Complex-Valued GNNs for Distributed Basis-Invariant Control of Planar Systems

arXiv cs.LG / 4/6/2026

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Key Points

Distributed GNN-based controllers often break down in GPS/compass-denied settings because they rely on nodes observing data in compatible local bases.
The paper introduces a complex-valued GNN parametrization that is globally invariant to arbitrary local frame choices by representing 2D geometric features and basis-to-basis transformations in the complex domain.
It uses complex-valued linear layers and phase-equivariant activation functions to ensure that, when expressed in a fixed global frame, the learned policies are strictly invariant to local frames.
Experiments on an imitation-learning flocking task indicate improved data efficiency, tracking performance, and generalization compared with a real-valued baseline.
The approach aims to make learned distributed control policies more robust to sensor and coordinate-frame inconsistencies common in challenging navigation environments.

Abstract

Graph neural networks (GNNs) are a well-regarded tool for learned control of networked dynamical systems due to their ability to be deployed in a distributed manner. However, current distributed GNN architectures assume that all nodes in the network collect geometric observations in compatible bases, which limits the usefulness of such controllers in GPS-denied and compass-denied environments. This paper presents a GNN parametrization that is globally invariant to choice of local basis. 2D geometric features and transformations between bases are expressed in the complex domain. Inside each GNN layer, complex-valued linear layers with phase-equivariant activation functions are used. When viewed from a fixed global frame, all policies learned by this architecture are strictly invariant to choice of local frames. This architecture is shown to increase the data efficiency, tracking performance, and generalization of learned control when compared to a real-valued baseline on an imitation learning flocking task.