Sparse Approximation Property and Stable Recovery of Sparse Signals From Noisy Measurements

摘要

In this correspondence, we introduce a sparse approximation property of order $s$ for a measurement matrix ${bf A}$:${Vert{bf x}_{s}Vert}_{2}leq DVert{bf Ax}Vert_{2}+beta{(sigma_{s}({bf x}))/sqrt{s}}quad{hbox{for all}} {bf x},$where ${bf x}_{s}$ is the best $s$ -sparse approximation of the vector ${bf x}$ in $ell^{2}$, $sigma_{s}({bf x})$ is the $s$-sparse approximation error of the vector ${bf x}$ in $ell^{1}$ , and $D$ and $beta$ are positive constants. The sparse approximation property for a measurement matrix can be thought of as a weaker version of its restricted isometry property and a stronger version of its null space property. In this correspondence, we show that the sparse approximation property is an appropriate condition on a measurement matrix to consider stable recovery of any compressible signal from its noisy measurements. In particular, we show that any compressible signal can be stably recovered from its noisy measurements via solving an $ell^{1}$-minimization problem if the measurement matrix has the sparse appr-n-noximation property with $betain(0,1)$, and conversely the measurement matrix has the sparse approximation property with $betain(0,infty)$ if any compressible signal can be stably recovered from its noisy measurements via solving an $ell^{1}$ -minimization problem.