Stable phase retrieval with low-redundancy frames

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 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.Comment: 12 pages AMSLaTeX, 1 figur

Frames, Graphs and Erasures

Two-uniform frames and their use for the coding of vectors are the main subject of this paper. These frames are known to be optimal for handling up to two erasures, in the sense that they minimize the largest possible error when up to two frame coefficients are set to zero. Here, we consider various numerical measures for the reconstruction error associated with a frame when an arbitrary number of the frame coefficients of a vector are lost. We derive general error bounds for two-uniform frames when more than two erasures occur and apply these to concrete examples. We show that among the 227 known equivalence classes of two-uniform (36,15)-frames arising from Hadamard matrices, there are 5 that give smallest error bounds for up to 8 erasures.Comment: 28 pages LaTeX, with AMS macros; v.3: fixed Thm 3.6, added comment, Lemma 3.7 and Proposition 3.8, to appear in Lin. Alg. App