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 notation="LaTeX">$\text {y}_{\text {i}}=| {a}_{i}^{ \mathsf {H}} }|$ , where notation="LaTeX">$\{ {a}_{\text {i}}\}_{\text {i}=1}^{ {m}}$ are measurement vectors. Spectral method is widely used for initialization in many phase algorithms. The quality spectral can have major impact on overall algorithm. In this paper, we focus model {A}=[ {a}_{1},\ldots, {a}_{ {m}}]^{ {H}}$ has orthonormal columns, and study under asymptotic setting {m}, {n}\to \infty $ with {m}/ \delta \in (1,\infty)$ . We use expectation propagation framework characterize performance Haar distributed matrices. Our numerical results confirm that predictions EP accurate not-only matrices, but also realistic Fourier based models (e.g. coded diffraction model). main findings paper following: 1) There exists threshold notation="LaTeX">$\delta (denoted as _{ \mathrm {weak}}$ ) below which cannot produce meaningful estimate. show {weak}}=2$ column-orthonormal model. contrast, previous by Mondelli Montanari {weak}}=1$ i.i.d. Gaussian 2) optimal design coincides model, latter was recently introduced Luo, Alghamdi Lu.