Performance bound of the intensity-based model for noisy phase retrieval

The aim of noisy phase retrieval is to estimate a signal $$\mathbf{x}$_0\in $\mathbb{C}$^d$from$m$noisyintensitymeasurements$b_j=\left\lvert \langle $\mathbf{a}$j,$\mathbf{x}$0 \rangle \right\rvert^2+\eta_j, ; j=1,\ldots,m$, where ${\mathbf{a}}_j\in {\mathbb{C}}^d$ are known measurement vectors and $\eta =({\eta }_1,\dots ,{\eta }_m{)}^{\mathrm{\top }}\in {\mathbb{R}}^m$ is a noise vector. A commonly used model for estimating ${\mathbf{x}}_0$ is the intensity-based model $\widehat{$\mathbf{x}$}:=\mbox{argmin}{$\mathbf{x}$ \in $\mathbb{C}$^d} \sum{j=1}^m \big(\left\lvert \langle $\mathbf{a}$j,$\mathbf{x}$ \rangle \right\rvert^2-b_j \big)^2$. Although one has already developed many efficient algorithms to solve the intensity-based model, there are very few results about its estimation performance. In this paper, we focus on the estimation performance of the intensity-based model and prove that the error bound satisfies $\min{\theta\in $\mathbb{R}$}|\widehat{$\mathbf{x}$}-e^{i\theta}$\mathbf{x}$_0|_2 \lesssim \min\Big{\frac{\sqrt{|\eta|_2}}{{m}^{1/4}}, \frac{|\eta|_2}{| $\mathbf{x}$_0|_2 \cdot \sqrt{m}}\Big}$undertheassumptionof$m \gtrsim d$and$$\mathbf{a}$_j, j=1,\ldots,m,$ being Gaussian random vectors. We also show that the error bound is sharp. For the case where ${\mathbf{x}}_0$ is a $s$-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 are the first theoretical guarantees for the intensity-based model and its sparse version. Our proofs employ Mendelson’s small ball method which can deliver an effective lower bound on a nonnegative empirical process.