We study the phase synchronization problem with measurements $Y=z^*z^{*H}+\sigma W\in\mathbb{C}^{n\times n}$, 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 $\ell_2$ 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.
翻译:我们研究了以美元=zz ⁇ H ⁇ H ⁇ gmaW\gma_mathbb{C ⁇ n\timen}(美元=美元)测量的阶段同步问题, 美元=美元( 美元=美元) 复合单位模量矢量为美元, 美元=美元( 美元) 随机矩阵值为美元( 美元) 。 假设每个条目都以概率 $( jk}) 观察到美元。 我们证明, 在平方 $( ell_ 2$) 下估算 $( 美元) 美元( 0. 1 o (1))\ frac =gmax% 2 ⁇ 2p} 下, 最低限值为 $( 1+1)\ frac=gma=2 ⁇ 2p} 。 我们还表明, 通用功率方法和最大可能性的估量值均值都达到了错误( $(1+(1))\ frac\ gmasc=2 ⁇ 2p} 。 因此, $( $) 是这一问题的精确的微缩错误错误。 我们的大小分析涉及对它的统计属性的精确定性。