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 the problem of recovering an unknown vector x ε Rn from measurements of the form yi = aTi x2 (for i = 1..., m), 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. However, without making additional assumptions on the random variables aij - for example on the behavior of their small ball probabilities - it may happen some vectors x cannot be uniquely recovered. We show that for any sub-Gaussian distribution V, with no additional assumptions, it is still possible to recover most vectors x. More precisely, one can recover those vectors x that are not too peaky in the sense that at most a constant fraction of their mass is concentrated on any one coordinate. The recovery guarantees in this paper are for the PhaseLift algorithm, a tractable convex program based on a matrix formulation of the problem. We prove uniform recovery of all not too peaky vectors from m = 0(n) measurements, in the presence of noise. This extends previous work on PhaseLift by Candès and Li [8].

AB - We study the problem of recovering an unknown vector x ε Rn from measurements of the form yi = aTi x2 (for i = 1..., m), 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. However, without making additional assumptions on the random variables aij - for example on the behavior of their small ball probabilities - it may happen some vectors x cannot be uniquely recovered. We show that for any sub-Gaussian distribution V, with no additional assumptions, it is still possible to recover most vectors x. More precisely, one can recover those vectors x that are not too peaky in the sense that at most a constant fraction of their mass is concentrated on any one coordinate. The recovery guarantees in this paper are for the PhaseLift algorithm, a tractable convex program based on a matrix formulation of the problem. We prove uniform recovery of all not too peaky vectors from m = 0(n) measurements, in the presence of noise. This extends previous work on PhaseLift by Candès and Li [8].

UR - http://www.scopus.com/inward/record.url?scp=84941034115&partnerID=8YFLogxK

U2 - 10.1109/SAMPTA.2015.7148966

DO - 10.1109/SAMPTA.2015.7148966

M3 - Conference contribution

AN - SCOPUS:84941034115

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

SP - 622

EP - 626

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 -