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 $\boldx_*$ in $\R^n$, which we wish to recover using sampling vectors $\bolda_1,\ldots,\bolda_m$, and measurements $y_1,\ldots,y_m$, which are related by the equation $f(\inprod{\bolda_i,\boldx_*}) = y_i$. Here, $f$ is an unknown link function satisfying a positive correlation with the quadratic function. This problem was recently analyzed in \cite{Wang2016a}, which 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 $\boldx_*$ efficiently given geometric prior information other than sparsity.