  Group synchronization aims to recover the group elements from their noisy
pairwise measurements. It has found many applications in community detection,
clock synchronization, and joint alignment problem. This paper focuses on the
orthogonal group synchronization which is often used in cryo-EM and computer
vision. However, it is generally NP-hard to retrieve the group elements by
finding the least squares estimator. In this work, we first study the
semidefinite programming (SDP) relaxation of the orthogonal group
synchronization and its tightness, i.e., the SDP estimator is exactly equal to
the least squares estimator. Moreover, we investigate the performance of the
Burer-Monteiro factorization in solving the SDP relaxation by analyzing its
corresponding optimization landscape. We provide deterministic sufficient
conditions which guarantee: (i) the tightness of SDP relaxation; (ii)
optimization landscape arising from the Burer-Monteiro approach is benign,
i.e., the global optimum is exactly the least squares estimator and no other
spurious local optima exist. Our result provides a solid theoretical
justification of why the Burer-Monteiro approach is remarkably efficient and
effective in solving the large-scale SDPs arising from orthogonal group
synchronization. We perform numerical experiments to complement our theoretical
analysis, which gives insights into future research directions.
