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.

翻译：我们考虑去中心化网络中的一致性问题，重点关注一个作为非凸约束集的紧致子流形。通过利用紧致子流形的近端光滑性（该性质包含了局部单点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回缩之间的联系，我们证明：当网络具有令人满意的连通水平时，采用单位步长的黎曼梯度下降算法具有局部线性收敛性。此外，基于紧致子流形的几何特性，我们证明在非凸一致性问题的解集周围一个明确刻画的邻域内，一种类似于凸性的正则性条件（称为受限割线不等式）总是成立。通过利用该受限割线不等式，并在去中心化网络上施加较弱的连通性要求，我们在考虑适当初始化和步长的条件下，对黎曼梯度下降的线性收敛性进行了全面分析。更进一步，若网络连通性良好，我们证明由近端光滑性赋予的局部利普希茨连续性是受限割线不等式成立的充分条件，从而有助于局部误差界的建立。我们相信所建立的结果将在更一般的近端光滑集上的一致性问题中得到更多应用。通过数值实验验证了我们的理论发现。