Stable phase retrieval with low-redundancy frames
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We investigate the recovery of vectors from magnitudes of frame coefficients when the frames have a low redundancy, meaning a small number of frame vectors compared to the dimension of the Hilbert space. We first show that for complex vectors in d dimensions, 4d−4 suitably chosen frame vectors are sufficient to uniquely determine each signal, up to an overall unimodular constant, from the magnitudes of its frame coefficients. Then we discuss the effect of noise and show that 8d−4 frame vectors provide a stable recovery if part of the frame coefficients is bounded away from zero. In this regime, perturbing the magnitudes of the frame coefficients by noise that is sufficiently small results in a recovery error that is at most proportional to the noise level.
KeywordsMagnitude measurements Trigonometric polynomials Roots of complex polynomials Newton’s identitites
Mathematics Subject Classifications (2010)15A29 42C15 30C15
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