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 $m/n = \delta\in(1,\infty)$, and obtain a simple
expression for the overlap between the spectral estimator and the true signal
vector.

本研究旨在探究相位恢复系统中采用随机正交向量矩阵的谱方法启动的局部搜索算法。我们在渐进的情况下，获得了谱估计器和真实信号向量之间的最大重叠程度的简单表达式。