Phase retrieval without small-ball probability assumptions: Recovery guarantees for phaselift

We study the problem of recovering an unknown vector x ε R
ⁿ
from measurements of the form y
_i
= |a
^T
_i
x|
²
(for i = 1,..., m), where the vectors a
_i
ε R
ⁿ
are chosen independently at random, with each coordinate a
_ij
ε R being chosen independently from a fixed sub-Gaussian distribution D. However, without making additional assumptions on the random variables a
_ij
- 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].