  Community detection and orthogonal group synchronization are both fundamental
problems with a variety of important applications in science and engineering.
In this work, we consider the joint problem of community detection and
orthogonal group synchronization which aims to recover the communities and
perform synchronization simultaneously. To this end, we propose a simple
algorithm that consists of a spectral decomposition step followed by a
blockwise column pivoted QR factorization (CPQR). The proposed algorithm is
efficient and scales linearly with the number of edges in the graph. We also
leverage the recently developed `leave-one-out' technique to establish a
near-optimal guarantee for exact recovery of the cluster memberships and stable
recovery of the orthogonal transforms. Numerical experiments demonstrate the
efficiency and efficacy of our algorithm and confirm our theoretical
characterization of it.
