Stable recovery of complex dictionary-sparse signals from phaseless measurements

TL;DR

This paper establishes conditions under which complex dictionary-sparse signals can be stably reconstructed from phaseless measurements using $ ext{l}_1$-analysis minimization, extending previous real-signal results to complex signals.

Contribution

The paper introduces a new $ ext{l}_1$-dictionary restricted isometry property ($ ext{l}_1$-DRIP) for complex signals and demonstrates stable recovery via $ ext{l}_1$-analysis minimization.

Findings

Complex signals can be recovered under $ ext{l}_1$-DRIP.

Generalization to $ ext{l}_q$-analysis minimization for $0<q extless=1$.

Theoretical guarantees for stable reconstruction from phaseless measurements.

Abstract

Dictionary-sparse phase retrieval, which is also known as phase retrieval with redundant dictionary, aims to reconstruct an original dictionary-sparse signal from its measurements without phase information. It is proved that if the measurement matrix $A$ satisfies null space property (NSP)/strong dictionary restricted isometry property (S-DRIP), then the dictionary-sparse signal can be exactly/stably recovered from its magnitude-only measurements up to a global phase. However, the S-DRIP holds only for real signals. Hence, in this paper, we mainly study the stability of the $\ell_1$-analysis minimization and its generalized $\ell_q\;(0<q\leq1)$-analysis minimization for the recovery of complex dictionary-sparse signals from phaseless measurements. First, we introduce a new $l_1$-dictionary restricted isometry property ($\ell_1$-DRIP) for rank-one and dictionary-sparse matrices, and show that complex dictionary-sparse signals can be stably recovered by magnitude-only measurements via $\ell_1$-analysis minimization provided that the quadratic measurement map $\mathcal{A}$ satisfies $\ell_1$-DRIP. Then, we generalized the $\ell_1$-DRIP condition under the framework of $\ell_q\;(0<q\leq1)$-analysis minimization.