On Phase Transition of Compressed Sensing in the Complex Domain

TL;DR

This paper empirically investigates the phase transition behavior of complex-valued compressed sensing using $ ext{l}_1$ minimization, revealing a phase transition above that of real-valued CS and aligning with block-sparse CS theory.

Contribution

It provides the first empirical evaluation of phase transition in complex-valued CS, extending phase transition theory beyond real-valued and block-sparse cases.

Findings

Empirical phase transition of CVCS is above real CS.

CVCS phase transition coincides with block-sparse CS.

Application of ONE-L1 algorithms enables phase transition evaluation.

Abstract

The phase transition is a performance measure of the sparsity-undersampling tradeoff in compressed sensing (CS). This letter reports our first observation and evaluation of an empirical phase transition of the $\ell_1$ minimization approach to the complex valued CS (CVCS), which is positioned well above the known phase transition of the real valued CS in the phase plane. This result can be considered as an extension of the existing phase transition theory of the block-sparse CS (BSCS) based on the universality argument, since the CVCS problem does not meet the condition required by the phase transition theory of BSCS but its observed phase transition coincides with that of BSCS. Our result is obtained by applying the recently developed ONE-L1 algorithms to the empirical evaluation of the phase transition of CVCS.