Almost lossless analog compression without phase information - complex case

We extend the recently proposed information-theoretic framework for phase retrieval [1] to the complex case. Specifically, we consider the problem of recovering an unknown random vector x ∈ ℂⁿ up to an overall phase factor from ⌊Rn⌋ phaseless measurements with compression rate R and derive a general achievability bound for R. Although phase retrieval is known not to extend straightforwardly from the real to the complex case, our bound on the compression rate turns out to be conceptually similar to the one derived for real-valued signals [1]. For x being s-sparse our results imply that 2s phaseless measurements are sufficient to recover x up to an overall phase factor irrespectively of the specific distribution of x. The best known recovery threshold for deterministic complex-valued s-sparse vectors is 4s - 2 so far.