Uniform Exact Reconstruction of Sparse Signals and Low-rank Matrices from Phase-Only Measurements

TL;DR

This paper proves that complex sparse signals and low-rank matrices can be exactly and uniformly reconstructed from phase-only measurements using near optimal Gaussian measurements, extending previous results and ensuring stability under noise.

Contribution

It establishes the first uniform exact recovery guarantees for complex signals in phase-only compressive sensing, employing RIP and covering arguments.

Findings

Uniform exact recovery for complex signals and matrices from phase measurements.

Recovery is stable under bounded noise.

Adding Gaussian dither enables full reconstruction with norm information.

Abstract

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.