Phase retrieval refers to the problem of recovering a signal x⋆∈Cn {x}_{\star }\in \mathbb {C}^{n} from its phaseless measurements yi=|aHix⋆|\text {y}_{\text {i}}=| {a}_{i}^{ \mathsf {H}} {x}_{\star }| , where {ai}mi=1\{ {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=[a1,…,am]H {A}=[ {a}_{1},\ldots, {a}_{ {m}}]^{ \mathsf {H}} has orthonormal columns, and study the spectral initialization under the asymptotic setting m,n→∞ {m}, {n}\to \infty with m/n→δ∈(1,∞) {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 δ\delta (denoted as δweak\delta _{ \mathrm {weak}} ) below which the spectral method cannot produce a meaningful estimate. We show that δweak=2\delta _{ \mathrm {weak}}=2 for the column-orthonormal model. In contrast, previous results by Mondelli and Montanari show that δweak=1\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.

中文翻译：

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相位检索的谱法：期望传播视角


相位检索是指从信号 x⋆εCn {x}_{\star }\in \mathbb {C}^{n} 的无相测量值 yi=|aHix⋆|\text {y}_{ 中恢复信号的问题\text {i}}=| {a}_{i}^{ \mathsf {H}} {x}_{\star }| ，其中 {ai}mi=1\{ {a}_{\text {i}}\}_{\text {i}=1}^{ {m}} 是测量向量。谱法在许多相位检索算法中广泛用于初始化。谱初始化的质量会对整体算法产生重大影响。在本文中，我们关注模型，其中 A=[a1,…,am]H {A}=[ {a}_{1},\ldots, {a}_{ {m}}]^{ \mathsf {H}}具有正交列，并研究渐近设置下的谱初始化 m,n→∞ {m}, {n}\to \infty 且 m/n→δε(1,∞) {m}/ { n}\to \delta \in (1,\infty) 。我们使用期望传播框架来表征 Haar 分布矩阵的谱初始化的性能。我们的数值结果证实，EP 方法的预测不仅对于 Haar 分布矩阵是准确的，而且对于基于傅立叶的现实模型（例如编码衍射模型）也是准确的。本文的主要发现如下： 1）δ\delta 存在一个阈值（表示为 δweak\delta _{ \mathrm {weak}} ），低于该阈值谱方法无法产生有意义的估计。我们证明对于列正交模型，δweak=2\delta _{ \mathrm {weak}}=2。相比之下，Mondelli 和 Montanari 之前的结果表明，对于 iid 高斯模型，δweak=1\delta _{ \mathrm {weak}}=1。 2）谱法的优化设计与独立同分布高斯模型​​的优化设计一致，后者是由Luo、Alghamdi和Lu最近提出的。