66 research outputs found
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
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