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Phase retrieval refers to algorithmic methods for recovering a signal fromits phaseless measurements. Local search algorithms that work directly on thenon-convex formulation of the problem have been very popular recently. Due tothe nonconvexity of the problem, the success of these local search algorithmsdepends heavily on their starting points. The most widely used initializationscheme is the spectral method, in which the leading eigenvector of adata-dependent matrix is used as a starting point. Recently, the performance ofthe spectral initialization was characterized accurately for measurementmatrices with independent and identically distributed entries. This paper aimsto obtain the same level of knowledge for isotropically randomcolumn-orthogonal matrices, which are substantially better models for practicalphase retrieval systems. Towards this goal, we consider the asymptotic settingin 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 simpleexpression for the overlap between the spectral estimator and the true signalvector.

Analysis of Spectral Methods for Phase Retrieval with Random Orthogonal Matrices