  Random matrices are widely used in sparse recovery problems, and the relevant
properties of matrices with i.i.d. entries are well understood. The current
paper discusses the recently introduced Restricted Eigenvalue (RE) condition,
which is among the most general assumptions on the matrix, guaranteeing
recovery. We prove a reduction principle showing that the RE condition can be
guaranteed by checking the restricted isometry on a certain family of
low-dimensional subspaces. This principle allows us to establish the RE
condition for several broad classes of random matrices with dependent entries,
including random matrices with subgaussian rows and non-trivial covariance
structure, as well as matrices with independent rows, and uniformly bounded
entries.
