频道以及亚欧库西分布类中包含的迷幻和噪音的所有分布式分布式。 这个类别包括高斯分布式的分布式功能, 但也有许多其它实际相关的案例, 如AMiddidalton 噪音和光线性部分。 此外, 光线性和噪音中可能存在关联性, 因此, 所展示的结果也会降低, Oral- dreal rodreal roal roal 。 例如, 我们的流流流流流流流流流和电流流流流流流流流流流流流流 和流流流流的计算中不具有这种分析中的任何偏差 。