In Group Synchronization, one attempts to recover a collection of unknown group elements from noisy measurements of their pairwise differences. Several important problems in vision and data analysis reduce to group synchronization over various compact groups. Spectral Group Synchronization is a commonly used, robust algorithm for solving group synchronization problems, which relies on diagonalization of a block matrix whose blocks are matrix representations of the measured pairwise differences. Assuming uniformly distributed measurement errors, we present a rigorous analysis of the accuracy and noise sensitivity of spectral group synchronization algorithms over any compact group. We identify a noise threshold above which the performance of the algorithm completely breaks down. Below the threshold, we calculate an asymptotically exact formula for the accuracy, up to the rounding error, as a function of the noise level. We also provide a consistent risk estimate, allowing practitioners to estimate the method's accuracy from available measurements.

在“组同步”中，一种尝试是通过对成对的差异进行嘈杂的测量来恢复一组未知的组元素。视觉和数据分析中的几个重要问题简化为各个紧凑组之间的组同步。频谱组同步是解决组同步问题的常用，鲁棒算法，它依赖于块矩阵的对角化，该块矩阵的块是所测量的成对差的矩阵表示。假设测量误差均匀分布，我们对任何紧凑群上的频谱群同步算法的准确性和噪声敏感性进行了严格的分析。我们确定了一个噪声阈值，超过该阈值，该算法的性能将完全崩溃。低于阈值 我们根据噪声水平计算出精度（直到舍入误差）的渐近精确公式。我们还提供一致的风险估算，使从业人员可以从可用的测量方法中估算方法的准确性。