Sparse Phase Retrieval via Sparse PCA Despite Model Misspecification: A Simplified and Extended Analysis

We consider the problem of high-dimensional misspecified phase retrieval. This is where we have an $s$-sparse signal vector $\mathbf{x}_*$ in $\mathbb{R}^n$, which we wish to recover using sampling vectors $\textbf{a}_1,\ldots,\textbf{a}_m$, and measurements $y_1,\ldots,y_m$, which are related by the equation $f(\left<\textbf{a}_i,\textbf{x}_*\right>) = y_i$. Here, $f$ is an unknown link function satisfying a positive correlation with the quadratic function. This problem was analyzed in a recent paper by Neykov, Wang and Liu, who provided recovery guarantees for a two-stage algorithm with sample complexity $m = O(s^2\log n)$. In this paper, we show that the first stage of their algorithm suffices for signal recovery with the same sample complexity, and extend the analysis to non-Gaussian measurements. Furthermore, we show how the algorithm can be generalized to recover a signal vector $\textbf{x}_*$ efficiently given geometric prior information other than sparsity.