Agnostic Estimation for Misspecified Phase Retrieval Models
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Abstract
The goal of noisy high-dimensional phase retrieval is to estimate an s-sparse parameter 
from n realizations of the model 
. Based on this model, we propose a significant semi-parametric generalization called misspecified phase retrieval (MPR), in which 
) with unknown f and Cov(
. For example, MPR encompasses 
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 
. Our theory is backed up by thorough numerical results.
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