[Paper Review] Spectral Method for Phase Retrieval: an Expectation Propagation Perspective

Junjie Ma, Rishabh Dudeja+3 more|arXiv (Cornell University)|Mar 6, 2019

Advanced X-ray Imaging Techniques40 references4 citations

TL;DR

This paper introduces an expectation propagation (EP) framework to analyze spectral initialization in phase retrieval for orthonormal measurement matrices, revealing a phase transition threshold at δ=2 for meaningful recovery. It shows that the optimal spectral method design matches that of i.i.d. Gaussian models, with EP predictions accurately capturing performance across Haar and Fourier-based models.

ABSTRACT

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 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 $\mathbf{A}=[\mathbf{a}_1,\ldots,\mathbf{a}_m]^{\mathrm{H}}$ has orthonormal columns, and study the spectral initialization under the asymptotic setting $m,n o\infty$ with $m/n oδ\in(1,\infty)$. 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 $δ$ (denoted as $δ_{\mathrm{weak}}$) below which the spectral method cannot produce a meaningful estimate. We show that $δ_{\mathrm{weak}}=2$ for the column-orthonormal model. In contrast, previous results by Mondelli and Montanari show that $δ_{\mathrm{weak}}=1$ 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.

Motivation & Objective

• To characterize the performance of spectral initialization in phase retrieval under orthonormal measurement matrices with m,n→∞ and m/n→δ.
• To identify the critical threshold δ_weak below which spectral initialization fails to produce meaningful estimates.
• To validate the accuracy of expectation propagation (EP) predictions for both Haar-distributed and realistic Fourier-based measurement models.
• To determine the optimal spectral processing function T(y) that maximizes initialization quality under the orthonormal model.
• To compare the spectral method's performance threshold with prior results from i.i.d. Gaussian models, where δ_weak=1.

Proposed method

• Uses the expectation propagation (EP) framework to derive asymptotic performance predictions for spectral initialization in phase retrieval.
• Models the measurement matrix A as having orthonormal columns and assumes Haar-distributed entries in the asymptotic regime.
• Derives state evolution equations for the cosine similarity between the true signal and spectral estimate via EP approximation.
• Introduces a nonlinear processing function T(y) in the data matrix D = A^H diag{T(y_i)} A to improve initialization quality.
• Applies Chebyshev’s association inequality and monotonicity analysis to prove convergence and optimality of the EP-based performance bounds.
• Validates EP predictions numerically on both Haar-distributed and coded diffraction pattern (Fourier-based) models.

Experimental results

Research questions

• RQ1 What is the critical threshold δ_weak below which spectral initialization fails to produce a meaningful estimate for orthonormal measurement matrices?
• RQ2 How does the performance of spectral initialization depend on the choice of nonlinear processing function T(y) in the data matrix?
• RQ3 Can the expectation propagation (EP) framework accurately predict the performance of spectral initialization in non-Gaussian measurement models like coded diffraction patterns?
• RQ4 Does the optimal design of the spectral method under orthonormal matrices coincide with that for i.i.d. Gaussian matrices?
• RQ5 What is the relationship between the monotonicity of the processing function T(y) and the convergence of the EP-based state evolution?

Key findings

• A phase transition occurs at δ=2 for orthonormal measurement matrices, below which spectral initialization cannot produce a meaningful estimate, i.e., δ_weak=2.
• The optimal processing function T(y) for the orthonormal model matches that recently proposed by Luo, Alghamdi, and Lu for the i.i.d. Gaussian model.
• The EP framework accurately predicts spectral initialization performance not only for Haar-distributed matrices but also for realistic Fourier-based models such as the coded diffraction pattern.
• The cosine similarity between the true signal and spectral estimate is maximized when the processing function T(y) is increasing and satisfies specific monotonicity conditions derived via EP.
• Theoretical analysis confirms that ψ1(μ) > ψ2(μ) for all μ∈(0,1) when T(y)=T⋆(y)=1−1/(δy²), indicating improved performance under optimal design.
• Numerical results validate that the predicted threshold δ=2 is sharp and that EP-based predictions closely match empirical performance across diverse measurement models.