Title: Sparse Phase Retrieval via Sparse PCA Despite Model Misspecification: A Simplified and Extended Analysis,
Abstract: 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.