Sparse recovery properties of statistical RIP matrices

Compressive sampling is a technique of recovering sparse N-dimensional signals from low-dimensional sketches, i.e., their linear images in R^m, m ≪ N. The main question associated with this technique is construction of linear operators that allow faithful recovery of the signal from its sketch. The most frequently used sufficient condition for robust recovery is the near-isometry property of the operator when restricted to k-sparse signals. In this paper we study performance of standard sparse recovery algorithms in the situation when the sampling matrices satisfy statistical isometry properties. Namely, we show it is possible to recover a sparse signal from its sketch with high probability using the basis pursuit algorithm. Moreover, the same statistical isometry conditions are sufficient for robust model selection with the Lasso algorithm. Finally we show that matrices with the needed properties can be constructed from binary error-correcting codes.