Subspace Phase Retrieval

In this paper, we propose a novel algorithm, termed Subspace Phase Retrieval (SPR), which can accurately recover any $n$-dimensional $k$-sparse signal from $\mathcal O(k\log n)$ magnitude-only Gaussian samples. This offers a significant improvement over some existing results that require $\mathcal O(k^2 \log n)$ samples. We also present a geometrical analysis for a subproblem, where we recover the sparse signal given that at least one support index of this signal is identified already. It is shown that with high probability, $\mathcal O(k\log k)$ magnitude-only Gaussian samples ensure i) that all local minima of our objective function are clustered around the expected global minimum within arbitrarily small distances, and ii) that all critical points outside of this region have at least one negative curvature. When the input signal is nonsparse (i.e., $k = n$), our result indicates an analogous geometric property with $\mathcal O(n \log n)$ samples. This affirmatively answers the open question by Sun-Qu-Wright [1].