A Unified Approach to Synchronization Problems Over Subgroups of the Orthogonal Group

The problem of synchronization over a group $\mathcal{G}$ aims to estimate collection elements $G^*_1, \dots, G^*_n \in \mathcal{G}$ based on noisy observations subset all pairwise ratios the form $G^*_i {G^*_j}^{-1}$. Such has gained much attention recently and finds many applications across wide range scientific engineering areas. In this paper, we consider class problems in which is closed subgroup orthogonal group. This covers that arise practice. Our contribution fivefold. First, propose unified approach for solving problems, consists suitable initialization step an iterative refinement generalized power method, show it enjoys strong theoretical guarantee estimation error under certain assumptions group, measurement graph, noise, initialization. Second, formulate two geometric conditions are required by our they hold various practically relevant subgroups closely related error-bound geometry -- important notion optimization. Third, verify graph noise standard random matrix models. Fourth, classic metric entropy, develop analyze novel spectral-type estimator. Finally, via extensive numerical experiments proposed non-convex outperforms existing approaches terms computational speed, scalability, and/or error.