Compressed sensing - a look beyond linear programming

Recently, significant attention in compressed sensing has been focused on basis pursuit, exchanging the cardinality operator with the l₁-norm, which leads to a linear formulation. Here, we want to look beyond using the l₁-norm in two ways: investigating non-linear solutions of higher complexity, but closer to the original problem for one, and improving known low complexity solutions based on matching pursuit using rollout concepts. Our simulation results concur with previous findings that once x is "sparse enough", many algorithms find the correct solution, but for averagely sparse problems we find that the l₁-norm often does not converge to the correct solution - in fact being outperformed by matching pursuit based algorithms at lower complexity. The non-linear algorithm we suggest has increased complexity, but shows superior performance in this setting.