Towards improving ℓ1 optimization in compressed sensing

Recently, theoretically analyzed the success of a polynomial ℓ₁-optimization algorithm in solving an under-determined system of linear equations. In a large dimensional and statistical context proved that if the number of equations (measurements in the compressed sensing terminology) in the system is proportional to the length of the unknown vector then there is a sparsity (number of non-zero elements of the unknown vector) also proportional to the length of the unknown vector such that ℓ₁-optimization succeeds in solving the system. In this paper, we consider the same problem while additionally assuming that the locations of a fraction of non-zero elements are a priori known. We provide a performance analysis of a modified ℓ₁-optimization. As expected, the obtained recoverable sparsity proportionality constants improve on the equivalent ones that can be obtained if no information about the non-zero elements location is available. This effectively means that, for certain sparsity levels where ℓ₁-optimization is unsuccessful, instead of solving exactly an under-determined system of linear equations one may be interested in searching for approximate algorithms that only recover location of the fraction of the non-zero part of the unknown vector.