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 complex-valued 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, our sampling complexity is nearly optimal as it is very close to the fundamental limit $\mathcal O(k \log \frac{n}{k})$ for sparse phase retrieval.

翻译：近年来，相位恢复在统计学、应用数学和光学工程等多个领域受到了广泛关注。本文提出一种高效算法——子空间相位恢复（SPR），该算法能够在仅给定$\mathcal O(k\log^3 n)$个幅度高斯采样的情况下，精确恢复$n$维$k$稀疏复值信号。与许多需要$\mathcal O(k^2 \log n)$或更多采样的现有方法相比，该算法实现了显著改进。此外，其采样复杂度接近稀疏相位恢复的基本极限$\mathcal O(k \log \frac{n}{k})$，因而近乎最优。