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The stability of sparse signal reconstruction with respect to measurement noise is investigated in this paper. We design efficient algorithms to verify the sufficient condition for unique ℓ1 sparse recovery. One of our algorithms produces comparable results with the state-of-the-art technique and performs orders of magnitude faster. We show that the ℓ1 -constrained minimal singular value (ℓ1-CMSV) of the measurement matrix determines, in a very concise manner, the recovery performance of ℓ1-based algorithms such as the Basis Pursuit, the Dantzig selector, and the LASSO estimator. Compared to performance analysis involving the Restricted Isometry Constant, the arguments in this paper are much less complicated and provide more intuition on the stability of sparse signal recovery. We show also that, with high probability, the subgaussian ensemble generates measurement matrices with ℓ1-CMSVs bounded away from zero, as long as the number of measurements is relatively large. To compute the ℓ1-CMSV and its lower bound, we design two algorithms based on the interior point algorithm and the semidefinite relaxation.