Computable quantification of the stability of sparse signal reconstruction

The ℓ₁-constrained minimal singular value (ℓ₁-CMSV) of the sensing matrix is shown to determine, in a concise and tight manner, the recovery performance of ℓ₁-based algorithms such as Basis Pursuit, the Dantzig selector, and the LASSO estimator. Several random measurement ensembles are shown to have ℓ₁-CMSVs bounded away from zero with high probability, as long as the number of measurements is relatively large. Three algorithms based on projected gradient method and interior point algorithm are developed to compute ℓ₁-CMSV. A lower bound of the ℓ₁-CMSV is also available by solving a semi-definite programming problem.