# Stability estimates for phase retrieval from discrete Gabor measurements

## TL;DR

This paper investigates the stability of phase retrieval from discrete Gabor measurements, demonstrating that semi-global reconstruction can achieve polynomially bounded stability constants in finite-dimensional, bandlimited signal settings.

## Contribution

It provides the first evidence that semi-global phase retrieval from discrete Gabor measurements in finite dimensions can have stability constants that grow only polynomially with dimension.

## Key findings

- Stability constant scales polynomially with dimension
- Semi-global regime enables stable phase retrieval for bandlimited signals
- Utilizes recent reconstruction formulae for analysis

## Abstract

Phase retrieval refers to the problem of recovering some signal (which is often modelled as an element of a Hilbert space) from phaseless measurements. It has been shown that in the deterministic setting phase retrieval from frame coefficients is always unstable in infinite-dimensional Hilbert spaces [7] and possibly severely ill-conditioned in finite-dimensional Hilbert spaces [7]. Recently, it has also been shown that phase retrieval from measurements induced by the Gabor transform with Gaussian window function is stable under a more relaxed semi-global phase recovery regime based on atoll functions [1]. In finite dimensions, we present first evidence that this semi-global reconstruction regime allows one to do phase retrieval from measurements of bandlimited signals induced by the discrete Gabor transform in such a way that the corresponding stability constant only scales like a low order polynomial in the space dimension. To this end, we utilise reconstruction formulae which have become common tools in recent years [6,12,18,20].