The $ \ell_1 $-analysis with redundant dictionary in phase retrieval

TL;DR

This paper investigates the recovery of signals from magnitude-only measurements using an -analysis model with redundant dictionaries, providing new theoretical guarantees for exact and stable recovery.

Contribution

It introduces a null space property and a new S-DRIP property of measurement matrices, ensuring exact and stable recovery in phase retrieval with redundant dictionaries.

Findings

Null space property guarantees exact recovery in noiseless case.

S-DRIP property ensures stable recovery for nearly sparse signals.

Theoretical analysis extends phase retrieval guarantees to overcomplete dictionaries.

Abstract

This article presents new results concerning the recovery of a signal from magnitude only measurements where the signal is not sparse in an orthonormal basis but in a redundant dictionary. To solve this phaseless problem, we analyze the $ \ell_1 $-analysis model. Firstly we investigate the noiseless case with presenting a null space property of the measurement matrix under which the $ \ell_1 $-analysis model provide an exact recovery. Secondly we introduce a new property (S-DRIP) of the measurement matrix. By solving the $ \ell_1 $-analysis model, we prove that this property can guarantee a stable recovery of real signals that are nearly sparse in highly overcomplete dictionaries.