We propose a general framework for solving the group synchronization problem, where we focus on the setting of adversarial or uniform corruption and sufficiently small noise. Specifically, we apply a novel message passing procedure that uses cycle consistency information in order to estimate the corruption levels of group ratios and consequently solve the synchronization problem in our setting. We first explain why the group cycle consistency information is essential for effectively solving group synchronization problems. We then establish exact recovery and linear convergence guarantees for the proposed message passing procedure under a deterministic setting with adversarial corruption. These guarantees hold as long as the ratio of corrupted cycles per edge is bounded by a reasonable constant. We also establish the stability of the proposed procedure to sub-Gaussian noise. We further establish exact recovery with high probability under a common uniform corruption model.

我们提出了一个解决组同步问题的通用框架，其中我们专注于对抗性或统一损坏和足够小的噪声的设置。具体来说，我们应用了一种新的消息传递程序，该程序使用循环一致性信息来估计组比率的损坏程度，从而解决我们设置中的同步问题。我们首先解释为什么组周期一致性信息对于有效解决组同步问题至关重要。然后，我们在具有对抗性损坏的确定性设置下为提议的消息传递过程建立精确的恢复和线性收敛保证。只要每条边的损坏周期比率受合理常数的限制，这些保证就成立。我们还建立了所提出的程序对亚高斯噪声的稳定性。我们在通用的统一腐败模型下进一步建立了高概率的精确恢复。