Error Bound of Empirical ℓ2 Risk Minimization for Noisy Standard and Generalized Phase Retrieval Problems

In this paper, we study the estimation performance of empirical ℓ2 risk minimization (ERM) in noisy (standard) phase retrieval (NPR) given by yk = |αk^*x0|2+ηk, or noisy generalized phase retrieval (NGPR) formulated as yk = x0^*Akx0 + ηk, where x0∈mathbbKd is the desired signal, n is the sample size, η= (η1,...,ηn)^top is the noise vector. We establish new error bounds under different noise patterns, and our proofs are valid for both mathbbK=ℝ and mathbbK=mathbbC. In NPR under arbitrary noise vector η, we derive a new error bound Obig(|η|_∞√fracdn + frac|1^topη|nbig), which is tighter than the currently known one Obig(frac|η|√nbig) in many cases. In NGPR, we show Obig(|η|frac√dnbig) for arbitrary η. In both problems, the bounds for arbitrary noise immediately give rise to tildeO(√fracdn) 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 tildeObig(big[√fracdnbig]1-1/lbig) 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 Akk=1n in NGPR is more robust to biased noise than the rank-1 frame αkαk^*k=1n in NPR. Extensive experimental results are presented to illustrate our theoretical findings.