# Fourier Phase Retrieval with Extended Support Estimation via Deep Neural Network

## TL;DR

This paper introduces a deep neural network approach for sparse Fourier phase retrieval that estimates an extended support set to improve signal reconstruction accuracy with low computational complexity.

## Contribution

The paper proposes a novel DNN-based method to estimate an extended support set for sparse phase retrieval, enhancing accuracy and efficiency over existing methods.

## Key findings

- Outperforms local search-based greedy methods in accuracy.
- Achieves lower computational complexity.
- Demonstrates superior performance in numerical experiments.

## Abstract

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