标题： 在紧致子流形上达成共识 显示英文标题

标题： Achieving Consensus over Compact Submanifolds

摘要： 我们考虑在去中心化网络中的共识问题，重点关注一个作为非凸约束集的紧子流形。 通过利用紧子流形的近似光滑性，该性质包括局部单例性质和流形上投影算子的局部利普希茨连续性，并建立投影算子与一般再映射之间的联系，我们证明了如果网络具有足够的连通性，那么具有单位步长的黎曼梯度下降方法具有局部线性收敛性。 此外，基于紧子流形的几何特性，我们证明了一个类似凸性的正则性条件，称为受限割线不等式，在非凸共识问题解集的一个显式表征邻域内始终成立。 通过利用这个受限割线不等式，并对去中心化网络施加较弱的连通性要求，我们对黎曼梯度下降的线性收敛性进行了全面分析，同时考虑了适当的初始化和步长。 此外，如果网络连接良好，我们证明了由近似光滑性赋予的局部利普希茨连续性是受限割线不等式的充分条件，从而有助于局部误差界。 我们认为，我们建立的结果将在更一般的近似光滑集合上的共识问题中找到更多应用。 进行了数值实验以验证我们的理论结果。 显示英文摘要

摘要： 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.