Subspace Phase Retrieval

TL;DR

This paper introduces Subspace Phase Retrieval (SPR), an efficient algorithm for accurately recovering sparse complex signals from magnitude-only Gaussian samples, achieving near-optimal sampling complexity with strong empirical performance.

Contribution

The paper proposes SPR, a novel algorithm that improves sparse phase retrieval by reducing sample complexity and providing theoretical guarantees for accurate signal recovery.

Findings

SPR recovers signals with $ ilde{O}(k^2 ext{log} n)$ samples under certain conditions.

Exact recovery is possible with $ ilde{O}(k ext{log} n)$ samples when the signal's energy is concentrated.

Numerical experiments show SPR outperforms existing phase retrieval algorithms.

Abstract

In recent years, phase retrieval has received much attention in statistics, applied mathematics and optical engineering. In this paper, we propose an efficient algorithm, termed Subspace Phase Retrieval (SPR), which can accurately recover an $n$-dimensional $k$-sparse complex-valued signal $\x$ given its $\Omega(k^2\log n)$ magnitude-only Gaussian samples if the minimum nonzero entry of $\x$ satisfies $|x_{\min}| = \Omega(\|\x\|/\sqrt{k})$. Furthermore, if the energy sum of the most significant $\sqrt{k}$ elements in $\x$ is comparable to $\|\x\|^2$, the SPR algorithm can exactly recover $\x$ with $\Omega(k \log n)$ magnitude-only samples, which attains the information-theoretic sampling complexity for sparse phase retrieval. Numerical Experiments demonstrate that the proposed algorithm achieves the state-of-the-art reconstruction performance compared to existing ones.