The Cahill-Casazza-Daubechies problem on H\"older stable phase retrieval

TL;DR

This paper investigates the stability of phase retrieval in infinite-dimensional Hilbert spaces, demonstrating conditions under which Lipschitz stability can or cannot be achieved, thus bridging the understanding between finite and infinite-dimensional cases.

Contribution

The paper provides the first examples of subsets in infinite-dimensional Hilbert spaces that either satisfy or fail Lipschitz stable phase retrieval, addressing a key open question.

Findings

Examples of subsets with Lipschitz stability

Examples of subsets without Lipschitz stability

Clarification of conditions for stability in infinite dimensions

Abstract

Phase retrieval using a frame for a finite-dimensional Hilbert space is known to always be Lipschitz stable. However, phase retrieval using a frame or a continuous frame for an infinite-dimensional Hilbert space is always unstable. In order to bridge the gap between the finite and infinite dimensional phenomena, Cahill-Casazza-Daubechies (Trans.Amer.Math.Soc. 2016) gave a construction of a family of nonlinear subsets of an infinite-dimensional Hilbert space where phase retrieval could be performed with a H\"older stability estimate. They then posed the question of whether these subsets satisfied Lipschitz stable phase retrieval. We solve this problem both by giving examples which fail Lipschitz stability and by giving examples which satisfy Lipschitz stability.