Stable phase retrieval from perturbed samples in shift-invariant spaces

Phase retrieval in a shift-invariant space can also be viewed as a phaseless sampling and reconstruction problem. A fundamental question in sampling theory is the stability of the sampling operator, particularly in the presence of noise and perturbations. In this paper, we investigate the stability of phase retrieval for signals in shift-invariant spaces when the sampling points are subject to small perturbations. Employing the Cheeger constant, we establish stable phase retrieval for nonseparable signals in shift-invariant spaces. We then show that sampling sets guaranteeing stable phase retrieval remain effective under some perturbations, specifically, when the sampling set Γ is perturbed and the sampling locations are not exactly known. Since not all signals in a shift-invariant space are phase retrievable, we further extend our analysis to demonstrate the stable phase retrieval of any signal in the shift-invariant space, but up to a sign factor rather than a global one.