Error Bound of Empirical $\ell_2$ Risk Minimization for Noisy Standard and Generalized Phase Retrieval Problems

A noisy generalized phase retrieval (NGPR) problem refers to a problem of estimating $x_0 \in \mathbb{C}^d$ by noisy quadratic samples $\big\{x_0^*A_kx_0+\eta_k\big\}_{k=1}^n$ where $A_k$ is a Hermitian matrix and $\eta_k$ is a noise scalar. When $A_k=\alpha_k\alpha_k^*$ for some $\alpha_k\in\mathbb{C}^d$, it reduces to a standard noisy phase retrieval (NPR) problem. The main aim of this paper is to study the estimation performance of empirical $\ell_2$ risk minimization in both problems when $A_k$ in NGPR, or $\alpha_k$ in NPR, is drawn from sub-Gaussian distribution. Under different kinds of noise patterns, we establish error bounds that can imply approximate reconstruction and these results are new in the literature. In NGPR, we show the bounds are of $O\big(\frac{||\eta||}{\sqrt{n}}\big)$ and $O\big(||\eta||_\infty \sqrt{\frac{d}{n}}\big)$ for general noise, and of $O\big(\sqrt{\frac{d\log n}{n}}\big)$ and $O\big(\sqrt{\frac{d(\log n)^2}{n}}\big)$ for random noise with sub-Gaussian and sub-exponential tail respectively, where $\| \eta \|$ and $\| \eta \|_{\infty}$ are the 2-norm and sup-norm of the noise vector of $\eta_k$. Under heavy-tailed noise, by truncating response outliers we propose a robust estimator that possesses an error bound with slower convergence rate. On the other hand, we obtain in NPR the bound is of $O\big(\sqrt{\frac{d\log n}{n}}\big)$ and $O\big(\sqrt{\frac{d(\log n)^2}{n}}\big)$) for sub-Gaussian and sub-exponential noise respectively, which is essentially tighter than the existing bound $O\big(\frac{||\eta||_2}{\sqrt{n}}\big)$. Although NGPR involving measurement matrix $A_k$ is more computationally demanding than NPR involving measurement vector $\alpha_k$, our results reveal that NGPR exhibits stronger robustness than NPR under biased and deterministic noise. Experimental results are presented to confirm and demonstrate our theoretical findings.