In the presence of heterogeneous data, where randomly rotated objects fall into multiple underlying categories, it is challenging to simultaneously classify them into clusters and synchronize them based on pairwise relations. This gives rise to the joint problem of community detection and synchronization. We propose a series of semidefinite relaxations, and prove their exact recovery when extending the celebrated stochastic block model to this new setting where both rotations and cluster identities are to be determined. Numerical experiments demonstrate the efficacy of our proposed algorithms and confirm our theoretical result which indicates a sharp phase transition for exact recovery.

翻译：在有多种数据的情况下,随机旋转的物体可归入多个基本类别,同时将它们分为组群,并根据对称关系同步,这是很困难的。这引起了社区探测和同步的共同问题。 我们提出了一系列半无限期的放松措施,并在将已知的随机切换区块模型扩展至确定旋转和组群特性的新环境时证明它们准确恢复。 数字实验显示了我们拟议算法的功效,证实了我们的理论结果,表明精确恢复的急剧阶段过渡。