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

In such situations it is common to postulate a linear "working" model, and search for a d-dimensional signal vector β

The goal of noisy high-dimensional phase retrieval is to estimate an $s$-sparse parameter $\boldsymbol{\beta}^*\in \mathbb{R}^d$ from $n$ realizations of the model $Y = (\boldsymbol{X}^{\top} \boldsymbol{\beta}^*)^2 + \varepsilon$. Based on this model, we propose a significant semi-parametric generalization called misspecified phase retrieval (MPR), in which $Y = f(\boldsymbol{X}^{\top}\boldsymbol{\beta}^*, \varepsilon)$ with unknown $f$ and $\operatorname{Cov}(Y, (\boldsymbol{X}^{\top}\boldsymbol{\beta}^*)^2) > 0$. For example, MPR encompasses $Y = h(|\boldsymbol{X}^{\top} \boldsymbol{\beta}^*|) + \varepsilon$ 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 $\boldsymbol{\beta}^*$. Our theory is backed up by thorough numerical results.