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Group synchronization refers to estimating a collection of group elementsfrom the noisy pairwise measurements. Such a nonconvex problem has receivedmuch attention from numerous scientific fields including computer vision,robotics, and cryo-electron microscopy. In this paper, we focus on theorthogonal group synchronization problem with general additive noise modelsunder incomplete measurements, which is much more general than the commonlyconsidered setting of complete measurements. Characterizations of theorthogonal group synchronization problem are given from perspectives ofoptimality conditions as well as fixed points of the projected gradient ascentmethod which is also known as the generalized power method (GPM). It is wellworth noting that these results still hold even without generative models. Inthe meantime, we derive the local error bound property for the orthogonal groupsynchronization problem which is useful for the convergence rate analysis ofdifferent algorithms and can be of independent interest. Finally, we prove thelinear convergence result of the GPM to a global maximizer under a generaladditive noise model based on the established local error bound property. Ourtheoretical convergence result holds under several deterministic conditionswhich can cover certain cases with adversarial noise, and as an example wespecialize it to the setting of the Erd\"os-R\'enyi measurement graph andGaussian noise.

Orthogonal Group Synchronization with Incomplete Measurements: Error Bounds and Linear Convergence of the Generalized Power Method