Fourier Phase Retrieval with Extended Support Estimation via Deep Neural Network

We consider the problem of sparse phase retrieval from Fourier transform magnitudes to recover $k$-sparse signal vector and its support $\mathcal{T}$. We exploit extended support estimate $\mathcal{E}$ of size larger than $k$ satisfying $\mathcal{E} \supseteq \mathcal{T}$, obtained by a trained deep neural network (DNN). To make the DNN learnable, we let the DNN provide $\mathcal{E}$ as a union of equivalent solutions of $\mathcal{T}$ by utilizing modulo Fourier invariances. Note that $\mathcal{E}$ can be estimated with fast running time via the DNN and the support $\mathcal{T}$ can be found enough in the DNN output $\mathcal{E}$ rather than in the full index set by applying the hard thresholding to $\mathcal{E}$. Thus, the DNN-based extended support estimation improves the reconstruction performance of the signal with a low complexity dependent on $k$. Numerical results support our claim such that the proposed scheme has a superior performance with a lower complexity compared to the local search-based greedy sparse phase retrieval method and a state-of-the-art variant of the Fienup method.