Recovery Conditions in Weighted Sparse Phase Retrieval via Weighted ℓq(0<q≤1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell _q\, (0<q\le 1)$$\end{document} Minimization

In this paper, we generalize the conditions for the exact or stable recovery of weighted k-sparse signals in weighted sparse phase retrieval in our previous work [11] from the weighted ℓ1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell _1$$\end{document} minimization to the weighted ℓq(0<q≤1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell _q\, (0<q\le 1)$$\end{document} minimization in a broad sense. Specifically, we first present that the weighted null space property (WNSP) is a sufficient and necessary condition to guarantee the exact recovery of a weighted k-sparse signal from its noiseless phaseless measurements via the weighted ℓq(0<q≤1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell _q\, (0<q\le 1)$$\end{document} minimization in both the real and complex cases. In addition, we establish a general strong weighted restricted isometry property (SWRIP) condition for the stable recovery of a weighted k-sparse signal from its noisy phaseless measurements via the weighted ℓq(0<q≤1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell _q\, (0<q\le 1)$$\end{document} minimization in the real case.