We study the performance of the spectral method for the phase synchronization problem with additive Gaussian noises and incomplete data. The spectral method utilizes the leading eigenvector of the data matrix followed by a normalization step. We prove that it achieves the minimax lower bound of the problem with a matching leading constant under a squared $\ell_2$ loss. This shows that the spectral method has the same performance as more sophisticated procedures including maximum likelihood estimation, generalized power method, and semidefinite programming, when consistent parameter estimation is possible. To establish our result, we first have a novel choice of the population eigenvector, which enable us to establish the exact recovery of the spectral method when there is no additive noise. We then develop a new perturbation analysis toolkit for the leading eigenvector and show it can be well-approximated by its first-order approximation with a small $\ell_2$ error. We further extend our analysis to establish the exact minimax optimality of the spectral method for the orthogonal group synchronization.

翻译：我们用添加高斯噪音和不完全的数据来研究相片同步问题的光谱方法的性能。光谱方法使用数据矩阵的先导导体,然后是一个正常化步骤。我们证明,它达到了问题的最低偏角,在折方美元/ ell_ 2美元损失下有一个匹配的前导常数。这显示,光谱方法的性能与更复杂的程序相同,包括最大可能性估计、通用功率法和在有可能进行一致的参数估计时的半确定性编程。为了确定我们的结果,我们首先对人口源进行了新的选择,从而能够在没有添加噪音的情况下确定光谱方法的确切恢复情况。我们随后为主要的导体开发了一个新的扰动分析工具,并表明它能够与第一个波段的近近近点和一个小的 $/ ell_ 2美元的误差十分接近。我们进一步扩展了我们的分析,以确定光谱方法对于整组同步的精确微速率优化性。