The paper aims to study the performance of the amplitude-based model \newline $\widehat{\mathbf x} \in {\rm argmin}_{{\mathbf x}\in \mathbb{C}^d}\sum_{j=1}^m\left(|\langle {\mathbf a}_j,{\mathbf x}\rangle|-b_j\right)^2$, 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：本文旨在研究振幅模型 \newline $\widehat{\mathbf x} \in {\rm argmin}_{{\mathbf x}\in \mathbb{C}^d}\sum_{j=1}^m\left(|\langle {\mathbf a}_j,{\mathbf x}\rangle|-b_j\right)^2$ 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。该模型在相位恢复以及绝对值整流神经网络中均有应用。过去几十年间，人们已开发出多种高效算法求解该模型。然而，在含噪条件下的复数情形中，关于估计性能的结果极为稀少。本文针对含噪复数相位恢复问题，给出了振幅模型的理论保证。具体而言，我们证明：当测量向量 ${\mathbf a}_j\in \mathbb{C}^d$（$j=1,\ldots,m$）为独立同分布的复数次高斯随机向量且 $m\gtrsim d$ 时，高概率成立 $\min_{\theta\in[0,2\pi)}\|\widehat{\mathbf x}-\exp(\mathrm{i}\theta)\cdot{\mathbf x}_0\|_2 \lesssim \frac{\|{\mathbf \eta}\|_2}{\sqrt{m}}$。其中 ${\mathbf \eta}=(\eta_1,\ldots,\eta_m)\in \mathbb{R}^m$ 为噪声向量，且无需对分布作任何假设。进一步，我们证明该重构误差是紧的。对于目标信号 ${\mathbf x}_0\in \mathbb{C}^{d}$ 为稀疏的情况，我们为非线性约束 $\ell_1$ 最小化模型建立了类似结果。为此，我们利用了一种针对同时低秩与稀疏矩阵空间上算子的强限制等距性质。