The paper aims to study the performance of the amplitude-based model \newline $\widehat{\mathbf x} \in argmin{{\mathbf x}\in \mathbb{C}^d}\sum_{j=1}^m\left(|\langle {\mathbf a}_j,{\mathbf x}\rangle|-b_j\right)^2$, where $b_j:=|\langle {\mathbf a}_j,{\mathbf x}_0\rangle|+\eta_j$ and ${\mathbf x}_0\in \mathbb{C}^d$ is a target signal. The model is raised in phase retrieval as well as in absolute value rectification neural networks. Many efficient algorithms have been developed to solve it in the past decades. {However, there are very few results available regarding the estimation performance in the complex case under noisy conditions.} In this paper, {we present a theoretical guarantee on the amplitude-based model for the noisy complex phase retrieval problem}. Specifically, we show that $\min_{\theta\in[0,2\pi)}\|\widehat{\mathbf x}-\exp(\mathrm{i}\theta)\cdot{\mathbf x}_0\|_2 \lesssim \frac{\|{\mathbf \eta}\|_2}{\sqrt{m}}$ holds with high probability provided the measurement vectors ${\mathbf a}_j\in \mathbb{C}^d,$ $j=1,\ldots,m,$ are {i.i.d.} complex sub-Gaussian random vectors and $m\gtrsim d$. Here ${\mathbf \eta}=(\eta_1,\ldots,\eta_m)\in \mathbb{R}^m$ is the noise vector without any assumption on the distribution. Furthermore, we prove that the reconstruction error is sharp. For the case where the target signal ${\mathbf x}_0\in \mathbb{C}^{d}$ is sparse, we establish a similar result for the nonlinear constrained $\ell_1$ minimization model. { To accomplish this, we leverage a strong version of restricted isometry property for an operator on the space of simultaneous low-rank and sparse matrices.}

翻译：本文旨在研究基于幅度的模型 \newline $\widehat{\mathbf x} \in argmin{{\mathbf x}\in \mathbb{C}^d}\sum_{j=1}^m\left(|\langle {\mathbf a}_j,{\mathbf x}\rangle|-b_j\right)^2$ 的性能，其中 $b_j:=|\langle {\mathbf a}_j,{\mathbf x}_0\rangle|+\eta_j$，且 ${\mathbf x}_0\in \mathbb{C}^d$ 为目标信号。该模型源自相位恢复问题以及绝对值整流神经网络。过去几十年间，已有许多高效算法被提出以求解该模型。然而，在噪声条件下复情形的估计性能研究结果极为有限。本文中，我们针对带噪复相位恢复问题下的幅度模型提供了理论保证。具体而言，我们证明：当测量向量 ${\mathbf a}_j\in \mathbb{C}^d,$ $j=1,\ldots,m$ 为独立同分布复次高斯随机向量且 $m\gtrsim d$ 时， $\min_{\theta\in[0,2\pi)}\|\widehat{\mathbf x}-\exp(\mathrm{i}\theta)\cdot{\mathbf x}_0\|_2 \lesssim \frac{\|{\mathbf \eta}\|_2}{\sqrt{m}}$ 以高概率成立。这里 ${\mathbf \eta}=(\eta_1,\ldots,\eta_m)\in \mathbb{R}^m$ 为噪声向量，其分布无需任何假设。此外，我们证明了该重构误差是紧的。针对目标信号 ${\mathbf x}_0\in \mathbb{C}^{d}$ 为稀疏情形，我们为非线性的约束 $\ell_1$ 最小化模型建立了类似结论。为此，我们利用了一个针对同时低秩与稀疏矩阵空间上算子的强版本受限等距性质。