Sparse Phase Retrieval with Redundant Dictionary via $\ell_q (0<q\le 1)$-Analysis Model

TL;DR

This paper extends the theoretical framework for sparse phase retrieval with redundant dictionaries by introducing $\,\ell_q$-analysis models, providing new null space and restricted isometry conditions for exact and stable recovery.

Contribution

It generalizes existing conditions to the $\,\ell_q$-analysis model, enabling improved recovery guarantees in sparse phase retrieval with redundant dictionaries.

Findings

Introduces NSP variants for $\,\ell_q$-analysis model in noiseless case.

Analyzes S-DRIP conditions for stable recovery in noisy scenarios.

Extends theoretical guarantees to a broader class of $\,\ell_q$ models.

Abstract

Sparse phase retrieval with redundant dictionary is to reconstruct the signals of interest that are (nearly) sparse in a redundant dictionary or frame from the phaseless measurements via the optimization models. Gao [7] presented conditions on the measurement matrix, called null space property (NSP) and strong dictionary restricted isometry property (S-DRIP), for exact and stable recovery of dictionary-$k$-sparse signals via the $\ell_1$-analysis model for sparse phase retrieval with redundant dictionary, respectively, where, in particularly, the S-DRIP of order $tk$ with $t>1$ was derived. In this paper, motivated by many advantages of the $\ell_q$ minimization with $0<q\leq1$, e.g., reduction of the number of measurements required, we generalize these two conditions to the $\ell_q$-analysis model. Specifically, we first present two NSP variants for exact recovery of dictionary-$k$-sparse signals via the $\ell_q$-analysis model in the noiseless scenario. Moreover, we investigate the S-DRIP of order $tk$ with $0<t<\frac{4}{3}$ for stable recovery of dictionary-$k$-sparse signals via the $\ell_q$-analysis model in the noisy scenario, which will complement the existing result of the S-DRIP of order $tk$ with $t\geq2$ obtained in [4].