Sparse Phase Retrieval With Partial Convolutional Measurements

In this paper, we consider the sparse phase retrieval problem, that is, recovering an unknown
$s$
-sparse signal
$\boldsymbol {x}\in \mathbb {R}^{n}$
from intensity-only measurements. Specifically, we focus on the problem of recovering
$\boldsymbol {x}$
from the observations that are cyclically convoluted with a kernel
$\boldsymbol {a}\in \mathbb {R}^{n}$
, denoted as
$\boldsymbol {y}=|R_{I}(\boldsymbol {a}\circledast \boldsymbol {x})|^{2}$
. Here,
$R_{I}: \mathbb {R}^{n}\rightarrow \mathbb {R}^{|I|}$
represents a random subsampled operator that is restricted to the index set
$I$
. This model is motivated by real-world applications in optics and communications. We provide that if
$\boldsymbol {a}$
is a random Gaussian vector and the number of subsampled measurements is on the order of
$s\text {polylog}(n)$
, one can recover
$\boldsymbol {x}$
up to a global phase provided that the initialization estimator is around
$\boldsymbol {x}$
. It is the first theoretical result that discusses the behavior of sparse convolutional phase retrieval under physically realistic measurements, as opposed to independent Gaussian measurements.