Sparse Recovery with Random Matrices

Abstract

In this chapter, the restricted isometry property, which guarantees the uniform recovery of sparse vectors via a variety of methods, is proved to hold with high probability for subgaussian random matrices provided the number of rows (i.e., measurements) scales like the sparsity times a logarithmic factor. For Gaussian matrices, precise estimates for the required number of measurements (including optimal or at least small values of the constants) are given both in the setting of nonuniform recovery and of uniform recovery. In the latter case, this is first done via an estimate of the restricted isometry constants and then directly through the null space property. Finally, a close relation between the restricted isometry property and the Johnson–Lindenstrauss lemma is uncovered.