A Spectral Method for Joint Community Detection and Orthogonal Group Synchronization

TL;DR

This paper introduces an efficient spectral algorithm for simultaneously solving community detection and orthogonal group synchronization, providing near-optimal recovery guarantees and demonstrating strong empirical performance.

Contribution

It presents a novel spectral method combining decomposition and QR factorization for joint community detection and synchronization, with theoretical guarantees and scalable implementation.

Findings

Algorithm achieves near-optimal exact recovery.

Method scales linearly with graph size.

Numerical results confirm theoretical guarantees.

Abstract

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.