Exact Minimax Estimation for Phase Synchronization

We study the phase synchronization problem with measurements Y=z^*z*H+σ W∈mathbbCn× n, where z^* is an n-dimensional complex unit-modulus vector and W is a complex-valued Gaussian random matrix. It is assumed that each entry Yjk is observed with probability p. We prove that the minimax lower bound of estimating z^* under the squared ℓ2 loss is (1-o(1))fracσ22p. We also show that both generalized power method and maximum likelihood estimator achieve the error bound (1+o(1))fracσ22p. Thus, fracσ22p is the exact asymptotic minimax error of the problem. Our upper bound analysis involves a precise characterization of the statistical property of the power iteration. The lower bound is derived through an application of van Trees' inequality.