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