A new condition for stable recovery of sparse signals from noisy measurements

The null space property and the restricted isometry property for a measurement matrix are two basic properties in compressive sampling, and are closely related to the sparse approximation. In this paper, we introduce the sparse approximation property of order s for a measurement matrix A: ‖xs‖2 ≤ D‖Ax‖2 + β σs(x) √ s for all x, where xs is the best s-sparse approximation of the vector x in `, σs(x) is the s-sparse approximation error of the vector x in `, and D and β are positive constants. In this paper, we show that the sparse approximation property of a measurement matrix can be thought of as a weaker version of the restricted isometry property and a stronger version of the null space property, and could be an appropriate condition 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 `-minimization problem if the measurement matrix has the sparse approximation property with β ∈ (0, 1), and conversely the measurement matrix has the sparse approximation property with β ∈ (0,∞) if any compressible signal can be stably recovered from its noisy measurements via solving an `minimization problem.