  In phase-only compressive sensing (PO-CS), our goal is to recover
low-complexity signals (e.g., sparse signals, low-rank matrices) from the phase
of complex linear measurements. While perfect recovery of signal direction in
PO-CS was observed quite early, the exact reconstruction guarantee for a fixed,
real signal was recently done by Jacques and Feuillen [IEEE Trans. Inf. Theory,
67 (2021), pp. 4150-4161]. However, two questions remain open: the uniform
recovery guarantee and exact recovery of complex signal. In this paper, we
almost completely address these two open questions. We prove that, all complex
sparse signals or low-rank matrices can be uniformly, exactly recovered from a
near optimal number of complex Gaussian measurement phases. By recasting PO-CS
as a linear compressive sensing problem, the exact recovery follows from
restricted isometry property (RIP). Our approach to uniform recovery guarantee
is based on covering arguments that involve a delicate control of the (original
linear) measurements with overly small magnitude. To work with complex signal,
a different sign-product embedding property and a careful rescaling of the
sensing matrix are employed. In addition, we show an extension that the uniform
recovery is stable under moderate bounded noise. We also propose to add
Gaussian dither before capturing the phases to achieve full reconstruction with
norm information. Experimental results are reported to corroborate and
demonstrate our theoretical results.
