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Suppose that $\mathbf{y}=\lvert A\mathbf{x_0}\rvert+\eta$ where $\mathbf{x_0}\in \mathbb{R}^d$ is the target signal and $\eta\in \mathbb{R}^m$ is a noisevector. The aim of phase retrieval is to estimate $\mathbf{x_0}$ from$\mathbf{y}$. A popular model for estimating $\mathbf{x_0} $ is the nonlinearleast square $ \widehat{\mathbf{x}}:={\rm argmin}_{\mathbf{x}} \| \lvert A\mathbf{x}\rvert-\mathbf{y}\|_2$. One already develops many efficientalgorithms for solving the model, such as the seminal error reductionalgorithm. In this paper, we present the estimation performance of the modelwith proving that $\|\widehat{\mathbf{x}}-\mathbf{x_0} \|\lesssim{\|\eta\|_2}/{\sqrt{m}}$ under the assumption of $A$ being a Gaussian randommatrix. We also prove the reconstruction error ${\|\eta\|_2}/{\sqrt{m}}$ issharp. For the case where $\mathbf{x_0}$ is sparse, we study the estimationperformance of both the nonlinear Lasso of phase retrieval and itsunconstrained version. Our results are non-asymptotic, and we do not assume anydistribution on the noise $\eta$. To the best of our knowledge, our resultsrepresent the first theoretical guarantee for the nonlinear least square andfor the nonlinear Lasso of phase retrieval.

The estimation performance of nonlinear least squares for phase retrieval