Sparse Phase Retrieval via Truncated Amplitude Flow

This paper develops a novel algorithm, termed
SPARse Truncated Amplitude flow
(SPARTA), to reconstruct a sparse signal from a small number of magnitude-only measurements. It deals with what is also known as sparse phase retrieval (PR), which is
NP-hard
in general and emerges in many science and engineering applications. Upon formulating sparse PR as an amplitude-based nonconvex optimization task, SPARTA works iteratively in two stages: In stage one, the support of the underlying sparse signal is recovered using an analytically well-justified rule, and subsequently a sparse orthogonality-promoting initialization is obtained via power iterations restricted on the support; and in the second stage, the initialization is successively refined by means of hard thresholding based gradient-type iterations. SPARTA is a simple yet effective, scalable, and fast sparse PR solver. On the theoretical side, for any
$n$
-dimensional
$k$
-sparse (
$k\ll n$
) signal
$\boldsymbol {x}$
with minimum (in modulus) nonzero entries on the order of
$(1/\sqrt{k})\Vert \boldsymbol {x}\Vert _2$
, SPARTA recovers the signal exactly (up to a global unimodular constant) from about
$k^2\log n$
random Gaussian measurements with high probability. Furthermore, SPARTA incurs computational complexity on the order of
$k^2n\log n$
with total runtime proportional to the time required to read the data, which improves upon the state of the art by at least a factor of
$k$
. Finally, SPARTA is robust against additive noise of bounded support. Extensive numerical tests corroborate markedly improved recovery performance and speedups of SPARTA relative to existing alternatives.