In recent years, phase retrieval has received much attention in many fields including statistics, applied mathematics and optical engineering. In this paper, we propose an efficient algorithm, termed Subspace Phase Retrieval (SPR), which can accurately recover a $n$-dimensional $k$-sparse signal given its $\mathcal O(k\log^3 n)$ magnitude-only Gaussian samples. This offers a significant improvement over many existing methods that require $\mathcal O(k^2 \log n)$ or more samples. Also, the proposed sampling complexity is nearly optimal as it is very close to the fundamental limit $\mathcal O(k \log \frac{n}{k})$ for the sparse phase retrieval problem.

翻译：近年来，相位恢复在统计学、应用数学和光学工程等诸多领域受到广泛关注。本文提出了一种高效算法，称为子空间相位恢复（SPR），该算法能够在仅获得 $\mathcal O(k\log^3 n)$ 个幅度高斯样本的前提下，精确恢复 $n$ 维 $k$ 稀疏信号。与现有许多需要 $\mathcal O(k^2 \log n)$ 或更多样本的方法相比，本算法实现了显著改进。此外，所提出的采样复杂度接近最优，因为它非常接近稀疏相位恢复问题的理论极限 $\mathcal O(k \log \frac{n}{k})$。