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We consider the consensus problem in a decentralized network, focusing on a compact submanifold that acts as a nonconvex constraint set. By leveraging the proximal smoothness of the compact submanifold, which encompasses the local singleton property and the local Lipschitz continuity of the projection operator on the manifold, and establishing the connection between the projection operator and general retraction, we show that the Riemannian gradient descent with a unit step size has locally linear convergence if the network has a satisfactory level of connectivity. Moreover, based on the geometry of the compact submanifold, we prove that a convexity-like regularity condition, referred to as the restricted secant inequality, always holds in an explicitly characterized neighborhood around the solution set of the nonconvex consensus problem. By leveraging this restricted secant inequality and imposing a weaker connectivity requirement on the decentralized network, we present a comprehensive analysis of the linear convergence of the Riemannian gradient descent, taking into consideration appropriate initialization and step size. Furthermore, if the network is well connected, we demonstrate that the local Lipschitz continuity endowed by proximal smoothness is a sufficient condition for the restricted secant inequality, thus contributing to the local error bound. We believe that our established results will find more application in the consensus problems over a more general proximally smooth set. Numerical experiments are conducted to validate our theoretical findings., Comment: 25 pages

With a computationally efficient approximation of the second-order information, natural gradient methods have been successful in solving large-scale structured optimization problems. We study the natural gradient methods for the large-scale decentralized optimization problems on Riemannian manifolds, where the local objective function defined by the local dataset is of a log-probability type. By utilizing the structure of the Riemannian Fisher information matrix (RFIM), we present an efficient decentralized Riemannian natural gradient descent (DRNGD) method. To overcome the communication issue of the high-dimension RFIM, we consider a class of structured problems for which the RFIM can be approximated by a Kronecker product of two low-dimension matrices. By performing the communications over the Kronecker factors, a high-quality approximation of the RFIM can be obtained in a low cost. We prove that DRNGD converges to a stationary point with the best-known rate of $\mathcal{O}(1/K)$. Numerical experiments demonstrate the efficiency of our proposed method compared with the state-of-the-art ones. To the best of our knowledge, this is the first Riemannian second-order method for solving decentralized manifold optimization problems., Comment: 17 pages

This paper proposes two innovative vector transport operators, leveraging the Cayley transform, for the generalized Stiefel manifold embedded with a non-standard metric. Specifically, it introduces the differentiated retraction and an approximation of the Cayley transform to the differentiated matrix exponential. These vector transports are demonstrated to satisfy the Ring-Wirth non-expansive condition under non-standard metrics, and one of them is also isometric. Building upon the novel vector transport operators, we extend the modified Polak-Ribi$\grave{e}$re-Polyak (PRP) conjugate gradient method to the generalized Stiefel manifold. Under a non-monotone line search condition, we prove our algorithm globally converges to a stationary point. The efficiency of the proposed vector transport operators is empirically validated through numerical experiments involving generalized eigenvalue problems and canonical correlation analysis.