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Phase retrieval refers to the problem of recovering a signal$\mathbf{x}_{\star}\in\mathbb{C}^n$ from its phaseless measurements$y_i=|\mathbf{a}_i^{\mathrm{H}}\mathbf{x}_{\star}|$, where$\{\mathbf{a}_i\}_{i=1}^m$ are the measurement vectors. Many popular phaseretrieval algorithms are based on the following two-step procedure: (i)initialize the algorithm based on a spectral method, (ii) refine the initialestimate by a local search algorithm (e.g., gradient descent). The quality ofthe spectral initialization step can have a major impact on the performance ofthe overall algorithm. In this paper, we focus on the model where themeasurement matrix $\mathbf{A}=[\mathbf{a}_1,\ldots,\mathbf{a}_m]^{\mathrm{H}}$has orthonormal columns, and study the spectral initialization under theasymptotic setting $m,n\to\infty$ with $m/n\to\delta\in(1,\infty)$. We use theexpectation propagation framework to characterize the performance of spectralinitialization for Haar distributed matrices. Our numerical results confirmthat the predictions of the EP method are accurate for not-only Haardistributed matrices, but also for realistic Fourier based models (e.g. thecoded diffraction model). The main findings of this paper are the following:  (1) There exists a threshold on $\delta$ (denoted as$\delta_{\mathrm{weak}}$) below which the spectral method cannot produce ameaningful estimate. We show that $\delta_{\mathrm{weak}}=2$ for thecolumn-orthonormal model. In contrast, previous results by Mondelli andMontanari show that $\delta_{\mathrm{weak}}=1$ for the i.i.d. Gaussian model.  (2) The optimal design for the spectral method coincides with that for thei.i.d. Gaussian model, where the latter was recently introduced by Luo,Alghamdi and Lu.

Spectral Method for Phase Retrieval: an Expectation Propagation Perspective