A signal recovery guarantee with Restricted Isometry Property and Null Space Property for weighted $\ell_1$ minimization

Abstract: Signal reconstruction is a crucial aspect of compressive sensing. In weighted cases, there are two common types of weights. In order to establish a unified framework for handling various types of weights, the sparse function is introduced. By employing this sparse function, a generalized form of the weighted null space property is developed, which is sufficient and necessary to exact recovery through weighted $\ell_1$ minimization. This paper will provide a new recovery guarantee called $\omega$-RIP-NSP with the weighted $\ell_1$ minimization, combining the weighted null space property and the weighted restricted isometry property. The new recovery guarantee only depends on the kernel of matrices and provides robust and stable error bounds. The third aim is to explain the relationships between $\omega$-RIP, $\omega$-RIP-NSP and $\omega$-NSP. $\omega$-RIP is obviously stronger than $\omega$-RIP-NSP by definition. We show that $\omega$-RIP-NSP is stronger than the weighted null space property by constructing a matrix that satisfies the weighted null space property but not $\omega$-RIP-NSP.