Compressive Phase Retrieval via Reweighted Amplitude Flow

TL;DR

This paper introduces CRAF, a new algorithm for sparse signal reconstruction from magnitude-only measurements, combining spectral initialization and reweighted gradient iterations, achieving exact recovery with high probability.

Contribution

The paper proposes a novel two-stage algorithm, CRAF, that improves spectral initialization and recovery performance in compressive phase retrieval.

Findings

CRAF achieves exact recovery with high probability.

Sample complexity matches state-of-the-art methods.

Spectral initialization significantly enhances performance.

Abstract

The problem of reconstructing a sparse signal vector from magnitude-only measurements (a.k.a., compressive phase retrieval), emerges naturally in diverse applications, but it is NP-hard in general. Building on recent advances in nonconvex optimization, this paper puts forth a new algorithm that is termed compressive reweighted amplitude flow and abbreviated as CRAF, for compressive phase retrieval. Specifically, CRAF operates in two stages. The first stage seeks a sparse initial guess via a new spectral procedure. In the second stage, CRAF implements a few hard thresholding based iterations using reweighted gradients. When there are sufficient measurements, CRAF provably recovers the underlying signal vector exactly with high probability under suitable conditions. Moreover, its sample complexity coincides with that of the state-of-the-art procedures. Finally, substantial simulated tests showcase remarkable performance of the new spectral initialization, as well as improved exact recovery relative to competing alternatives.