2022/09/12 by Anderson Ye Zhang, Zhang, Anderson Ye · 1 citation
Computer Science · Materials Science · #Blind Source Separation Techniques #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT) #Nonlinear Dynamics and Pattern Formation #Phase-change materials and chalcogenides #Spectral Theory (math.SP) #Statistics Theory (math.ST)
paper · pdf · doi:10.48550/arxiv.2209.04962
openalex publication_date 2022/09/12 · openalex created_date 2022/09/14 · openalex updated_date 2026/07/28
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 ℓ2 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 ℓ2 error. We further extend our analysis to establish the exact minimax optimality of the spectral method for the orthogonal group synchronization.