Spectral Method for Phase Retrieval: an Expectation Propagation Perspective

2019/03/06 by Junjie Ma, Rishabh Dudeja, Ma, Junjie +7 · 3 citations
Physics and Astronomy · Computer Science · Environmental Science · #Advanced X-ray Imaging Techniques #Optical measurement and interference techniques #Soil Geostatistics and Mapping

paper · pdf · doi:10.48550/arxiv.1903.02505

Abstract

Phase retrieval refers to the problem of recovering a signal x_⋆∈ℂⁿ from its phaseless measurements y_i=|a_i^Hx_⋆|, where \a_i\_i=1^m are the measurement vectors. Many popular phase retrieval algorithms are based on the following two-step procedure: (i) initialize the algorithm based on a spectral method, (ii) refine the initial estimate by a local search algorithm (e.g., gradient descent). The quality of the spectral initialization step can have a major impact on the performance of the overall algorithm. In this paper, we focus on the model where the measurement matrix A=[a₁,…,a_m]^H has orthonormal columns, and study the spectral initialization under the asymptotic setting m,n→∞ with m/n→δ∈(1,∞). 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 δ (denoted as δ_weak) below which the spectral method cannot produce a meaningful estimate. We show that δ_weak=2 for the column-orthonormal model. In contrast, previous results by Mondelli and Montanari show that δ_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.

Citations

• PhaseLift: Exact and Stable Signal Recovery from Magnitude Measurements via Convex Programming

Cited by

• Fast Escape, Slow Convergence: Learning Dynamics of Phase Retrieval under Power-Law Data
• Spectral Gradient Descent Mitigates Anisotropy-Driven Misalignment: A Case Study in Phase Retrieval
• Structured Random Models for Phase Retrieval with Optical Diffusers

Related

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• A Spectral Estimation Framework for Phase Retrieval via Bregman Divergence Minimization