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, as long as consistent parameter estimation is possible. To establish our result, we first have a novel choice of the population eigenvector, which enables 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：本文研究了谱方法在含加性高斯噪声及不完整数据下的相位同步问题中的性能表现。谱方法利用数据矩阵的主特征向量后接归一化步骤。我们证明该方法在平方$\ell_2$损失下达到了问题的极小极大下界，且匹配首项常数。这一结果表明，只要一致参数估计可行，谱方法就能达到与最大似然估计、广义幂法和半定规划等更复杂方法相同的性能。为确立该结果，我们首先提出一种新颖的总体特征向量选择方案，从而在无加性噪声条件下建立谱方法的精确恢复性。随后我们开发了一套针对主特征向量的新扰动分析工具包，证明其可通过一阶近似实现良好逼近且$\ell_2$误差极小。我们将分析进一步扩展至正交群同步场景，建立了谱方法的精确极小极大最优性。