The Cahill-Casazza-Daubechies problem on Hölder stable phase retrieval
Daniel Freeman, Mitchell A. Taylor
TL;DR
The paper addresses whether Hölder-stable phase retrieval on the Cahill–Casazza–Daubechies subsets can be strengthened to Lipschitz stability. It first proves a general counterexample showing Lipschitz stability fails on the CCD-type sets $\mathcal{B}_\gamma(R)$, highlighting a fundamental limitation in infinite dimensions. It then provides constructive positive results: real and complex settings where Lipschitz stable phase retrieval holds on appropriately chosen subsets, extending the CCD framework. Finally, it extends the analysis to multidimensional models with tail decay, demonstrating robust Lipschitz stability under structured decompositions that localize mass in a finite-dimensional subspace. The results delineate the delicate boundary between stability and instability in infinite-dimensional phase retrieval and offer practical guidance for stable recovery in localized or subspace-restricted scenarios.
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 Hölder 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.