Fast and robust compressive phase retrieval with sparse-graph codes

In this paper, we tackle the compressive phase retrieval problem in the presence of noise. The noisy compressive phase retrieval problem is to recover a K-sparse complex signal s ∈ ℂ
ⁿ
, from a set of m noisy quadratic measurements: y
_i
= |a
_i
^H
s|
²
+ w
_i
; where a
_i
^H
∈ ℂ
ⁿ
is the ith row of the measurement matrix A ∈ ℂ
^m×n
, and w
_i
is the additive noise to the ith measurement. We consider the regime where K = βn
^δ
, δ ∈ (0; 1). We use the architecture of PhaseCode algorithm [1], and robustify it using two schemes: the almost-linear scheme and the sublinear scheme. We prove that with high probability, the almost-linear scheme recovers s with sample complexity
¹
Θ(K log(n)) and computational complexity Θ(n log(n)), and the sublinear scheme recovers s with sample complexity Θ(K log
³
(n)) and computational complexity Θ(K log
³
(n)). To the best of our knowledge, this is the first scheme that achieves sublinear computational complexity for compressive phase retrieval problem. Finally, we provide simulation results that support our theoretical contributions.