%0 Journal Article
%J SampTA
%D 2015
%T Phase Retrieval Without Small-Ball Probability Assumptions: Stability and Uniqueness
%A Felix Krahmer
%A Yi-Kai Liu
%X We study stability and uniqueness for the phase retrieval problem. That is, we ask when is a signal x ∈ R n stably and uniquely determined (up to small perturbations), when one performs phaseless measurements of the form yi = |a T i x| 2 (for i = 1, . . . , N), where the vectors ai ∈ R n are chosen independently at random, with each coordinate aij ∈ R being chosen independently from a fixed sub-Gaussian distribution D. It is well known that for many common choices of D, certain ambiguities can arise that prevent x from being uniquely determined. In this note we show that for any sub-Gaussian distribution D, with no additional assumptions, most vectors x cannot lead to such ambiguities. More precisely, we show stability and uniqueness for all sets of vectors T ⊂ R n which are not too peaky, in the sense that at most a constant fraction of their mass is concentrated on any one coordinate. The number of measurements needed to recover x ∈ T depends on the complexity of T in a natural way, extending previous results of Eldar and Mendelson [12].
%B SampTA
%P 411-414
%8 2015/05/25
%G eng
%U http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=7148923&tag=1