Non-Convex Phase Retrieval from STFT Measurements

TL;DR

This paper addresses the challenge of recovering signals from phaseless short-time Fourier transform measurements, proposing novel initialization and optimization methods that are robust and effective even with limited redundancy and noise.

Contribution

It introduces a new approach for phase retrieval from STFT measurements, including eigenvector-based initialization and two non-convex optimization algorithms, with theoretical guarantees and empirical validation.

Findings

Eigenvector initialization aids in accurate signal recovery.

Proposed algorithms converge even with low measurement redundancy.

Methods demonstrate robustness to noise in experiments.

Abstract

The problem of recovering a one-dimensional signal from its Fourier transform magnitude, called Fourier phase retrieval, is ill-posed in most cases. We consider the closely-related problem of recovering a signal from its phaseless short-time Fourier transform (STFT) measurements. This problem arises naturally in several applications, such as ultra-short laser pulse characterization and ptychography. The redundancy offered by the STFT enables unique recovery under mild conditions. We show that in some cases the unique solution can be obtained by the principal eigenvector of a matrix, constructed as the solution of a simple least-squares problem. When these conditions are not met, we suggest using the principal eigenvector of this matrix to initialize non-convex local optimization algorithms and propose two such methods. The first is based on minimizing the empirical risk loss function, while the second maximizes a quadratic function on the manifold of phases. We prove that under appropriate conditions, the proposed initialization is close to the underlying signal. We then analyze the geometry of the empirical risk loss function and show numerically that both gradient algorithms converge to the underlying signal even with small redundancy in the measurements. In addition, the algorithms are robust to noise.