Phase retrieval refers to algorithmic methods for recovering a signal from its phaseless measurements. Local search algorithms that work directly on the non-convex formulation of the problem have been very popular recently. Due to the nonconvexity of the problem, the success of these local search algorithms depends heavily on their starting points. The most widely used initialization scheme is the spectral method, in which the leading eigenvector of a data-dependent matrix is used as a starting point. Recently, the performance of the spectral initialization was characterized accurately for measurement matrices with independent and identically distributed entries. This paper aims to obtain the same level of knowledge for isotropically random column-orthogonal matrices, which are substantially better models for practical phase retrieval systems. Towards this goal, we consider the asymptotic setting in which the number of measurements $m$, and the dimension of the signal, $n$, diverge to infinity with $m/n = δ\in(1,\infty)$, and obtain a simple expression for the overlap between the spectral estimator and the true signal vector.
On the Construction of Quasi-Binary and Quasi-Orthogonal Matrices over Finite Fields
正交和准正交矩阵在数字图像处理、数字和无线通信、密码学以及计算机科学和编码理论的许多其他领域中具有长期的使用历史。使用正交矩阵的实际好处来自于这样的事实，即通过简单地使用正交矩阵的转置来避免逆矩阵的计算。本文引入了有限域上的一类新的矩阵族，我们称之为\emph{Quasi-Binary and Quasi-Orthogonal Matrices}。我们称矩阵为准二进制矩阵，因为矩阵只有两个元素$a, b \in \mathbb{F}_q$，但这些元素不是$0$和$1$。此外，我们称它们为准正交的原因是因为它们的逆不是通过简单的转置获得的，而是需要额外的操作：用另外两个值$c$和$d$替换$a$和$b$。本文给出了任意有限域，特别是特征为2的有限域的值$a, b, c$和$d$之间的一个简单关系。我们的构造是基于循环拉丁矩形的关联矩阵，所提出的算法的效率来自于避免矩阵矩阵或矩阵向量乘法。
