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

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We consider the problem of high-dimensional misspecified phase retrieval. This is where we have an s-sparse signal vector x* in Rⁿ, which we wish to recover using sampling vectors a₁,,aₘ, and measurements y₁,,yₘ, which are related by the equation f(<aᵢ,x*>) = yᵢ. 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(s² 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 x* efficiently given geometric prior information other than sparsity.