On the Local Linear Rate of Consensus on the Stiefel Manifold

Coordinated group behavior arising from purely local interactions has been successfully modeled with distributed consensus-seeking dynamics, where the local behavior is aimed at minimizing the disagreement with neighboring peers. However, it has been recently shown that when constrained by a manifold geometry, distributed consensus-seeking dynamics may ultimately fail to converge to a global consensus state. In this article, we study discrete-time consensus-seeking dynamics on the Stiefel manifold and identify conditions on the network topology to ensure convergence to a global consensus state. We further prove a (local) linear convergence rate to the consensus state that is on par with the well-known rate in the Euclidean space. These results have implications for consensus applications constrained by manifold geometry, such as synchronization and collective motion, and they can be used for convergence analysis of decentralized Riemannian optimization on the Stiefel Manifold.