Almost Lossless Analog Compression without Phase Information

TL;DR

This paper establishes an information-theoretic framework for phase retrieval, showing that phaseless measurements can achieve nearly lossless analog compression, matching the bounds of full phase measurements.

Contribution

It introduces a theoretical bound for phase retrieval, demonstrating that phase information can be effectively ignored without significant loss in compression efficiency.

Findings

Achievability bound for phase retrieval matches that of full phase measurements.

Phaseless linear measurements are as effective as measurements with phase information.

Ignoring phase reduces ambiguity to an overall sign factor.

Abstract

We propose an information-theoretic framework for phase retrieval. Specifically, we consider the problem of recovering an unknown n-dimensional vector x up to an overall sign factor from m=Rn phaseless measurements with compression rate R and derive a general achievability bound for R. Surprisingly, it turns out that this bound on the compression rate is the same as the one for almost lossless analog compression obtained by Wu and Verd\'u (2010): Phaseless linear measurements are as good as linear measurements with full phase information in the sense that ignoring the sign of m measurements only leaves us with an ambiguity with respect to an overall sign factor of x.