Spectral Method for Phase Retrieval: An Expectation Propagation Perspective.

Phase retrieval refers to the problem of recovering a signal $ {x}_{\star }\in \mathbb {C}^{n}$ from its phaseless measurements $\text {y}_{\text {i}}=| {a}_{i}^{ \mathsf {H}} {x}_{\star }|$ , where $\{ {a}_{\text {i}}\}_{\text {i}=1}^{ {m}}$ are the measurement vectors. Spectral method is widely used for initialization in many phase retrieval algorithms. The quality of spectral initialization can have a major impact on the overall algorithm. In this paper, we focus on the model where $ {A}=[ {a}_{1},\ldots, {a}_{ {m}}]^{ \mathsf {H}}$ has orthonormal columns, and study the spectral initialization under the asymptotic setting $ {m}, {n}\to \infty $ with $ {m}/ {n}\to \delta \in (1,\infty)$. We use the expectation propagation framework to characterize the performance of spectral initialization for Haar distributed matrices. Our numerical results confirm that the predictions of the EP method are accurate for not-only Haar distributed matrices, but also for realistic Fourier based models (e.g. the coded 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 a meaningful estimate. We show that $\delta _{ \mathrm {weak}}=2$ for the column-orthonormal model. In contrast, previous results by Mondelli and Montanari 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 the i.i.d. Gaussian model, where the latter was recently introduced by Luo, Alghamdi and Lu. [ABSTRACT FROM AUTHOR]