Subspace Phase Retrieval

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.