Spectral Method for Phase Retrieval: An Expectation Propagation Perspective
Spectral Method for Phase Retrieval: An Expectation Propagation Perspective
复制标题
DOI：
10.1109/tit.2021.3049172
复制
发表时间：
2019-03
影响因子：
2.5
通讯作者：
Junjie Ma;Rishabh Dudeja;Ji Xu;A. Maleki;Xiaodong Wang
中科院分区：
文献类型：
--
作者：
Junjie Ma;Rishabh Dudeja;Ji Xu;A. Maleki;Xiaodong Wang
相位恢复是指从无相位测量中恢复信号$ {x}_{\star }\in \mathbb {C}^{n}$的问题$\text {y}_{\text {i}}=| {a}_{i}^{ \mathsf {H}} {x}_{\star }|$，其中$\{ {a}_{\text {i}}\}_{\text {i}=1}^{ {m}}$是测量向量。在许多相位恢复算法中，谱法被广泛用于初始化。谱初始化的质量会对整个算法产生重大影响。本文主要研究了$ {A}=[ {a}_{1},\ldots, {a}_{ {m}}]^{ \mathsf {H}}$具有正交列的模型，并利用$ {m}/ {n}\to \delta \in (1,\infty)$研究了渐近设置$ {m}, {n}\to \infty $下的谱初始化。我们使用期望传播框架来表征Haar分布矩阵谱初始化的性能。我们的数值结果证实了EP方法的预测不仅对Haar分布矩阵是准确的，而且对现实的基于傅立叶的模型（例如编码衍射模型）也是准确的。本文的主要发现如下：1)在$\delta $（记为$\delta _{ \mathrm {weak}}$）上存在一个阈值，低于该阈值，谱法无法产生有意义的估计。我们证明了$\delta _{ \mathrm {weak}}=2$对于列标准正交模型。相比之下，Mondelli和Montanari先前的结果表明，对于i.i.d高斯模型$\delta _{ \mathrm {weak}}=1$。2)光谱方法的优化设计与i.i.d高斯模型的优化设计一致，后者是最近由Luo， Alghamdi和Lu引入的。
Phase retrieval refers to the problem of recovering a signal $ {x}_{\star }\in \mathbb {C}^{n}$ from its phaseless measurements $\text {y}_{\text {i}}=| {a}_{i}^{ \mathsf {H}} {x}_{\star }|$ , where $\{ {a}_{\text {i}}\}_{\text {i}=1}^{ {m}}$ 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}=[ {a}_{1},\ldots, {a}_{ {m}}]^{ \mathsf {H}}$ has orthonormal columns, and study the spectral initialization under the asymptotic setting $ {m}, {n}\to \infty $ with $ {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 $\delta $ (denoted as $\delta _{ \mathrm {weak}}$ ) below which the spectral method cannot produce a meaningful estimate. We show that $\delta _{ \mathrm {weak}}=2$ for the column-orthonormal model. In contrast, previous results by Mondelli and Montanari show that $\delta _{ \mathrm {weak}}=1$ 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.