A Spectral Method for Joint Community Detection and Orthogonal Group Synchronization
This addresses a combined graph and matrix recovery problem relevant to network analysis and signal processing applications, representing an incremental advancement in spectral methods.
The paper tackles the joint problem of community detection and orthogonal group synchronization by proposing a spectral algorithm with blockwise CPQR factorization, achieving near-optimal guarantees for exact community recovery and stable orthogonal transform recovery with linear scaling relative to graph edges.
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.