Robust instance optimal phase-only compressed sensing

Phase-only compressed sensing (PO-CS) concerns the recovery of sparse signals from the phases of complex measurements. Recent results show that sparse signals in the standard sphere (Formula presented) can be exactly recovered from complex Gaussian phases by a linearization procedure, which recasts PO-CS as linear compressed sensing and then applies (quadratically constrained) basis pursuit to obtain (Formula presented). This paper focuses on the instance optimality and robustness of (Formula presented). First, we strengthen the non-uniform instance optimality of Jacques and Feuillen (2021, IEEE Trans. Inf. Theory, 67, 4150–4161) to a uniform one over the entire signal space. We show the existence of some universal constant (Formula presented) such that (Formula presented) holds for all  (Formula presented) in the unit Euclidean sphere, where (Formula presented) is the (Formula presented) distance of (Formula presented) to its closest (Formula presented) -sparse signal. This is achieved by showing that the new sensing matrices corresponding to all approximately sparse signals simultaneously satisfy restricted isometry property. Second, we investigate the estimator’s robustness to noise and corruption. We show that dense noise with entries bounded by some small (Formula presented), appearing either prior or posterior to retaining the phases, increments (Formula presented) by (Formula presented). This is near-optimal (up to log factors) for any algorithm. On the other hand, adversarial corruption, which changes an arbitrary (Formula presented) -fraction of the measurements to any phase-only values, increments (Formula presented) by (Formula presented). We demonstrate the tightness of this result via a partial analysis under suboptimal noise parameter and numerical evidence, while showing that the impact of sparse corruption can be eliminated: to this end, we propose an extended linearization approach that can exactly recover (Formula presented) from the corrupted phases. The developments are then combined to yield a robust instance optimal guarantee that resembles the standard one in linear compressed sensing.