  We investigate the sparse recovery problem of reconstructing a
high-dimensional non-negative sparse vector from lower dimensional linear
measurements. While much work has focused on dense measurement matrices, sparse
measurement schemes are crucial in applications, such as DNA microarrays and
sensor networks, where dense measurements are not practically feasible. One
possible construction uses the adjacency matrices of expander graphs, which
often leads to recovery algorithms much more efficient than $\ell_1$
minimization. However, to date, constructions based on expanders have required
very high expansion coefficients which can potentially make the construction of
such graphs difficult and the size of the recoverable sets small.
  In this paper, we construct sparse measurement matrices for the recovery of
non-negative vectors, using perturbations of the adjacency matrix of an
expander graph with much smaller expansion coefficient. We present a necessary
and sufficient condition for $\ell_1$ optimization to successfully recover the
unknown vector and obtain expressions for the recovery threshold. For certain
classes of measurement matrices, this necessary and sufficient condition is
further equivalent to the existence of a "unique" vector in the constraint set,
which opens the door to alternative algorithms to $\ell_1$ minimization. We
further show that the minimal expansion we use is necessary for any graph for
which sparse recovery is possible and that therefore our construction is tight.
We finally present a novel recovery algorithm that exploits expansion and is
much faster than $\ell_1$ optimization. Finally, we demonstrate through
theoretical bounds, as well as simulation, that our method is robust to noise
and approximate sparsity.
