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：在异构数据中，随机旋转的物体可能属于多个潜在类别，如何同时将其分类为不同簇并基于成对关系进行同步，是一项具有挑战性的任务。这引出了社区检测与同步的联合问题。我们提出了一系列半定松弛方法，并在将著名的随机块模型扩展至同时需确定旋转与簇标识的新场景时，证明了其精确恢复能力。数值实验验证了我们所提算法的有效性，并证实了理论结果所揭示的精确恢复的尖锐相变现象。