  Rotation group $\mathcal{SO}(d)$ synchronization is an important inverse
problem and has attracted intense attention from numerous application fields
such as graph realization, computer vision, and robotics. In this paper, we
focus on the least-squares estimator of rotation group synchronization with
general additive noise models, which is a nonconvex optimization problem with
manifold constraints. Unlike the phase/orthogonal group synchronization, there
are limited provable approaches for solving rotation group synchronization.
First, we derive improved estimation results of the least-squares/spectral
estimator, illustrating the tightness and validating the existing relaxation
methods of solving rotation group synchronization through the optimum of
relaxed orthogonal group version under near-optimal noise level for exact
recovery. Moreover, departing from the standard approach of utilizing the
geometry of the ambient Euclidean space, we adopt an intrinsic Riemannian
approach to study orthogonal/rotation group synchronization. Benefiting from a
quotient geometric view, we prove the positive definite condition of quotient
Riemannian Hessian around the optimum of orthogonal group synchronization
problem, and consequently the Riemannian local error bound property is
established to analyze the convergence rate properties of various Riemannian
algorithms. As a simple and feasible method, the sequential convergence
guarantee of the (quotient) Riemannian gradient method for solving
orthogonal/rotation group synchronization problem is studied, and we derive its
global linear convergence rate to the optimum with the spectral initialization.
All results are deterministic without any probabilistic model.
