Agnostic estimation for misspecified phase retrieval models

Abstract

The goal of noisy high-dimensional phase retrieval is to estimate an s-sparse parameter β∗∈ℝd from n realizations of the model Y=(X⊤β∗)2+ε. Based on this model, we propose a significant semi-parametric generalization called misspecified phase retrieval (MPR), in which Y=f(X⊤β∗,ε) with unknown f and Cov(Y,(X⊤β∗)2)>0. For example, MPR encompasses Y=h(|X⊤β∗|)+ε with increasing h as a special case. Despite the generality of the MPR model, it eludes the reach of most existing semi-parametric estimators. In this paper, we propose an estimation procedure, which consists of solving a cascade of two convex programs and provably recovers the direction of β∗. Our theory is backed up by thorough numerical results.