Agnostic Estimation for Misspecified Phase Retrieval Models

The goal of noisy high-dimensional phase retrieval is to estimate an s -sparse parameter beta* is an element of R-d from n realizations of the model Y = (X(sic)beta*)(2) + epsilon. Based on this model, we propose a significant semi-parametric generalization called misspecified phase retrieval (MPR), in which Y = f(X(sic)beta*, epsilon) with unknown f and Cov(Y; (X>beta*)(2)) > 0. For example, MPR encompasses Y = h((X(sic)vertical bar beta*vertical bar) + epsilon 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 beta*. Our theory is backed up by thorough numerical results.