论文标题
确切的最小值估计相同步
Exact Minimax Estimation for Phase Synchronization
论文作者
论文摘要
我们研究了与测量的相同步问题$ y = z^*z^{*h}+σw\ in \ mathbb {c}^{n \ times n} $，其中$ z^*$是$ n $ n $ dibemential-dimensional-dimensional complect unit-modulus vector，$ w $ $ w $ $ w $是复杂的配值速率均匀的随机速率。假定每个条目$ y_ {jk} $在概率$ p $中观察到。我们证明，在平方$ \ ell_2 $损失下估算$ z^*$的最小值是$（1-o（1））\ frac {σ^2} {2p} $。我们还表明，通用功率方法和最大似然估计器达到了错误绑定$（1+O（1））\ frac {σ^2} {2p} $。因此，$ \ frac {σ^2} {2p} $是问题的确切渐近最小值错误。我们的上限分析涉及对功率迭代的统计特性的精确表征。下限是通过货车不平等的应用得出的。
We study the phase synchronization problem with measurements $Y=z^*z^{*H}+σW\in\mathbb{C}^{n\times n}$, 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 $(1-o(1))\frac{σ^2}{2p}$. We also show that both generalized power method and maximum likelihood estimator achieve the error bound $(1+o(1))\frac{σ^2}{2p}$. Thus, $\frac{σ^2}{2p}$ 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.
