Agnostic Estimation for Misspecified Phase Retrieval Models

The goal of noisy high-dimensional phase retrieval is to estimate an 8-sparse parameter beta* is an element of R-d from n realizations of the model Y = (X-inverted perpendicular beta*)(2) + epsilon. Based on this model, we propose a significant semi-parametric generalization called misspecified phase retrieval (MPR), in which Y = f (X-inverted perpendicular beta*, epsilon) with unknown f and Cov(Y, (X-inverted perpendicular beta*)(2)) > 0. For example, MPR encompasses Y = h(vertical bar X-inverted perpendicular 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*. Furthermore, we prove that our procedure is minimax optimal over the class of MPR models. Interestingly, our minimax analysis characterizes the statistical price of misspecifying the link function in phase retrieval models. Our theory is backed up by thorough numerical results.