Topological neural networks (TNNs) are information processing architectures that model representations from data lying over topological spaces (e.g., simplicial or cell complexes) and allow for decentralized implementation through localized communications over different neighborhoods. Existing TNN architectures have not yet been considered in realistic communication scenarios, where channel effects typically introduce disturbances such as fading and noise. This paper aims to propose a novel TNN design, operating on regular cell complexes, that performs over-the-air computation, incorporating the wireless communication model into its architecture. Specifically, during training and inference, the proposed method considers channel impairments such as fading and noise in the topological convolutional filtering operation, which takes place over different signal orders and neighborhoods. Numerical results illustrate the architecture's robustness to channel impairments during testing and the superior performance with respect to existing architectures, which are either communication-agnostic or graph-based.

翻译：拓扑神经网络（TNNs）是一种信息处理架构，它能够对位于拓扑空间（例如单纯复形或胞腔复形）上的数据进行表征建模，并允许通过不同邻域上的局部化通信实现去中心化部署。现有的TNN架构尚未在现实的通信场景中得到充分考虑，而信道效应通常会引入诸如衰落和噪声等干扰。本文旨在提出一种在规则胞腔复形上运行的新型TNN设计，该设计执行空中计算，并将无线通信模型纳入其架构之中。具体而言，在训练和推理过程中，所提方法在拓扑卷积滤波操作中考虑了衰落和噪声等信道损伤，该操作在不同信号阶次和邻域上进行。数值结果验证了该架构在测试阶段对信道损伤的鲁棒性，并展示了其相对于现有架构（无论是忽略通信影响的架构还是基于图的架构）的优越性能。