We propose a robust and efficient approach to the problem of compressive
phase retrieval in which the goal is to reconstruct a sparse vector from the
magnitude of a number of its linear measurements. The proposed framework relies
on constrained sensing vectors and a two-stage reconstruction method that
consists of two standard convex programs that are solved sequentially.
In recent years, various methods are proposed for compressive phase
retrieval, but they have suboptimal sample complexity or lack robustness
guarantees. The main obstacle has been that there is no straightforward convex
relaxations for the type of structure in the target. Given a set of
underdetermined measurements, there is a standard framework for recovering a
sparse matrix, and a standard framework for recovering a low-rank matrix.
However, a general, efficient method for recovering a jointly sparse and
low-rank matrix has remained elusive.
Deviating from the models with generic measurements, in this paper we show
that if the sensing vectors are chosen at random from an incoherent subspace,
then the low-rank and sparse structures of the target signal can be effectively
decoupled. We show that a recovery algorithm that consists of a low-rank
recovery stage followed by a sparse recovery stage will produce an accurate
estimate of the target when the number of measurements is
$\mathsf{O}(k\,\log\frac{d}{k})$, where $k$ and $d$ denote the sparsity level
and the dimension of the input signal. We also evaluate the algorithm through
numerical simulation.

本文提出了一种鲁棒且高效的压缩相位恢复方法，通过收集稀疏向量的多个线性测量值的幅值，利用约束感知向量和两阶段重建方法来重构目标信号，在随机不连贯子空间中选择感知向量后，通过低秩恢复阶段和稀疏恢复阶段的策略来准确地估算目标信号，该算法的测量数级别达到 O (k log (d/k))，在数值模拟中得到了验证。

约束感知向量的高效压缩相位恢复

Efficient Compressive Phase Retrieval with Constrained Sensing Vectors

In phase retrieval, the goal is to recover a signal
$\mathbf{x}\in\mathbb{C}^N$ from the magnitudes of linear measurements
$\mathbf{Ax}\in\mathbb{C}^M$. While recent theory has established that
$M\approx 4N$ intensity measurements are necessary and sufficient to recover
generic $\mathbf{x}$, there is great interest in reducing the number of
measurements through the exploitation of sparse $\mathbf{x}$, which is known as
compressive phase retrieval. In this work, we detail a novel, probabilistic
approach to compressive phase retrieval based on the generalized approximate
message passing (GAMP) algorithm. We then present a numerical study of the
proposed PR-GAMP algorithm, demonstrating its excellent phase-transition
behavior, robustness to noise, and runtime. Our experiments suggest that
approximately $M\geq 2K\log_2(N/K)$ intensity measurements suffice to recover
$K$-sparse Bernoulli-Gaussian signals for $\mathbf{A}$ with i.i.d Gaussian
entries and $K\ll N$. Meanwhile, when recovering a 6k-sparse 65k-pixel
grayscale image from 32k randomly masked and blurred Fourier intensity
measurements at 30~dB measurement SNR, PR-GAMP achieved an output SNR of no
less than 28~dB in all of 100 random trials, with a median runtime of only 7.3
seconds. Compared to the recently proposed CPRL, sparse-Fienup, and GESPAR
algorithms, our experiments suggest that PR-GAMP has a superior phase
transition and orders-of-magnitude faster runtimes as the sparsity and problem
dimensions increase.

本研究提出了一种基于广义近似消息传递算法 (GAMP) 的新型压缩相位恢复方法 (PR-GAMP)，并通过实验表明，该算法在减少测量数量、回收信号、抗噪声和运行时间等方面表现出了优越性。