Spectral Method for Phase Retrieval: An Expectation Propagation Perspective

Abstract

Phase retrieval refers to the problem of recovering a signal <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$ {x}_{\star }\in \mathbb {C}^{n}$ </tex-math></inline-formula> from its phaseless measurements <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\text {y}_{\text {i}}=| {a}_{i}^{ \mathsf {H}} {x}_{\star }|$ </tex-math></inline-formula> , where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\{ {a}_{\text {i}}\}_{\text {i}=1}^{ {m}}$ </tex-math></inline-formula> 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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$ {A}=[ {a}_{1},\ldots, {a}_{ {m}}]^{ \mathsf {H}}$ </tex-math></inline-formula> has orthonormal columns, and study the spectral initialization under the asymptotic setting <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$ {m}, {n}\to \infty $ </tex-math></inline-formula> with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$ {m}/ {n}\to \delta \in (1,\infty)$ </tex-math></inline-formula> . 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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\delta $ </tex-math></inline-formula> (denoted as <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\delta _{ \mathrm {weak}}$ </tex-math></inline-formula> ) below which the spectral method cannot produce a meaningful estimate. We show that <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\delta _{ \mathrm {weak}}=2$ </tex-math></inline-formula> for the column-orthonormal model. In contrast, previous results by Mondelli and Montanari show that <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\delta _{ \mathrm {weak}}=1$ </tex-math></inline-formula> 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.