Exact Minimax Estimation for Phase Synchronization
Exact Minimax Estimation for Phase Synchronization
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DOI：
10.1109/tit.2021.3112712
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发表时间：
2020-10
影响因子：
2.5
通讯作者：
Chao Gao;A. Zhang
中科院分区：
文献类型：
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作者：
Chao Gao;A. Zhang
我们研究了测量值${Y}= {z}^{\ast} {z}^{\ast{\mathrm {H}}}+\sigma {W}\in \mathbb {C}^{n}\times {n}$的相位同步问题，其中${z}^{\ast}$是一个${n}$维复单位模向量，${W}$是一个复值高斯随机矩阵。假设每个条目${Y}_{jk}$以${p}$的概率被观察到。证明了在$\ell _{2}$损失平方下估计${z}^{\ast}$的极大极小下界为$(1- {o}(1))\frac {\sigma ^{2}}{2p}$。我们还证明了广义幂方法和极大似然估计都能达到误差界$(1+ {o}(1))\frac {\sigma ^{2}}{2p}$。因此，$\frac {\sigma ^{2}}{2p}$是问题的精确渐近极大极小误差。我们的上界分析涉及对幂次迭代的统计性质的精确描述。下界是通过范树不等式的应用推导出来的。
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.