Sparse Phase Retrieval via Sparse PCA Despite Model Misspecification: A Simplified and Extended Analysis
This work provides an incremental improvement for researchers in signal processing and machine learning by simplifying and extending existing algorithms for sparse phase retrieval under model misspecification.
The paper tackles the problem of high-dimensional misspecified phase recovery by simplifying a two-stage algorithm to a single stage, achieving signal recovery with sample complexity m = O(s^2 log n) and extending it to non-Gaussian measurements and geometric priors.
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.