Agnostic estimation for misspecified phase retrieval models

Matey Neykov, Zhaoran Wang, Han Liu

Research output: Contribution to journalArticlepeer-review

Abstract

The goal of noisy high-dimensional phase retrieval is to estimate an s-sparse parameter β∗∈ Rd from n realizations of the model Y = (XTβ∗)2 + ∈. Based on this model, we propose a significant semi-parametric generalization called misspecified phase retrieval (MPR), in which Y = f(XTβ∗, ∈) with unknown f and Cov(Y, (XTβ∗)2) > 0. For example, MPR encompasses Y = h(|XTβ∗|) + ∈ 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 β∗. 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.

Original languageEnglish (US)
Pages (from-to)1-39
Number of pages39
JournalJournal of Machine Learning Research
Volume21
StatePublished - Jun 2020

ASJC Scopus subject areas

  • Software
  • Control and Systems Engineering
  • Statistics and Probability
  • Artificial Intelligence

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