Performance bounds of the intensity-based estimators for noisy phase retrieval

The aim of noisy phase retrieval is to estimate a signal $x_0\in C^d$ from m noisy intensity measurements $b_j={|〈a_j,x_0〉|}^2+{\eta }_j,j=1,\dots ,m$, where $a_j\in C^d$ are known measurement vectors and $\eta ={({\eta }_1,\dots ,{\eta }_m)}^{\top }\in R^m$ is a noise vector. A commonly used estimator for $x_0$ is to minimize the intensity-based loss function, i.e., $\hat{x}:={\text{argmin}}_{x\in C^d}{\sum }_{j=1}^m{({|〈a_j,x〉|}^2-b_j)}^2$. Although many algorithms for solving the intensity-based estimator have been developed, there are very few results about its estimation performance. In this paper, we focus on the performance of the intensity-based estimator and prove that the error bound satisfies ${\min }_{\theta \in R}{\Vert \hat{x}-e^{i\theta }{x}_0\Vert }_2\lesssim \min \{\frac{\sqrt{{\Vert \eta \Vert }_2}}{m^{1/4}},\frac{{\Vert \eta \Vert }_2}{{\Vert x_0\Vert }_2\cdot \sqrt{m}}\}$ under the assumption of $m\gtrsim d$ and $a_j\in C^d,j=1,\dots ,m$, being complex Gaussian random vectors. We also show that the error bound is rate optimal when $m\gtrsim d\log m$. In the case where $x_0$ is an s-sparse signal, we present a similar result under the assumption of $m\gtrsim s\log (ed/s)$. To the best of our knowledge, our results provide the first theoretical guarantees for both the intensity-based estimator and its sparse version.