Stable phase retrieval and perturbations of frames
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A frame (x_j)_j∈ J for a Hilbert space H is said to do phase retrieval if for all distinct vectors x,y∈ H the magnitude of the frame coefficients (|⟨ x, x_j⟩|)_j∈ J and (|⟨ y, x_j⟩|)_j∈ 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∈ 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.
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