Group synchronization asks to recover group elements from their pairwise measurements. It has found numerous applications across various scientific disciplines. In this work, we focus on orthogonal and permutation group synchronization which are widely used in computer vision such as object matching and structure from motion. Among many available approaches, the spectral methods have enjoyed great popularity due to their efficiency and convenience. We will study the performance guarantees of the spectral methods in solving these two synchronization problems by investigating how well the computed eigenvectors approximate each group element individually. We establish our theory by applying the recent popular leave-one-out technique and derive a block-wise performance bound for the recovery of each group element via eigenvectors. In particular, for orthogonal group synchronization, we obtain a near-optimal performance bound for the group recovery in presence of additive Gaussian noise. For permutation group synchronization under random corruption, we show that the widely-used two-step procedure (spectral method plus rounding) can recover all the group elements exactly if the SNR (signal-to-noise ratio) is close to the information theoretical limit. Our numerical experiments confirm our theory and indicate a sharp phase transition for the exact group recovery.

组同步要求从成对测量中恢复组元素。它已在各个科学学科中找到了许多应用。在这项工作中，我们专注于正交和置换组同步，它们广泛用于计算机视觉，例如对象匹配和运动结构。在许多可用的方法中，光谱方法由于其效率和便利性而广受欢迎。我们将通过研究计算的特征向量单独逼近每个组元素的程度来研究谱方法在解决这两个同步问题中的性能保证。我们通过应用最近流行的留一法来建立我们的理论，并推导出一个分块通过特征向量恢复每个组元素的性能界限。特别是，对于正交组同步，我们在存在加性高斯噪声的情况下获得了组恢复的近乎最佳性能界限。对于随机损坏下的置换群同步，我们证明了如果 SNR（信噪比）接近信息理论极限，广泛使用的两步过程（频谱法加舍入）可以准确地恢复所有群元素. 我们的数值实验证实了我们的理论，并表明了精确的群恢复的急剧相变。