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We study the performance of the spectral method for the phase synchronizationproblem with additive Gaussian noises and incomplete data. The spectral methodutilizes the leading eigenvector of the data matrix followed by a normalizationstep. We prove that it achieves the minimax lower bound of the problem with amatching leading constant under a squared $\ell_2$ loss. This shows that thespectral method has the same performance as more sophisticated proceduresincluding maximum likelihood estimation, generalized power method, andsemidefinite programming, as long as consistent parameter estimation ispossible. To establish our result, we first have a novel choice of thepopulation eigenvector, which enables us to establish the exact recovery of thespectral method when there is no additive noise. We then develop a newperturbation analysis toolkit for the leading eigenvector and show it can bewell-approximated by its first-order approximation with a small $\ell_2$ error.We further extend our analysis to establish the exact minimax optimality of thespectral method for the orthogonal group synchronization.

Exact Minimax Optimality of Spectral Methods in Phase Synchronization and Orthogonal Group Synchronization