Sensitivity analysis in RIPless compressed sensing

Sensitivity analysis in optimization theory explores how the solution to a particular optimization problem changes as the objective function or constraints of such optimization problem perturb. A recent and yet important class of optimization problems is the framework of compressed sensing where the objective is to find the sparsest solution to an under-determined and possibly noisy system of linear equations. In this paper, we show that by utilizing some tools in sensitivity analysis, namely Invariant Support Sets (ISS), one can improve certain developed results in the field of compressed sensing. More specifically, we show that in a noiseless and RIP-less setting [11], the recovery process of a compressed sensing framework is a binary event in the sense that either all vectors with the same support and sign pattern can be recovered from their compressive samples or none can be estimated correctly. However, in a noisy and RIP-less setting, recovering only one signal from its limited noisy samples guarantees that there exist signals (possibly even with different supports and sign patterns) and noise vectors that shall be recovered with good accuracies by using Lasso.