Stable phase retrieval and perturbations of frames

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.