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 complex signal from $\mathcal O(k\log n)$ magnitude-only samples. This offers a significant improvement over some existing results that require $\mathcal O(k^2 \log n)$ samples. We also present a favorable geometric property for the subproblem where we are concerned with the recovery of a sparse signal given that at least one support index of the signal is known already. Specifically, $\mathcal O(k\log k)$ magnitude-only samples ensure i) that all local minima are clustered around the expected global minimum within arbitrarily small distances, and ii) that all the 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 under $\mathcal O(n \log n)$. This affirmatively answers the open question by Sun-Qu-Wright~[1].