The performance of the amplitude-based model for complex phase retrieval

Abstract

Abstract This paper aims to study the performance of the amplitude-based model $\widehat{{\boldsymbol x}} \in \mathop{\mathrm{argmin}}\limits _{{\boldsymbol x}\in \mathbb{C}^{d}}\sum _{j=1}^{m}\left (|\langle{\boldsymbol a}_{j},{\boldsymbol x}\rangle |-b_{j}\right )^{2}$, where $b_{j}:=|\langle{\boldsymbol a}_{j},{\boldsymbol x}_{0}\rangle |+\eta _{j}$ and ${\boldsymbol x}_{0}\in \mathbb{C}^{d}$ 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 $\min _{\theta \in [0,2\pi )}\|\widehat{{\boldsymbol x}}-\exp (\mathrm{i}\theta )\cdot{\boldsymbol x}_{0}\|_{2} \lesssim \frac{\|{\boldsymbol \eta }\|_{2}}{\sqrt{m}}$ holds with high probability provided the measurement vectors ${\boldsymbol a}_{j}\in \mathbb{C}^{d},$ $j=1,\ldots ,m,$ are i.i.d. complex sub-Gaussian random vectors and $m\gtrsim d$. Here ${\boldsymbol \eta }=(\eta _{1},\ldots ,\eta _{m})\in \mathbb{R}^{m}$ 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 ${\boldsymbol x}_{0}\in \mathbb{C}^{d}$ 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.