Achieving Local Consensus Over Compact Submanifolds

Decentralized optimization often relies on achieving consensus among disparate agents. This article addresses the consensus problem in decentralized networks, focusing on the challenges posed by a nonconvex compact submanifold constraint. We identify conditions on network topology that facilitate local linear convergence to global consensus, where the achieved linear rate matches that of the Euclidean setting. Central to our analysis are the convex-like properties, specifically proximal smoothness and the restricted secant inequality, which form the foundation of our theoretical framework. These results will be useful for the design and analysis of decentralized manifold optimization algorithms. Numerical experiments are conducted to validate our theoretical findings.