In the context of the compressed sensing problem, we propose a new ensemble
of sparse random matrices which allow one (i) to acquire and compress a
{\rho}0-sparse signal of length N in a time linear in N and (ii) to perfectly
recover the original signal, compressed at a rate {\alpha}, by using a message
passing algorithm (Expectation Maximization Belief Propagation) that runs in a
time linear in N. In the large N limit, the scheme proposed here closely
approaches the theoretical bound {\rho}0 = {\alpha}, and so it is both optimal
and efficient (linear time complexity). More generally, we show that several
ensembles of dense random matrices can be converted into ensembles of sparse
random matrices, having the same thresholds, but much lower computational
complexity.Comment: 7 pages, 6 figure
