Misspecified nonconvex statistical optimization for sparse phase retrieval

Zhuoran Yang, Lin F. Yang, Ethan X. Fang, Tuo Zhao, Zhaoran Wang, Matey Neykov

Abstract: Existing nonconvex statistical optimization theory and methods crucially rely on the correct specification of the underlying “true” statistical models. To address this issue, we take a first step towards taming model misspecification by studying the high-dimensional sparse phase retrieval problem with misspecified link functions. In particular, we propose a simple variant of the thresholded Wirtinger flow algorithm that, given a proper initialization, linearly converges to an estimator with optimal statistical accuracy for a broad family of unknown link functions. We further provide extensive numerical experiments to support our theoretical findings.