Gabor phase retrieval is severely ill-posed

TL;DR

This paper investigates the stability issues in Gabor phase retrieval, revealing that stability deteriorates exponentially with increasing subspace dimension and that common priors do not regularize the problem effectively.

Contribution

It demonstrates the severe ill-posedness of Gabor phase retrieval in finite-dimensional subspaces and shows that typical priors do not improve stability.

Findings

Stability constant scales at least quadratically exponentially with dimension

Gabor phase retrieval is severely ill-posed in finite-dimensional settings

Common priors like sparsity or smoothness do not regularize the problem

Abstract

The problem of reconstructing a function from the magnitudes of its frame coefficients has recently been shown to be never uniformly stable in infinite-dimensional spaces [5]. This result also holds for frames that are possibly continuous [2]. On the other hand, the problem is always stable in finite-dimensional settings. A prominent example of such a phase retrieval problem is the recovery of a signal from the modulus of its Gabor transform. In this paper, we study Gabor phase retrieval and ask how the stability degrades on a natural family of finite-dimensional subspaces of the signal domain $L^2(\mathbb{R})$. We prove that the stability constant scales at least quadratically exponentially in the dimension of the subspaces. Our construction also shows that typical priors such as sparsity or smoothness promoting penalties do not constitute regularization terms for phase retrieval.