Stability of Phase Retrievable Frames

TL;DR

This paper investigates the stability of phase retrievable frames under perturbations, establishing bounds that preserve phase retrievability and confirming the stability of recent constructions.

Contribution

It proves that phase retrievability is stable under perturbations and reduces a key conjecture to a stability problem for non-phase-retrievable frames.

Findings

Perturbation bounds for phase retrievable frames are established.

Recent phase retrieval constructions are shown to be stable.

The critical cardinality conjecture is reduced to a stability problem.

Abstract

In this paper we study the property of phase retrievability by redundant sysems of vectors under perturbations of the frame set. Specifically we show that if a set $\fc$ of $m$ vectors in the complex Hilbert space of dimension n allows for vector reconstruction from magnitudes of its coefficients, then there is a perturbation bound $\rho$ so that any frame set within $\rho$ from $\fc$ has the same property. In particular this proves the recent construction in \cite{BH13} is stable under perturbations. By the same token we reduce the critical cardinality conjectured in \cite{BCMN13a} to proving a stability result for non phase-retrievable frames.