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We consider the problem of decentralized nonconvex optimization over acompact submanifold, where each local agent's objective function defined by thelocal dataset is smooth. Leveraging the powerful tool of proximal smoothness,we establish local linear convergence of the projected gradient descent methodwith a unit step size for solving the consensus problem over the nonconvexcompact submanifold. This serves as the basis for designing and analyzingdecentralized algorithms on manifolds. Subsequently, we propose twodecentralized methods: the decentralized projected Riemannian gradient descent(DPRGD) and the decentralized projected Riemannian gradient tracking (DPRGT).We establish their convergence rates of $\mathcal{O}(1/\sqrt{K})$ and$\mathcal{O}(1/K)$, respectively, to reach a stationary point. To the best ofour knowledge, DPRGT is the first decentralized algorithm to achieve exactconvergence for solving decentralized optimization over a compact submanifold.Beyond the linear convergence results on the consensus, two key tools developedin the proof are the Lipschitz-type inequality of the projection operator andthe Riemannian quadratic upper bound for smooth functions on the compactsubmanifold, which could be of independent interest. Finally, we demonstratethe effectiveness of our proposed methods compared to state-of-the-art onesthrough numerical experiments on eigenvalue problems and low-rank matrixcompletion.

Decentralized projected Riemannian gradient method for smooth optimization on compact submanifolds