Agnostic Estimation for Misspecified Phase Retrieval Models

The goal of noisy high-dimensional phase retrieval is to estimate an s-sparse parameter β∗ ∈ R 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.