2022/05/27 by Junren Chen, Michael K. Ng, Chen, Junren +1 · 1 citation
Physics and Astronomy · #Advanced X-ray Imaging Techniques #FOS: Computer and information sciences #Machine Learning (cs.LG) #Machine Learning (stat.ML)
paper · pdf · doi:10.48550/arxiv.2205.13827
openalex publication_date 2022/05/27 · openalex created_date 2022/06/13 · openalex updated_date 2026/07/28
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=ℂ. In NPR under arbitrary noise vector η, we derive a new error bound O(‖η‖_∞√((d)/(n)) + (|1^\topη|)/(n)), which is tighter than the currently known one O((‖η‖)/(√(n))) in many cases. In NGPR, we show O(‖η‖(√(d))/(n)) for arbitrary η. In both problems, the bounds for arbitrary noise immediately give rise to O(√((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 O([√((d)/(n))]1-1/l) 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 \Ak\k=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.