Stable phase retrieval and perturbations of frames

TL;DR

This paper investigates the stability of phase retrieval in frames for Hilbert spaces, providing new bounds on how small perturbations affect the stability constant, independent of dimension and frame size.

Contribution

It introduces quantitative bounds on the impact of perturbations on the stability constant for phase retrieval, enhancing understanding of robustness in high-dimensional settings.

Findings

Stability constants are affected by perturbations independently of dimension.

Small perturbations preserve phase retrieval with quantifiable bounds.

New bounds improve robustness analysis of frames in phase retrieval.

Abstract

A frame $(x_j)_{j\in J}$ for a Hilbert space $H$ is said to do phase retrieval if for all distinct vectors $x,y\in H$ the magnitude of the frame coefficients $(|\langle x, x_j\rangle|)_{j\in J}$ and $(|\langle y, x_j\rangle|)_{j\in J}$ distinguish $x$ from $y$ (up to a unimodular scalar). A frame which does phase retrieval is said to do $C$-stable phase retrieval if the recovery of any vector $x\in H$ from the magnitude of the frame coefficients is $C$-Lipschitz. It is known that if a frame does stable phase retrieval then any sufficiently small perturbation of the frame vectors will do stable phase retrieval, though with a slightly worse stability constant. We provide new quantitative bounds on how the stability constant for phase retrieval is affected by a small perturbation of the frame vectors. These bounds are significant in that they are independent of the dimension of the Hilbert space and the number of vectors in the frame.