Misspecified Phase Retrieval with Generative Priors

TL;DR

This paper addresses phase retrieval with model misspecification using generative priors, proposing a two-step method that achieves near-optimal statistical rates and outperforms existing techniques in image experiments.

Contribution

It introduces a novel two-step approach for misspecified phase retrieval with generative priors, providing theoretical guarantees and empirical improvements.

Findings

Achieves a statistical rate of rac{rac{(k ext{log}L) imes( ext{log}m)}{m}

Performs comparably or better than existing methods on image datasets

Provides theoretical analysis under model misspecification

Abstract

In this paper, we study phase retrieval under model misspecification and generative priors. In particular, we aim to estimate an $n$-dimensional signal $\mathbf{x}$ from $m$ i.i.d.~realizations of the single index model $y = f(\mathbf{a}^T\mathbf{x})$, where $f$ is an unknown and possibly random nonlinear link function and $\mathbf{a} \in \mathbb{R}^n$ is a standard Gaussian vector. We make the assumption $\mathrm{Cov}[y,(\mathbf{a}^T\mathbf{x})^2] \ne 0$, which corresponds to the misspecified phase retrieval problem. In addition, the underlying signal $\mathbf{x}$ is assumed to lie in the range of an $L$-Lipschitz continuous generative model with bounded $k$-dimensional inputs. We propose a two-step approach, for which the first step plays the role of spectral initialization and the second step refines the estimated vector produced by the first step iteratively. We show that both steps enjoy a statistical rate of order $\sqrt{(k\log L)\cdot (\log m)/m}$ under suitable conditions. Experiments on image datasets are performed to demonstrate that our approach performs on par with or even significantly outperforms several competing methods.