Exact Minimax Estimation for Phase Synchronization
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We study the phase synchronization problem with measurements Y=z^*z^*H+σ W∈ℂ^n× 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 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))σ^2/2p. We also show that both generalized power method and maximum likelihood estimator achieve the error bound (1+o(1))σ^2/2p. Thus, σ^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.
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