Compressed sensing with sparse, structured matrices

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.