In phase-only compressive sensing (PO-CS), our goal is to recover low-complexity signals (e.g., sparse signals, low-rank matrices) from the phase of complex linear measurements. While perfect recovery of signal direction in PO-CS was observed quite early, the exact reconstruction guarantee for a fixed, real signal was recently done by Jacques and Feuillen [IEEE Trans. Inf. Theory, 67 (2021), pp. 4150-4161]. However, two questions remain open: the uniform recovery guarantee and exact recovery of complex signal. In this paper, we almost completely address these two open questions. We prove that, all complex sparse signals or low-rank matrices can be uniformly, exactly recovered from a near optimal number of complex Gaussian measurement phases. By recasting PO-CS as a linear compressive sensing problem, the exact recovery follows from restricted isometry property (RIP). Our approach to uniform recovery guarantee is based on covering arguments that involve a delicate control of the (original linear) measurements with overly small magnitude. To work with complex signal, a different sign-product embedding property and a careful rescaling of the sensing matrix are employed. In addition, we show an extension that the uniform recovery is stable under moderate bounded noise. We also propose to add Gaussian dither before capturing the phases to achieve full reconstruction with norm information. Experimental results are reported to corroborate and demonstrate our theoretical results.

翻译：在相位压缩感知（PO-CS）中，我们的目标是从复线性测量的相位中恢复低复杂度信号（如稀疏信号、低秩矩阵）。尽管很早就观察到PO-CS中信号方向的完美恢复，但针对固定实信号的精确重构保证最近才由Jacques和Feuillen完成[IEEE Trans. Inf. Theory, 67 (2021), pp. 4150-4161]。然而，两个问题仍未解决：均匀恢复保证与复信号的精确恢复。本文几乎完全解决了这两个开放性问题。我们证明，所有复稀疏信号或低秩矩阵均可从接近最优数量的复高斯测量相位中均匀、精确地恢复。通过将PO-CS转化为线性压缩感知问题，精确恢复可由限制等距性质（RIP）保证。我们的均匀恢复保证方法基于覆盖论证，涉及对幅值过小的（原始线性）测量的精细控制。为处理复信号，我们采用了不同的符号积嵌入性质，并对感知矩阵进行仔细的重新缩放。此外，我们证明了该方法在中度有界噪声下具有稳定均匀恢复的扩展性质。我们还提出在捕获相位前加入高斯抖动，以利用范数信息实现完全重构。实验结果验证并展示了我们的理论结论。