Topological Neural Networks over the Air

TL;DR

Topological neural networks extend learning to data defined on higher-order topological spaces, but practical deployments face wireless channel impairments. The paper introduces AirTNN, which embeds the channel model into topological filters over regular cell complexes via over-the-air computation (AirTF) that aggregates information across two neighborhoods (lower and upper). By training with channel realizations, AirTNN learns weights that are robust to fading and noise, and its layer equation takes the form Y = act( sum_{p=0}^{P} X^{(u,p)} W_p^{(u)} + sum_{p=0}^{P} X^{(d,p)} W_p^{(d)} ). Numerical results on a source localization task show that AirTNN outperforms AirGNN and baseline TNN/GNN architectures under realistic wireless conditions, validating the approach for distributed processing of topological data in communication networks.

Abstract

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.

Figures (3)

• Figure 1: Upper (a) and lower (c) communication networks induced by the cell complex built on the input network (b).
• Figure 2: Accuracy versus $\delta$, with a fixed SNR of 20 dB.
• Figure 3: Accuracy versus SNR, with a fixed $\delta$ equal to 1.