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：在纯相位压缩感知（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（例如稀疏信号、低秩矩阵）。尽管PO-CS中信号方向的完美恢复早已被观察到，但针对固定实信号的精确重建保证是最近由Jacques和Feuillen完成的 [IEEE Trans. Inf. Theory, 67 (2021), pp. 4150-4161]。然而，仍有两个问题悬而未决：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遗留问题。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且精确地恢复。通过将PO-CS重新表述为线性压缩感知问题，精确恢复可依据受限等距性质（RIP）得出。我们的均匀恢复保证方法基于覆盖论证，其中需对幅值过小的（原始线性）测量值进行精细控制。为处理复信号，我们采用了不同的符号-乘积嵌入性质以及对感知矩阵的谨慎重缩放。此外，我们证明该均匀恢复在中等有界噪声下具有稳定性。我们还提出在采集相位前加入高斯抖动，以实现包含范数信息的完整重建。实验结果验证并展示了我们的理论成果。