Exact Minimax Estimation for Phase Synchronization

We study the phase synchronization problem with measurements {Y}= {z}^{\ast} {z}^{\ast{\mathrm {H}}}+\sigma {W}\in \mathbb {C}^{n}\times {n} , where {z}^{\ast} is an {n} -dimensional complex unit-modulus vector and {W} is a complex-valued 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}^{\ast} under the squared \ell _{2} loss is (1- {o}(1))\frac {\sigma ^{2}}{2p} . We also show that both generalized power method and maximum likelihood estimator achieve the error bound (1+ {o}(1))\frac {\sigma ^{2}}{2p} . Thus, \frac {\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.