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

In this paper, we study the estimation performance of empirical $\ell_2$ risk minimization (ERM) in noisy (standard) phase retrieval (NPR) given by $y_k = |\alpha_k^*x_0|^2+\eta_k$, or noisy generalized phase retrieval (NGPR) formulated as $y_k = x_0^*A_kx_0 + \eta_k$, where $x_0\in\mathbb{K}^d$ is the desired signal, $n$ is the sample size, $\eta= (\eta_1,...,\eta_n)^\top$ is the noise vector. We establish new error bounds under different noise patterns, and our proofs are valid for both $\mathbb{K}=\mathbb{R}$ and $\mathbb{K}=\mathbb{C}$. In NPR under arbitrary noise vector $\eta$, we derive a new error bound $O\big(\|\eta\|_\infty\sqrt{\frac{d}{n}} + \frac{|\mathbf{1}^\top\eta|}{n}\big)$, which is tighter than the currently known one $O\big(\frac{\|\eta\|}{\sqrt{n}}\big)$ in many cases. In NGPR, we show $O\big(\|\eta\|\frac{\sqrt{d}}{n}\big)$ for arbitrary $\eta$. In both problems, the bounds for arbitrary noise immediately give rise to $\tilde{O}(\sqrt{\frac{d}{n}})$ for sub-Gaussian or sub-exponential random noise, with some conventional but inessential assumptions (e.g., independent or zero-mean condition) removed or weakened. In addition, we make a first attempt to ERM under heavy-tailed random noise assumed to have bounded $l$-th moment. To achieve a trade-off between bias and variance, we truncate the responses and propose a corresponding robust ERM estimator, which is shown to possess the guarantee $\tilde{O}\big(\big[\sqrt{\frac{d}{n}}\big]^{1-1/l}\big)$ in both NPR, NGPR. All the error bounds straightforwardly extend to the more general problems of rank-$r$ matrix recovery, and these results deliver a conclusion that the full-rank frame $\{A_k\}_{k=1}^n$ in NGPR is more robust to biased noise than the rank-1 frame $\{\alpha_k\alpha_k^*\}_{k=1}^n$ in NPR. Extensive experimental results are presented to illustrate our theoretical findings.