Exact Minimax Estimation for Phase Synchronization

study the phase synchronization problem with measurements Y = z* z*(H) + sigma W is an element of C-nxn, where z* is an n-dimensional complex unit-modulus vector and W is a complexvalued Gaussian random matrix. It is assumed that each entry Y-jk is observed with probability p. We prove that the minimax lower bound of estimating z* under the squared l(2) loss is (1-o(1)) sigma(2)/2p. We also show that both generalized power method and maximum likelihood estimator achieve the error bound (1+o(1)) sigma(2)/2p. Thus, sigma(2)/2p 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.