Uniform Exact Reconstruction of Sparse Signals and Low-rank Matrices from Phase-Only Measurements
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The reconstruction of low-complexity, particularly sparse signal from phase of complex linear measurements is known as phase-only compressive sensing (PO-CS). 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. In addition, we show an extension that the uniform recovery is stable under moderate bounded noise, as well as propose a simple way to incorporate norm reconstruction into PO-CS, with the aid of Gaussian dither. Experimental results are reported to corroborate and demonstrate our theoretical results.
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