Agnostic Estimation for Misspecified Phase Retrieval Models

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.