论文标题
在相试验中构建最佳光谱方法
Construction of optimal spectral methods in phase retrieval
论文作者
论文摘要
我们考虑了阶段检索问题，在该问题中，观察者希望从（可能是嘈杂的）观察到$ | | | | | | | \mathbfφ\ Mathbf {x}}^}^\ star | $ yMathbfφ$ n $ size $ size $ size $ size $ size $ size $ size $ size $ size $ size $ size $ size $ size $ size $ size $ size $ size $ size $ size $ size $ size $ size $ size $ size $ sizes我们考虑一个\ emph {高维}设置，其中$ n，m \ to \ infty $带有$ m/n = \ mathcal {o}（1）$，以及一大类（可能相关的）随机矩阵$ \mathbfφ$和观察渠道。光谱方法是一种强大的工具，可以以低计算成本以低计算成本来获得信号$ \ mathbf {x}^\ star $的近似观测值，然后可以用作后续算法的初始化。在本文中，我们扩展并统一了有关阶段检索问题的光谱方法的先前结果和方法。更准确地说，我们结合了通话算法的线性化和\ emph {bethe hessian}的分析，这是一种经典的统计物理工具。使用此工具箱，我们以自动化的方式显示了如何得出任意通道噪声和直率的矩阵$ \mathbfφ$的最佳光谱方法（即没有对任何超参数或预处理功能进行优化的情况）。
We consider the phase retrieval problem, in which the observer wishes to recover a $n$-dimensional real or complex signal $\mathbf{X}^\star$ from the (possibly noisy) observation of $|\mathbfΦ \mathbf{X}^\star|$, in which $\mathbfΦ$ is a matrix of size $m \times n$. We consider a \emph{high-dimensional} setting where $n,m \to \infty$ with $m/n = \mathcal{O}(1)$, and a large class of (possibly correlated) random matrices $\mathbfΦ$ and observation channels. Spectral methods are a powerful tool to obtain approximate observations of the signal $\mathbf{X}^\star$ which can be then used as initialization for a subsequent algorithm, at a low computational cost. In this paper, we extend and unify previous results and approaches on spectral methods for the phase retrieval problem. More precisely, we combine the linearization of message-passing algorithms and the analysis of the \emph{Bethe Hessian}, a classical tool of statistical physics. Using this toolbox, we show how to derive optimal spectral methods for arbitrary channel noise and right-unitarily invariant matrix $\mathbfΦ$, in an automated manner (i.e. with no optimization over any hyperparameter or preprocessing function).
