The $ \ell_1 $-analysis with redundant dictionary in phase retrieval
Provides theoretical foundations for phase retrieval of non-sparse signals in overcomplete dictionaries, addressing a gap in existing theory.
The paper establishes theoretical guarantees for recovering signals that are sparse in a redundant dictionary from magnitude-only measurements using the ℓ1-analysis model, proving exact recovery under a null space property and stable recovery under a new S-DRIP property.
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.