TY - GEN

T1 - Phase retrieval without small-ball probability assumptions

T2 - 11th International Conference on Sampling Theory and Applications, SampTA 2015

AU - Krahmer, Felix

AU - Liu, Yi Kai

N1 - Publisher Copyright:
© 2015 IEEE.

PY - 2015/7/2

Y1 - 2015/7/2

N2 - We study stability and uniqueness for the phase retrieval problem. That is, we ask when is a signal x ε Rn stably and uniquely determined (up to small perturbations), when one performs phaseless measurements of the form yi = aTix2 (for i = 1..., N), where the vectors ai ε Rn are chosen independently at random, with each coordinate aij ε 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 ⊂ Rn 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 ε T depends on the complexity of T in a natural way, extending previous results of Eldar and Mendelson [12].

AB - We study stability and uniqueness for the phase retrieval problem. That is, we ask when is a signal x ε Rn stably and uniquely determined (up to small perturbations), when one performs phaseless measurements of the form yi = aTix2 (for i = 1..., N), where the vectors ai ε Rn are chosen independently at random, with each coordinate aij ε 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 ⊂ Rn 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 ε T depends on the complexity of T in a natural way, extending previous results of Eldar and Mendelson [12].

UR - http://www.scopus.com/inward/record.url?scp=84941044807&partnerID=8YFLogxK

U2 - 10.1109/SAMPTA.2015.7148923

DO - 10.1109/SAMPTA.2015.7148923

M3 - Conference contribution

AN - SCOPUS:84941044807

T3 - 2015 International Conference on Sampling Theory and Applications, SampTA 2015

SP - 411

EP - 414

BT - 2015 International Conference on Sampling Theory and Applications, SampTA 2015

PB - Institute of Electrical and Electronics Engineers Inc.

Y2 - 25 May 2015 through 29 May 2015

ER -