  In recent years, phase retrieval has received much attention in statistics,
applied mathematics and optical engineering. In this paper, we propose an
efficient algorithm, termed Subspace Phase Retrieval (SPR), which can
accurately recover an $n$-dimensional $k$-sparse complex-valued signal $\x$
given its $\Omega(k^2\log n)$ magnitude-only Gaussian samples if the minimum
nonzero entry of $\x$ satisfies $|x_{\min}| = \Omega(\|\x\|/\sqrt{k})$.
Furthermore, if the energy sum of the most significant $\sqrt{k}$ elements in
$\x$ is comparable to $\|\x\|^2$, the SPR algorithm can exactly recover $\x$
with $\Omega(k \log n)$ magnitude-only samples, which attains the
information-theoretic sampling complexity for sparse phase retrieval. Numerical
Experiments demonstrate that the proposed algorithm achieves the
state-of-the-art reconstruction performance compared to existing ones.
